Dipartimento di Informatica - Università degli studi di Torino

1. CondLean 2.0: a Theorem Prover for standard Conditional Logics Nicola Olivetti – Gian Luca Pozzato Dipartimento di Informatica - Università degli studi di Torino

2. Outline • Brief introduction of Conditional Logics • Sequent calculi SeqS for some standard conditional logics • List of results, in order to obtain a decision procedure for conditional logics and the reformulation BSeqS • CondLean 2.0: a SICStus Prolog implementation of sequent calculi BSeqS • Future work and references

3. 1 Conditional logics

4. Conditional logics • Conditional logics have a long history • Recently, they have been used in some branches of artificial intelligence, such as: • non-monotonic reasoning (for example, prototypical reasoning and default reasoning); • belief revision; • deductive databases; • representation of counterfactuals.

5. Conditional logics Syntax • Conditional logic is an extension of classical logic by the conditional operator. • We consider a language L over a set ATM of propositional variables. Formulas of L are obtained applying the classical connectives and the conditional operator  to the propositional variables.

6. Conditional logics Semantics • We consider the selection function semantics; the model is a triple:

7. Conditional logics Semantics • We consider the selection function semantics; the model is a triple: < W, f, [ ] >

8. Conditional logics Semantics • We consider the selection function semantics; the model is a triple: < W, f, [ ] > - W is a non-empty set of items called worlds;

9. Conditional logics Semantics • We consider the selection function semantics; the model is a triple: < W, f, [ ] > - W is a non-empty set of items called worlds; - f is a function f: W x 2W 2W , called the selection function;

10. Conditional logics Semantics • We consider the selection function semantics; the model is a triple: < W, f, [ ] > - W is a non-empty set of items called worlds; - f is a function f: W x 2W 2W , called the selection function; - [ ] is an evaluation function [ ] : ATM 2W.

11. Conditional logics Semantics • The selection function f (w, [A]) selects the worlds “closest”to w given the information A.

12. Conditional logics Semantics • [ ] assigns to an atomic formula P the set of worlds where P is true; [ ] is also extended to complex formulas as follows : • [  ] =  • [ A  B ] = (W - [ A ]) [ B ] • [ A  B ] = {w  W | f (w, [ A ]) [ B ]} • A conditional formula A  B is true in a world w if B is true in all the worlds “closest” to w given the information A.

13. Conditional logics Semantics • We say that a formula A is valid in a model Mif [ A ] = W. A formula A is valid if it is valid in every model M.

14. Conditional logics System CK • The semantics above characterizesthe minimal normal conditional logic CK, which is axiomatized as follows:

15. Conditional logics - System CK All the tautologies of the classical propositional logic are CK axioms; modus ponens: RCEA: RCK: A A  B B A  B (A C)  (B C) (A1 A2 …  An) B (C A1 C A2 …  C An)  (C B)

16. Conditional logics Systems CK{+MP}{+ID} With some properties of the selection function, we have the following extensions: System Axiom Selection function property

17. Conditional logics Systems CK{+MP}{+ID} With some properties of the selection function, we have the following extensions: System Axiom Selection function property CK+ID f (x, [ A ]) [ A ] A A

18. Conditional logics Systems CK{+MP}{+ID} With some properties of the selection function, we have the following extensions: System Axiom Selection function property CK+ID f (x, [ A ]) [ A ] A A CK+MP w [ A ]  w  f (w, [ A ]) (A B)  (A B)

19. Conditional logics Systems CK{+MP}{+ID} With some properties of the selection function, we have the following extensions: System Axiom Selection function property CK+ID f (x, [ A ]) [ A ] A A CK+MP w [ A ]  w  f (w, [ A ]) (A B)  (A B) w [ A ]  w  f (w, [ A ]) (A B)  (A B) CK+MP+ID f (x, [ A ]) [ A ] A A

20. 2 Sequent Calculi SeqS

21. Sequent Calculi SeqS • In [OlivettiSchwind01] sequent calculi for conditional logics CK{+MP}{+ID} called SeqS, where S={CK, ID, MP, ID+MP}, are introduced. • These calculi use transition formulas and labels, in a similar way to [Viganò00] and [Gabbay96].

22. Sequent Calculi SeqS • A sequent is a pair < ,  >, written as usual as  ; •  and  are multisets of formulas; we have two kinds of formulas:

23. Sequent Calculi SeqS • A sequent is a pair < ,  >, written as usual as  ; •  and  are multisets of formulas; we have two kinds of formulas: Labelled formulas, like x: A;

24. Sequent Calculi SeqS • A sequent is a pair < ,  >, written as usual as  ; •  and  are multisets of formulas; we have two kinds of formulas: transition formulas, like . Labelled formulas, like x: A; A x y

25. Sequent Calculi SeqS • A sequent is a pair < ,  >, written as usual as  ; •  and  are multisets of formulas; we have two kinds of formulas: transition formulas, like . Labelled formulas, like x: A; A x y • A labelled formula x: A represents that the formula Ais true in the world x.

26. Sequent Calculi SeqS • A sequent is a pair < ,  >, written as usual as  ; •  and  are multisets of formulas; we have two kinds of formulas: transition formulas, like . Labelled formulas, like x: A; A x y • A labelled formula x: A represents that the formula Ais true in the world x. A • A transition formula represents that y  f ( x, [ A ] ). x y

27. Sequent Calculi SeqS Theorem (soundness and completeness of SeqS):  is valid iff it is derivable in SeqS.

28. 3 How to obtain a decision procedure

29. How to obtain a decision procedure • SeqS calculi have the following contraction rules:  , F, F , F, F  (ContrR) (ContrL)  , F , F 

30. How to obtain a decision procedure • SeqS calculi have the following contraction rules:  , F, F , F, F  (ContrR) (ContrL)  , F , F  • In backward proof search, the contraction rules add a formula in the premise; all the other rules are analytic. • In order to obtain a decision procedure, it is essential to control the application of the contraction rules.

31. How to obtain a decision procedure In [OlivettiSchwind01] it is shown that: the contraction rules can be eliminated in SeqCK SeqCK is complete without (ContrL) e (ContrR); so we have a decision procedure for CK (all the SeqCK’s rules are analytic).

32. How to obtain a decision procedure • In [Pozzato03]it is shown that:

33. How to obtain a decision procedure • In [Pozzato03]it is shown that: 1. SeqID is complete without the contraction rules.

34. How to obtain a decision procedure • In [Pozzato03]it is shown that: 1. SeqID is complete without the contraction rules. 2. SeqMP and SeqID+MP are NOT complete without the contraction rules. • This analysis is inspired by the work made by Luca Viganò for modal logic T [Viganò00]

35. How to obtain a decision procedure • One can control the application of the contraction rules as follows:

36. How to obtain a decision procedure • One can control the application of the contraction rules as follows: 2.1. SeqMP and SeqID+MP are complete without the (ContrR) rule.

37. How to obtain a decision procedure • One can control the application of the contraction rules as follows: 2.1. SeqMP and SeqID+MP are complete without the (ContrR) rule. 2.2. In SeqMP and SeqID+MP one needs to apply the (ContrL) rule at most one time on every conditional formula x: A  B in every branch of the proof tree. • Here are the BSeqS calculi presented in [OlivettiPozzatoSchwind04]:

38. How to obtain a decision procedure • The (L) rule is “split” in three rules, to keep into account of the necessary application of (ContrL).

39. How to obtain a decision procedure • The (L) rule is “split” in three rules, to keep into account of the necessary application of (ContrL). 1. The first rule decomposes the principal formula x: A  B adding a copy of the formula in the multiset CondContr: If x: A  B K K  { x: A  B } | CondContr  { x: A  B } | , y: B  A x y K  { x: A  B } | CondContr  { x: A  B } |  , (L)1 K | CondContr | , x: A  B 

40. How to obtain a decision procedure • The (L) rule is “split” in three rules, to keep into account of the necessary application of (ContrL). 1. The second rule is applied if x: A  Bhas already been contracted in that branch (i.e. belongs to K); it decomposes the principal formula x: A  B without adding any copy of it: If x: A  B K K | CondContr | , y: B  A x y K | CondContr |  , (L)2 K | CondContr | , x: A  B 

41. How to obtain a decision procedure • The (L) rule is “split” in three rules, to keep into account of the necessary application of (ContrL). 2. The third rule decomposes a contracted formulax: A  B in CondContr, without adding a copy of it: A x y K  { x: A  B } | CondContr |   K  { x: A  B } | CondContr | , y: B  (L)3 K  { x: A  B } | CondContr  { x: A  B } |  

42. How to obtain a decision procedure • We have improved SeqS calculi presented in [OlivettiPozzato03], where the rule (L) was split in two rules; • Reduced number of application of (implicit) contraction in each branch: better performances • Improved version of the graphical user interface • Many features inherited from CondLean

43. 4 Design of CondLean 2.0

44. Design of CondLean 2.0 • CondLean 2.0 is a Prolog implementation of BSeqS calculi; it is written in SICStus Prolog and it is inspired by leanTAP, introduced by Beckert and Posegga in [BeckertPosegga96]. • The program comprises a set of clauses, each one of them represents a sequent rule or axiom; the proof search is provided for free by the mere depth-first search mechanism of Prolog.

45. Design of CondLean 2.0 • CondLean 2.0 is a Prolog implementation of BSeqS calculi; it is written in SICStus Prolog and it is inspired by leanTAP, introduced by Beckert and Posegga in [BeckertPosegga96]. • The program comprises a set of clauses, each one of them represents a sequent rule or axiom; the proof search is provided for free by the mere depth-first search mechanism of Prolog. • The sequent calculi are implemented by the predicate • prove(Sigma, Delta, Labels) • This predicate succeeds if and only if the sequent   is derivable in SeqS, where Sigma e Delta are the lists representing multisets  and , and Labels is the list of labels introduced in that branch .

46. Design of CondLean 2.0 • Each clause of predicate prove implements one axiom or rule of BSeqS. • The clauses of prove are ordered to postpone the application of the branching rules.

47. Design of CondLean 2.0 • Each clause of predicate prove implements one axiom or rule of BSeqS. • The clauses of prove are ordered to postpone the application of the branching rules. Example 1: clause implementing (AX) axiom; both the antecedent and the consequent contain the same complex formula F: prove([_,_,ComplexSigma],[_,_,ComplexDelta],_):- member(F,ComplexSigma), member(F,ComplexDelta),!. (AX) , F , F

48. Design of CondLean 2.0 Example 2: clause implementing (R): prove([LitSigma,TransSigma,ComplexSigma], A , , y: B x y (R)  , x: A  B [LitDelta,TransDelta,ComplexDelta],Labels): select([X,A => B], ComplexDelta,ResComplexDelta),!, createLabels(Y,Labels), put([Y,B], LitDelta, ResComplexDelta, NewLitDelta, NewComplexDelta), prove([LitSigma, [[X,A,Y]|TransSigma], ComplexSigma],[NewLitDelta,TransDelta, NewComplexDelta],[Y|Labels]).

49. Design of CondLean 2.0 • For systems BSeqMP and BSeqID+MP the predicate prove has two additional arguments: • prove(K, CondContr, Sigma, Delta, Labels) • K and CondContr are the auxiliary sets of BSeqS calculi, used to control the application of (L)

50. Design of CondLean 2.0 Example 3: clause implementing (L)1: K  { x: A  B } | CondContr  { x: A  B } | , y: B  A x y K  { x: A  B } | CondContr  { x: A  B } |  , (L)1 K | CondContr | , x: A  B  prove(K,CondContr,[LS,TS,CS],[LD,TD,CD],Labels):- select([X,A => B],CS,ResCS), \+member([X,A => B],K), select(Y,Labels), put([Y,B],LS,ResCS,NewLS,NewCS), prove([[X,A => B]|K],[[X,A => B]|CondContr], [NewLS,TS,NewCS],[LD,TD,CD],Labels), prove([[X,A => B]|K],[[X,A => B]|CondContr], [LS,TS,ResCS],[LD,[[X,A,Y]|TD],CD],Labels).