a goal-directed calculus for standard conditional logicspozzato/oldws/cilc 2007 a.pdf · a...

A Goal-Directed Calculus forStandard Conditional Logics

Nicola Olivetti1 and Gian Luca Pozzato2

1 LSIS - UMR CNRS 6168 Universite Paul Cezanne (Aix-Marseille 3), Avenue

Escadrille Normandie-Niemen 13397 Marseille Cedex 20 - France

[email protected] Dipartimento di Informatica - Universita degli Studi di Torino, corso Svizzera 185 -

10149 Turin - Italy

[email protected]

A Goal-Directed Calculus for Standard Conditional Logics – p. 1

Introduction

• Proof methods for conditional logics


Introduction


• Goal-directed calculi for conditional logics


Introduction


• Goal-directed calculi for conditional logics

• Simple implementation in SICStus Prolog


Outline

• Introduction to CL


Outline


• Sequent calculi SeqS’


Outline



• Goal-directed calculi U S’


Outline



• Goal-directed calculi U S’

• GOALDUCK: implementation of U S’


Conditional Logics


Introduction to Conditional Logics

• Extensions of classical logic obtained by adding the

conditional operator ⇒



• Extensions of classical logic obtained by adding the

conditional operator ⇒

• Long history (Stalnaker and Lewis early 70s)

• Formalization of hypothetical reasoning : “if A were the

case then B”

• Formalization of counterfactual sentences : conditionals

of the form “if A were the case then B would be the

case", where A is false



• Recently, conditional logics have found applications in

several areas of artificial intelligence and knowledge

representation



• Recently, conditional logics have found applications in

several areas of artificial intelligence and knowledge

representation

• Formalization of knowledge update and revision [Gra98]

• Correspondence between AGM revision and conditional

logics

• Ramsey test : A ⇒ B “holds" in a knowledge base K if

and only if B “holds" in the knowledge base K

revised/updated with A

• (RT) K ⊢ A ⇒ B iff K ◦ A ⊢ B (◦ ≡ revision/update

operator)



• modelization of hypothetical queries in deductive databases

and logic programming [GGM+00]

• modelization of causal inference and reasoning about action

execution in planning [Sch99, GS04]

• modelization of actions and causations

[Pea99, Pea00, GP98]



• modelization of hypothetical queries in deductive databases

and logic programming [GGM+00]

• modelization of causal inference and reasoning about action

execution in planning [Sch99, GS04]

• modelization of actions and causations

[Pea99, Pea00, GP98]

• very few proof methods (and even less theorem provers)


Conditional Logics

• Propositional conditional language L defined from:


Conditional Logics


• a set of propositional variables ATM


Conditional Logics



• symbols of false ⊥ and true ⊤


Conditional Logics




• set of connectives ¬, →, ∨, ∧, ⇒


Conditional Logics





• Formulas:


Conditional Logics





• Formulas:

• atomic formulas: ⊥, ⊤, and the propositional variables of

ATM


Conditional Logics





• Formulas:


ATM

• if A is a formula, then ¬A is a complex formula


Conditional Logics





• Formulas:


ATM

• if A is a formula, then ¬A is a complex formula

• if A and B are formulas, A → B, A ∨ B, A ∧ B, and A ⇒

B are complex formulas


Conditional Logics

• Example : (P → R) → (Q ⇒ ⊥)

• Example : (P ⇒ Q) ⇒ R


Conditional Logics

• Example : (P → R) → (Q ⇒ ⊥)


• Possible world structures (as in modal logics)

• The conditional operator is a sort of modality indexed by a

formula of the same language


Conditional Logics

• Example : (P → R) → (Q ⇒ ⊥)





• Lack of a universally accepted semantics

• Two most popular semantics:

• selection function semantics

• sphere semantics


Selection function model

M = 〈W, f, [ ]〉

• W is a non empty set of items called worlds

• f is the selection function

f : W × 2W −→ 2W

• [ ] is the evaluation function :

• [⊥] = ∅

• [A → B]=(W - [A]) ∪ [B]

• [A ⇒ B]={w ∈ W | f (w, [A]) ⊆ [B]}


Conditional Logics

• Intuitive meaning: given a world w and a formula A, f

selects the worlds most “similar” to w given A


Conditional Logics



• A ⇒ B holds in w if B holds in all worlds selected by f for w

and A


Conditional Logics




and A

• A ⇒ B holds in w if B holds in f(w,A)


Conditional Logics




and A


• Normality condition : f(w, [A]) rather than f(w,A)


Conditional Logics




and A


• Normality condition : f(w, [A]) rather than f(w,A)

• Equivalent to: if [A] = [A′], then f(w,A) = f(w,A′)


Conditional Logics

• Semantics of the basic normal conditional system CK

• CK is the fundamental system (as K for modal logic)

• Formulas valid in CK are valid in every selection function

model

• Other conditional systems are obtained by assuming further

properties on the selection function

System Axiom Model condition

ID A ⇒ A f(w, [A]) ⊆ [A]

MP (A ⇒ B) → (A → B) w ∈ [A] → w ∈ f(w, [A])


Sequent Calculi SeqS’


Proof methods for Conditional Logics

• Very few proof methods for conditional logics




• Sequent calculi SeqS’ for:





• the basic system CK





• the basic system CK

• extensions with ID and MP



• Labelled deductive system




• Labels used to represent possibile worlds





• 2 kinds of labelled formulas:

• world formulas x : A

• transition formulas xA

−→ y





• 2 kinds of labelled formulas:

• world formulas x : A


−→ y

• representing:

• A holds in the world x

• y ∈ f(x, [A])



• A sequent is a pair

〈Γ,∆〉

Γ ⊢ ∆

• Γ and ∆ are multisets of labelled formulas



• A sequent is a pair

〈Γ,∆〉

Γ ⊢ ∆

• Γ and ∆ are multisets of labelled formulas

• Intuitive meaning of Γ ⊢ ∆:

• every model that satisfies all labelled formulas of Γ in the

respective worlds (specified by the labels) satisfies at least

one of the labelled formulas of ∆ (in those worlds)


(AX) ! , x : P ! ", x : P (P " ATM )

(A!) ! , x : ! " " (A!) ! " ", x : !

! ! ", x : A(¬R)

! ! ", x : ¬A(¬L) ! , x : A ! "! , x : ¬A ! "

(!L)! , x : A ! B " "

! , x : A, x : B ! "(!R)

! ! ", x : A " B

! ! ", x : A ! ! ", x : B

(!L)! , x : A ! " ! , x : B ! "

! , x : A ! B " "

! ! ", x : A, x : B(!R)

! ! ", x : A " B

(! L)! , x : A! B " "

! ! ", x : A ! , x : B ! "(! R)

! ! ", x : A" B

! , x : A ! ", x : B

(! L)! , x : A! B " "

! , x : A! B " ", x A#$ y ! , x : A! B , y : B " "(! R)

! ! ", x : A" B

! , x A!" y # ", y : B

! , x A!" y # ", x B!" y

u : A ! u : B u : B ! u : A(EQ)

! ! ", x A"# x , x : A

! ! ", x A"# x(MP )

! , x A!" y # "(ID) ! , x A!" y , y : A # "

Goal-directed calculi U S’


Uniform Proofs

• SeqS’ calculi used to develop goal-directed proof

procedures for conditional logics

• paradigm of Uniform Proofs [MNPS91, GO00]

• extensions of logic programming based on conditional logics

• generalization of conventional logic programming


Uniform Proofs

• SeqS’ calculi used to develop goal-directed proof

procedures for conditional logics

• paradigm of Uniform Proofs [MNPS91, GO00]

• extensions of logic programming based on conditional logics

• generalization of conventional logic programming

• Sequent Γ ⊢ G where:

• Γ represents the “program” or “database”

• G is the “goal” whose proof is searched


Uniform Proofs

• Basic idea: backward proof of Γ ⊢ G is driven by the goal G


Uniform Proofs


• G is stepwise decomposed according to its logical structure

by the rules of the calculus, until its atomic constituents are

reached


Uniform Proofs




reached

• The connectives in G can be interpreted operationally as

search instructions


Uniform Proofs




reached

• The connectives in G can be interpreted operationally as

search instructions

• To prove an atomic goal Q, one looks in Γ for one “clause”

whose head matches with Q and tries to prove the “body” of

the clause


Uniform Proofs

• Not every provable sequent admits a uniform proof

• One must identify a (significant) fragment of the langauge

allowing uniform proofs


Uniform Proofs




• We distinguish:

• database formulas (D-formulas)


−→ y

• formulas that can be asked as goals (G-formulas)


Uniform Proofs




• We distinguish:

• database formulas (D-formulas)


−→ y

• formulas that can be asked as goals (G-formulas)

• Goal-directed calculi for conditional logics: U S’


Uniform Proofs

• Fragment of L allowing uniform proofs:

• D = G → Q | A ⇒ D

• G = Q | ⊤ | G ∧ G | G ∨ G | A ⇒ G

• A = Q | A ∧ A | A ∨ A


Uniform Proofs

• Fragment of L allowing uniform proofs:

• D = G → Q | A ⇒ D

• G = Q | ⊤ | G ∧ G | G ∨ G | A ⇒ G

• A = Q | A ∧ A | A ∨ A

• database Γ = D-formulas + transition formulas


Calculi U S’

• U S’ calculi are sound and complete for the selected

fragment


Calculi U S’

• U S’ calculi are sound and complete for the selected

fragment

• Example : reading of a conditional A ⇒ B

• “if the current state is updated with A then B holds”

• (A might be an action) “as an effect of A, B holds” or

“having performed A, B holds”


Calculi U S’

• formula of type A: either an atomic formula or a boolean

combination of conjunctions and disjunctions of atomic

formulas

• Flat(x : A): flats the conjunction/disjunction into sets of

atomic formulas and SG ∈ Flat(x : G)}

• Flat(x : Q) = {{x : Q}}, with Q ∈ ATM

• Flat(x : F ∨ G) = Flat(x : F ) ∪ Flat(x : G)

• Flat(x : F ∧ G) = {SF ∪ SG | SF ∈ Flat(x : F )}


(U trans) ! !GD xA"# y ! Flat(u : A!) !GD u : A and Flat(u : A) !GD u : A! if x

A!"# y $ !

(U!) ! "GD x : !

if y : A1 ! A2 ! . . .! An ! (G " Q)

(U ax) ! !GD x : Q if x : Q " !

(U !) ! "GD x : G1 !G2 ! ! "GD x : G1 or ! "GD x : G2

(U prop) ! !GD x : Q ! ! !GD y A1"# x1 ,! !GD x1A2"# x2 , . . . ,! !GD xn!1

An"# x ,! !GD x : G

(U !) ! "GD x : G1 !G2 ! ! "GD x : G1 and ! "GD x : G2

! !

and x != y

(U !)CK ! "GD x : A! G ! !, xA#$ y "GD y : G (y new)

(U !)ID ! "GD x : A! G ! !, xA#$ y, Flat(y : A) "GD y : G (y new)

(U trans)MP ! !GD xA"# x ! ! !GD x : A

GOALDUCK: implementation of U S’

• GOALDUCK: simple implementation of the calculi U S’

• SICStus Prolog program of only eight clauses: each one of

them implements a rule of U S’






• Goal-directed proof search implemented by a predicate

prove(Goal,Gamma,Trans,Labels).






• Goal-directed proof search implemented by a predicate

prove(Goal,Gamma,Trans,Labels).

which succeeds iff G, represented by Goal, derives from

the KB Γ, represented by the list of world formulas Gamma

and by the list of transition formulas Trans



prove([X,A => G],Gamma,Trans,Labels):-

generateLabel(Y,Labels),

prove([Y,G],Gamma,[[X,A,Y]|Trans],[Y|Labels]).

prove([X,Q],Gamma,Trans,Labels):-

member([Y,F=>(G->Q)],Gamma),atom(Q),

prove([X,G],Gamma,Trans,Labels),extract(F,List),

proveList(X,Y,List,Gamma,Trans,Labels).



prove([X,A,Y], ,Trans, ):-

member([X,AP,Y],Trans),

flat(A,FlattenedA),flat(AP,FlattenedAP),

proveFlat([x,A],[],[],[x],x,FlattenedAP),

proveFlat([x,AP],[],[],[x],x,FlattenedA).


Performances of GOALDUCK

Theorem prover positive answers negative answers timeouts

CondLean 47 0 53

GOALDUCK 67 16 17

• 100 goal

• 20 propositional variables in the database, only 3 in the

goals

• 200 formulas in the database

• time limit: 100ms


Conclusions and FutureWork


Conclusions

• We have provided goal-directed calculi for normal

conditional logics: U S’


Conclusions



• Sound and complete wrt to the semantics, if the language

considered is restricted to a specific fragment allowing

uniform proofs


Conclusions





uniform proofs

• GOALDUCK: simple implementation in SICStus Prolog


Conclusions





uniform proofs

• GOALDUCK: simple implementation in SICStus Prolog

• no other goal-directed calculi/theorem provers in the

literature


Future Work

• Further investigation could lead to the development of

extensions of logic programming based on conditional logics


Future Work



• We aim to develop goal-directed calculi for other conditional

logics (CEM, CS), in order to extend GOALDUCK to support

all of them


Future Work





all of them

• Broader language allowing uniform proofs


Future Work





all of them

• Broader language allowing uniform proofs

• Increase GOALDUCK’s performances by experimenting

standard refinements and heuristics


Thank you!!!!!!


References

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and M. L. Sapino. Conditional reasoning in logic

programming. Journal of Logic Programming,

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[GO00] D. M. Gabbay and N. Olivetti. Goal-directed Proof

Theory. Kluwer Academic Publishers, 2000.

[GP98] D. Galles and J. Pearl. An axiomatic character-

ization of causal counterfactuals. Foundation of

Science, 3(1):151–182, 1998.

[Gra98] G. Grahne. Updates and counterfactuals. Journal

of Logic and Computation, 8(1):87–117, 1998.

[GS04] L. Giordano and C. Schwind. Conditional logic

of actions and causation. Artificial Intelligence,

157(1-2):239–279, 2004.

[MNPS91] D. Miller, G. Nadathur, F. Pfenning, and A. Scedrov.

Uniform proofs as a foundation for logic program-

ming. In Annal of Pure and Applied Logic, 51(1-

2):125–157, 1991.

32-1

[Pea99] J. Pearl. Reasoning with cause and effect.

In T. Dean, editor, Proceedings of IJCAI 1999

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den, 1999. Morgan Kaufmann.

[Pea00] J. Pearl. Causality: Model, Reasoning and

Inference. Cambridge University Press, 2000.

[Sch99] C. B. Schwind. Causality in action theories.

Electronic Transactions on Artificial Intelligence

(ETAI), 3(A):27–50, 1999.

32-2

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