an introduction to normal multimodal logics: interaction...

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An Introduction to Normal Multimodal Logics:Interaction Axioms, Prefixed Tableau Calculus,

some [un]Decidability Results, and Applications

Matteo Baldoni

Dipartimento di Informatica - Universita` degli Studi di TorinoC.so Svizzera, 185 - I-10149 Torino (Italy)

e-mail: [email protected]: http://www.di.unito.it/~baldoni

Genova, 3 maggio 2000 An Introduction toNormal Multimodal Logics

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In the presentation ...

• an introduction to Modal and Multimodal Logics

• a tableau calculus for a wide class of normal multimodal logics(inclusion [Fariñas del Cerro and Penttonen, 1988] and incestual[Catach, 1988] multimodal logics) modular w.r.t. the axiomsystems;

• some (un)decidability results for the class of inclusion andincestual multimodal logics;

• an application of inclusion modal logics to logic programming: thelogic programming languages NemoLOG and DyLOG.

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3

(Mono)Modal Logic


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Modal Logics

Knowledge Beliefs

Actions Dynamic changes Time

Modal logics are suitable to deal with reasoning aboutdistributed knowledge

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5

The Modal Operator “ “

ϕno truth-functional

This means that themeaning of thisformula does notdepend only on thetruth-value of itssubformulae.

This means that ϕ isnot only true butnecessarily true, it istrue independentlyfrom the scenario(or state, world, etc.)

It qualifies the truth value of ϕ


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The Modal Operator “ “

ϕϕ is believed

ϕ is known

ϕ is necessarilytrue

ϕ is true in anypossible scenario

ϕ is always true

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7

The Modal Operator “ “: Kripke semantics

w

ϕ1w

kw

jw

nw

ϕ

ϕ

ϕϕ¬

ϕwM ,

ϕiwM ,ii wRww :∀

if and only if

ϕ1,wM ϕnwM ,

ϕkwM ,

M

accessibilityrelation


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w

1w

kw

jw

nw

=M

,,W R V

ϕ

ϕ

ϕϕ¬

ϕ

[Hughes and Cresswell, 1996; Fitting, 1993]

Kripkeinterpretation

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Axiomatization

• all axiom schemas for the propositional calculus;

)()(: ψϕψϕ ⊃⊃⊃K• the axiom schema:

• the necessitation rule of inference:

if I can infer ϕ then I can infer ϕ

• the modus ponens rule of inference;

• some other properties...


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Properties for the Modal Operator “ “

w

1w

2w

3w

ϕϕ ⊃:4

Transitivity(positive introspection)

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11


w

1w

2w

3w

ϕϕ ⊃:B

Simmetry


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w

1w

2w

3w

ϕϕ ⊃:T

Reflexivity

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13


w

1w

2wϕϕ ⊃:D

Seriality

3w


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w

1w

2wϕϕ ⊃:5

3w

Euclideanness(negative introspection)

¬¬≡

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15

Multimodal Logic


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Multimodal Operators

ϕaw

ϕ][a

1w

kw

jw

nw

ϕ

ψ

ψM

a

a

b

bϕ

ψ][b

• more than one modaloperator

[Halpern and Moses, 1992]

• they are named by meansof labels

ab

ab

• “a” often identifies thename of an agent

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The Multimodal Operator “[a]“

ϕ][aϕ is believed

by “a”ϕ is known

by “a”

ϕ is necessarilytrue for theagent “a”

ϕ is true in anypossible scenario

of “a”

ϕ is true afterexecuting

the action “a”


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w

1w

kw

jw

nw

=M,,,, LW R1 V

ϕ

ϕ

ϕϕ¬

ϕ

[Genesereth and Nilsson, 1989]

Rn

a1

an

an

an

an

a1

a1

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Multimodal systems [Catach, 1991]

• Complex modalities (obtained by composing modal operators ofdifferent types).

• Several modal aspects can be captured at the same time (e.g.,knowledge and time, knowledge and beliefs, beliefs and actions,etc.).

• They allow agent situations to be designed:– different ways of reasoning;– different ways of interacting between each other.

• Properties of modalities as set of axioms.


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An example: The fox and the raven (1)

… the fox tries to capture the raven’s cheese, in order to do so thefox charmes the raven ...

][always

][ fox

][praise

][sing

it represents what the fox believes

it represents the action in which thefox prises the raven

it represents the action in which theraven sings

it expresses the facts that are alwaystrue after executing any actions

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ϕϕ ⊃][:)( alwaysalwaysT

ϕϕ ]][[][:)(4 alwaysalwaysalwaysalways ⊃ϕϕ ]][[][:),(4 alwayspraisealwayspraisealwaysM ⊃

ϕϕ ]][[][:),(4 alwayssingalwayssingalwaysM ⊃

][always

][ fox

][praise

][sing

axiomatized only by K



axiomatized by


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)(]][[ ravencharmedpraisefox

))()(](][[ cheesedroppedsingravencharmedpraisefox ⊃

)(]][[ cheesedroppedsingpraisefox

the fox believes that if it praises theraven, then the raven is charmed

the fox believes that in any momentif the raven is charmed then it ispossible that the raven sings and sodrops the cheese

the fox believes that after praisingthe raven may sing and so it dropsthe cheese

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An example: the friends puzzle (1)

Two friends, John and Peter, have an appointment ...

[ ]j

• modality to represent what Peter’s wife believes:

• modalities to represent what John and Peter know: [ ]p

[ ( )]w p

T p p( ):[ ]ϕ ϕ⊃4( ):[ ] [ ][ ]p p p pϕ ϕ⊃ S p4( )

T j j( ):[ ]ϕ ϕ⊃4( ):[ ] [ ][ ]j j j jϕ ϕ⊃ S j4( )

• interaction axioms between Peter, Peter’s wife, and John:

P p j p j j p( , ):[ ][ ] [ ][ ]ϕ ϕ⊃

I w p p w p p( ( ), ):[ ( )] [ ]ϕ ϕ⊃

- if Peter knows that John knows something, then John knows that Peterknows that thing:

- if Peter’s wife believes something, then Peter believes the same thing:

K j( )


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An example: the friends puzzle (2)

... does each of the two friends know that the other one knows that hehas an appointment?

• Peter’s wife believes that if Peter knows the time of their appointment, thenJohn knows that too:

• Peter knows the time of the appointment and that John knows the place oftheir appointment:

• Peter knows that if John knows the place and the time of their appointment,then John knows that he has an appointment:

timep ][[ ][ ]j l

[ ( )]([ ] [ ] )ti j ti

[ ][ ]( )j l ti i t t

[ ][ ] [ ][ ]j i t t j i t t

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Interaction axioms:Inclusion Modal Logics


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Inclusion Modal Logics

• We are interested in the class of inclusion multimodal logics

[Fariñas del Cerro and Penttonen, 1988]

• They are characterized by set of logical axioms of the form

[ ][ ]...[ ] [ ][ ]...[ ] ( , )t t t s s s n mn m1 2 1 2 0 0ϕ ϕ⊃ > ≥

• Motivations:– non-homogeneous– interaction axioms– they have interesting computational properties

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Inclusion Modal Logics: examples

T t t( ):[ ]ϕ ϕ⊃

4( ):[ ] [ ][ ]t t t tϕ ϕ⊃

I t t t t( , ' ):[ ] [ ' ]ϕ ϕ⊃

4M t t t t t( , ' ):[ ] [ ' ][ ]ϕ ϕ⊃P t t t t t t( , ' ):[ ][ ' ] [ ' ][ ]ϕ ϕ⊃

D t t t( ):[ ]ϕ ϕ⊃

5( ): [ ]t t t tϕ ϕ⊃

[ ][ ]...[ ] [ ][ ]...[ ]t t t s s sn m1 2 1 2ϕ ϕ⊃

B t t t( ): [ ]ϕ ϕ⊃

• reflexivity• transitivity

• inclusion

• mutual trans.• persistency

• seriality• simmetry• euclideanness


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ϕmsss L21 ϕnttt L21⊃

Inclusion ModalLogics: possible-worlds semantics

{ }W t MOD Vt, ,ℜ ∈

• W is a set of “worlds”;• the ’s are the accessibility

relations, one for each modality;• V is a valuation function.

ℜ

ℜ ℜ ℜ ⊇ℜ ℜ ℜt t t s s sn m1 2 1 2o o o o o o... ...

1

' 1'

t t

[ ][ ]...[ ] [ ][ ]...[ ]t t t s s sn m1 2 1 2ϕ ϕ⊃

ϕmsss L21 ϕnttt L21

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Inclusion Modal Logics: examples

T t t( ):[ ]ϕ ϕ⊃

4( ):[ ] [ ][ ]t t t tϕ ϕ⊃

I t t t t( , ' ):[ ] [ ' ]ϕ ϕ⊃

4M t t t t t( , ' ):[ ] [ ' ][ ]ϕ ϕ⊃P t t t t t t( , ' ):[ ][ ' ] [ ' ][ ]ϕ ϕ⊃

• reflexivity• transitivity

• inclusion

• mutual trans.• persistency

It ⊇ℜ

ttt ℜℜ⊇ℜ o

'tt ℜ⊇ℜ

ttt ℜℜ⊇ℜ o'

tttt ℜℜ⊇ℜℜ oo ''

D t t t( ):[ ]ϕ ϕ⊃

5( ): [ ]t t t tϕ ϕ⊃B t t t( ): [ ]ϕ ϕ⊃

• seriality• simmetry• euclideanness

ℜ ℜ ℜ ⊇ℜ ℜ ℜt t t s s sn m1 2 1 2o o o o o o... ...


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t t t s s sn m1 2 1 2... ...→[ ][ ]...[ ] [ ][ ]...[ ]t t t s s sn m1 2 1 2ϕ ϕ⊃

a aa→a→ ε

[ ] [ ][ ]a a aϕ ϕ⊃[ ]a ϕ ϕ⊃

Example:

• [Fariñas del Cerro & Penttonen, 88];

but

• no proof method (a part of axiomsystems);

• no (un)decidability results ofrestricted subclasses.

• for simulating the behaviour ofgrammars;

• undecidability result;

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Proof Theory:• a prefixed tableau calculus

to deal in a uniform way withall logics in the class byusing directly thecharacterizing axioms asrewriting rules

(Un)Decidability:• about some subclasses

defined on the analogy withthe grammar productions ofrewriting systems (eg.context-sensitive, context-free, right-regular).

A tool for


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Proof theory: A Tableaux Calculus

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Axiom system vs other calculi

• Easy and intuitive.

• It is not an appropiate choice for automatization.

• ‘‘Subformula principle’’ (everything one needs in order to proveor disprove a formula is contained in the formula itself):– resolution;– sequent calculi;– tableau calculi.

• Relatively few works on this topics.


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Proof theory : a prefixed tableau calculus

[Fariñas del Cerro and Enjalbert, 1989;...]

tableaux methods

translation methods

resolution methods

[Ohlbach, 1991;Auffray and Enjalbert, 1992;Gasquet, 1994; ...]

Prefixed tableaux:[Fitting, 1983; Nerode, 1989;Catach, 1991; Massacci, 1994;Goré, 1995; Governatori, 1995;Cunningham and Pitt, 1996,Beckert and Goré, 1997;Fariñas del Cerro et al., 1998; ... ]

[Fitting, 1983; ...]

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Proof Theory : a tableau calculusIt is an attempt to build an interpretation in which a given formulais satisfiable; i.e. a refutation method.

• it does not require any normal forms;

• tableau calculi have a strong relationship with the semanticsissue, then they are easier and more natural to develop especiallyfor non-classical logics for which, generally, the semantics isknown better than the computational properties;

• tableau methods can supply a return answer.

[Fitting, 83; Massacci, 94; Goré, 95; ]


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Proof theory: a prefixed tableau calculus

It is a labeled tree where each node consists of:

a prefixed signed formulae, or of an accessibility relation formula

w T: ϕprefix

(constant)sign

formula

w wtρ 'prefix prefix

label: name ofaccessibility relation

They describe agraph

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w ww T tw T

T t

tρϕϕ

': [ ]':

[ ]

w F tw Tw w

F t

t

: [ ]':

'

[ ]ϕϕ

ρ

w Tw F

T:':¬

¬ϕϕ

w Fw T

F:':¬

¬ϕϕ

⊃⊃ F

FwTw

Fw

ψϕψϕ

::

)(:⊃⊃ T

TwFwTw

ψϕψϕ::

)(:w

w'

tw wtρ '

T t[ ]ϕ

Tϕ

w

w'

tw wtρ '

F t[ ]ϕ

Fϕ

They describe a calculus for K(t) !


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ρρρ '... 111wwww

msms −

wm−1

w n' −1

w1

s1w wsρ 1 1

smw wm sm−1ρ '

w'1

t1w wtρ1 1'

tnw wn sn

' '−1 ρ

[ ][ ]...[ ] [ ][ ]...[ ]t t t s s sn m1 2 1 2ϕ ϕ⊃1'1

ww tρ wwntn ρ1' −

...

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Graph vs path representation

)]([:1 paqcpaFi ⟩⟨⊃⟩⟨∧ w1

w2

( .)≡1( . )≡11c

( . )≡11a

( . . )≡11 1a b

a

c

b[ ][ ] [ ]a b cϕ ϕ⊃Axiom:

i

1 12 13 14 15 116 11 1

.: ([ ] ).: [ ].:.:. .:. . .:

?

F a p c q a pT a pT c qF a pTqTq

c

a b

∧⟨ ⟩ ⊃⟨ ⟩

⟨ ⟩⟨ ⟩ The subprefix

does not occur onthe branch!

11. a

Tpw :8 2

paTi ][:2qcTi ⟩⟨:3paFi ⟩⟨:4

Tqw :5 1

16 wi cρ 2wi aρ 12 ww bρ

Fpw :7 2

×


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Graph vs path representation

×

( .)≡1( . )≡11c( . )≡11a( . )≡11b

[ ] [ ][ ] [ ]a cb cϕ ϕϕ ϕ⊃⊃Axioms: i

1 12 13 14 15 116 117 118 119 11

.: ([ ] ).: [ ].:.:. .:. .:. .:. .:. .:

?

F a p c q b pT a pT c qF b pTqTqTqFpTp

c

a

b

a

b

∧⟨ ⟩ ⊃⟨ ⟩

⟨ ⟩⟨ ⟩

The subprefix ,and must beidentified!

11. a11. c

11. b

a

b

w1

c

)]([:1 pbqcpaFi ⟩⟨⊃⟩⟨∧paTi ][:2qcTi ⟩⟨:3pbFi ⟩⟨:4

Tqw :5 1

16 wi cρ 1wi aρ 1wi bρ

Fpw :7 1

Tpw :8 1

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(Un)Decidability results


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Decidability

The class of inclusion modal logics is undecidable

Completeness of the tableaucalculus implies the semi-decidability of the inclusion modallogics.

Thue Logics are undecidable:word problem - satisfiability

[ ][ ]...[ ] [ ][ ]...[ ]t t t s s sn m1 2 1 2ϕ ϕ⇔t t t s s sn m1 2 1 2... ...↔

Yes! No!

ϕ is valid in a given IML?

[Fariñas del Cerro and Penttonen, 88]

But it is possible to define a decisionprocedure which works for the wholeclass of propositional inclusionmodal logics?

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Decidability for Modal Logics

• Finite Model Property (f.m.p.): a modal system L has the f.m.p. ifand only if each non-theorem of L is false in some finite model.

• Filtration method by [Fisher and Ladner, 1979].

• Each of fifteen normal system obtained by D, T, B, 4, 5 isdecidable (has the f.m.p.) [Chellas, 1980].

• A decision procedure based on a tableau system (prefixed tableau[Fitting, 1983]).


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(Un)Decidability results

t t t s s sn m1 2 1 2... ...→[ ][ ]...[ ] [ ][ ]...[ ]t t t s s sn m1 2 1 2ϕ ϕ⊃

t t t s s sn m1 2 1 2... ...↔

t t t s s s n mn m1 2 1 2... ... ( )→ ≤

t s s st V s V T

m

i

→∈ ∈ ∪

1 2 ..., ( ) *

t s s s st V s T s V T

m m

i m m

→∈ ∈ ∈ ∪

−

<

1 2 1.. ., * , ( ) * Thue systems

Unrestricted grammars

Context sensitive grammars

Context-free grammars

Right-regular grammars

U

U

U

U

D

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Undecidability resultsUnrestricted and Thue grammars:

[ ] [ ][ ]...[ ]S p s s s pm⊃ 1 2S s s sG m⇒* ...1 2

G V T P S= ( , , , )

has a tableau proofiff

...

s1

sm

s2

S

S

s1 s2

T S p[ ]F s s s pm[ ][ ]...[ ]1 2

F s s pm[ ]...[ ]2

F s s pm[ ]...[ ]3

F s pm[ ]

FpTp

sm...


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Undecidability resultsContext sensitive, Context-free and deterministic grammars:

G V V T T P S= ∪ ∪( , , , )1 2 1 2

{ }P P P S t S S t t T= ∪ ∪ → → ∈1 2 ,G V T P S1 1 1 1 1= ( , , , ) G V T P S2 2 2 2 2= ( , , , )

t1ti

tn

... ...

t1 ti tn... ...

......

sm

s2

s1

t1ti tn

... ...

...

t1ti tn

... ...

...

S1 S2

T S[ ]1

FT

F S p2

T t S tT ⟨ ⟩ ⟨ ⟩( [ ] )S1

S2

s1 sms2 ...

L G L G( ) ( )1 2∩ ≠ ∅ has a tableauproof

iff pSpS ⟩⟨⊃⊃ 21][)][( qtSqtTt ⟩⟨∧⟩⟨∧ ∈

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47

A Decision Procedure?

[ ] [ ][ ]a b aϕ ϕ⊃[ ] [ ]a bϕ ϕ⊃

)][(:1 pbapbFi ⟩⟨⊃⟩⟨

w1i w2

w3

a

a a

a a a

b b b ...

...

pbTi ⟩⟨:2pbaFi ][:3 ⟩⟨

Tpw :4 1

115 wiwi ab ρρ

pbFw ][:6 1

Fpw :7 2

212218 wiwwww aab ρρρ

pbTw ][:9 2

Fpw :10 3

331323211 wiwwwwww aaab ρρρρ


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Incestual Modal Logics

• We are interested in the class of incestual multimodal logics

[Catach, 1988]

• They are characterized by set of logical axioms of the form

• Motivations:– non-homogeneous– interaction axioms

⟨ ⟩ ⊃ ⟨ ⟩a b c d[ ] [ ]ϕ ϕModal operators can be labeled bycomplex parameters built up fromatomic labels by means of union“U” and composition “;”

[ ][ ' ] [ ][ ' ]t t t t

[ ' ] [ ] [ ' ]t t t t

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49

Some well-known incestual axioms

• seriality:• simmetry:• transitivity• euclideanness:• determinism:• density:• mutual ser.:• relative incl.:• persistency:• ...

5( ): [ ]t t t tϕ ϕ⊃

B t t t( ): [ ]ϕ ϕ⊃

D t t t t( , ' ):[ ] 'ϕ ϕ⊃[ ] ([ ' ] [ "] )t t tϕ ϕ ϕ⊃ ⊃

⟨ ⟩ ⊃ ⟨ ⟩a b c d[ ] [ ]ϕ ϕ

δ ϕ ϕ( ): [ ]t t t⊃De t t t t( ): ϕ ϕ⊃

P t t t t t t( , ' ):[ ][ ' ] [ ' ][ ]ϕ ϕ⊃

D t t t( ):[ ]ϕ ϕ⊃ ⟨ = ⟩ = ⊃ = ⟨ = ⟩a b t c d tε ϕ ε ϕ[ ] [ ]

4( ):[ ] [ ][ ]t t t tϕ ϕ⊃ ⟨ = ⟩ = ⊃ = ⟨ = ⟩a b t c t t dε ϕ ε ϕ[ ] [ ; ]


50

IMLs: possible-worlds semantics

{ }W t MOD Vt, ,ℜ ∈• W is a set of “worlds”;• the ’s are the accessibility

relations, one for each modality;• V is a valuation function.

ℜt

ℜ ℜ ⊇ℜ ℜ− −b d a co o1 1

ℜ ℜ

ℜb ℜd

⟨ ⟩ ⊃ ⟨ ⟩a b c d[ ] [ ]ϕ ϕ

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51

An example: the three wise men puzzle

[ ]( ( ) ( ) ( ))any ws a ws b ws c∨ ∨

[ ]( ( ) [ ] ( ))any ws X Y ws X⊃[ ]( ( ) [ ] ( ))any ws X Y ws X¬ ⊃ ¬

T any any( ) [ ]ϕ ϕ⊃

4( ) [ ] [ ][ ]any any any anyϕ ϕ⊃

I any X any X( , ) [ ] [ ]ϕ ϕ⊃

5( , ) [ ] [ ] [ ]X Y X Y X¬ ⊃ ¬ϕ ϕ

4M X Y X Y X( , ) [ ] [ ][ ]ϕ ϕ⊃

• At least one of the wise men has a white spot

• Whenever one of them has (not) a white spot, the others know this fact.

• [any] is a weak commonknowledge operator:

• whenever a wise men does (not)know something, the others knowthat he does (not) know that thing:


52

Incestual Modal Logics:tableau calculus (main rules)

ℜ ℜ

ℜb ℜd

w w w ww w w w

a c

b d

ρ ρρ ρ

ρ' "

' * " *

w ww ww w

t t

t

t

ρρρ

ρα; '

'

'' '

' ' 'w w

w w w wt t

t t

ρρ ρ

ρβ∪ '

'

'' '

∈ℜ ∪t t ' ∈ℜ ∪t t '

∈ℜ t t; '

∈ℜt '

∈ℜ tw' '

∈ℜt '∈ℜ t

27


53

An example of a tableau proof)][)]"[]'([:1 pcpbpbaFi ⊃∧⟩⟨

i

b b' ' '∪

b'

a

w1

w3

c

w2( ' ' )b

ε

a b b c[ ' "] [ ]∪ ⊃ϕ ε ϕAxiom:

)]"[]'([:2 pbpbaTi ∧⟩⟨

pcFi ][:3

)]"[]'([:4 1 pbpbTw ∧

15 wi aρ

pbTw ]'[:6 1

pbTw ]"[:7 1

Fpw :8 2

19 wi cρ

323"'110 wwww bb ερρ ∪

3'111 wwa bρ 3''111 wwb bρ

Tpwa :12 3 Tpwb :12 3

× ×


54

Decidability: some subclasses of IMLs

D DU U

U U

UIMLs

⟨ ⟩ ⊃ ⟨ ⟩a b c d[ ] [ ]ϕ ϕ

ConfLs[ ] [ ]b c dϕ ϕ⊃ ⟨ ⟩⟨ ⟩ ⊃ ⟨ ⟩a b d[ ]ϕ ϕ

DLs⟨ ⟩ ⊃a cϕ ϕ[ ]

SimLs⟨ ⟩ ⊃a b[ ]ϕ ϕϕ ϕ⊃ ⟨ ⟩[ ]c d

GLs[ ] [ ]b cϕ ϕ⊃⟨ ⟩ ⊃ ⟨ ⟩a dϕ ϕ

SerLs[ ]b dϕ ϕ⊃ ⟨ ⟩

EuLs

⟨ ⟩ ⊃ ⟨ ⟩a c dϕ ϕ[ ]⟨ ⟩ ⊃a b c[ ] [ ]ϕ ϕ

U[ ] [ ]b⟨ ⟩ ⟨ ⟩d

TLs

by means of thetableau calculus

by simulatingGrammar Logics

by means of thetableau calculus

by reducing the word problem tothe satisfiability problem

28


55

Applications: Logic Programming Extensions


56

Modal Extentions of Logic Programming

• Modal extensions of logic programming join tools for formalizingand reasoning about temporal and epistemic knowledge withdeclarative features of logic programming languages.

• They support the “context abstraction”, which allows to describedynamic and context-dependent properties of certain problems ina natural and problem-oriented way.

• Goal Directed Proof Procedure: a sound and complete operationalsemantics with respect to declarative semantics (filling the gap!).

• Lots of proposals [Orgun and Ma, 1994; Fisher and Owens, 1993].

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57

Grammar logics: NemoLOG

• beliefs, knowledges, actions, ...;

• tools for software engeneering (e.g. modularity, readability,reusability, hierarchical dependecies, inheritance, etc.);

• parametric w.r.t. the properties of multimodal operators;

• a proof procedure that can deal in a uniform way with alllogics in the class (it uses directly the characterizingaxioms as rewriting rules ...).


58

Inclusion axiom clauses:

A program is a pair:

• Ds is a set of extendedclauses;

• Ax is a set of inclusion axiomclauses.

Ds Ax,

[ ][ ]([ ] [ ][ ] [ ][ ] )t t t a t t b t t c1 2 5 6 7 3 4∧ ⊃

[ ]([ ] [ ][ ] ) [ ]t t a t t b t c1 2 3 4 5∧ ∧

[t ][t ] [t ]1 2 3

]c[t]b),][t[t]a,]([t[t 54321

]b)][t[t,]a-[t:]c][t]([t][t[t 7654321

[ ][ ] [ ]t t t1 2 3

What is a program in NemoLOG ?

Goals:

Clauses:

30


59

it is a framework for studyingand developing extensions oflogic programming suitable to

NemoLOG:SomeApplications

structure knowledge andperform epistemicreasoning

introduce operators forstructuring logicprograms

describe inheritance in ahierarchy of classes(modules)

reason about actions


60

NemoLOG: an example

[export][animal]{ mode(walk).mode(run) :- no_of_legs(X), X >= 2.mode(gallop) :- no_of_legs(X), X >= 4. }

[export][bird]{ mode(fly).no_of_legs(2).

covering(feather). }

[export][tweety]{owner(fred). }

[export][horse]{no_of_legs(4). covering(hair). }

[ ] [ ]i l bi d [ ] [ ]animal horse→

[ ] [ ]bi d t t

Goals:?- [bird]mode(run).YES!?- [X]mode(fly).YES! X = bird or X = tweety

31


61

Grammar logics: DyLOG

• a multimodal language for reasoning about dynamicdomains (the effects of actions in a dinamically changingworld) in a logic programming setting;

• a ‘‘programming language Prolog-like’’ for actions: a way forcomposing actions by defining conditional or iterativeactions (GOLOG, Transaction Logic);

• the procedures that define complex actions are representedby means a set of inclusion axioms;

• a goal directed proof procedure (it uses directly thecharacterizing axioms as rewriting rules ...).


62

[ ]( [ ] )always Fs a F⊃

What is a program in DyLOG ?A program is a pair:

• is a set of simple actionclau-ses and procedureclauses;

• Obs is a set of initial obser-vations.

( )Π,ObsΠ[ ]( )always Fs a F⊃

[ ]( )always Fs F⊃

p p p pn1 2 0L ϕ ϕ⊃

F

Simple action clauses:

Procedures:

Observations:

Goals:

• action laws

• causal laws

• precondition laws

p p p Fsn1 2 L

Answers: a state!a a am1 2L

32


63

DyLOG: an example of program

( ) ? ( , ) ( , ( ,N move Y X stack B Y stack N Y> − ⊃ −0 1 1ϕ ϕ

ϕ ϕ⊃ stack X( , )0

pickup X putdown X Y move X Y( ) ( , ) ( , )ϕ ϕ⊃

[ ]( ( , ) [ ( )] ( ))always on X Y pickup X clear Y⊃[ ]( ( ) ( ) )always clear X pickup X true⊃

[ ]( ( , ) ( ))always on X Y clear Y¬ ⊃

[ ]( [ ( , )] ( , ))always true putdonw X Y on X Y⊃[ ]( ( , ) ( ) ( , ) )always X Y wider Y X clear Y putdonw X Y true≠ ∧ ∧ ⊃

[ ]( ( ) ( , ))always clear Y on X Y¬ ⊃

stack a clear b( , ) ( )2


64

Conclusions and future works• Implementation vs uniformity

[Governatori, 1995; Cunningham and Pitt, 1996,Beckert and Goré,1997]

• Extension of the tableau calculus to include dynamic logic

• Complexity of decidable classes

– Applications in Logic Programming: Epistemic reasoning;

- NemoLOG: An object-oriented logic language with state;

- DyLOG+: sensing actions and conditional plans;

• Thanks to Laura Giordano, Alberto Martelli, and Viviana Patti.

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65

References

• M. Baldoni, Normal Multimodal Logics: Automatic Deduction and LogicProgramming Extension. PhD thesis, Dipartimento di Informatica, Universita` degliStudi di Torino, Italy, 1998. Available at http://www.di.unito.it/~baldoni.

• M. Baldoni. Normal Multimodal Logics with Interaction Axioms. In D. Basin, M.D`Agostino, D. M. Gabbay, S. Matthews, and L. Vigano`, editors, LabelledDeduction, Kluwer Accademic Publishers, May 2000.

• M. Baldoni, L. Giordano, and A. Martelli. A Tableau Calculus for Multimodal Logicsand some (Un)Decidability Results. In H. de Swart, editor, Proc. of the InternationalConference on Analytic Tableaux and Related Methods, TABLEAUX'98, volume1397 of LNAI, pages 44-59. Springer-Verlag, 1998.

• M. Baldoni, L. Giordano, and A. Martelli. A Modal Extension of Logic Programming:Modularity, Beliefs and Hypothetical Reasoning. In Journal of Logic andComputation, 6(5):596-635, 1998.

• M. Baldoni, L. Giordano, A. Martelli, and V. Patti. An abductive Procedure forReasoning about Actions in Modal Logic Programming. In Proc. of the 2ndInternational Workshop on Non-Monotonic Extensions of Logic Programming,NMELP’96, vol. 1216 of LNAI, pages 132-150, Springer-Verlag, 1997.


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References

• M. Baldoni, L. Giordano, A. Martelli, and V. Patti. A Modal Programming Languagefor Representing Complex Actions. In A. Bonner, B. Freitag, and L. Giordano,editors, Proc. 1998 JICSLP’98 Post-Conference Workshop on Transaction andChange in Logic Databases, DYNAMICS ‘98, pages 1-15, Manchester, 1998.

• M. Fisher and R. Owens. An Introduction to Executable Modal and TemporalLogics. In Proc. of the IJCAI’93 Workshop on Executable Modal and TemporalLogics, volume 897 of LNAI, pages 1-20. Springer-Verlag, 1993.

• M. Orgun and W. Ma. An overview of temporal and modal logic programming. In D.Gabbay and H. Ohlbach, editors, Proc. of the First International Conference onTemporal Logic, volume 827 of LNAI, pages 445-479. Springer-Verlag, 1994.

• M. Genereseth and N. Nilsson. Logical Foundations of Artificial Intelligence.Chapter 9. Morgan Kaufmann, 1987.

• L. Catach. Normal Multimodal Logics. In Proc. of the 7th National Conference onArtificial Intelligence, AAAI’88, volume 2, pages 491-495. Morgan Kaufmann, 1988.

• L. Catach. TABLEAUX: A General Theorem Prover for Modal Logics. Journal ofAutomated Reasoning, 7(4):489-510, 1991.

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67

References

• M. Chellas. Modal Logic: an Introduction. Cambridge University Press, 1980.• M. Fitting. Proof Methods for Modal and Intuitionistic Logics, volume 160 of

Synthese library. D. Reidel, Dordrecht, Holland, 1983.• M. Fitting. Basic Modal Logic. In D. Gabbay, C. J. Hogger, J. A. Robinson, editors,

Handbook of Logic in Artificial Intelligence and Logic Programming, volume 1,pages 365-448. Oxfors Science Publications, 1983.

• A. Nerode. Some Lectures on Modal Logic. In F. L. Bauer, editor, Logic, Algebra,and Computation, volume 79 of NATO ASI Series. Springer-Verlag, 1989.or ModalLogics of Knowledge and Belief. Artificial Intelligence, 54:319-379, 1992.

• J. Y. Halpern and Y. Moses. A Guide to Completenes and Complexity f• G. E. Hughes and M. J. Cresswell. A New Introduction to Modal Logic. Routledge,

1996.• M. D’Agostino, D. M. Gabbay, R. Hähnle, and J. Possegga, editors. Handbook of

Tableau Methods. Kluwer Academic Publishers, 1999.• N. Olivetti. Algorithmic Proof Theory for Non-Classical and Modal Logics. PhD

thesis, Dipartimento di Informatica, Universita` degli Studi di Torino, 1995.

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