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Dependence and Independence in Logic Erich Grädel Tutorial at Logic Colloquium , Helsinki, August Erich Grädel Dependence and Independence in Logic

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Page 1: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Dependence and Independence in Logic

Erich Grädel

Tutorial at Logic Colloquium , Helsinki, August

Erich Grädel Dependence and Independence in Logic

Page 2: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Part I: Dependence. Independence, Team Semantics

Motivation and historical remarks

Dependence and independence as atomic properties

Logics of dependence and independence. Team semantics

Expressive power

Negation

Erich Grädel Dependence and Independence in Logic

Page 3: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Dependence and independence

What does it mean that “x depends on y” or that “x and y are independent” ?

Compare this to statements of a more common form such as “x divides y”.

To make sense of this we fix a structure A in which the notion of divisibility is

well-defined and an assignment s ∶ {x , y}→ A and use Tarski-style semanticsto determine whether A ⊧s “x divides y”.

Dependence and independence are notions of a different kind!�ey manifest

themselves not in single assignments, but only in larger amounts of data:

- tables or relations

- sets of plays in a game (e.g. strategies)

- sets of assignments.

A set of assigments (all with the same domain of variables) is called a team.

Erich Grädel Dependence and Independence in Logic

Page 4: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Dependence and independence

What does it mean that “x depends on y” or that “x and y are independent” ?

Compare this to statements of a more common form such as “x divides y”.

To make sense of this we fix a structure A in which the notion of divisibility is

well-defined and an assignment s ∶ {x , y}→ A and use Tarski-style semanticsto determine whether A ⊧s “x divides y”.

Dependence and independence are notions of a different kind!�ey manifest

themselves not in single assignments, but only in larger amounts of data:

- tables or relations

- sets of plays in a game (e.g. strategies)

- sets of assignments.

A set of assigments (all with the same domain of variables) is called a team.

Erich Grädel Dependence and Independence in Logic

Page 5: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Dependence and independence

What does it mean that “x depends on y” or that “x and y are independent” ?

Compare this to statements of a more common form such as “x divides y”.

To make sense of this we fix a structure A in which the notion of divisibility is

well-defined and an assignment s ∶ {x , y}→ A and use Tarski-style semanticsto determine whether A ⊧s “x divides y”.

Dependence and independence are notions of a different kind!�ey manifest

themselves not in single assignments, but only in larger amounts of data:

- tables or relations

- sets of plays in a game (e.g. strategies)

- sets of assignments.

A set of assigments (all with the same domain of variables) is called a team.

Erich Grädel Dependence and Independence in Logic

Page 6: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Dependence and independence

What does it mean that “x depends on y” or that “x and y are independent” ?

Compare this to statements of a more common form such as “x divides y”.

To make sense of this we fix a structure A in which the notion of divisibility is

well-defined and an assignment s ∶ {x , y}→ A and use Tarski-style semanticsto determine whether A ⊧s “x divides y”.

Dependence and independence are notions of a different kind!�ey manifest

themselves not in single assignments, but only in larger amounts of data:

- tables or relations

- sets of plays in a game (e.g. strategies)

- sets of assignments.

A set of assigments (all with the same domain of variables) is called a team.

Erich Grädel Dependence and Independence in Logic

Page 7: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Dependence and independence

What does it mean that “x depends on y” or that “x and y are independent” ?

Compare this to statements of a more common form such as “x divides y”.

To make sense of this we fix a structure A in which the notion of divisibility is

well-defined and an assignment s ∶ {x , y}→ A and use Tarski-style semanticsto determine whether A ⊧s “x divides y”.

Dependence and independence are notions of a different kind!�ey manifest

themselves not in single assignments, but only in larger amounts of data:

- tables or relations

- sets of plays in a game (e.g. strategies)

- sets of assignments.

A set of assigments (all with the same domain of variables) is called a team.

Erich Grädel Dependence and Independence in Logic

Page 8: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Logics of dependence and independence

Henkin, Enderton, Walkoe, . . . : partially ordered (or Henkin-) quantifiers

Blass and Gurevich: correspondence to Σ (and thus NP)

Hintikka and Sandu: Independence-friendly (IF) logic with explicit

dependencies of quantifiers on each other

Semantics in terms of games with imperfect information

Claim: Impossibility of model-theoretic (compositional) semantics

never made precise, never proved

Hodges: model-theoretic semantics for IF-logic

Difference to Tarski semantics: a formula is not evaluated against a single

assignment but against a set of assignments

�is kind of semantics is an important achievement of independent interest.

�e full potential of this innovation has still not been fully appreciated.

Erich Grädel Dependence and Independence in Logic

Page 9: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Logics of dependence and independence

Henkin, Enderton, Walkoe, . . . : partially ordered (or Henkin-) quantifiers

Blass and Gurevich: correspondence to Σ (and thus NP)

Hintikka and Sandu: Independence-friendly (IF) logic with explicit

dependencies of quantifiers on each other

Semantics in terms of games with imperfect information

Claim: Impossibility of model-theoretic (compositional) semantics

never made precise, never proved

Hodges: model-theoretic semantics for IF-logic

Difference to Tarski semantics: a formula is not evaluated against a single

assignment but against a set of assignments

�is kind of semantics is an important achievement of independent interest.

�e full potential of this innovation has still not been fully appreciated.

Erich Grädel Dependence and Independence in Logic

Page 10: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Logics of dependence and independence

Henkin, Enderton, Walkoe, . . . : partially ordered (or Henkin-) quantifiers

Blass and Gurevich: correspondence to Σ (and thus NP)

Hintikka and Sandu: Independence-friendly (IF) logic with explicit

dependencies of quantifiers on each other

Semantics in terms of games with imperfect information

Claim: Impossibility of model-theoretic (compositional) semantics

never made precise, never proved

Hodges: model-theoretic semantics for IF-logic

Difference to Tarski semantics: a formula is not evaluated against a single

assignment but against a set of assignments

�is kind of semantics is an important achievement of independent interest.

�e full potential of this innovation has still not been fully appreciated.

Erich Grädel Dependence and Independence in Logic

Page 11: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Logics of dependence and independence

Henkin, Enderton, Walkoe, . . . : partially ordered (or Henkin-) quantifiers

Blass and Gurevich: correspondence to Σ (and thus NP)

Hintikka and Sandu: Independence-friendly (IF) logic with explicit

dependencies of quantifiers on each other

Semantics in terms of games with imperfect information

Claim: Impossibility of model-theoretic (compositional) semantics

never made precise, never proved

Hodges: model-theoretic semantics for IF-logic

Difference to Tarski semantics: a formula is not evaluated against a single

assignment but against a set of assignments

�is kind of semantics is an important achievement of independent interest.

�e full potential of this innovation has still not been fully appreciated.

Erich Grädel Dependence and Independence in Logic

Page 12: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Henkin quantifiers

φ ∶= ( ∀x ∃y∀u ∃v

) Pxyuv

Intuitively, this says that for all x , u there exist y, v such that Pxyuv holds, andmoreover, the choice of y only depends on the value of x and the choice of vonly depends on the value of u.

Precise semantics can be given either in terms of Skolem functions, or in terms

of games of imperfect information.

(A, P) ⊧ φ if there exist functions f , g ∶ A→ A such that P(a, f a, c, gc) holdsfor all a, c ∈ A.

Erich Grädel Dependence and Independence in Logic

Page 13: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Henkin quantifiers

φ ∶= ( ∀x ∃y∀u ∃v

) Pxyuv

Intuitively, this says that for all x , u there exist y, v such that Pxyuv holds, andmoreover, the choice of y only depends on the value of x and the choice of vonly depends on the value of u.

Precise semantics can be given either in terms of Skolem functions, or in terms

of games of imperfect information.

(A, P) ⊧ φ if there exist functions f , g ∶ A→ A such that P(a, f a, c, gc) holdsfor all a, c ∈ A.

Erich Grädel Dependence and Independence in Logic

Page 14: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Henkin quantifiers

φ ∶= ( ∀x ∃y∀u ∃v

) Pxyuv

Intuitively, this says that for all x , u there exist y, v such that Pxyuv holds, andmoreover, the choice of y only depends on the value of x and the choice of vonly depends on the value of u.

Precise semantics can be given either in terms of Skolem functions, or in terms

of games of imperfect information.

(A, P) ⊧ φ if there exist functions f , g ∶ A→ A such that P(a, f a, c, gc) holdsfor all a, c ∈ A.

Erich Grädel Dependence and Independence in Logic

Page 15: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Henkin quantifiers and NP

Simple formulae with Henkin quantifiers express NP-complete problems

A graph G = (V , E) is -colourable if, and only if,

G ⊧ ( ∀x ∃y∀u ∃v

)(y, v ∈ {, , } ∧ (x = u → y = v) ∧ (Exu → y ≠ v))

�eorem (Blass, Gurevich)

Henkin quantfiers over first-order formulae capture NP

Erich Grädel Dependence and Independence in Logic

Page 16: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Independence-friendly logic

IF-logic is first-order logic where quantifiers are annotated by independencies

Quantification rule: If φ is a formula, x is a variable, andW is a finite set of

variables, then the expressions (∃x/W)φ and (∀x/W)φ are also formulae.

Game-theoretic semantics: In the evaluation game for (Qx/W)φ, the valuefor x must be chosen independently from the the values of the variables inW .

At two positions ((Qx/W)φ, s) and (Qx/W)φ, s′) such that s(y) ≠ s′(y)only for variables inW , the same value for x must be chosen.

Erich Grädel Dependence and Independence in Logic

Page 17: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

IF-logic and existential second-order logic

A graph G = (V , E) admits a perfect matching if, and only if,

G ⊧ ∀x∀y(∃u/{y})(∃v/{x , u})((x = y → u = v) ∧ (u = y → v = x) ∧ Exu)

Skolem semantics. Replace existential quantifiers (∃x/W) by Skolemfunctions whose arguments are the variables outsideW .

�e formula for the perfect matching becomes

(∃ f )(∃g)∀x∀y((x = y → f x = g y) ∧ ( f x = y → g y = x) ∧ Ex f x)

which is equivalent to (∃ f )∀x( f f x = x ∧ Ex f x)

�eorem. In expressive power, IF-logic is equivalent to existential

second-order logic and thus captures NP on finite structures.

Erich Grädel Dependence and Independence in Logic

Page 18: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

IF-logic and existential second-order logic

A graph G = (V , E) admits a perfect matching if, and only if,

G ⊧ ∀x∀y(∃u/{y})(∃v/{x , u})((x = y → u = v) ∧ (u = y → v = x) ∧ Exu)

Skolem semantics. Replace existential quantifiers (∃x/W) by Skolemfunctions whose arguments are the variables outsideW .

�e formula for the perfect matching becomes

(∃ f )(∃g)∀x∀y((x = y → f x = g y) ∧ ( f x = y → g y = x) ∧ Ex f x)

which is equivalent to (∃ f )∀x( f f x = x ∧ Ex f x)

�eorem. In expressive power, IF-logic is equivalent to existential

second-order logic and thus captures NP on finite structures.

Erich Grädel Dependence and Independence in Logic

Page 19: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

IF-logic and existential second-order logic

A graph G = (V , E) admits a perfect matching if, and only if,

G ⊧ ∀x∀y(∃u/{y})(∃v/{x , u})((x = y → u = v) ∧ (u = y → v = x) ∧ Exu)

Skolem semantics. Replace existential quantifiers (∃x/W) by Skolemfunctions whose arguments are the variables outsideW .

�e formula for the perfect matching becomes

(∃ f )(∃g)∀x∀y((x = y → f x = g y) ∧ ( f x = y → g y = x) ∧ Ex f x)

which is equivalent to (∃ f )∀x( f f x = x ∧ Ex f x)

�eorem. In expressive power, IF-logic is equivalent to existential

second-order logic and thus captures NP on finite structures.

Erich Grädel Dependence and Independence in Logic

Page 20: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

IF-logic and existential second-order logic

A graph G = (V , E) admits a perfect matching if, and only if,

G ⊧ ∀x∀y(∃u/{y})(∃v/{x , u})((x = y → u = v) ∧ (u = y → v = x) ∧ Exu)

Skolem semantics. Replace existential quantifiers (∃x/W) by Skolemfunctions whose arguments are the variables outsideW .

�e formula for the perfect matching becomes

(∃ f )(∃g)∀x∀y((x = y → f x = g y) ∧ ( f x = y → g y = x) ∧ Ex f x)

which is equivalent to (∃ f )∀x( f f x = x ∧ Ex f x)

�eorem. In expressive power, IF-logic is equivalent to existential

second-order logic and thus captures NP on finite structures.

Erich Grädel Dependence and Independence in Logic

Page 21: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Logics of dependence and independence

Väänänen: Dependence logic

rather than stating dependencies or independencies as annotations of

quantifiers, treat dependence as an atomic statement

model-theoretic semantics in terms of sets of assignments (team semantics)

Grädel, Väänänen: Independence logic

based on independence rather than dependence

dependence as a special case of a general form of independence

Galliani: Logics based on other dependency properties

Erich Grädel Dependence and Independence in Logic

Page 22: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Logics of dependence and independence

Väänänen: Dependence logic

rather than stating dependencies or independencies as annotations of

quantifiers, treat dependence as an atomic statement

model-theoretic semantics in terms of sets of assignments (team semantics)

Grädel, Väänänen: Independence logic

based on independence rather than dependence

dependence as a special case of a general form of independence

Galliani: Logics based on other dependency properties

Erich Grädel Dependence and Independence in Logic

Page 23: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Logics of dependence and independence

Väänänen: Dependence logic

rather than stating dependencies or independencies as annotations of

quantifiers, treat dependence as an atomic statement

model-theoretic semantics in terms of sets of assignments (team semantics)

Grädel, Väänänen: Independence logic

based on independence rather than dependence

dependence as a special case of a general form of independence

Galliani: Logics based on other dependency properties

Erich Grädel Dependence and Independence in Logic

Page 24: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Dependence atoms

Dependence atoms are expressions =(x , y).

Semantics: Let A be a structure and X a team of assignments s ∶ V → A.

A ⊧X =(x , y) if y depends on x in A and X.

�is means that for all s, s′ ∈ X,n⋀i=

s(xi) = s′(xi) Ô⇒ s(y) = s′(y)

Expanded form of dependence atoms =(x , y) saying that all variables of ydepend on x.

�us dependence is understood as functional dependence, which is the

strongest form of dependence.�e values of x, . . . xn in X determine thevalue of y as surely as x and y determine x + y and x ⋅ y in elementaryarithmetic. Weaker forms of dependence are definable from that.

Erich Grädel Dependence and Independence in Logic

Page 25: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Dependence atoms

Dependence atoms are expressions =(x , y).

Semantics: Let A be a structure and X a team of assignments s ∶ V → A.

A ⊧X =(x , y) if y depends on x in A and X.

�is means that for all s, s′ ∈ X,n⋀i=

s(xi) = s′(xi) Ô⇒ s(y) = s′(y)

Expanded form of dependence atoms =(x , y) saying that all variables of ydepend on x.

�us dependence is understood as functional dependence, which is the

strongest form of dependence.�e values of x, . . . xn in X determine thevalue of y as surely as x and y determine x + y and x ⋅ y in elementaryarithmetic. Weaker forms of dependence are definable from that.

Erich Grädel Dependence and Independence in Logic

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Armstrong’s axioms for dependence

() =(x , x)Anything is functionally dependent of itself.

() If =(x , y), x ⊆ x′ and y′ ⊆ y, then =(x′, y′).Functional dependence is preserved by increasing input data and

decreasing output data.

() If =(x , y) and x′ and y′ are permutations of x and y, then =(x′, y′).Functional dependence does not look at the order of the variables.

() If =(x , y) and =(y, z), then =(x , z).Functional dependences can be transitively composed.

Completeness: If T is a finite set of dependence atoms of the form =(x , y) forvarious x and y, then =(x , y) follows from T according to Armstrong’s rules if,and only if, every team that satisfies T also satisfies =(x , y).

Erich Grädel Dependence and Independence in Logic

Page 27: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Independence: informal discussion

Independence is a more complicated notion than dependence.

Strongest form of logical independence of x and y:

Every conceivable pattern of values for (x , y) occurs.

Knowing one of x and y gives no information about the other.

Caution. Probability theory has its own concept of independence: two

random variables are independent if observing one does not affect the

probabilities of the other. Logical independence is in harmony with

probabilistic independence, but does not pay attention to how o�en a pattern

occurs.

Erich Grädel Dependence and Independence in Logic

Page 28: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Example for independence

Suppose you want gather experimental data by throwing balls of various size

and masses from the Leaning Tower of Pisa to observe how size and mass

influence the time of descent. Setting up the experiment you may want to

make sure that

�e size of the ball is independent from its mass

How to make sure of this? Vary sizes and masses so that if one size is chosen

for one mass it is also chosen for all other masses (and vice versa).�is would

eliminate any dependence between size and mass and the experiment would

genuinely tell us something relevant about the time of descent.

Erich Grädel Dependence and Independence in Logic

Page 29: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Example for independence

Suppose you want gather experimental data by throwing balls of various size

and masses from the Leaning Tower of Pisa to observe how size and mass

influence the time of descent. Setting up the experiment you may want to

make sure that

�e size of the ball is independent from its mass

How to make sure of this? Vary sizes and masses so that if one size is chosen

for one mass it is also chosen for all other masses (and vice versa).�is would

eliminate any dependence between size and mass and the experiment would

genuinely tell us something relevant about the time of descent.

Erich Grädel Dependence and Independence in Logic

Page 30: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Independence atoms: simple case

Definition. A team X satisfies the atom x � y if

(∀s, s′ ∈ X)(∃s′′ ∈ X)(s′′(x) = s(x) ∧ s′′(y) = s′(y))

Suppose you know X and you know that s is some assignment in X.You want to gather information about s(y). What you know is thats(y) ∈ {a ∶ a = s′(y) for some s′ ∈ X}.

Suppose that I tell you s(x) where x � y.You cannot infer anything new about s(y). Indeed, for all potential values afor s(y) there is an assignment s′′ ∈ X with s′′(x) = s(x) and s′′(y) = a.

Erich Grädel Dependence and Independence in Logic

Page 31: Dependence and Independence in Logicgraedel/lc15.pdf · Henkin,Enderton,Walkoe,...:partiallyordered(orHenkin-)quantifiers BlassandGurevich:correspondenceto ... Erich Grädel Dependence

Independence atoms: simple case

Definition. A team X satisfies the atom x � y if

(∀s, s′ ∈ X)(∃s′′ ∈ X)(s′′(x) = s(x) ∧ s′′(y) = s′(y))

Suppose you know X and you know that s is some assignment in X.You want to gather information about s(y). What you know is thats(y) ∈ {a ∶ a = s′(y) for some s′ ∈ X}.

Suppose that I tell you s(x) where x � y.You cannot infer anything new about s(y). Indeed, for all potential values afor s(y) there is an assignment s′′ ∈ X with s′′(x) = s(x) and s′′(y) = a.

Erich Grädel Dependence and Independence in Logic

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Independence atoms: simple case

Definition. A team X satisfies the atom x � y if

(∀s, s′ ∈ X)(∃s′′ ∈ X)(s′′(x) = s(x) ∧ s′′(y) = s′(y))

Suppose you know X and you know that s is some assignment in X.You want to gather information about s(y). What you know is thats(y) ∈ {a ∶ a = s′(y) for some s′ ∈ X}.

Suppose that I tell you s(x) where x � y.You cannot infer anything new about s(y). Indeed, for all potential values afor s(y) there is an assignment s′′ ∈ X with s′′(x) = s(x) and s′′(y) = a.

Erich Grädel Dependence and Independence in Logic

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Constants

Consider a team of data obtained by throwing objects of the same size but

different masses from the leaning tower of Pisa.

Galileo:�e time of descent is independent of the mass.

Special theory of relativity:

Einstein:�e speed of light is independent of the observer’s state of motion.

A special case of total independence is when one of the variables is a constant !

Proposition. A constant variable is independent from every other variable

including itself: =(x) ⇐⇒ x � x

Erich Grädel Dependence and Independence in Logic

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Constants

Consider a team of data obtained by throwing objects of the same size but

different masses from the leaning tower of Pisa.

Galileo:�e time of descent is independent of the mass.

Special theory of relativity:

Einstein:�e speed of light is independent of the observer’s state of motion.

A special case of total independence is when one of the variables is a constant !

Proposition. A constant variable is independent from every other variable

including itself: =(x) ⇐⇒ x � x

Erich Grädel Dependence and Independence in Logic

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Constants

Consider a team of data obtained by throwing objects of the same size but

different masses from the leaning tower of Pisa.

Galileo:�e time of descent is independent of the mass.

Special theory of relativity:

Einstein:�e speed of light is independent of the observer’s state of motion.

A special case of total independence is when one of the variables is a constant !

Proposition. A constant variable is independent from every other variable

including itself: =(x) ⇐⇒ x � x

Erich Grädel Dependence and Independence in Logic

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Constants

Consider a team of data obtained by throwing objects of the same size but

different masses from the leaning tower of Pisa.

Galileo:�e time of descent is independent of the mass.

Special theory of relativity:

Einstein:�e speed of light is independent of the observer’s state of motion.

A special case of total independence is when one of the variables is a constant !

Proposition. A constant variable is independent from every other variable

including itself: =(x) ⇐⇒ x � x

Erich Grädel Dependence and Independence in Logic

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Independence atoms: general case

�e independence atom x � y, and also its extension to independency atomsx � y on tuples of variables, are special forms of a more general atom

x �z y

saying that the variables x are completely independent from y for any constantvalue of z.

Definition. A team X satisfies the atom x �z y if for any pair of assignmentss, s′ ∈ X with s(z) = s′(z) there is a third assignment s′′ ∈ X with- s′′(z) = s(z) = s′(z)- s′′(x) = s(x)- s′′(y) = s′(y).

Erich Grädel Dependence and Independence in Logic

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Dependence versus independence

How do dependence atoms and independence atoms compare with respect to

expressive power?

Lemma =(z, x) logically implies x �z y

Lemma x �z y logically implies =(z, x ∩ y)

Corollary =(z, x) ⇐⇒ x �z x

So dependence is a special case of (the general form of) independence

Erich Grädel Dependence and Independence in Logic

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Dependence versus independence

How do dependence atoms and independence atoms compare with respect to

expressive power?

Lemma =(z, x) logically implies x �z y

Lemma x �z y logically implies =(z, x ∩ y)

Corollary =(z, x) ⇐⇒ x �z x

So dependence is a special case of (the general form of) independence

Erich Grädel Dependence and Independence in Logic

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Dependence versus independence

How do dependence atoms and independence atoms compare with respect to

expressive power?

Lemma =(z, x) logically implies x �z y

Lemma x �z y logically implies =(z, x ∩ y)

Corollary =(z, x) ⇐⇒ x �z x

So dependence is a special case of (the general form of) independence

Erich Grädel Dependence and Independence in Logic

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Dependence versus independence

How do dependence atoms and independence atoms compare with respect to

expressive power?

Lemma =(z, x) logically implies x �z y

Lemma x �z y logically implies =(z, x ∩ y)

Corollary =(z, x) ⇐⇒ x �z x

So dependence is a special case of (the general form of) independence

Erich Grädel Dependence and Independence in Logic

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Dependence versus independence

How do dependence atoms and independence atoms compare with respect to

expressive power?

Lemma =(z, x) logically implies x �z y

Lemma x �z y logically implies =(z, x ∩ y)

Corollary =(z, x) ⇐⇒ x �z x

So dependence is a special case of (the general form of) independence

Erich Grädel Dependence and Independence in Logic

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Inclusion, exclusion, and all that

Besides dependence and independence, there are other interesting atomic

properties of teams. One source of such properties is database dependency

theory.

Inclusion dependencies:

⊧X (x ⊆ y) ∶⇐⇒ (∀s ∈ X)(∃s′ ∈ X)(s(x) = s′(y))

Exclusion dependencies:

⊧X (x ∣ y) ∶⇐⇒ (∀s ∈ X)(∀s′ ∈ X)(s(x) ≠ s′(y))

Equiextension:

⊧X (x & y) ∶⇐⇒ {s(x) ∶ s ∈ X} = {s(y) ∶ s ∈ X}

Erich Grädel Dependence and Independence in Logic

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Inclusion, exclusion, and all that

Besides dependence and independence, there are other interesting atomic

properties of teams. One source of such properties is database dependency

theory.

Inclusion dependencies:

⊧X (x ⊆ y) ∶⇐⇒ (∀s ∈ X)(∃s′ ∈ X)(s(x) = s′(y))

Exclusion dependencies:

⊧X (x ∣ y) ∶⇐⇒ (∀s ∈ X)(∀s′ ∈ X)(s(x) ≠ s′(y))

Equiextension:

⊧X (x & y) ∶⇐⇒ {s(x) ∶ s ∈ X} = {s(y) ∶ s ∈ X}

Erich Grädel Dependence and Independence in Logic

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Inclusion, exclusion, and all that

Besides dependence and independence, there are other interesting atomic

properties of teams. One source of such properties is database dependency

theory.

Inclusion dependencies:

⊧X (x ⊆ y) ∶⇐⇒ (∀s ∈ X)(∃s′ ∈ X)(s(x) = s′(y))

Exclusion dependencies:

⊧X (x ∣ y) ∶⇐⇒ (∀s ∈ X)(∀s′ ∈ X)(s(x) ≠ s′(y))

Equiextension:

⊧X (x & y) ∶⇐⇒ {s(x) ∶ s ∈ X} = {s(y) ∶ s ∈ X}

Erich Grädel Dependence and Independence in Logic

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Inclusion, exclusion, and all that

Besides dependence and independence, there are other interesting atomic

properties of teams. One source of such properties is database dependency

theory.

Inclusion dependencies:

⊧X (x ⊆ y) ∶⇐⇒ (∀s ∈ X)(∃s′ ∈ X)(s(x) = s′(y))

Exclusion dependencies:

⊧X (x ∣ y) ∶⇐⇒ (∀s ∈ X)(∀s′ ∈ X)(s(x) ≠ s′(y))

Equiextension:

⊧X (x & y) ∶⇐⇒ {s(x) ∶ s ∈ X} = {s(y) ∶ s ∈ X}

Erich Grädel Dependence and Independence in Logic

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Logics of dependence and independence

Combine the atoms stating dependencies and/or independencies with the

common logical operators, such as connectives and quantifiers, to obtain

full-fledged logics for reasoning about dependence and independence.

Dependence logic: FO + dependence atoms =(x , y)

Independence logic: FO + independence atoms x �z y

Inclusion logic: FO + inclusion atoms x ⊆ y

and so on.

All these logics require team semantics.

What precisely does it mean that, A ⊧X ψ(x) ?

Erich Grädel Dependence and Independence in Logic

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Logics of dependence and independence

Combine the atoms stating dependencies and/or independencies with the

common logical operators, such as connectives and quantifiers, to obtain

full-fledged logics for reasoning about dependence and independence.

Dependence logic: FO + dependence atoms =(x , y)

Independence logic: FO + independence atoms x �z y

Inclusion logic: FO + inclusion atoms x ⊆ y

and so on.

All these logics require team semantics.

What precisely does it mean that, A ⊧X ψ(x) ?

Erich Grädel Dependence and Independence in Logic

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Team semantics for first-order logic

A ⊧X ψ(y) ⇐⇒ A ⊧s ψ(y) for all s ∈ X

A ⊧X ψ ∧ φ ∶⇐⇒ for all s ∈ X, A ⊧s ψ and A ⊧s φ

A ⊧X ψ ∨ φ ∶⇐⇒ for all s ∈ X, A ⊧s ψ or A ⊧s φ

A ⊧X ∃yψ ∶⇐⇒ for all s ∈ X there exist a ∈ Awith A ⊧s[y↦a] ψ

A ⊧X ∀yψ ∶⇐⇒ for all s ∈ X and all a ∈ A, A ⊧s[y↦a] ψ

Erich Grädel Dependence and Independence in Logic

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Team semantics for first-order logic

A ⊧X ψ(y) ⇐⇒ A ⊧s ψ(y) for all s ∈ X

A ⊧X ψ ∧ φ ∶⇐⇒ for all s ∈ X, A ⊧s ψ and A ⊧s φ

A ⊧X ψ ∨ φ ∶⇐⇒ for all s ∈ X, A ⊧s ψ or A ⊧s φ

A ⊧X ∃yψ ∶⇐⇒ for all s ∈ X there exist a ∈ Awith A ⊧s[y↦a] ψ

A ⊧X ∀yψ ∶⇐⇒ for all s ∈ X and all a ∈ A, A ⊧s[y↦a] ψ

Erich Grädel Dependence and Independence in Logic

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Team semantics: inductive definition

For ψ(y) ∈ FO ∶ A ⊧X ψ(y) ⇐⇒ A ⊧s ψ(y) for all s ∈ X

A ⊧X ψ ∧ φ ∶⇐⇒ for all s ∈ X, A ⊧s ψ and A ⊧s φ

A ⊧X ψ ∨ φ ∶⇐⇒ for all s ∈ X, A ⊧s ψ or A ⊧s φ

A ⊧X ∃yψ ∶⇐⇒ for all s ∈ X there exist a ∈ Awith A ⊧s[y↦a] ψ

A ⊧X ∀yψ ∶⇐⇒ for all s ∈ X and all a ∈ A, A ⊧s[y↦a] ψ

Erich Grädel Dependence and Independence in Logic

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Team semantics: inductive definition

For ψ(y) ∈ FO ∶ A ⊧X ψ(y) ⇐⇒ A ⊧s ψ(y) for all s ∈ X

A ⊧X ψ ∧ φ ∶⇐⇒ A ⊧X ψ and A ⊧X φ

A ⊧X ψ ∨ φ ∶⇐⇒ for all s ∈ X, A ⊧s ψ or A ⊧s φ

A ⊧X ∃yψ ∶⇐⇒ for all s ∈ X there exist a ∈ Awith A ⊧s[y↦a] ψ

A ⊧X ∀yψ ∶⇐⇒ for all s ∈ X and all a ∈ A, A ⊧s[y↦a] ψ

Erich Grädel Dependence and Independence in Logic

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Team semantics: inductive definition

For ψ(y) ∈ FO ∶ A ⊧X ψ(y) ⇐⇒ A ⊧s ψ(y) for all s ∈ X

A ⊧X ψ ∧ φ ∶⇐⇒ A ⊧X ψ and A ⊧X φ

A ⊧X ψ ∨ φ ∶⇐⇒ X = Y ∪ Z such that A ⊧Y ψ and A ⊧Z φ

A ⊧X ∃yψ ∶⇐⇒ for all s ∈ X there exist a ∈ Awith A ⊧s[y↦a] ψ

A ⊧X ∀yψ ∶⇐⇒ for all s ∈ X and all a ∈ A, A ⊧s[y↦a] ψ

Notice that (φ ∨ φ) is, in general, not equivalent to φ

An example from dependence logic:

=(y)means that the value of y is constant in the given team=(y) ∨ =(y)means that y takes at most two values in the given team

Erich Grädel Dependence and Independence in Logic

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Team semantics: inductive definition

For ψ(y) ∈ FO ∶ A ⊧X ψ(y) ⇐⇒ A ⊧s ψ(y) for all s ∈ X

A ⊧X ψ ∧ φ ∶⇐⇒ A ⊧X ψ and A ⊧X φ

A ⊧X ψ ∨ φ ∶⇐⇒ X = Y ∪ Z such that A ⊧Y ψ and A ⊧Z φ

A ⊧X ∃yψ ∶⇐⇒ for all s ∈ X there exist a ∈ Awith A ⊧s[y↦a] ψ

A ⊧X ∀yψ ∶⇐⇒ for all s ∈ X and all a ∈ A, A ⊧s[y↦a] ψ

Notice that (φ ∨ φ) is, in general, not equivalent to φ

An example from dependence logic:

=(y)means that the value of y is constant in the given team=(y) ∨ =(y)means that y takes at most two values in the given team

Erich Grädel Dependence and Independence in Logic

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Team semantics: inductive definition

For ψ(y) ∈ FO ∶ A ⊧X ψ(y) ⇐⇒ A ⊧s ψ(y) for all s ∈ X

A ⊧X ψ ∧ φ ∶⇐⇒ A ⊧X ψ and A ⊧X φ

A ⊧X ψ ∨ φ ∶⇐⇒ X = Y ∪ Z such that A ⊧Y ψ and A ⊧Z φ

A ⊧X ∃yψ ∶⇐⇒ for all s ∈ X there exist a ∈ Awith A ⊧s[y↦a] ψ

A ⊧X ∀yψ ∶⇐⇒ for all s ∈ X and all a ∈ A, A ⊧s[y↦a] ψ

Notice that (φ ∨ φ) is, in general, not equivalent to φ

An example from dependence logic:

=(y)means that the value of y is constant in the given team=(y) ∨ =(y)means that y takes at most two values in the given team

Erich Grädel Dependence and Independence in Logic

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Team semantics: inductive definition

For ψ(y) ∈ FO ∶ A ⊧X ψ(y) ⇐⇒ A ⊧s ψ(y) for all s ∈ X

A ⊧X ψ ∧ φ ∶⇐⇒ A ⊧X ψ and A ⊧X φ

A ⊧X ψ ∨ φ ∶⇐⇒ X = Y ∪ Z such that A ⊧Y ψ and A ⊧Z φ

A ⊧X ∃yψ ∶⇐⇒ ∃F ∶ X → P(A) ∖ {∅} such that A ⊧X[y↦F] ψ

A ⊧X ∀yψ ∶⇐⇒ for all s ∈ X and all a ∈ A, A ⊧s[y↦a] ψ

Choose (for every s ∈ X) an arbitrary non-empty set of witnesses for ∃x . . .rather than just a single witness: lax semantics as opposed to strict semantics.

For FO and dependence logic the difference is immaterial

For stronger logics, only lax semantics guarantees the locality principle:

A ⊧X φ ⇐⇒ A ⊧X↾free(φ) φ

Erich Grädel Dependence and Independence in Logic

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Team semantics: inductive definition

For ψ(y) ∈ FO ∶ A ⊧X ψ(y) ⇐⇒ A ⊧s ψ(y) for all s ∈ X

A ⊧X ψ ∧ φ ∶⇐⇒ A ⊧X ψ and A ⊧X φ

A ⊧X ψ ∨ φ ∶⇐⇒ X = Y ∪ Z such that A ⊧Y ψ and A ⊧Z φ

A ⊧X ∃yψ ∶⇐⇒ ∃F ∶ X → P(A) ∖ {∅} such that A ⊧X[y↦F] ψ

A ⊧X ∀yψ ∶⇐⇒ for all s ∈ X and all a ∈ A, A ⊧s[y↦a] ψ

Choose (for every s ∈ X) an arbitrary non-empty set of witnesses for ∃x . . .rather than just a single witness: lax semantics as opposed to strict semantics.

For FO and dependence logic the difference is immaterial

For stronger logics, only lax semantics guarantees the locality principle:

A ⊧X φ ⇐⇒ A ⊧X↾free(φ) φ

Erich Grädel Dependence and Independence in Logic

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Team semantics: inductive definition

For ψ(y) ∈ FO ∶ A ⊧X ψ(y) ⇐⇒ A ⊧s ψ(y) for all s ∈ X

A ⊧X ψ ∧ φ ∶⇐⇒ A ⊧X ψ and A ⊧X φ

A ⊧X ψ ∨ φ ∶⇐⇒ X = Y ∪ Z such that A ⊧Y ψ and A ⊧Z φ

A ⊧X ∃yψ ∶⇐⇒ ∃F ∶ X → P(A) ∖ {∅} such that A ⊧X[y↦F] ψ

A ⊧X ∀yψ ∶⇐⇒ A ⊧X[y↦A] ψ

For sentences we define: A ⊧ ψ ∶⇐⇒ A ⊧{∅} ψ

Notice that we cannot reasonably replace {∅} by ∅ since the empty teamsatisfies all formulae: A ⊧∅ ψ for all ψ

Erich Grädel Dependence and Independence in Logic

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Team semantics: inductive definition

For ψ(y) ∈ FO ∶ A ⊧X ψ(y) ⇐⇒ A ⊧s ψ(y) for all s ∈ X

A ⊧X ψ ∧ φ ∶⇐⇒ A ⊧X ψ and A ⊧X φ

A ⊧X ψ ∨ φ ∶⇐⇒ X = Y ∪ Z such that A ⊧Y ψ and A ⊧Z φ

A ⊧X ∃yψ ∶⇐⇒ ∃F ∶ X → P(A) ∖ {∅} such that A ⊧X[y↦F] ψ

A ⊧X ∀yψ ∶⇐⇒ A ⊧X[y↦A] ψ

For sentences we define: A ⊧ ψ ∶⇐⇒ A ⊧{∅} ψ

Notice that we cannot reasonably replace {∅} by ∅ since the empty teamsatisfies all formulae: A ⊧∅ ψ for all ψ

Erich Grädel Dependence and Independence in Logic

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Example: Defining -SAT in dependence logic

Represent an instance φ = ⋀mi=(Xi ∨ Xi ∨ Xi) of -SAT by a team

Zφ = {(i , j, X , σ) ∶ in clause i at position j, the variable X appears with parity σ}

Example: �e formula φ = (X ∨ ¬X ∨ X) ∧ (X ∨ X ∨ ¬X), is describedby the team

clause position variable parity

X +

X -

X +

X +

X +

X -

Proposition. φ is satisfiable if, and only if, the team Zφ is a model of

=(clause,position) ∨ =(clause, position) ∨ =(variable,parity)

Erich Grädel Dependence and Independence in Logic

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Example: Defining -SAT in dependence logic

Represent an instance φ = ⋀mi=(Xi ∨ Xi ∨ Xi) of -SAT by a team

Zφ = {(i , j, X , σ) ∶ in clause i at position j, the variable X appears with parity σ}

Example: �e formula φ = (X ∨ ¬X ∨ X) ∧ (X ∨ X ∨ ¬X), is describedby the team

clause position variable parity

X +

X -

X +

X +

X +

X -

Proposition. φ is satisfiable if, and only if, the team Zφ is a model of

=(clause,position) ∨ =(clause, position) ∨ =(variable,parity)

Erich Grädel Dependence and Independence in Logic

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From team semantics to Tarski semantics

A team X of assignments s ∶ V → A can be represented as a relationrel(X) ⊆ A∣V ∣.

�e translation to Tarski semantics requires that we go to existential

second-order logic Σ.

Proposition. Every formula ψ(x, . . . , xn) in dependence or independencelogic, with vocabulary τ, can be translated into a Σ-sentence ψ∗ of vocabularyτ ∪ {R} such that

A ⊧X ψ(x) ⇐⇒ (A, rel(X)) ⊧ ψ∗

Indeed this holds for any extension of FO by atomic properties of teams that

are first-order expressible (on the relation describing the team).

For sentences, dependence logic, and thus also independence logic coincides

in expressive power with Σ.

Erich Grädel Dependence and Independence in Logic

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From team semantics to Tarski semantics

A team X of assignments s ∶ V → A can be represented as a relationrel(X) ⊆ A∣V ∣.

�e translation to Tarski semantics requires that we go to existential

second-order logic Σ.

Proposition. Every formula ψ(x, . . . , xn) in dependence or independencelogic, with vocabulary τ, can be translated into a Σ-sentence ψ∗ of vocabularyτ ∪ {R} such that

A ⊧X ψ(x) ⇐⇒ (A, rel(X)) ⊧ ψ∗

Indeed this holds for any extension of FO by atomic properties of teams that

are first-order expressible (on the relation describing the team).

For sentences, dependence logic, and thus also independence logic coincides

in expressive power with Σ.

Erich Grädel Dependence and Independence in Logic

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�e translation from dependence logic into Σ

Construct translation ψ(x, . . . , xn) ↦ ψ∗(R) such that

A ⊧X ψ(x) ⇐⇒ (A, rel(X)) ⊧ ψ∗

- first-order literals α(x) are translated into ∀x(Rx → α(x))- dependency atoms =(xi , . . . , xik , xi) are translated into

∀x∀y(Rx ∧ Ry ∧k⋀j=

(xi j = yi j)→ (xi = yi))

- (ψ(x) ∧ φ(x))∗ ∶= ψ∗(R) ∧ φ∗(R)- (ψ(x) ∨ φ(x))∗ ∶= ∃S∃T(∀x(Rx → (Sx ∨ Tx)) ∧ ψ∗(S) ∧ φ∗(T))- (∀yψ)∗ ∶= ∃S(∀x∀y(Rx → Sxy) ∧ ψ∗(S))- (∃yψ)∗ ∶= ∃S(∀x∃y(Rx → Sxy) ∧ ψ∗(S))

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�e translation from dependence logic into Σ

Construct translation ψ(x, . . . , xn) ↦ ψ∗(R) such that

A ⊧X ψ(x) ⇐⇒ (A, rel(X)) ⊧ ψ∗

- first-order literals α(x) are translated into ∀x(Rx → α(x))

- dependency atoms =(xi , . . . , xik , xi) are translated into

∀x∀y(Rx ∧ Ry ∧k⋀j=

(xi j = yi j)→ (xi = yi))

- (ψ(x) ∧ φ(x))∗ ∶= ψ∗(R) ∧ φ∗(R)- (ψ(x) ∨ φ(x))∗ ∶= ∃S∃T(∀x(Rx → (Sx ∨ Tx)) ∧ ψ∗(S) ∧ φ∗(T))- (∀yψ)∗ ∶= ∃S(∀x∀y(Rx → Sxy) ∧ ψ∗(S))- (∃yψ)∗ ∶= ∃S(∀x∃y(Rx → Sxy) ∧ ψ∗(S))

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�e translation from dependence logic into Σ

Construct translation ψ(x, . . . , xn) ↦ ψ∗(R) such that

A ⊧X ψ(x) ⇐⇒ (A, rel(X)) ⊧ ψ∗

- first-order literals α(x) are translated into ∀x(Rx → α(x))- dependency atoms =(xi , . . . , xik , xi) are translated into

∀x∀y(Rx ∧ Ry ∧k⋀j=

(xi j = yi j)→ (xi = yi))

- (ψ(x) ∧ φ(x))∗ ∶= ψ∗(R) ∧ φ∗(R)- (ψ(x) ∨ φ(x))∗ ∶= ∃S∃T(∀x(Rx → (Sx ∨ Tx)) ∧ ψ∗(S) ∧ φ∗(T))- (∀yψ)∗ ∶= ∃S(∀x∀y(Rx → Sxy) ∧ ψ∗(S))- (∃yψ)∗ ∶= ∃S(∀x∃y(Rx → Sxy) ∧ ψ∗(S))

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�e translation from dependence logic into Σ

Construct translation ψ(x, . . . , xn) ↦ ψ∗(R) such that

A ⊧X ψ(x) ⇐⇒ (A, rel(X)) ⊧ ψ∗

- first-order literals α(x) are translated into ∀x(Rx → α(x))- dependency atoms =(xi , . . . , xik , xi) are translated into

∀x∀y(Rx ∧ Ry ∧k⋀j=

(xi j = yi j)→ (xi = yi))

- (ψ(x) ∧ φ(x))∗ ∶= ψ∗(R) ∧ φ∗(R)

- (ψ(x) ∨ φ(x))∗ ∶= ∃S∃T(∀x(Rx → (Sx ∨ Tx)) ∧ ψ∗(S) ∧ φ∗(T))- (∀yψ)∗ ∶= ∃S(∀x∀y(Rx → Sxy) ∧ ψ∗(S))- (∃yψ)∗ ∶= ∃S(∀x∃y(Rx → Sxy) ∧ ψ∗(S))

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�e translation from dependence logic into Σ

Construct translation ψ(x, . . . , xn) ↦ ψ∗(R) such that

A ⊧X ψ(x) ⇐⇒ (A, rel(X)) ⊧ ψ∗

- first-order literals α(x) are translated into ∀x(Rx → α(x))- dependency atoms =(xi , . . . , xik , xi) are translated into

∀x∀y(Rx ∧ Ry ∧k⋀j=

(xi j = yi j)→ (xi = yi))

- (ψ(x) ∧ φ(x))∗ ∶= ψ∗(R) ∧ φ∗(R)- (ψ(x) ∨ φ(x))∗ ∶= ∃S∃T(∀x(Rx → (Sx ∨ Tx)) ∧ ψ∗(S) ∧ φ∗(T))

- (∀yψ)∗ ∶= ∃S(∀x∀y(Rx → Sxy) ∧ ψ∗(S))- (∃yψ)∗ ∶= ∃S(∀x∃y(Rx → Sxy) ∧ ψ∗(S))

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�e translation from dependence logic into Σ

Construct translation ψ(x, . . . , xn) ↦ ψ∗(R) such that

A ⊧X ψ(x) ⇐⇒ (A, rel(X)) ⊧ ψ∗

- first-order literals α(x) are translated into ∀x(Rx → α(x))- dependency atoms =(xi , . . . , xik , xi) are translated into

∀x∀y(Rx ∧ Ry ∧k⋀j=

(xi j = yi j)→ (xi = yi))

- (ψ(x) ∧ φ(x))∗ ∶= ψ∗(R) ∧ φ∗(R)- (ψ(x) ∨ φ(x))∗ ∶= ∃S∃T(∀x(Rx → (Sx ∨ Tx)) ∧ ψ∗(S) ∧ φ∗(T))- (∀yψ)∗ ∶= ∃S(∀x∀y(Rx → Sxy) ∧ ψ∗(S))

- (∃yψ)∗ ∶= ∃S(∀x∃y(Rx → Sxy) ∧ ψ∗(S))

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�e translation from dependence logic into Σ

Construct translation ψ(x, . . . , xn) ↦ ψ∗(R) such that

A ⊧X ψ(x) ⇐⇒ (A, rel(X)) ⊧ ψ∗

- first-order literals α(x) are translated into ∀x(Rx → α(x))- dependency atoms =(xi , . . . , xik , xi) are translated into

∀x∀y(Rx ∧ Ry ∧k⋀j=

(xi j = yi j)→ (xi = yi))

- (ψ(x) ∧ φ(x))∗ ∶= ψ∗(R) ∧ φ∗(R)- (ψ(x) ∨ φ(x))∗ ∶= ∃S∃T(∀x(Rx → (Sx ∨ Tx)) ∧ ψ∗(S) ∧ φ∗(T))- (∀yψ)∗ ∶= ∃S(∀x∀y(Rx → Sxy) ∧ ψ∗(S))- (∃yψ)∗ ∶= ∃S(∀x∃y(Rx → Sxy) ∧ ψ∗(S))

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Expressive power of dependence logic: formulae

Downwards Closure: If A ⊧X ψ and Y ⊆ X, then A ⊧Y ψ

Hence, the sentences ψ∗ have to be downwards monotone:If (A, R) ⊧ ψ∗ and S ⊆ R then (A, S) ⊧ ψ∗.

�us dependence logic is a strict fragment of existential second-order logic.

�eorem (Kontinen and Väänänen )

�e expressive power of formulae ψ(x, . . . , xn) of dependence logic isprecisely that of existential second-order sentences with the predicate for the

team occurring only negatively.

While sentences of dependence logic can express all NP-properties of finite

structures, only the downwards closed NP-properties of teams are definable by

open formulae of dependence logic.

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Expressive power of dependence logic: formulae

Downwards Closure: If A ⊧X ψ and Y ⊆ X, then A ⊧Y ψ

Hence, the sentences ψ∗ have to be downwards monotone:If (A, R) ⊧ ψ∗ and S ⊆ R then (A, S) ⊧ ψ∗.

�us dependence logic is a strict fragment of existential second-order logic.

�eorem (Kontinen and Väänänen )

�e expressive power of formulae ψ(x, . . . , xn) of dependence logic isprecisely that of existential second-order sentences with the predicate for the

team occurring only negatively.

While sentences of dependence logic can express all NP-properties of finite

structures, only the downwards closed NP-properties of teams are definable by

open formulae of dependence logic.

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Expressive power of dependence logic: formulae

Downwards Closure: If A ⊧X ψ and Y ⊆ X, then A ⊧Y ψ

Hence, the sentences ψ∗ have to be downwards monotone:If (A, R) ⊧ ψ∗ and S ⊆ R then (A, S) ⊧ ψ∗.

�us dependence logic is a strict fragment of existential second-order logic.

�eorem (Kontinen and Väänänen )

�e expressive power of formulae ψ(x, . . . , xn) of dependence logic isprecisely that of existential second-order sentences with the predicate for the

team occurring only negatively.

While sentences of dependence logic can express all NP-properties of finite

structures, only the downwards closed NP-properties of teams are definable by

open formulae of dependence logic.

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Expressive power of dependence logic: formulae

Downwards Closure: If A ⊧X ψ and Y ⊆ X, then A ⊧Y ψ

Hence, the sentences ψ∗ have to be downwards monotone:If (A, R) ⊧ ψ∗ and S ⊆ R then (A, S) ⊧ ψ∗.

�us dependence logic is a strict fragment of existential second-order logic.

�eorem (Kontinen and Väänänen )

�e expressive power of formulae ψ(x, . . . , xn) of dependence logic isprecisely that of existential second-order sentences with the predicate for the

team occurring only negatively.

While sentences of dependence logic can express all NP-properties of finite

structures, only the downwards closed NP-properties of teams are definable by

open formulae of dependence logic.

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�e translation from independence logic into Σ

Construct translation ψ(x, . . . , xn) ↦ ψ∗(R) such that

A ⊧X ψ(x) ⇐⇒ (A, rel(X)) ⊧ ψ∗

- first-order literals α(x) are translated into ∀x(Rx → α(x))- independence atoms xi �xk x j are translated into

∀x∀y(Rx ∧ Ry ∧ (xk = yk)→ ∃z(Rz ∧ (zk = xk) ∧ (zi = xi) ∧ (z j = y j)))

- (ψ(x) ∧ φ(x))∗ ∶= ψ∗(R) ∧ φ∗(R)- (ψ(x) ∨ φ(x))∗ ∶= ∃S∃T(∀x(Rx↔ (Sx ∨ Tx)) ∧ ψ∗(S) ∧ φ∗(T))- (∀yψ)∗ ∶= ∃S(∀x∀y(Rx↔ Sxy) ∧ ψ∗(S))- (∃yψ)∗ ∶= ∃S(∀x∃y(Rx↔ Sxy) ∧ ψ∗(S))

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Expressive power of independence logic

Contrary to dependence logic, inclusion logic and independence logic are not

downwards closed.

For instance x � y is not preserved under subteams.

Inclusion logic is closed under union. Independence logic is not.

● Exclusion logic ≡ dependence logic.

● Inclusion logic and dependence logic are incomparable.

● FO + inclusion + exclusion ≡ independence logic.

● Independence logic coincides with Σ also wrt the

expressive power of open formulae.

Corollary. All NP-properties of teams are definable in independence logic.

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Expressive power of independence logic

Contrary to dependence logic, inclusion logic and independence logic are not

downwards closed.

For instance x � y is not preserved under subteams.

Inclusion logic is closed under union. Independence logic is not.

● Exclusion logic ≡ dependence logic.

● Inclusion logic and dependence logic are incomparable.

● FO + inclusion + exclusion ≡ independence logic.

● Independence logic coincides with Σ also wrt the

expressive power of open formulae.

Corollary. All NP-properties of teams are definable in independence logic.

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Negation

We have permitted negation only applied to first-order atoms. But we can

define an operator ψ ↦ ψ¬, taking the negation normal form of ¬ψ (with theconvention that negated dependency atoms are true only in the empty team)

Negation is not a semantic operation, contrary to disjunction, conjunction,

and quantifiers.

Define the semantic extension of ψ on A as [[ψ]]A ∶= {X ∶ A ⊧X ψ}.

When we know [[ψ]]A and [[φ]]A we can easily compute [[φ ∧ ψ]]A and[[φ ∨ ψ]]A (without even knowing the syntax of ψ and φ). Analogousobservations for quantifiers.

However, knowing [[ψ]]A does not provide much knowledge about [[ψ¬]]A.

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Negation

We have permitted negation only applied to first-order atoms. But we can

define an operator ψ ↦ ψ¬, taking the negation normal form of ¬ψ (with theconvention that negated dependency atoms are true only in the empty team)

Negation is not a semantic operation, contrary to disjunction, conjunction,

and quantifiers.

Define the semantic extension of ψ on A as [[ψ]]A ∶= {X ∶ A ⊧X ψ}.

When we know [[ψ]]A and [[φ]]A we can easily compute [[φ ∧ ψ]]A and[[φ ∨ ψ]]A (without even knowing the syntax of ψ and φ). Analogousobservations for quantifiers.

However, knowing [[ψ]]A does not provide much knowledge about [[ψ¬]]A.

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Negation

We have permitted negation only applied to first-order atoms. But we can

define an operator ψ ↦ ψ¬, taking the negation normal form of ¬ψ (with theconvention that negated dependency atoms are true only in the empty team)

Negation is not a semantic operation, contrary to disjunction, conjunction,

and quantifiers.

Define the semantic extension of ψ on A as [[ψ]]A ∶= {X ∶ A ⊧X ψ}.

When we know [[ψ]]A and [[φ]]A we can easily compute [[φ ∧ ψ]]A and[[φ ∨ ψ]]A (without even knowing the syntax of ψ and φ). Analogousobservations for quantifiers.

However, knowing [[ψ]]A does not provide much knowledge about [[ψ¬]]A.

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Negation and interpolation

We only consider structures with at least wo elements. Two formulae ψ and φare contradictory if [[ψ]]A ∩ [[φ]]A = {∅} for any structure A.

Proposition. (Kontinen and Väänänen) For any two contradictory formulae

ψ and φ of dependence logic there is a formula ϑ such that [[ϑ]]A = [[ψ]]A and[[ϑ¬]]A = [[φ]]A for all A.

In this form, this is not true in general for logics with team semantics. For

instance independence logic contains ∀x(x ⊆ y) and =(y) which arecontradictory but we shall see that they have no such interpolant.

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Negation and interpolation

We only consider structures with at least wo elements. Two formulae ψ and φare contradictory if [[ψ]]A ∩ [[φ]]A = {∅} for any structure A.

Proposition. (Kontinen and Väänänen) For any two contradictory formulae

ψ and φ of dependence logic there is a formula ϑ such that [[ϑ]]A = [[ψ]]A and[[ϑ¬]]A = [[φ]]A for all A.

In this form, this is not true in general for logics with team semantics. For

instance independence logic contains ∀x(x ⊆ y) and =(y) which arecontradictory but we shall see that they have no such interpolant.

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Strongly contradictory formulae

Two formulae ψ and φ are strongly contradictory if X ∩ Y = ∅ for all teamsX ,Y and all structures A such that A ⊧X ψ and A ⊧Y φ.

Lemma. Every formula is strongly contradictory to its negation.

Corollary. Only a pair of strongly contradictory formulae ψ, φ can have aninterpolant ϑ with ψ ≡ ϑ and φ ≡ ϑ¬.

�e formulae ∀x(x ⊆ y) and =(y) are contradictory (on structures with atleast two elements) but have no interpolant, since they are not strongly

contradictory. Indeed, let sa be the assignment y ↦ a.�enX = {sa ∶ a ∈ A} ∈ [[∀x(x ⊆ y)]]A. Further, {sa} ∈ [[=(y)]]A for all a ∈ A.However, X ∩ {sa} = {sa} ≠ ∅.

Being strongly contradictory is a necessary condition for having an

interpolant. It turns out that it is also sufficient.

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Completely undetermined sentences

A sentence is completely undetermined if neither the sentence itself, nor its

negation is true on any structure with more than one element.

Examples: ∀x =(x) or ∀x∃y(x�y ∧ x = y)

Proposition. Let L be any logic with team semantics, closed under first-orderoperations, containing a completely undetermined sentence ��.�en for anyψ, φ ∈ L, the following are equivalent() �ere is a formula η such that ψ ⊧ η and φ ⊧ η¬

() �ere is a formula ϑ such that ψ ≡ ϑ and φ ≡ ϑ¬

Proof. ()⇒ (): Take η = ϑ.

()⇒ (): Take ϑ ∶= (ψ ∨ ��) ∧ ((φ ∨ ��)¬ ∨ η).�enϑ ≡ ψ ∧ (� ∨ η) ≡ ψ ∧ η ≡ ψϑ¬ = (ψ ∨ ��)¬ ∨ ((φ ∨ ��) ∧ η¬) ≡ � ∨ (φ ∧ η¬) ≡ φ ∧ η¬ ≡ φ.

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-closed formulae

A formula ψ -closed if whenever A ⊧X ψ then also A ⊧{s} ψ for all s ∈ X.

All formulae of dependence logic are -closed. Further independence atoms

(and in fact all purely existential formulae of independence logic) are -closed.

Lemma. Let L be a logic with team semantics, that is closed under first-orderoperations and translatable into Σ.�en for every formula ψ ∈ L there exists aformula ψ↓ in dependence logic such that the teams satisfying ψ↓ are exactlythe subteams of the teams satisfying ψ.

Proof. Translate ψ into ψ∗(Y) ∈ Σ and letφ(X) = ∃Y(∀x(Xx → Yx) ∧ ψ∗(Y)). Since X appears only negatively inφ(X), it can be translated into an equivalent formula ψ↓.

Notice that ψ ⊧ ψ↓ and that ψ↓ is -closed. Further ψ and φ are stronglycontradictory if, and only if, ψ↓ and φ↓ are contradictory (and hence stronglycontradictory).

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�e Interpolation�eorem

Let L be a logic with team semantics, which contains FO and can be embeddedinto Σ, and which has a totally undetermined sentence.

�eorem. For any two strongly contradictory formulae ψ, φ from L, thereexists a formula ϑ in L such that ψ ≡ ϑ and φ ≡ ϑ¬.

Let ∃Rψ̃(R, X) and ∃Sφ̃(S , X) be Σ-translations of ψ↓ and φ↓.�enψ̃(R, X) ⊧ (¬φ̃(S , X) ∨ X = ∅) is a valid first-order implication.

By Craig’s Interpolation�eorem there is a first-order sentence η̃(X) such thatψ̃(R, X) ⊧ η̃(X) and (φ̃(S , X) ∧ X ≠ ∅) ⊧ ¬η̃(X).

Let η(x) ∶= η̃[Xz/z = x]. For any team {s} of size one, we have (A, {s}) ⊧ η̃ if,and only if, A ⊧{s} η.

�en ψ ⊧ η and φ ⊧ ¬η (and hence an interpolant ϑ exists)

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Part II: Model-Checking Games

Model-checking games for first-order logic

Second-order reachability games

Model-checking games for logics with team semantics

Examples

Complexity

Deterministic versus nondeterministic strategies

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Model-Checking Games

�e model checking problem for a logic L (with classical Tarski-semantics)

Given: structure A

formula ψ(x) ∈ Lassignment s ∶ free(ψ)→ A

Question: A ⊧s ψ ?

Reduce model checking problem A ⊧s ψ to strategy problem for modelchecking game G(A,ψ, s), played by– Falsifier (also called Player ), and

– Verifier (also called Player ), such that

A ⊧s ψ ⇐⇒ Verifier has winning strategy for G(A,ψ, s)

Ô⇒ Model checking via construction of winning strategies

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Model-Checking Games

�e model checking problem for a logic L (with classical Tarski-semantics)

Given: structure A

formula ψ(x) ∈ Lassignment s ∶ free(ψ)→ A

Question: A ⊧s ψ ?

Reduce model checking problem A ⊧s ψ to strategy problem for modelchecking game G(A,ψ, s), played by– Falsifier (also called Player ), and

– Verifier (also called Player ), such that

A ⊧s ψ ⇐⇒ Verifier has winning strategy for G(A,ψ, s)

Ô⇒ Model checking via construction of winning strategies

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Model-Checking Games

�e model checking problem for a logic L (with classical Tarski-semantics)

Given: structure A

formula ψ(x) ∈ Lassignment s ∶ free(ψ)→ A

Question: A ⊧s ψ ?

Reduce model checking problem A ⊧s ψ to strategy problem for modelchecking game G(A,ψ, s), played by– Falsifier (also called Player ), and

– Verifier (also called Player ), such that

A ⊧s ψ ⇐⇒ Verifier has winning strategy for G(A,ψ, s)

Ô⇒ Model checking via construction of winning strategies

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Model-checking game for first-order logic

�e game G(A,ψ, t) for a structure A and ψ(x) ∈ FO.

Positions: (φ, s) φ is a subformula of ψ and s ∶ free(φ)→ A

Verifier moves:

(φ ∨ φ, s)→ (φi , s ↾ free(φi)) (i = , )(∃xφ, s)→ (φ, s[x ↦ a]) (a ∈ A)

Falsifier moves

(φ ∧ φ, s)→ (φi , s ↾ free(φi)) (i = , )(∀xφ, s)→ (φ, s[x ↦ a]) (a ∈ A)

Terminal positions: φ atomic / negated atomic

Verifier

Falsifierwins at (φ, s) ⇐⇒ A

⊧s/⊧s

φ

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Reachability games

Game graph G = (V ,V,V, T , I, E), V = V ∪ V ∪ T , E ⊆ V × V

Player moves from positions v ∈ V,Player moves from v ∈ V,T is the set of terminal positions.

I is the set of initial positions

Moves are along edges.

Hence plays are finite or infinite paths through the graph

Winning condition Win ⊆ T : at a terminal position v ∈ T ,Player has won if v ∈Win and Player has won if v ∈ T ∖Win.Infinite plays are draws.

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Winning strategies

A (nondeterministic) strategy of Player for G, defined onW ⊆ V , is asubgraph S = (W , F) ⊆ (V , E) such that

() if v ∈ V ∩W then vF ≠ ∅,

() if v ∈ V ∩W then vF = vE.

�e strategy S = (W , F) is winning from X ⊆ I if X ⊆W and all plays that

start at some v ∈ X and are consistent with S reach a winning terminalposition w ∈Win.

Given a reachability game game (G ,Win) on a finite game graph, one cancompute in linear time O(∣V ∣ + ∣E∣)

the maximal subset X ⊆ I from which Player can win, and

a winning strategy from X.

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Reachability games for logics with team semantics

Reduce model checking problem A ⊧X ψ to a reachability game G(A, X ,ψ).

Positions: (φ,Y) Y is now a team of assignments s ∶ free(φ)→ A

Falsifier moves:

(φ ∧ φ,Y)→ (φi ,Y ↾ free(φi)) (i = , )(∀xφ,Y)→ (φ,Y[x ↦ A])

Verifier moves:

(∃xφ,Y)→ (φ,Y[x ↦ F]) for some F ∶ Y → (P(A) ∖ {∅})

Protocol for disjunctions:

At (φ ∨ φ,Y) Verifier selects Y,Y such that Y ∪ Y = Y .Falsifier selects i ∈ {, } and moves to (φi ,Yi ↾ free(φi))

Terminal positions: φ atomic / negated atomic

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Reachability games for logics with team semantics

Reduce model checking problem A ⊧X ψ to a reachability game G(A, X ,ψ).

Positions: (φ,Y) Y is now a team of assignments s ∶ free(φ)→ A

Falsifier moves:

(φ ∧ φ,Y)→ (φi ,Y ↾ free(φi)) (i = , )(∀xφ,Y)→ (φ,Y[x ↦ A])

Verifier moves:

(∃xφ,Y)→ (φ,Y[x ↦ F]) for some F ∶ Y → (P(A) ∖ {∅})

Protocol for disjunctions:

At (φ ∨ φ,Y) Verifier selects Y,Y such that Y ∪ Y = Y .Falsifier selects i ∈ {, } and moves to (φi ,Yi ↾ free(φi))

Terminal positions: φ atomic / negated atomic

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Do reachability games provide a satisfactory game-theoretic

analysis for logics with team semantics?

�e constructed reachability game really is the model-checking game for the

translated formula into existential second-order logic.

�e size of this model is exponentially larger than the first-order game.

It is not clear how to extract complexity results for model-checking problems

out of these games

�e game for the negated formula is not obtained by a simple dualization

operation or a role switch of players

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Do reachability games provide a satisfactory game-theoretic

analysis for logics with team semantics?

�e constructed reachability game really is the model-checking game for the

translated formula into existential second-order logic.

�e size of this model is exponentially larger than the first-order game.

It is not clear how to extract complexity results for model-checking problems

out of these games

�e game for the negated formula is not obtained by a simple dualization

operation or a role switch of players

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Do reachability games provide a satisfactory game-theoretic

analysis for logics with team semantics?

�e constructed reachability game really is the model-checking game for the

translated formula into existential second-order logic.

�e size of this model is exponentially larger than the first-order game.

It is not clear how to extract complexity results for model-checking problems

out of these games

�e game for the negated formula is not obtained by a simple dualization

operation or a role switch of players

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Do reachability games provide a satisfactory game-theoretic

analysis for logics with team semantics?

�e constructed reachability game really is the model-checking game for the

translated formula into existential second-order logic.

�e size of this model is exponentially larger than the first-order game.

It is not clear how to extract complexity results for model-checking problems

out of these games

�e game for the negated formula is not obtained by a simple dualization

operation or a role switch of players

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Second-order reachability games

Game graph: G = (V ,V,V, T , I, E)

Winning condition for Player : Win ⊆ P(T).

Does Player have a (nondeterministic) strategy such that the set of terminal

positions that are reachable by a play from I that is consistent with the strategybelongs to Win ?

A consistent winning strategy of Player from X is a strategyS = (W , F) ⊆ (V , E) such that

() W is the set of nodes that are reachable from X via edges in F

() W ∩ T ∈Win.

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Complexity

�eorem. �e problem whether a given game graph G with a compactdescription for Win admits a consistent winning strategy for Player , is

NP-complete.

�e problem is obviously in NP: guess a subgraph, and verify that it is a

consistent winning strategy.

NP-hardness by reduction from SAT. Given a CNF-formula ψ consider theobvious game G(ψ), where Player can move from the initial position ψ toclauses, and Player from clauses to their literals, with

T ∶= {(C ,Y) ∶ C is a clause of ψ,Y ∈ C}Win ∶= {U ⊆ T ∶ U contains no conflicting pair (C ,Y), (C′,Y)}

Player has a consistent winning strategy for G(ψ) ⇐⇒ ψ is satisfiable

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Complexity

�eorem. �e problem whether a given game graph G with a compactdescription for Win admits a consistent winning strategy for Player , is

NP-complete.

�e problem is obviously in NP: guess a subgraph, and verify that it is a

consistent winning strategy.

NP-hardness by reduction from SAT. Given a CNF-formula ψ consider theobvious game G(ψ), where Player can move from the initial position ψ toclauses, and Player from clauses to their literals, with

T ∶= {(C ,Y) ∶ C is a clause of ψ,Y ∈ C}Win ∶= {U ⊆ T ∶ U contains no conflicting pair (C ,Y), (C′,Y)}

Player has a consistent winning strategy for G(ψ) ⇐⇒ ψ is satisfiable

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Complexity

�eorem. �e problem whether a given game graph G with a compactdescription for Win admits a consistent winning strategy for Player , is

NP-complete.

�e problem is obviously in NP: guess a subgraph, and verify that it is a

consistent winning strategy.

NP-hardness by reduction from SAT. Given a CNF-formula ψ consider theobvious game G(ψ), where Player can move from the initial position ψ toclauses, and Player from clauses to their literals, with

T ∶= {(C ,Y) ∶ C is a clause of ψ,Y ∈ C}Win ∶= {U ⊆ T ∶ U contains no conflicting pair (C ,Y), (C′,Y)}

Player has a consistent winning strategy for G(ψ) ⇐⇒ ψ is satisfiable

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Model-checking game for logics with team semantics

Consider FO together with a collection of atomic properties of teams.

Appropriate model checking games are obtained as follows:

Take the same game graphs G(A,ψ) as for FO with Tarski-semantics(making sure that distinct occurrences of the same subformula are

represented by distinct nodes).

�e set of initial positions is I ∶= {(ψ, s) ∶ s ∶ free(ψ)→ A}.

�e winning condition is a second-order reachability condition.

�is imposes consistency conditions on the admissible strategies.

�e resulting games G(A,ψ), where Verifier may use only consistentstrategies, can be viewed as games of imperfect information.

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Model-checking game for logics with team semantics

Consider FO together with a collection of atomic properties of teams.

Appropriate model checking games are obtained as follows:

Take the same game graphs G(A,ψ) as for FO with Tarski-semantics(making sure that distinct occurrences of the same subformula are

represented by distinct nodes).

�e set of initial positions is I ∶= {(ψ, s) ∶ s ∶ free(ψ)→ A}.

�e winning condition is a second-order reachability condition.

�is imposes consistency conditions on the admissible strategies.

�e resulting games G(A,ψ), where Verifier may use only consistentstrategies, can be viewed as games of imperfect information.

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Second-order reachability games for model checking

Let (V ,V,V, T , I, E) be the game graph of a model-checking game G(A,ψ).Recall that V = {(φ, s) ∶ φ is a subformula of ψ, s ∶ free(φ)→ A}

Teams defined by strategies: For any subset U ⊆ V and any subformula φ of ψ

Team(W , φ) ∶= {s ∶ (φ, s) ∈W}

Second-order reachability condition Win ⊆ P(T):T = {(φ, s) ∈ V ∶ φ is an atomic or negated atomic subformula of ψ}Win ∶= {U ⊆ T ∶ A ⊧Team(U ,φ) φ for every atomic/negated atomic φ}.

A consistent winning strategy for Player (Verifier) S = (W , F) thus has toguarantee that A ⊧Team(W ,φ) φ for every atomic or negated atomic formula φ.

Notice that this refers to the entire set of terminal positions that are reachable

by plays that are consistent with the strategy.

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Second-order reachability games for model checking

Let (V ,V,V, T , I, E) be the game graph of a model-checking game G(A,ψ).Recall that V = {(φ, s) ∶ φ is a subformula of ψ, s ∶ free(φ)→ A}

Teams defined by strategies: For any subset U ⊆ V and any subformula φ of ψ

Team(W , φ) ∶= {s ∶ (φ, s) ∈W}

Second-order reachability condition Win ⊆ P(T):T = {(φ, s) ∈ V ∶ φ is an atomic or negated atomic subformula of ψ}Win ∶= {U ⊆ T ∶ A ⊧Team(U ,φ) φ for every atomic/negated atomic φ}.

A consistent winning strategy for Player (Verifier) S = (W , F) thus has toguarantee that A ⊧Team(W ,φ) φ for every atomic or negated atomic formula φ.

Notice that this refers to the entire set of terminal positions that are reachable

by plays that are consistent with the strategy.

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Second-order reachability games for model checking

Let (V ,V,V, T , I, E) be the game graph of a model-checking game G(A,ψ).Recall that V = {(φ, s) ∶ φ is a subformula of ψ, s ∶ free(φ)→ A}

Teams defined by strategies: For any subset U ⊆ V and any subformula φ of ψ

Team(W , φ) ∶= {s ∶ (φ, s) ∈W}

Second-order reachability condition Win ⊆ P(T):T = {(φ, s) ∈ V ∶ φ is an atomic or negated atomic subformula of ψ}Win ∶= {U ⊆ T ∶ A ⊧Team(U ,φ) φ for every atomic/negated atomic φ}.

A consistent winning strategy for Player (Verifier) S = (W , F) thus has toguarantee that A ⊧Team(W ,φ) φ for every atomic or negated atomic formula φ.

Notice that this refers to the entire set of terminal positions that are reachable

by plays that are consistent with the strategy.

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Correctness of the model checking games

�e consistency condition for winning strategies translates from the atomic

formulae to all positions of the game.

If S = (W , F) is a consistent winning strategy for G(A,ψ) then, for allsubformulae φ of ψ we have that A ⊧Team(S ,φ) φ.

�eorem.

A ⊧X ψ ⇐⇒ Verifier has a consistent winning strategy S for G(A,ψ)with Team(S ,ψ) = X.

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Correctness of the model checking games

�e consistency condition for winning strategies translates from the atomic

formulae to all positions of the game.

If S = (W , F) is a consistent winning strategy for G(A,ψ) then, for allsubformulae φ of ψ we have that A ⊧Team(S ,φ) φ.

�eorem.

A ⊧X ψ ⇐⇒ Verifier has a consistent winning strategy S for G(A,ψ)with Team(S ,ψ) = X.

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Complexity

Recall that the problem whether a given game graph G with a compactdescription for Win admits a consistent winning strategy for Player , is

NP-complete.

�e size of a model checking game G(A,ψ) on a finite structure A is boundedby ∣ψ∣ ⋅ ∣A∣width(ψ).

�eorem. Let L be any extension of first-order logic by atomic formulae onteams that can be evaluated in polynomial time.�en the model-checking

problem for L on finite structures is in N. For formulae of boundedwidth, the model-checking problem is in NP.

In fact, the problem is N-complete, even in the case where A is just

the set {, } and X = {∅}.

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Complexity

Recall that the problem whether a given game graph G with a compactdescription for Win admits a consistent winning strategy for Player , is

NP-complete.

�e size of a model checking game G(A,ψ) on a finite structure A is boundedby ∣ψ∣ ⋅ ∣A∣width(ψ).

�eorem. Let L be any extension of first-order logic by atomic formulae onteams that can be evaluated in polynomial time.�en the model-checking

problem for L on finite structures is in N. For formulae of boundedwidth, the model-checking problem is in NP.

In fact, the problem is N-complete, even in the case where A is just

the set {, } and X = {∅}.

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Complexity

Recall that the problem whether a given game graph G with a compactdescription for Win admits a consistent winning strategy for Player , is

NP-complete.

�e size of a model checking game G(A,ψ) on a finite structure A is boundedby ∣ψ∣ ⋅ ∣A∣width(ψ).

�eorem. Let L be any extension of first-order logic by atomic formulae onteams that can be evaluated in polynomial time.�en the model-checking

problem for L on finite structures is in N. For formulae of boundedwidth, the model-checking problem is in NP.

In fact, the problem is N-complete, even in the case where A is just

the set {, } and X = {∅}.

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Example: -colourability

X(G) = {s ∶ (edge,node)↦ (e , u) ∶ e is an edge of G and u ∈ e}

ψ(edge,node) ∶= ∃colour((=(colour) ∨ =(colour) ∨ =(colour))∧=(node,colour) ∧ =(edge,colour,node)).

Claim. G is -colourable ⇐⇒ V ∪ E ⊧X(G) ψ(edge,node)

A consistent winning strategy S = (W , F) with Team(S ,ψ) = X(G) selects, forevery assignment s ∶ (edge,node)↦ (e , u) at least one colour c(s).

First conjunct: at most three colours are used

Second conjunct: the colour of (e , u) only depends on the node u

Final conjunct: the edge and the colour determine the node, i.e. a different

colour is assigned to (e , u) and (e , v) for every edge e = {u, v}.

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Example: -colourability

X(G) = {s ∶ (edge,node)↦ (e , u) ∶ e is an edge of G and u ∈ e}

ψ(edge,node) ∶= ∃colour((=(colour) ∨ =(colour) ∨ =(colour))∧=(node,colour) ∧ =(edge,colour,node)).

Claim. G is -colourable ⇐⇒ V ∪ E ⊧X(G) ψ(edge,node)

A consistent winning strategy S = (W , F) with Team(S ,ψ) = X(G) selects, forevery assignment s ∶ (edge,node)↦ (e , u) at least one colour c(s).

First conjunct: at most three colours are used

Second conjunct: the colour of (e , u) only depends on the node u

Final conjunct: the edge and the colour determine the node, i.e. a different

colour is assigned to (e , u) and (e , v) for every edge e = {u, v}.

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Inclusion logic

Inclusion logic: FO + inclusion atoms x ⊆ y.

Recall that A ⊧X x ⊆ y if for all s ∈ X there exists a t ∈ X with t(y) = s(x).

Inclusion logic is an interesting variant of a dependence logic, with somewhat

unusual properties.

Contrary to dependence logic, it is not closed under subteams, but it is closed

under unions of teams: If A ⊧X ψ and A ⊧Y ψ then also A ⊧X∪Y ψ

Inclusion logic is closely related to (a fragment of) least fixed-point logic (LFP).

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Example: Infinite descending chains

(A, <) ⊧ ∃x∃y(y < x ∧ y ⊆ x) ⇐⇒ (A, <) is not well-founded

A consistent winning strategy S = (W , F) for this sentence just amounts to aselection of a team X of assignments s ∶ (x , y)↦ (a, b).

All atoms are reachable by the opponent. Hence Team(S , y < x) =Team(S , y ⊆ x) = X.�us, consistency of S means:

(A, <) ⊧X (y < x): For all s ∶ (x , y)↦ (a, b) in X, we have that b < a

(A, <) ⊧X (y ⊆ x): For all s ∶ (x , y)↦ (a, b) in X, we have anotherassignment t ∶ (x , y)↦ (b, c) in X.

Hence we have a consistent winning strategy in the model-checking game if,

and only if, (A, <) has an infinite descending chain.

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Example: Infinite descending chains

(A, <) ⊧ ∃x∃y(y < x ∧ y ⊆ x) ⇐⇒ (A, <) is not well-founded

A consistent winning strategy S = (W , F) for this sentence just amounts to aselection of a team X of assignments s ∶ (x , y)↦ (a, b).

All atoms are reachable by the opponent. Hence Team(S , y < x) =Team(S , y ⊆ x) = X.�us, consistency of S means:

(A, <) ⊧X (y < x): For all s ∶ (x , y)↦ (a, b) in X, we have that b < a

(A, <) ⊧X (y ⊆ x): For all s ∶ (x , y)↦ (a, b) in X, we have anotherassignment t ∶ (x , y)↦ (b, c) in X.

Hence we have a consistent winning strategy in the model-checking game if,

and only if, (A, <) has an infinite descending chain.

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Example: Infinite descending chains

(A, <) ⊧ ∃x∃y(y < x ∧ y ⊆ x) ⇐⇒ (A, <) is not well-founded

A consistent winning strategy S = (W , F) for this sentence just amounts to aselection of a team X of assignments s ∶ (x , y)↦ (a, b).

All atoms are reachable by the opponent. Hence Team(S , y < x) =Team(S , y ⊆ x) = X.�us, consistency of S means:

(A, <) ⊧X (y < x): For all s ∶ (x , y)↦ (a, b) in X, we have that b < a

(A, <) ⊧X (y ⊆ x): For all s ∶ (x , y)↦ (a, b) in X, we have anotherassignment t ∶ (x , y)↦ (b, c) in X.

Hence we have a consistent winning strategy in the model-checking game if,

and only if, (A, <) has an infinite descending chain.

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Example: Infinite descending chains

(A, <) ⊧ ∃x∃y(y < x ∧ y ⊆ x) ⇐⇒ (A, <) is not well-founded

A consistent winning strategy S = (W , F) for this sentence just amounts to aselection of a team X of assignments s ∶ (x , y)↦ (a, b).

All atoms are reachable by the opponent. Hence Team(S , y < x) =Team(S , y ⊆ x) = X.�us, consistency of S means:

(A, <) ⊧X (y < x): For all s ∶ (x , y)↦ (a, b) in X, we have that b < a

(A, <) ⊧X (y ⊆ x): For all s ∶ (x , y)↦ (a, b) in X, we have anotherassignment t ∶ (x , y)↦ (b, c) in X.

Hence we have a consistent winning strategy in the model-checking game if,

and only if, (A, <) has an infinite descending chain.

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Example: Infinite descending chains

(A, <) ⊧ ∃x∃y(y < x ∧ y ⊆ x) ⇐⇒ (A, <) is not well-founded

A consistent winning strategy S = (W , F) for this sentence just amounts to aselection of a team X of assignments s ∶ (x , y)↦ (a, b).

All atoms are reachable by the opponent. Hence Team(S , y < x) =Team(S , y ⊆ x) = X.�us, consistency of S means:

(A, <) ⊧X (y < x): For all s ∶ (x , y)↦ (a, b) in X, we have that b < a

(A, <) ⊧X (y ⊆ x): For all s ∶ (x , y)↦ (a, b) in X, we have anotherassignment t ∶ (x , y)↦ (b, c) in X.

Hence we have a consistent winning strategy in the model-checking game if,

and only if, (A, <) has an infinite descending chain.

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Example: Infinite descending chains

(A, <) ⊧ ∃x∃y(y < x ∧ y ⊆ x) ⇐⇒ (A, <) is not well-founded

A consistent winning strategy S = (W , F) for this sentence just amounts to aselection of a team X of assignments s ∶ (x , y)↦ (a, b).

All atoms are reachable by the opponent. Hence Team(S , y < x) =Team(S , y ⊆ x) = X.�us, consistency of S means:

(A, <) ⊧X (y < x): For all s ∶ (x , y)↦ (a, b) in X, we have that b < a

(A, <) ⊧X (y ⊆ x): For all s ∶ (x , y)↦ (a, b) in X, we have anotherassignment t ∶ (x , y)↦ (b, c) in X.

Hence we have a consistent winning strategy in the model-checking game if,

and only if, (A, <) has an infinite descending chain.

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Independence versus Henkin quantifiers

Independence atoms can be used to describe semantics of Henkin quantifiers.

φ ∶= ( ∀x ∃y∀u ∃v

) Pxyuv

(A, P) ⊧ φ if there exist functions f , g ∶ A→ A such that A ⊧ P(a, f a, c, gc)for all a, c ∈ A.

ψ ∶= ∀x∃y∀u∃v(xy�uv ∧ Pxyuv)

Game-theoretic semantics: (A, P) ⊧ ψ if there exist functionsF ∶ A→ P(A) ∖ {∅} and G ∶ A× A× A→ P(A) ∖ {∅} such that the team

XFG = {s ∶ (x , y, u, v)↦ (a, b, c, d) ∶ b ∈ F(a) and d ∈ G(a, b, c)}

satisfies xy�uv and Pxyuv

Claim. �e two formulae are equivalent.

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Independence versus Henkin quantifiers

Independence atoms can be used to describe semantics of Henkin quantifiers.

φ ∶= ( ∀x ∃y∀u ∃v

) Pxyuv

(A, P) ⊧ φ if there exist functions f , g ∶ A→ A such that A ⊧ P(a, f a, c, gc)for all a, c ∈ A.

ψ ∶= ∀x∃y∀u∃v(xy�uv ∧ Pxyuv)

Game-theoretic semantics: (A, P) ⊧ ψ if there exist functionsF ∶ A→ P(A) ∖ {∅} and G ∶ A× A× A→ P(A) ∖ {∅} such that the team

XFG = {s ∶ (x , y, u, v)↦ (a, b, c, d) ∶ b ∈ F(a) and d ∈ G(a, b, c)}

satisfies xy�uv and Pxyuv

Claim. �e two formulae are equivalent.

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Independence versus Henkin quantifiers

φ ∶= ( ∀x ∃y∀u ∃v

) Pxyuv ⊧ ψ ∶= ∀x∃y∀u∃v(xy�uv ∧ Pxyuv)

From Skolem functions f , g for φ we immediately get a consistent winningstrategy for ψ, witnessed by F(a) ∶= { f a} and G(a, b, c) ∶= {gc}.

�e team XFG then consists of all assignments (a, f a, c, gc) which clearlysatisfies Pxyuv and also satisfies the independence atom xy�uv since v isconstant for any fixed value for u.

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Independence versus Henkin quantifiers

ψ ∶= ∀x∃y∀u∃v(xy�uv ∧ Pxyuv) ⊧ φ ∶= ( ∀x ∃y∀u ∃v

) Pxyuv

Given a consistent winning strategy for ψ on A, witnessed by F and G,define Skolem functions for φ via a choice function ε on P(A).Set f a ∶= εF(a) and gc ∶= ε(⋃a∈AG(a, f a, c)).

Claim. P(a, f a, c, gc) for all a, c ∈ A.

Prove that, for all a, c, the assignment (a, f a, c, gc) belongs to XFG .

() Given a, c, there exists a′ ∈ A such that gc ∈ G(a′, f a′, c); hence(a′, f a′, c, gc) ∈ XFG .

() (a, f a, c, d) ∈ XFG for all d ∈ G(a, f a, c).

() By the independence atom we infer that XFG also contains (a, f a, c, gc).

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Deterministic versus nondeterministic strategies

Our notion of consistent winning strategies is nondeteministic:

A consistent winning strategy of Player for G and Win from X is a pairS = (W , F) ⊆ (V , E) with F ⊆ (W ×W) ∩ E such that

() W is the set of nodes that are reachable from X via edges in F

() if v ∈ V ∩W then vF ≠ ∅

() if v ∈ V ∩W then vF = vE

() W ∩ T ∈Win.

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Deterministic versus nondeterministic strategies

�e deterministic variant:

A deterministic consistent winning strategy of Player for G and Win from Xis a pair S = (W , F) ⊆ (V , E) with F ⊆ (W ×W) ∩ E such that

() W is the set of nodes that are reachable from X via edges in F

() if v ∈ V ∩W then ∣vF∣ =

() if v ∈ V ∩W then vF = vE

() W ∩ T ∈Win.

In most classical games, deterministic strategies are no less powerful than

nondeterministic ones. Is this also the case for second-order reachability

games?

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Why is the nondeterministic semantics the right one ?

Consider the formula ∃x(y ⊆ x ∧ z ⊆ x) which says�e team under consideration can be extended by values for x suchthat all values for y and z in the team occur als values for x

True for any team X: assign to x all values occurring for y and z in X.

But under deterministic (strict) semantics for (∃x) this not the case. LetX = {s} with s(y) ≠ s(z). By chosing just one witness for x, we cannot makethe formula true for this team.

So under deterministic semantics, this formula says something different.

So what?

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Why is the nondeterministic semantics the right one ?

Consider the formula ∃x(y ⊆ x ∧ z ⊆ x) which says�e team under consideration can be extended by values for x suchthat all values for y and z in the team occur als values for x

True for any team X: assign to x all values occurring for y and z in X.

But under deterministic (strict) semantics for (∃x) this not the case. LetX = {s} with s(y) ≠ s(z). By chosing just one witness for x, we cannot makethe formula true for this team.

So under deterministic semantics, this formula says something different.

So what?

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Why is the nondeterministic semantics the right one ?

Consider the formula ∃x(y ⊆ x ∧ z ⊆ x) which says�e team under consideration can be extended by values for x suchthat all values for y and z in the team occur als values for x

True for any team X: assign to x all values occurring for y and z in X.

But under deterministic (strict) semantics for (∃x) this not the case. LetX = {s} with s(y) ≠ s(z). By chosing just one witness for x, we cannot makethe formula true for this team.

So under deterministic semantics, this formula says something different.

So what?

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Why is the nondeterministic semantics the right one ?

Consider the formula ∃x(y ⊆ x ∧ z ⊆ x) which says�e team under consideration can be extended by values for x suchthat all values for y and z in the team occur als values for x

True for any team X: assign to x all values occurring for y and z in X.

But under deterministic (strict) semantics for (∃x) this not the case. LetX = {s} with s(y) ≠ s(z). By chosing just one witness for x, we cannot makethe formula true for this team.

So under deterministic semantics, this formula says something different.

So what?

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Why is the nondeterministic semantics the right one ?

Consider the formula ∃x(y ⊆ x ∧ z ⊆ x) and the team

X =y z u

Clearly X ⊧ ∃x(y ⊆ x ∧ z ⊆ x) even under deterministic semantics.

But under deterministic semantics X ↾ {y, z} /⊧ ∃x(y ⊆ x ∧ z ⊆ x)

Hence deterministic semantics violates the locality principle !

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Why is the nondeterministic semantics the right one ?

Consider the formula ∃x(y ⊆ x ∧ z ⊆ x) and the team

X =y z u

Clearly X ⊧ ∃x(y ⊆ x ∧ z ⊆ x) even under deterministic semantics.

But under deterministic semantics X ↾ {y, z} /⊧ ∃x(y ⊆ x ∧ z ⊆ x)

Hence deterministic semantics violates the locality principle !

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Why is the nondeterministic semantics the right one ?

Consider the formula ∃x(y ⊆ x ∧ z ⊆ x) and the team

X =y z u

Clearly X ⊧ ∃x(y ⊆ x ∧ z ⊆ x) even under deterministic semantics.

But under deterministic semantics X ↾ {y, z} /⊧ ∃x(y ⊆ x ∧ z ⊆ x)

Hence deterministic semantics violates the locality principle !

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Downwards Closure

In model-checking games for dependence logic, deterministic strategies

suffice.�e reason is that dependence logic is downwards closed for teams.

A winning condition Win ⊆ P(T) is downwards closed if U ∈Win and Z ⊆ Uimply Z ∈Win.

Proposition. Let Win be downwards closed.�en Player has a consistent

winning strategy for G and Win if, and only if, she has a deterministic one.

Is this sufficient condition also necessary?

No, but we can find a weaker condition, that is both necessary and sufficient

for guaranteeing the possibility to elminate nondeterministic strategies.

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Downwards Closure

In model-checking games for dependence logic, deterministic strategies

suffice.�e reason is that dependence logic is downwards closed for teams.

A winning condition Win ⊆ P(T) is downwards closed if U ∈Win and Z ⊆ Uimply Z ∈Win.

Proposition. Let Win be downwards closed.�en Player has a consistent

winning strategy for G and Win if, and only if, she has a deterministic one.

Is this sufficient condition also necessary?

No, but we can find a weaker condition, that is both necessary and sufficient

for guaranteeing the possibility to elminate nondeterministic strategies.

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Downwards Closure

In model-checking games for dependence logic, deterministic strategies

suffice.�e reason is that dependence logic is downwards closed for teams.

A winning condition Win ⊆ P(T) is downwards closed if U ∈Win and Z ⊆ Uimply Z ∈Win.

Proposition. Let Win be downwards closed.�en Player has a consistent

winning strategy for G and Win if, and only if, she has a deterministic one.

Is this sufficient condition also necessary?

No, but we can find a weaker condition, that is both necessary and sufficient

for guaranteeing the possibility to elminate nondeterministic strategies.

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Splits

A collection F ⊆ P(T) has a split if there exist U,U /∈ F such thatU ∪U ∈ F .

If F is downwards closed, then it has no splits.�e converse is not true.

�eorem. If Win ⊆ P(T) has no splits, then for any second-order reachabilitygame (G ,Win), Player has a consistent winning strategy, if and only if, shehas a deterministic one. Conversely, if Win ⊆ P(T) has a split, then thereexists a game graph G with T as its set of terminal nodes such that Player hasa consistent winning strategy for (G ,Win) but not a deterministic one.

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Splits

A collection F ⊆ P(T) has a split if there exist U,U /∈ F such thatU ∪U ∈ F .

If F is downwards closed, then it has no splits.�e converse is not true.

�eorem. If Win ⊆ P(T) has no splits, then for any second-order reachabilitygame (G ,Win), Player has a consistent winning strategy, if and only if, shehas a deterministic one. Conversely, if Win ⊆ P(T) has a split, then thereexists a game graph G with T as its set of terminal nodes such that Player hasa consistent winning strategy for (G ,Win) but not a deterministic one.

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Splits

A collection F ⊆ P(T) has a split if there exist U,U /∈ F such thatU ∪U ∈ F .

If F is downwards closed, then it has no splits.�e converse is not true.

�eorem. If Win ⊆ P(T) has no splits, then for any second-order reachabilitygame (G ,Win), Player has a consistent winning strategy, if and only if, shehas a deterministic one. Conversely, if Win ⊆ P(T) has a split, then thereexists a game graph G with T as its set of terminal nodes such that Player hasa consistent winning strategy for (G ,Win) but not a deterministic one.

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Inclusion logic and least fixed-point logic

For inclusion logic, the second-order reachability games describing the

model-checking problem can be translated into equivalent safety games.

Safety games also are the model-checking games for fixed-point formula, that

only make use of greatest fixed points: the posGFP-fragment of least

fixed-point logic LFP. Further, fundamental game-theoretic notions for safety

games, such as winning regions and traps can be defined in both posGFP and

inclusion logic.

By means of an interpretation argument one can then provide translations

between the two logics.

�eorem. (Galliani and Hella) Inclusion logic ≡ posGFP.

Corollary. On finite structures, inclusion logic ≡ LFP.In particular, on ordered finite structures, inclusion logic captures P.

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Inclusion logic and least fixed-point logic

For inclusion logic, the second-order reachability games describing the

model-checking problem can be translated into equivalent safety games.

Safety games also are the model-checking games for fixed-point formula, that

only make use of greatest fixed points: the posGFP-fragment of least

fixed-point logic LFP. Further, fundamental game-theoretic notions for safety

games, such as winning regions and traps can be defined in both posGFP and

inclusion logic.

By means of an interpretation argument one can then provide translations

between the two logics.

�eorem. (Galliani and Hella) Inclusion logic ≡ posGFP.

Corollary. On finite structures, inclusion logic ≡ LFP.In particular, on ordered finite structures, inclusion logic captures P.

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Inclusion logic and least fixed-point logic

For inclusion logic, the second-order reachability games describing the

model-checking problem can be translated into equivalent safety games.

Safety games also are the model-checking games for fixed-point formula, that

only make use of greatest fixed points: the posGFP-fragment of least

fixed-point logic LFP. Further, fundamental game-theoretic notions for safety

games, such as winning regions and traps can be defined in both posGFP and

inclusion logic.

By means of an interpretation argument one can then provide translations

between the two logics.

�eorem. (Galliani and Hella) Inclusion logic ≡ posGFP.

Corollary. On finite structures, inclusion logic ≡ LFP.In particular, on ordered finite structures, inclusion logic captures P.

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Part III: �e Quest for a Logic for P:

Fixed-Point Logic, Inclusion Logic, and Counting

Is there a logic for polynomial time?

Least fixed point logic

Inclusion logic versus fixed-point logic

Fixed-point logic with counting

Counting and Team semantics

�reshold games and traps

Inclusion logic with counting versus fixed-point logic with counting

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�e most important problem of Finite Model�eory

Is there a logic that captures P?

Informal definition: A logic L captures P if it defines precisely thoseproperties of finite structures that are decidable in polynomial time:

() For every sentence ψ ∈ L, the set of finite models of ψ is decidable inpolynomial time.

() For every P-property S of finite τ-structures, there is a sentenceψ ∈ L such that S = {A ∈ Fin(τ) ∶ A ⊧ ψ}.

�e precise definition is more subtle. It includes certain effectiveness

requirements to exclude pathological ‘solutions’.

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�e most important problem of Finite Model�eory

Is there a logic that captures P?

Informal definition: A logic L captures P if it defines precisely thoseproperties of finite structures that are decidable in polynomial time:

() For every sentence ψ ∈ L, the set of finite models of ψ is decidable inpolynomial time.

() For every P-property S of finite τ-structures, there is a sentenceψ ∈ L such that S = {A ∈ Fin(τ) ∶ A ⊧ ψ}.

�e precise definition is more subtle. It includes certain effectiveness

requirements to exclude pathological ‘solutions’.

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�e most important problem of Finite Model�eory

Is there a logic that captures P?

Informal definition: A logic L captures P if it defines precisely thoseproperties of finite structures that are decidable in polynomial time:

() For every sentence ψ ∈ L, the set of finite models of ψ is decidable inpolynomial time.

() For every P-property S of finite τ-structures, there is a sentenceψ ∈ L such that S = {A ∈ Fin(τ) ∶ A ⊧ ψ}.

�e precise definition is more subtle. It includes certain effectiveness

requirements to exclude pathological ‘solutions’.

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First-Order Logic

First-order logic (FO) is far too weak to capture P.

FO can express only local properties of finite structures

�eorems of Gaifman and Hanf

Global properties (e.g. planarity of graphs) are not expressible.

FO has no mechanism for recursion or unbounded iteration.

Transitive closures, reachability or termination properties, winning

regions in games, etc. are not FO-definable.

FO can only express properties in AC

AC is constant parallel time with polynomial hardware. In particular,

FO ⊆ L.

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Second-Order Logic

Second-order logic (SO) is (probably) too strong to capture P.

Fagin’s�eorem. Existential SO captures NP.

Corollary. SO captures the polynomial hierarchy.

�us SO captures polynomial time if, and only if, P = NP.

Monadic second-order logic is orthogonal to P:

On words, MSO captures the regular languages, and not all P-languages

are regular.

On graphs, MSO can express NP-complete properties, such as -colourability.

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Second-Order Logic

Second-order logic (SO) is (probably) too strong to capture P.

Fagin’s�eorem. Existential SO captures NP.

Corollary. SO captures the polynomial hierarchy.

�us SO captures polynomial time if, and only if, P = NP.

Monadic second-order logic is orthogonal to P:

On words, MSO captures the regular languages, and not all P-languages

are regular.

On graphs, MSO can express NP-complete properties, such as -colourability.

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Least fixed point logic LFP

Syntax. LFP extends FO by fixed point rule:

For every formula ψ(T , x . . . xk) ∈ LFP[τ ∪ {T}],T k-ary relation variable, occuring only positive in ψ,build formulae [lfpTx .ψ](x) and [gfpTx .ψ](x)

Semantics. On τ-structure A, ψ(T , x) defines monotone operator

ψA ∶ P(Ak)Ð→ P(Ak)T z→ {a ∶ (A, T) ⊧ ψ(T , a)}

A ⊧ [lfpTx .ψ(T , x)](a) ∶⇐⇒ a ∈ lfp(ψA)A ⊧ [gfpTx .ψ(T , x)](a) ∶⇐⇒ a ∈ gfp(ψA)

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LFP-definability for reachability and safety games

Reachability: Player has winning strategy for game G from position v⇐⇒

G = (V ,V,V, E) ⊧ [lfpWx . (Vx ∧ ∃y(Exy ∧Wy))∨ (Vx ∧ ∀y(Exy →Wy)](v)

Safety: Player can avoid losing G from position v⇐⇒

G = (V ,V,V, E) ⊧ [gfpWx . (Vx → ∃y(Exy ∧Wy))∧ (Vx → ∀y(Exy →Wy)](v)

On finite structure, the problem of computing winning regions for reachability

(or safety) games is complete for LFP (via quantifier-free reductions)

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LFP-definability for reachability and safety games

Reachability: Player has winning strategy for game G from position v⇐⇒

G = (V ,V,V, E) ⊧ [lfpWx . (Vx ∧ ∃y(Exy ∧Wy))∨ (Vx ∧ ∀y(Exy →Wy)](v)

Safety: Player can avoid losing G from position v⇐⇒

G = (V ,V,V, E) ⊧ [gfpWx . (Vx → ∃y(Exy ∧Wy))∧ (Vx → ∀y(Exy →Wy)](v)

On finite structure, the problem of computing winning regions for reachability

(or safety) games is complete for LFP (via quantifier-free reductions)

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LFP and polynomial time

�eorem (Immerman, Vardi)

On ordered finite structures, LFP captures P.

On arbitrary finite structures, LFP can express certain P-complete

problems (such as winning regions of reachability and safety games), but fails

to express all of P.

LFP is unable to count.

For instance the class of graphs with an even number of vertices is not

LFP-definable.

Further, LFP has a --law.

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LFP and polynomial time

�eorem (Immerman, Vardi)

On ordered finite structures, LFP captures P.

On arbitrary finite structures, LFP can express certain P-complete

problems (such as winning regions of reachability and safety games), but fails

to express all of P.

LFP is unable to count.

For instance the class of graphs with an even number of vertices is not

LFP-definable.

Further, LFP has a --law.

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�e gfp-fragment of LFP

�e fragment posGFP of least fixed-point logic can be defined in two

equivalent ways:

() posGFP is the closure of the set of formulae of form [gfpRx . φ(R, x)](x),where φ(Rx) is in FO, under disjunction, conjunction, quantifiers, andapplications of gfp.

() posGFP is the set of relations definable by simultaneous greatest fixed

points of systems of first-order formulae.

It is well-known that these two formulations are equivalent.

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�e gfp-fragment of LFP

�e fragment posGFP of least fixed-point logic can be defined in two

equivalent ways:

() posGFP is the closure of the set of formulae of form [gfpRx . φ(R, x)](x),where φ(Rx) is in FO, under disjunction, conjunction, quantifiers, andapplications of gfp.

() posGFP is the set of relations definable by simultaneous greatest fixed

points of systems of first-order formulae.

It is well-known that these two formulations are equivalent.

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�e alternation hierarchy

Fixed-point formulae become difficult to read and evaluate if they involve

more than (very) few alternations of least and greatest fixed points.

Alternation between lfp- and gfp-operations define a hierarchy analogous to

the Σ/Π hierarchies in first-order and second-order logic.

�e fragment posGFP is at the bottom level of the alternation hierarchy of

least fixed-point logic.

�eorem. (Immerman)

On finite structures the alternation hierarchy collapses: LFP ≡ posGFP.

However, this is not the case in general. For instance on (N,+, ⋅ ) thealternation hierarchy of LFP is strict.

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�e alternation hierarchy

Fixed-point formulae become difficult to read and evaluate if they involve

more than (very) few alternations of least and greatest fixed points.

Alternation between lfp- and gfp-operations define a hierarchy analogous to

the Σ/Π hierarchies in first-order and second-order logic.

�e fragment posGFP is at the bottom level of the alternation hierarchy of

least fixed-point logic.

�eorem. (Immerman)

On finite structures the alternation hierarchy collapses: LFP ≡ posGFP.

However, this is not the case in general. For instance on (N,+, ⋅ ) thealternation hierarchy of LFP is strict.

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Model-checking games for LFP and posGFP

�e model-checking games for LFP are parity games. It is, in the general case,

not known whether these can be solved in polynomial time.

�e model-checking games for posGFP are much simpler.�ey are

safety games in which all infinite plays are won by the Verifier.

In fact, also the model-checking games for inclusion logic can be translated

into safety games.

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Model-checking games for LFP and posGFP

�e model-checking games for LFP are parity games. It is, in the general case,

not known whether these can be solved in polynomial time.

�e model-checking games for posGFP are much simpler.�ey are

safety games in which all infinite plays are won by the Verifier.

In fact, also the model-checking games for inclusion logic can be translated

into safety games.

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Safety games for inclusion logic

Let G(A,ψ) be a second-order reachability game for a formula ψ ∈ Inc.We define an associated safety gameH(A,ψ) as follows:

From positions (x ⊆ y, s), Verifier can move to any position (x ⊆ y, t) suchthat t(y) = s(x). From there, Falsifier can make an arbitrary move upwards inthe game forest.

From a given set X ⊆ I of initial positions, Verifier must make sure to avoid- terminal positions (φ, s), where φ is a first-order literal with A ⊧s φ, and- initial positions outside X

Proposition. Every consistent winning strategy S = (W , F) from X forG(A,ψ) is also a winning strategy forH(A,ψ) that avoids I ∖ X, and viceversa.

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Safety games for inclusion logic

Let G(A,ψ) be a second-order reachability game for a formula ψ ∈ Inc.We define an associated safety gameH(A,ψ) as follows:

From positions (x ⊆ y, s), Verifier can move to any position (x ⊆ y, t) suchthat t(y) = s(x). From there, Falsifier can make an arbitrary move upwards inthe game forest.

From a given set X ⊆ I of initial positions, Verifier must make sure to avoid- terminal positions (φ, s), where φ is a first-order literal with A ⊧s φ, and- initial positions outside X

Proposition. Every consistent winning strategy S = (W , F) from X forG(A,ψ) is also a winning strategy forH(A,ψ) that avoids I ∖ X, and viceversa.

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Inclusion logic and least fixed-point logic

�eorem. (Galliani and Hella) For every formula φ(z) ∈ Inc one can constructa formula ψ(X , z) in posGFP, and vice versa, such that, for all A and all X

A ⊧X φ ⇐⇒ (A, X) ⊧ ∀z(Xz → ψ(X , z))�us the maximal team satisfying φ coincides with gfp(ψ).

For the case of sentences, ψ and φ are equivalent.

Corollary. For sentences, inclusion logic and posGFP have the same

expressive power.

Corollary. On finite structures, inclusion logic and LFP have the same

expressive power. In particular, on ordered finite structures, inclusion logic

captures P.

However, without a linear order, both logics fail to capture polynomial time.

In particular, neither LFP nor inclusion logic can express queries that involve

counting.

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Inclusion logic and least fixed-point logic

�eorem. (Galliani and Hella) For every formula φ(z) ∈ Inc one can constructa formula ψ(X , z) in posGFP, and vice versa, such that, for all A and all X

A ⊧X φ ⇐⇒ (A, X) ⊧ ∀z(Xz → ψ(X , z))�us the maximal team satisfying φ coincides with gfp(ψ).

For the case of sentences, ψ and φ are equivalent.

Corollary. For sentences, inclusion logic and posGFP have the same

expressive power.

Corollary. On finite structures, inclusion logic and LFP have the same

expressive power. In particular, on ordered finite structures, inclusion logic

captures P.

However, without a linear order, both logics fail to capture polynomial time.

In particular, neither LFP nor inclusion logic can express queries that involve

counting.

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Logic and counting: a difficult relationship

Counting is an important, computationally simple task that, however, is

somewhat problematic for logic.

Ravens, so we read, can only count up to seven. �ey can’t tell thedifference between two numbers greater than or equal to eight.First-order logic is much the same as ravens, except that the cutoffpoint is rather higher: it’s ω instead of . Wilfrid Hodges, Model�eory

In fact, for any fixed first-order sentence, the cutoff point is smaller than ω.Difficulties with counting are a central issue in finite model theory and

concern also logics that are much stronger than first-order logic.

E ∶= {A ∶ A has an even number of elements} is not definable infirst-order logic FO, least fixed-point logic LFP, or the infinitary logic Lω

∞ω

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Logic and counting: a difficult relationship

Counting is an important, computationally simple task that, however, is

somewhat problematic for logic.

Ravens, so we read, can only count up to seven. �ey can’t tell thedifference between two numbers greater than or equal to eight.First-order logic is much the same as ravens, except that the cutoffpoint is rather higher: it’s ω instead of . Wilfrid Hodges, Model�eory

In fact, for any fixed first-order sentence, the cutoff point is smaller than ω.Difficulties with counting are a central issue in finite model theory and

concern also logics that are much stronger than first-order logic.

E ∶= {A ∶ A has an even number of elements} is not definable infirst-order logic FO, least fixed-point logic LFP, or the infinitary logic Lω

∞ω

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Counting quantifiers

Add counting in one form or another to logical systems:

Simple variant of counting quantifiers, for fixed i ∈ N:∃≥ix: there exist at least i elements x such that . . .

Such quantifiers do not increase the expressive power of first-order logic, but

are nevertheless important for the study of bounded-variable logics.

Example. C ∶ two-variable first-order logic, with counting quantifiers ∃≥i .- C ≤ FO, but C /≤ FOk , for any fixed k ∈ ω- Sat(C) is decidable (Grädel, Otto, Rosen)

In many cases, and in particular in the quest for a logic for polynomial time, a

more general form of counting is needed, where counting parameters are

variables over a numeric domain.

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Counting quantifiers

Add counting in one form or another to logical systems:

Simple variant of counting quantifiers, for fixed i ∈ N:∃≥ix: there exist at least i elements x such that . . .

Such quantifiers do not increase the expressive power of first-order logic, but

are nevertheless important for the study of bounded-variable logics.

Example. C ∶ two-variable first-order logic, with counting quantifiers ∃≥i .- C ≤ FO, but C /≤ FOk , for any fixed k ∈ ω- Sat(C) is decidable (Grädel, Otto, Rosen)

In many cases, and in particular in the quest for a logic for polynomial time, a

more general form of counting is needed, where counting parameters are

variables over a numeric domain.

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Counting quantifiers over two-sorted structures

Expand a given (finite) structure A by a second numeric sort

A∗ = A ∪ (ω, <,+, ⋅, , ) or A∗ = A ∪ ({, . . . , ∣A∣}, <, , ∣A∣)

Consider two-sorted logics with point variables x , y, z, . . . over Aand number variables µ, ν, λ, . . . over the numeric sort

Add counting quantifiers ∃≥µx or counting terms #x[φ]Other counting quantifiers such as ∃=µx or ∃≤µx etc. can also be used

Examples.

E is defined by ∃µ∃ν(ν + ν = µ ∧ ∃=µx(x = x))

A graph G = (V , E) is regular if, and only if, G ⊧ ∀x∀y(#z[Exz] = #z[Eyz])

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Counting quantifiers over two-sorted structures

Expand a given (finite) structure A by a second numeric sort

A∗ = A ∪ (ω, <,+, ⋅, , ) or A∗ = A ∪ ({, . . . , ∣A∣}, <, , ∣A∣)

Consider two-sorted logics with point variables x , y, z, . . . over Aand number variables µ, ν, λ, . . . over the numeric sort

Add counting quantifiers ∃≥µx or counting terms #x[φ]Other counting quantifiers such as ∃=µx or ∃≤µx etc. can also be used

Examples.

E is defined by ∃µ∃ν(ν + ν = µ ∧ ∃=µx(x = x))

A graph G = (V , E) is regular if, and only if, G ⊧ ∀x∀y(#z[Exz] = #z[Eyz])

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Fixed-point logic with counting

(FP + C): Two-sorted fixed-point logic with counting

If the numeric sort is (ω, <,+, ⋅), then numeric variables must be explicitlyrestricted to take only polynomially bounded values.

Least, greatest, or inflationary fixed point operators defining formulae of the

form

[fpRxµ≤t .ψ(R, x , µ)](y, ν).

In the finite, it does not matter which kind of fixed-points. For us, it is

convenient to consider only greatest fixed points, since these can be directly

translated into existential second-order logic. To ensure monotonicity, use

only counting quantifiers ∃≥µ.

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Fixed-point logic with counting

(FP + C): Two-sorted fixed-point logic with counting

If the numeric sort is (ω, <,+, ⋅), then numeric variables must be explicitlyrestricted to take only polynomially bounded values.

Least, greatest, or inflationary fixed point operators defining formulae of the

form

[fpRxµ≤t .ψ(R, x , µ)](y, ν).

In the finite, it does not matter which kind of fixed-points. For us, it is

convenient to consider only greatest fixed points, since these can be directly

translated into existential second-order logic. To ensure monotonicity, use

only counting quantifiers ∃≥µ.

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Fixed-point logic with counting versus polynomial time

�eorem. (FP+C) ⊊ P (Cai, Fürer, Immerman )

Although (FP+C) fails to capture P it is the logic of reference in this area.

See survey by Dawar, SIGLOG-News,

Fixed-point logic with counting captures P on many interesting classes of

structures, such as

- trees and forests (Immerman, Lander)

- planar graphs and graphs of bounded genus (Grohe)

- structures of bounded tree-width (Grohe, Marino)

- all classes of graphs that exclude a minor (Grohe)

Current research investigates extensions of (FP+C), such as rank logic or

Choiceless Polynomial Time as candidates for a logic that captures P

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Fixed-point logic with counting versus polynomial time

�eorem. (FP+C) ⊊ P (Cai, Fürer, Immerman )

Although (FP+C) fails to capture P it is the logic of reference in this area.

See survey by Dawar, SIGLOG-News,

Fixed-point logic with counting captures P on many interesting classes of

structures, such as

- trees and forests (Immerman, Lander)

- planar graphs and graphs of bounded genus (Grohe)

- structures of bounded tree-width (Grohe, Marino)

- all classes of graphs that exclude a minor (Grohe)

Current research investigates extensions of (FP+C), such as rank logic or

Choiceless Polynomial Time as candidates for a logic that captures P

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Fixed-point logic with counting versus polynomial time

�eorem. (FP+C) ⊊ P (Cai, Fürer, Immerman )

Although (FP+C) fails to capture P it is the logic of reference in this area.

See survey by Dawar, SIGLOG-News,

Fixed-point logic with counting captures P on many interesting classes of

structures, such as

- trees and forests (Immerman, Lander)

- planar graphs and graphs of bounded genus (Grohe)

- structures of bounded tree-width (Grohe, Marino)

- all classes of graphs that exclude a minor (Grohe)

Current research investigates extensions of (FP+C), such as rank logic or

Choiceless Polynomial Time as candidates for a logic that captures P

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Fixed-point logic with counting versus polynomial time

�eorem. (FP+C) ⊊ P (Cai, Fürer, Immerman )

Although (FP+C) fails to capture P it is the logic of reference in this area.

See survey by Dawar, SIGLOG-News,

Fixed-point logic with counting captures P on many interesting classes of

structures, such as

- trees and forests (Immerman, Lander)

- planar graphs and graphs of bounded genus (Grohe)

- structures of bounded tree-width (Grohe, Marino)

- all classes of graphs that exclude a minor (Grohe)

Current research investigates extensions of (FP+C), such as rank logic or

Choiceless Polynomial Time as candidates for a logic that captures P

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Is counting relevant for logics with team semantics?

On the level of existential second-order logic (Σ), counting is available

anyway!

- Σ permits the introduction of a linear ordering < on the universe.

- A second sort is not needed. Represent µ by the µ-th element of <

- ∣X∣ ≥ µ if there is an injective function F ∶ {a, . . . , aµ}→ A

Dependence atoms =(x , y)may also be seen as a form of counting.

Hence counting is interesting in particular for logics that are weaker than Σ.

One motivation comes from the relationship between inclusion logic and LFP.

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Is counting relevant for logics with team semantics?

On the level of existential second-order logic (Σ), counting is available

anyway!

- Σ permits the introduction of a linear ordering < on the universe.

- A second sort is not needed. Represent µ by the µ-th element of <

- ∣X∣ ≥ µ if there is an injective function F ∶ {a, . . . , aµ}→ A

Dependence atoms =(x , y)may also be seen as a form of counting.

Hence counting is interesting in particular for logics that are weaker than Σ.

One motivation comes from the relationship between inclusion logic and LFP.

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Is counting relevant for logics with team semantics?

On the level of existential second-order logic (Σ), counting is available

anyway!

- Σ permits the introduction of a linear ordering < on the universe.

- A second sort is not needed. Represent µ by the µ-th element of <

- ∣X∣ ≥ µ if there is an injective function F ∶ {a, . . . , aµ}→ A

Dependence atoms =(x , y)may also be seen as a form of counting.

Hence counting is interesting in particular for logics that are weaker than Σ.

One motivation comes from the relationship between inclusion logic and LFP.

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Counting in team semantics

�ere are several ways to add counting constructs to team semantics.

Here, we consider two-sorted structures A∗ = A ∪ (ω, <,+, ⋅ ) andteams of two-sorted assignments s ∶ (x , µ)↦ (a, n)

Counting quantifiers:

A∗ ⊧X ∃≥µxφ ∶⇐⇒ A∗ ⊧X[x↦F] φ for some function F ∶ X → P(A)with ∣F(s)∣ ≥ s(µ) for all s ∈ X.

Counting quantifiers ∃≥µ preserve locality, closure under union anddownwards closure. Instead quantifiers ∃=µ or ∃≤µ are unsafe and may violatelocality.

Proposition. Counting quantifiers ∃≥µ are expressible by means of dependenceatoms: ∃≥µ yφ(x , y) ≡ (∀λ < µ)∃y(=(xy, λ) ∧ φ(x , y))

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Inclusion logic with counting

(Inc+C): First-order logic with team semantics, over two-sorted structures

A∗ = A ∪ (ω,+, ⋅), possibly equipped with a set of functions f ∶ Ak → ω, withinclusion atoms (xµ ⊆ yν) and counting quantifiers ∃≥µ.

Elementary properties: (Inc+C) is closed under union of teams

Question. Does the ‘equivalence’ between LFP and Inclusion Logic (on finite

structures) extend to a similar ‘equivalence’ between (FP+C) and (Inc + C)?

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Inclusion logic with Counting versus (FP+C)

�e Galliani-Hella�eorem indeed generalizes to the result that for

sentences, (FP+C) and inclusion logic with counting are equivalent.

Again, for open formulae, a more elaborate statement is needed.

�eorem. �ere are effective translations between formulae φ(xµ) of (Inc+C)and formulae ψ(X , xµ) in (FP+C), such that

A∗ ⊧X φ(xµ) ⇐⇒ (A∗, X) ⊧ ∀xµ(Xxµ → ψ(X , xµ))�us the maximal team satisfying φ coincides with gfp(ψ).

We prove this with a game-based approach.

As a first step, we define games that are appropriate for logics with counting.

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Inclusion logic with Counting versus (FP+C)

�e Galliani-Hella�eorem indeed generalizes to the result that for

sentences, (FP+C) and inclusion logic with counting are equivalent.

Again, for open formulae, a more elaborate statement is needed.

�eorem. �ere are effective translations between formulae φ(xµ) of (Inc+C)and formulae ψ(X , xµ) in (FP+C), such that

A∗ ⊧X φ(xµ) ⇐⇒ (A∗, X) ⊧ ∀xµ(Xxµ → ψ(X , xµ))�us the maximal team satisfying φ coincides with gfp(ψ).

We prove this with a game-based approach.

As a first step, we define games that are appropriate for logics with counting.

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Inclusion logic with Counting versus (FP+C)

�e Galliani-Hella�eorem indeed generalizes to the result that for

sentences, (FP+C) and inclusion logic with counting are equivalent.

Again, for open formulae, a more elaborate statement is needed.

�eorem. �ere are effective translations between formulae φ(xµ) of (Inc+C)and formulae ψ(X , xµ) in (FP+C), such that

A∗ ⊧X φ(xµ) ⇐⇒ (A∗, X) ⊧ ∀xµ(Xxµ → ψ(X , xµ))�us the maximal team satisfying φ coincides with gfp(ψ).

We prove this with a game-based approach.

As a first step, we define games that are appropriate for logics with counting.

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�reshold games

�reshold game: G = (V , E , t ∶ V → ω)We can assume that t(v) ≤ δ(v) + where δ(v) ∶= ∣vE∣

At node v in a play, Player selects X ⊆ vE with ∣X∣ ≥ t(v)Player choses w ∈ X and the play proceeds from w.

When a player cannot move, she loses.�is means that Player loses at all

nodes v with δ(v) < t(v) and Player loses at nodes v with t(v) = .

�reshold safety games: Infinite plays are won by Player

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Definability in threshold games

Winning regions of Player in threshold safety games G = (V , E , t ∶ V → ω)are definable in fixed-point logic with counting

Player wins G from v ⇐⇒ G ⊧ [gfpWx . ∃≥t(x)y(Exy ∧Wy)](v)

Further, the winning region for Player is the maximal teamW such that

G ⊧W ∃≥t(x)y(Exy ∧ y ⊆ x)

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Definability of I-traps

Let G = (V , E , t) be a threshold safety game with a set I ⊆ V of initialpositions. An I-trap in G is a set Z ⊆ I such that Player has a winningstrategy that, moreover, avoids I ∖ Z.

Definability of I-traps in both logics:

Z ⊆ I is an I-trap in G⇐⇒

(G , I, Z) ⊧ ∀x(Zx → [gfpWx . (Ix → Zx) ∧ ∃≥t(x)y(Exy ∧Wy)](x))⇐⇒

(G , I) ⊧Z Iz ∧ ∃x(z ⊆ x ∧ (¬Ix ∨ x ⊆ z) ∧ ∃≥t(x)y(Exy ∧ y ⊆ x))

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Definability of I-traps

Let G = (V , E , t) be a threshold safety game with a set I ⊆ V of initialpositions. An I-trap in G is a set Z ⊆ I such that Player has a winningstrategy that, moreover, avoids I ∖ Z.

Definability of I-traps in both logics:

Z ⊆ I is an I-trap in G⇐⇒

(G , I, Z) ⊧ ∀x(Zx → [gfpWx . (Ix → Zx) ∧ ∃≥t(x)y(Exy ∧Wy)](x))⇐⇒

(G , I) ⊧Z Iz ∧ ∃x(z ⊆ x ∧ (¬Ix ∨ x ⊆ z) ∧ ∃≥t(x)y(Exy ∧ y ⊆ x))

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�reshold games as model-checking games

For every formula ψ(x , µ) in (Inc+C) and every structure A∗, one can

construct a threshold safety game T (A∗,ψ) with the following properties:

- positions are (φ, s) where φ is a subformula of ψ and s is anassignment with domain free(φ).

- the set I of initial positions contains all pairs (ψ, s).Every team X for ψ defines the subset I(X) ∶= {(ψ, s) ∶ s ∈ X} of I

- A∗ ⊧X ψ(x , µ) ⇐⇒ I(X) is an I-trap in T (A∗,ψ)

Moreover, for every formula ψ(x , µ) there is a first-order interpretation Iψ thatinterprets the game T (A∗,ψ) in A∗.

Analogous statements hold for fixed-point logic with counting.

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�reshold games as model-checking games

For every formula ψ(x , µ) in (Inc+C) and every structure A∗, one can

construct a threshold safety game T (A∗,ψ) with the following properties:

- positions are (φ, s) where φ is a subformula of ψ and s is anassignment with domain free(φ).

- the set I of initial positions contains all pairs (ψ, s).Every team X for ψ defines the subset I(X) ∶= {(ψ, s) ∶ s ∈ X} of I

- A∗ ⊧X ψ(x , µ) ⇐⇒ I(X) is an I-trap in T (A∗,ψ)

Moreover, for every formula ψ(x , µ) there is a first-order interpretation Iψ thatinterprets the game T (A∗,ψ) in A∗.

Analogous statements hold for fixed-point logic with counting.

Erich Grädel Dependence and Independence in Logic

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�reshold games as model-checking games

For every formula ψ(x , µ) in (Inc+C) and every structure A∗, one can

construct a threshold safety game T (A∗,ψ) with the following properties:

- positions are (φ, s) where φ is a subformula of ψ and s is anassignment with domain free(φ).

- the set I of initial positions contains all pairs (ψ, s).Every team X for ψ defines the subset I(X) ∶= {(ψ, s) ∶ s ∈ X} of I

- A∗ ⊧X ψ(x , µ) ⇐⇒ I(X) is an I-trap in T (A∗,ψ)

Moreover, for every formula ψ(x , µ) there is a first-order interpretation Iψ thatinterprets the game T (A∗,ψ) in A∗.

Analogous statements hold for fixed-point logic with counting.

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Game-based translations between (Inc+C) and (FP+C)

Goal: Translate any ψ(xµ) in (Inc+C) to φ(X , xµ) in (FP+C) such thatA∗ ⊧X ψ(xµ) ⇐⇒ (A∗, X) ⊧ ∀x∀µ(Xxµ → φ(X , xµ))

Take a formula in (FP+C) that defines I-traps in threshold games:

T (A∗,ψ), Z ⊧ ∀x(Zx → itrap(Z , x)) ⇐⇒ Z is an I-trap in T (A∗,ψ).

�e interpretation Iψ defines a copy of the game T (A∗,ψ) inside A∗.

Further it maps formulae on the game back to formulae on A∗.

Iψ ∶ itrap(Z , x)↦ itrap∗(Y , yν).

If Y is the set of tuples in A∗ associated with Z ⊆ I, then(A∗,Y) ⊧ ∀yν(Y yν → itrap∗(Y , yν)) ⇐⇒ Z is an I-trap in T (A∗,ψ).

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Game-based translations between (Inc+C) and (FP+C)

Goal: Translate any ψ(xµ) in (Inc+C) to φ(X , xµ) in (FP+C) such thatA∗ ⊧X ψ(xµ) ⇐⇒ (A∗, X) ⊧ ∀x∀µ(Xxµ → φ(X , xµ))

Take a formula in (FP+C) that defines I-traps in threshold games:

T (A∗,ψ), Z ⊧ ∀x(Zx → itrap(Z , x)) ⇐⇒ Z is an I-trap in T (A∗,ψ).

�e interpretation Iψ defines a copy of the game T (A∗,ψ) inside A∗.

Further it maps formulae on the game back to formulae on A∗.

Iψ ∶ itrap(Z , x)↦ itrap∗(Y , yν).

If Y is the set of tuples in A∗ associated with Z ⊆ I, then(A∗,Y) ⊧ ∀yν(Y yν → itrap∗(Y , yν)) ⇐⇒ Z is an I-trap in T (A∗,ψ).

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Game-based translations between (Inc+C) and (FP+C)

Goal: Translate any ψ(xµ) in (Inc+C) to φ(X , xµ) in (FP+C) such thatA∗ ⊧X ψ(xµ) ⇐⇒ (A∗, X) ⊧ ∀x∀µ(Xxµ → φ(X , xµ))

Take a formula in (FP+C) that defines I-traps in threshold games:

T (A∗,ψ), Z ⊧ ∀x(Zx → itrap(Z , x)) ⇐⇒ Z is an I-trap in T (A∗,ψ).

�e interpretation Iψ defines a copy of the game T (A∗,ψ) inside A∗.

Further it maps formulae on the game back to formulae on A∗.

Iψ ∶ itrap(Z , x)↦ itrap∗(Y , yν).

If Y is the set of tuples in A∗ associated with Z ⊆ I, then(A∗,Y) ⊧ ∀yν(Y yν → itrap∗(Y , yν)) ⇐⇒ Z is an I-trap in T (A∗,ψ).

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Game-based translations between (Inc+C) and (FP+C)

If Y is the set of tuples in A∗ associated with Z ⊆ I, then(A∗,Y) ⊧ ∀yν(Y yν → itrap∗(Y , yν)) ⇐⇒ Z is an I-trap in T (A∗,ψ).

�e tuples describing positions (ψ, s) with s ∈ X are definable in (A∗, X).

We can thus massage itrap∗(Y , yν) into φ(X , xµ) in (FP+C) such that

(A∗, X) ⊧ ∀xµ(Xxµ → φ(X , xµ)) ⇐⇒ I(X) is an I-trap in T (A∗

,ψ)⇐⇒ A ⊧X ψ(xµ).

An analogous argument works for the translation of posGFP into Inc.

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Game-based translations between (Inc+C) and (FP+C)

If Y is the set of tuples in A∗ associated with Z ⊆ I, then(A∗,Y) ⊧ ∀yν(Y yν → itrap∗(Y , yν)) ⇐⇒ Z is an I-trap in T (A∗,ψ).

�e tuples describing positions (ψ, s) with s ∈ X are definable in (A∗, X).

We can thus massage itrap∗(Y , yν) into φ(X , xµ) in (FP+C) such that

(A∗, X) ⊧ ∀xµ(Xxµ → φ(X , xµ)) ⇐⇒ I(X) is an I-trap in T (A∗

,ψ)⇐⇒ A ⊧X ψ(xµ).

An analogous argument works for the translation of posGFP into Inc.

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Game-based translations between (Inc+C) and (FP+C)

If Y is the set of tuples in A∗ associated with Z ⊆ I, then(A∗,Y) ⊧ ∀yν(Y yν → itrap∗(Y , yν)) ⇐⇒ Z is an I-trap in T (A∗,ψ).

�e tuples describing positions (ψ, s) with s ∈ X are definable in (A∗, X).

We can thus massage itrap∗(Y , yν) into φ(X , xµ) in (FP+C) such that

(A∗, X) ⊧ ∀xµ(Xxµ → φ(X , xµ)) ⇐⇒ I(X) is an I-trap in T (A∗

,ψ)⇐⇒ A ⊧X ψ(xµ).

An analogous argument works for the translation of posGFP into Inc.

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Summary

�ere is a rich collection of logics of dependence and independence, on the

basis of different atomic dependency properties

All these logics are based on team semantics

In terms of expressive power, logics of dependence and independence are

equivalent to existential second-order logic or natural fragments of it.

Inclusion logic is equivalent to the posGFP fragment of fixed-point logic and

captures polynomial time on ordered finite structures

�e model-checking games for logics with team semantics are second-order

reachability games.�ese game provide evaluation procedures and complexity

results for these logics.

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Summary

In the case of inclusion logic, the games can be simplified to classical safety

games

�e equivalence between positive greatest fixed point logic and inclusion logic

li�s to an equivalence between corresponding counting extensions.�us

inclusion logic with counting is rather close to P.

Future work: Counting may become very relevant but also very difficult once

we shall move to multi-team semantics.

Erich Grädel Dependence and Independence in Logic