logics with probability operators l5:extensions and modi...

Logics with Probability OperatorsL5: Extensions and Modi�cations of Probability Logic

Zoran Ognjanovi¢ and Dragan Doder

Mathematical Institute SANU and IRIT, Universite Paul Sabatier

[email protected]

August 10, 2018

Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 1 / 33

Probability Logics: Beyond LPP1/LPP2 and FHM

1 Extensions:

First order logic (L4)Extending syntax with other probability operators (independence,qualitative probability,. . . ) (L4)Combining probability with other modalities (temporal, epistemic,dynamic,dots) � extending syntax and semantics

2 Modi�cations:

Changing the type of probability operators (conditional probability,expectation operator, lower and upper probability,. . . )Changing the range of probability measures (�nite, rational,nonstandard (in�nitesimals), complex-valued,. . . )

Probability Logics: Beyond LPP1/LPP2 and FHM

1 Extensions:

First order logic (L4)Extending syntax with other probability operators (independence,qualitative probability,. . . ) (L4)Combining probability with other modalities (temporal, epistemic,dynamic,dots) � extending syntax and semantics

2 Modi�cations:

Changing the type of probability operators (conditional probability,expectation operator, lower and upper probability,. . . )Changing the range of probability measures (�nite, rational,nonstandard (in�nitesimals), complex-valued,. . . )

Upper and lower probabilities

Halpern, Pucella.

A Logic for Reasoning about Upper Probabilities.

J. Artif. Intell. Res. 17: 57-81 (2002)

Savic, Doder, Ognjanovic.

Logics with lower and upper probability operators.

Int. J. Approx. Reasoning 88: 148-168 (2017)

Example

Set of probabilities P = {µα | α ∈ {0, 0.1, . . . , 0.7}}, where µα gives

green-event probability 0.3, blue-event probability α, and red-event

probability 0.7− α.

Example

Set of probabilities P = {µα | α ∈ {0, 0.1, . . . , 0.7}}, where µα gives

green-event probability 0.3, blue-event probability α, and red-event

probability 0.7− α.Ognjanovi¢, Doder (SANU, IRIT) Extensions of Probability Logics 9/8/2018 4 / 33

Example

P?(X ) = inf{µ(X ) | µ ∈ P} P?(X ) = sup{µ(X ) | µ ∈ P}

P?(X ) = 1− P?(X c)

Example

P?(X ) = inf{µ(X ) | µ ∈ P} P?(X ) = sup{µ(X ) | µ ∈ P}

P?(X ) = 1− P?(X c)

Example

P?(R) = 0, P?(R) = 0.7, P?(B) = 0, P?(B) = 0.7,P?(G ) = P?(G ) = 0.3.

Example

P?(R) = 0, P?(R) = 0.7,

P?(B) = 0, P?(B) = 0.7,P?(G ) = P?(G ) = 0.3.

Example

P?(R) = 0, P?(R) = 0.7, P?(B) = 0, P?(B) = 0.7,

P?(G ) = P?(G ) = 0.3.

Example

P?(R) = 0, P?(R) = 0.7, P?(B) = 0, P?(B) = 0.7,P?(G ) = P?(G ) = 0.3.

General remarks

Models:

uncertainty is modeled by a set of probabilities (on possible worlds)

Syntax:

P≥r is not appropriateµ1([α]) = 0.7, µ2([α]) = 0.9 in a world w , how about w |= P≥0.8α?We need L≥r and U≥r to speak about sets of probabilities (two simplefunctions that provide some information about a set)

Axiomatization:

How to formalize L and U? When do they correspond to P∗ and P∗,for some P?Can we describe them with a small number of properties that we canwrite down in our logic?Characterization result needed.

General remarks

Models:

Syntax:

Axiomatization:

General remarks

Models:

Syntax:

P≥r is not appropriate

µ1([α]) = 0.7, µ2([α]) = 0.9 in a world w , how about w |= P≥0.8α?We need L≥r and U≥r to speak about sets of probabilities (two simplefunctions that provide some information about a set)

Axiomatization:

General remarks

Models:

Syntax:

P≥r is not appropriateµ1([α]) = 0.7, µ2([α]) = 0.9 in a world w , how about w |= P≥0.8α?

We need L≥r and U≥r to speak about sets of probabilities (two simplefunctions that provide some information about a set)

Axiomatization:

General remarks

Models:

Syntax:

Axiomatization:

General remarks

Models:

Syntax:

Axiomatization:

How to formalize L and U? When do they correspond to P∗ and P∗,for some P?Can we describe them with a small number of properties that we canwrite down in our logic?

Characterization result needed.

General remarks

Models:

Syntax:

Axiomatization:

Theorem (Anger and Lembcke 1985)

Let W be a set, H an algebra of subsets of W , and f a function

f : H −→ [0, 1]. There exists a set P of probability measures such that

f = P? i� f satis�es the following three properties:

(1) f (∅) = 0,

(2) f (W ) = 1,

(3) for all natural numbers m, n, k and all subsets A1, . . . ,Am in H, if the

multiset {{A1, . . . ,Am}} is an (n, k)-cover of (A,W ), thenk + nf (A) ≤

∑mi=1 f (Ai ).

De�nition ((n, k)-cover)

A set A is said to be covered n times by a multiset {{A1, . . . ,Am}} of setsif every element of A appears in at least n sets from A1, . . . ,Am, i.e., for all

x ∈ A, there exists i1, . . . , in in {1, . . . ,m} such that for all j ≤ n, x ∈ Aij .

An (n, k)-cover of (A,W ) is a multiset {{A1, . . . ,Am}} that covers W ktimes and covers A n + k times.

Theorem (Anger and Lembcke 1985)

Let W be a set, H an algebra of subsets of W , and f a function

f : H −→ [0, 1]. There exists a set P of probability measures such that

f = P? i� f satis�es the following three properties:

(1) f (∅) = 0,

(2) f (W ) = 1,

(3) for all natural numbers m, n, k and all subsets A1, . . . ,Am in H, if the

multiset {{A1, . . . ,Am}} is an (n, k)-cover of (A,W ), thenk + nf (A) ≤

∑mi=1 f (Ai ).

De�nition ((n, k)-cover)

A set A is said to be covered n times by a multiset {{A1, . . . ,Am}} of setsif every element of A appears in at least n sets from A1, . . . ,Am, i.e., for all

x ∈ A, there exists i1, . . . , in in {1, . . . ,m} such that for all j ≤ n, x ∈ Aij .

An (n, k)-cover of (A,W ) is a multiset {{A1, . . . ,Am}} that covers W ktimes and covers A n + k times.

Possible-world semantics

Every world is equipped with

- an evaluation function on propositional letters, and

- one generalized probability space for each agent

De�nition

An ILUPP-structure is a tuple 〈W , LUP, υ〉, where:W is a nonempty set of worlds,

LUP assigns, to every w ∈W and every a ∈ Σ, a spaceLUP(w) = 〈W (w),H(w),P(w)〉, where:

∅ 6= W (w) ⊆W ,H(w) is an algebra of subsets of W (w)P(w) is a set of �nitely additive probability measures on H(w)

υ : W × P −→ {true, false}.

Possible-world semantics

Every world is equipped with

- an evaluation function on propositional letters, and

- one generalized probability space for each agent

De�nition

An ILUPP-structure is a tuple 〈W , LUP, υ〉, where:W is a nonempty set of worlds,

LUP assigns, to every w ∈W and every a ∈ Σ, a spaceLUP(w) = 〈W (w),H(w),P(w)〉, where:

∅ 6= W (w) ⊆W ,H(w) is an algebra of subsets of W (w)P(w) is a set of �nitely additive probability measures on H(w)

υ : W × P −→ {true, false}.

Satis�ability relation

Notation:

P?(w)([α]M,w ) = inf{µ([α]M,w ) | µ ∈ P(w)} andP?(w)([α]M,w ) = sup{µ([α]M,w ) | µ ∈ P(w)},

Satis�ability relation:

M,w |= U≥sα i� P?(w)([α]M,w ) ≥ s,

M,w |= L≥sα i� P?(w)([α]M,w ) ≥ s.

Satis�ability relation

Notation:

P?(w)([α]M,w ) = inf{µ([α]M,w ) | µ ∈ P(w)} andP?(w)([α]M,w ) = sup{µ([α]M,w ) | µ ∈ P(w)},

Satis�ability relation:

M,w |= U≥sα i� P?(w)([α]M,w ) ≥ s,

M,w |= L≥sα i� P?(w)([α]M,w ) ≥ s.

Axiom schemes

(1) all instances of the classical propositional tautologies

(2) U≤1α ∧ L≤1α

(3) U≤rα→ U<sα, s > r

(4) U<sα→ U≤sα

(5) (U≤r1α1 ∧ · · · ∧ U≤rmαm)→ U≤rα, ifα→

∨J⊆{1,...,m},|J|=k+n

∧j∈J αj and

∨J⊆{1,...,m},|J|=k

∧j∈J αj are

propositional tautologies, where r =∑m

i=1 ri−kn , n 6= 0

(6) ¬(U≤r1α1 ∧ · · · ∧ U≤rmαm), if∨

J⊆{1,...,m},|J|=k

∧j∈J αj is a

propositional tautology and∑m

i=1 ri < k

(7) L=1(α→ β)→ (U≥sα→ U≥sβ)

Inference Rules

(1) From α and α→ β infer β

(2) From α infer L≥1α

(3) From the set of premises

{α→ U≥s− 1kβ | k ≥ 1

infer α→ U≥sβ

(4) From the set of premises

{α→ L≥s− 1kβ | k ≥ 1

infer α→ L≥sβ.

Non-standard Ranges

Probabilities with Non-standard Ranges

Raskovic, Markovic, Ognjanovic.

A logic with approximate conditional probabilities that can model

default reasoning.

Int. J. Approx. Reasoning 49(1): 52-66 (2008)

Non-standard Ranges

System P

Kraus, Lehmann, Magidor.

Nonmonotonic Reasoning, Preferential Models and Cumulative Logics.

Artif. Intell. 44(1-2): 167-207 (1990)

α |∼ β � �if α, then generally β"

REF :α|∼α

; LLE :` α↔ β, α|∼γ

β|∼γ;

RW :` α→ β, γ|∼α

γ|∼β; AND :

α|∼β, α|∼γα|∼β ∧ γ

OR :α|∼γ, β|∼γα ∨ β|∼γ

; CM :α|∼β, α|∼γα ∧ β|∼γ

Non-standard Ranges

System P

Kraus, Lehmann, Magidor.

Nonmonotonic Reasoning, Preferential Models and Cumulative Logics.

Artif. Intell. 44(1-2): 167-207 (1990)

α |∼ β � �if α, then generally β"

REF :α|∼α

; LLE :` α↔ β, α|∼γ

β|∼γ;

RW :` α→ β, γ|∼α

γ|∼β; AND :

α|∼β, α|∼γα|∼β ∧ γ

OR :α|∼γ, β|∼γα ∨ β|∼γ

; CM :α|∼β, α|∼γα ∧ β|∼γ

Non-standard Ranges

Rational preferential relations

Lehmann, Magidor.

What does a Conditional Knowledge Base Entail?

Artif. Intell. 55(1): 1-60 (1992)

Rational relation = System P + Rational Monotonicity (RM)

RM :α |∼ γ, α |6∼ ¬β

α ∧ β |∼ γ

Nonstandard probabilistic semantics:

|∼ is rational i� there exists a neat �nitely additive non-standard probability

measure µ such that

|∼=|∼µ,

α |∼µ β i� µ(β|α) ≈ 1 or µ(α) = 0.

Non-standard Ranges

Rational preferential relations

Lehmann, Magidor.

What does a Conditional Knowledge Base Entail?

Artif. Intell. 55(1): 1-60 (1992)

Rational relation = System P + Rational Monotonicity (RM)

RM :α |∼ γ, α |6∼ ¬β

α ∧ β |∼ γ

Nonstandard probabilistic semantics:

|∼ is rational i� there exists a neat �nitely additive non-standard probability

measure µ such that

|∼=|∼µ,

α |∼µ β i� µ(β|α) ≈ 1 or µ(α) = 0.

Non-standard Ranges

Logic with non-standard probability values

Modeling α |∼ β � we need:

In syntax:

conditional probability operators

�approximately 1�

In semantics:

measures with non-standard values

Axiomatization:

Archimedean rule is not appropriate

we need a new rule to control the range

Non-standard Ranges

Syntax and Semantics

Hardy �eld Q(ε) is a recursive non-archimedean �eld which contains:

a �xed positive in�nitesimal ε

all standard rational numbers

Q(ε) = all rational functions of ε

eg. ε3+ε4

ε2−5ε6

countable

Semantics: the range of µ is [0, 1]Q(ε) � the unit interval of Q(ε).

Syntax:

CP≥s , CP≤s where s ∈ [0, 1]Q(ε)

CP≈r , where r ∈ [0, 1]Q

Abbreviations: P∗sα =def CP∗s(α,>), where ∗ ∈ {≥,≤,≈}No iterations, no mixing with classical formulas

Non-standard Ranges

Axioms

1 all ForC -instances of classical propositional tautologies

2 all ForP -instances of classical propositional tautologies

3 CP≥0(α, β)

4 CP≤s(α, β)→ CP<t(α, β), t > s

5 CP<s(α, β)→ CP≤s(α, β)

6 P≥1(α↔ β)→ (P=sα→ P=sβ)

7 P≤sα↔ P≥1−s¬α8 (P=sα ∧ P=tβ ∧ P≥1¬(α ∧ β))→ P=min(1,s+t)(α ∨ β)

9 P=0β → CP=1(α, β)

10 (P=tβ ∧ P=s(α ∧ β))→ CP=s/t(α, β), t 6= 0

11 CP≈r (α, β)→ CP≥r1(α, β), for every rational r1 ∈ [0, r)

12 CP≈r (α, β)→ CP≤r1(α, β), for every rational r1 ∈ (r , 1]

Non-standard Ranges

Inference rules

1 From ϕ and ϕ→ ψ infer ψ.

2 If α ∈ ForC , from α infer P≥1α.

3 From A→ P 6=sα, for every s ∈ [0, 1]Q(ε), infer A→ ⊥.4 For every r ∈ [0, 1]Q, from A→ CP≥r−1/n(α, β), for every integer

n ≥ 1/r , and A→ CP≤r+1/n(α, β) for every integer n ≥ 1/(1− r),infer A→ CP≈r (α, β).

Strong completeness for the class LPCP[0,1]Q(ε),≈2,Meas,Neat.

µ([α]) = s ⇔def T∗ ` P=sα.

(neat: If µ([α]) = 0, then there is no w in which α holds.)

Non-standard Ranges

Inference rules

1 From ϕ and ϕ→ ψ infer ψ.

2 If α ∈ ForC , from α infer P≥1α.

3 From A→ P 6=sα, for every s ∈ [0, 1]Q(ε), infer A→ ⊥.4 For every r ∈ [0, 1]Q, from A→ CP≥r−1/n(α, β), for every integer

n ≥ 1/r , and A→ CP≤r+1/n(α, β) for every integer n ≥ 1/(1− r),infer A→ CP≈r (α, β).

Strong completeness for the class LPCP[0,1]Q(ε),≈2,Meas,Neat.

µ([α]) = s ⇔def T∗ ` P=sα.

(neat: If µ([α]) = 0, then there is no w in which α holds.)

Temporal probability logics

Probability and Time

Temporal probability logics About temporal logic

Temporal operators

Basic:

©α: α has to hold at the next stateαUβ: α has to hold at least until β, which holds at the current or afuture moment

A � universal path operator (branching time)

Other:

G � always, F � sometime

E � existential path operator (branching time)

Temporal operators

Basic:

Other:

Temporal operators

Basic:

αUβ: α has to hold at least until β, which holds at the current or afuture moment

Other:

Temporal operators

Basic:

Other:

Temporal operators

Basic:

Other:

Temporal operators

Basic:

Other:

Semantics of LTL

Semantics for LTL � the set of paths Σ

σ = s0, s1, s2, . . .

si � the i-th time instance of σ � a subset of P,

Abbreviations:

σ≥i is the path si , si+1, si+2, . . .

σi is the state si .

Temporal probability logics How to combine them?

How to combine probabilistic and temporal logics

1 Temporal reasoning about probabilistic information

2 Probabilistic reasoning about temporal information

3 Modal approach - random nesting of both types of modalities

Examples:

1 Halpern, Pucella: A logic for reasoning about evidence (JAIR, 2006)

2 Grant, Parisi, Parker, Subrahmanian: An agm-style belief revision

mechanism for probabilistic spatio-temporal logics (AIJ, 2010)

3 Ognjanovic: Discrete linear-time probabilistic logics: Completeness,

decidability and complexity (JLC, 2006)

Examples:

The logic PLLTL

Doder, Ognjanovic. A Probabilistic Logic for Reasoningabout Uncertain Temporal Information. UAI 2015

We extend both:

Linear time logic LTL

Probabilistic logic (FHM)

We allow formulas like

�A will always hold"

�the probability that A will hold in next moment is at least the

probability that B will always hold"

The results:

Strongly complete axiomatization for countably additive semantics

Decidability: our logic is PSPACE-complete, no worse than LTL

The logic PLLTL

We extend both:

The results:

The logic PLLTL

We extend both:

The results:

The logic PLLTL Syntax and semantics

Syntax

Two types of formulas:

Certain knowledge: linear temporal formulas (ForLTL)

Uncertain knowledge: linear weighted formulas over ForLTL (ForP)

�if the probability that either p or q hold in this moment is equal to theprobability that p will hold in the next moment, then the probabilitythat q will always hold is at most one half"

For = ForLTL ∪ ForP .

Syntax

Semantics

M = 〈W ,H, µ, π〉:W � a nonempty set of worlds,

〈W ,H, µ〉 � a probability space, i.e.

H � an algebra of subsets of Wµ � a countably additive probability measure on H

π : W −→ Σ provides for each world w ∈W a path π(w).

Satis�ability relation � LTL formulas

The evaluation function v : Σ× ForLTL −→ {0, 1}:

- v(σ, αUβ) = 1 i� there is some i ∈ ω such that v(σ≥iβ) = 1, and for

each j ∈ ω, if 0 ≤ j < i then v(σ≥j , β) = 1.

M |= α i� v(π(w), α) = 1 for every w ∈W ,

Satis�ability relation � LTL formulas

The evaluation function v : Σ× ForLTL −→ {0, 1}:

- v(σ, αUβ) = 1 i� there is some i ∈ ω such that v(σ≥iβ) = 1, and for

each j ∈ ω, if 0 ≤ j < i then v(σ≥j , β) = 1.

M |= α i� v(π(w), α) = 1 for every w ∈W ,

Satis�ability relation � probabilistic formulas

M = 〈W ,H, µ, π〉- [α]M = {w ∈W | v(π(w), α) = 1}- M is measurable, if [α]M ∈ H for every α ∈ ForLTL

M |= P(α) ≥ r i� µ([α]M) ≥ r ,

Satis�ability relation � probabilistic formulas

M = 〈W ,H, µ, π〉- [α]M = {w ∈W | v(π(w), α) = 1}- M is measurable, if [α]M ∈ H for every α ∈ ForLTL

M |= P(α) ≥ r i� µ([α]M) ≥ r ,

On countable additivity

αUnβ ≡ αUβ + β has realization at least at time n

[αUβ]M =⋃n∈ω

[αUnβ]M

Countable additivity necessary!

T = {P(αUβ) = 1} ∪ {P(αUnβ) = 0 | n ∈ ω}

[αUβ]M =⋃n∈ω

[αUnβ]M

T = {P(αUβ) = 1} ∪ {P(αUnβ) = 0 | n ∈ ω}

[αUβ]M =⋃n∈ω

[αUnβ]M

T = {P(αUβ) = 1} ∪ {P(αUnβ) = 0 | n ∈ ω}

The logic PLLTL Completeness and Decidability

Axioms

1 All instances of classical propositional tautologies for both LTL and

probabilistic formulas.

2 3 standard LTL axioms

3 Probabilisitc axioms a la FHM

4 Axioms for reasoning about linear inequalities a la FHM

Inference rules

1 2 x Modus Ponens

3 From α infer P(α) = 1

4 From {γ → ¬(αUnβ) | n ∈ ω} infer γ → ¬(αUβ).

5 From {φ→ f ≥ r − 1n | n ∈ ω \ {0}} infer φ→ f ≥ r .

6 From {φ→ P(αUnβ) ≤ r | n ∈ ω} infer φ→ P(αUβ) ≤ r .

Completeness theorem

Theorem (Strong completeness)

A set of formulas T ⊆ For is consistent i� it is satis�able.

Proof. Henkin-like construction; extending T to a maximal consistent set

T ∗, and then using T ∗ to de�ne the canonical model.

Decidability and complexity

Theorem

The problem of deciding whether a formula of the logic PLLTL is satis�able

in a measurable structure from PLMeasLTL is PSPACE -complete. (no worse

than LTL)

logics with probability operators l5:extensions and modi...

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