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

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 Outline

1 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. Conditional logics

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. Conditional logics Syntax

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

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

- W is a non-empty set of items called worlds; Conditional logics Semantics We consider the selection function semantics; the model is a triple:

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

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

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

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

The semantics above characterizes the minimal normal conditional logic CK, which is axiomatized as follows: 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) (A 1  A 2  …  A n )  B (C  A 1  C  A 2  …  C  A n )  (C  B) Conditional logics - System CK

With some properties of the selection function, we have the following extensions: SystemAxiomSelection function property Conditional logics Systems CK{+MP}{+ID}

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

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

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

2 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]. Sequent Calculi SeqS

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

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

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

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

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

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

3 How to obtain a decision procedure

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

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

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

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

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

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

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

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. One can control the application of the contraction rules as follows: How to obtain a decision procedure 2.1. SeqMP and SeqID+MP are complete without the (ContrR) rule. Here are the BSeqS calculi presented in [OlivettiPozzatoSchwind04]:

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

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

1. The second rule is applied if x : A  B has 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: How to obtain a decision procedure The (  L) rule is “split” in three rules, to keep into account of the necessary application of (ContrL). If x: A  B  K K | CondContr | , x: A  B  K | CondContr |  , (L)2(L)2 xy A K | CondContr | , y: B 

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

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 How to obtain a decision procedure

4 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. Design of CondLean 2.0

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. 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.

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),!. , F , F ( AX ) 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 2: clause implementing (  R): prove([LitSigma,TransSigma,ComplexSigma],  , x: A  B , , y: B (R)(R) xy A Design of CondLean 2.0 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]). [LitDelta,TransDelta,ComplexDelta],Labels):

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)

Example 3: clause implementing (  L) 1 : 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). K | CondContr | , x: A  B  K  { x: A  B } | CondContr  { x: A  B } |  , (L)1(L)1 xy A K  { x: A  B } | CondContr  { x: A  B } | , y: B  prove(K,CondContr,[LS,TS,CS],[LD,TD,CD],Labels):- Design of CondLean 2.0

We present three different implmentations for our theorem provers: 1. Constant labels version; 2. Free-variables version; 3. Heuristic version. Design of CondLean 2.0

1. Constant labels version This version makes use of Prolog constants to represent SeqS’s labels, introdouced by the (  R) rule. Design of CondLean 2.0

When the (  L) clause is used to prove  , a backtracking point is introduced by the choice of a label y occurring in the two premises: , x: A  B   , (L)(L) xy A , y: B  Design of CondLean Constant labels version This version makes use of Prolog constants to represent SeqS’s labels, introdouced by the (  R) rule. If there are n labels to choose, the computation might succeed only after n-1 backtracking steps, with a significant loss of efficiency.

2. Free-variables version In this implementation, CondLean 2.0 makes use of Prolog variables to represent all the labels that can be used in an application of the (  L) clause. This solution is inspired to the free-variable tableaux introduced in [BeckertGorè97]. Design of CondLean 2.0

, x: A  B   , (L)(L) xV A , V: B  Each free variable will be then istantiated by Prolog’s pattern matching to apply either the (EQ) rule, or to close a branch with an axiom. Free variable Design of CondLean 2.0

To manage free variable domains we use the constraints (CLP); when a free variable V is introduced by the application of (  L), a constraint on its domain is added to the constraint store. The constraint solver (given for free by the clpfd library of SICStus Prolog) will control the consistency of the constraint store during the computation in a very efficient way. Design of CondLean 2.0

3. Heuristic version This implementation performs a “two-phase” computation: Design of CondLean 2.0

1. An incomplete theorem prover searches a derivation exploring a reduced search space, to check the validity of a sequent in a very small time; 3. Heuristic version This implementation performs a “two-phase” computation: Design of CondLean 2.0

2. In case of failure of phase 1, the free variable version is called to complete the computation. 3. Heuristic version This implementation performs a “two-phase” computation: Design of CondLean An incomplete theorem prover searches a derivation exploring a reduced search space, to check the validity of a sequent in a very small time;

2. In case of failure of phase 1, the free variable version is called to complete the computation. 1. An incomplete theorem prover searches a derivation exploring a reduced search space, to check the validity of a sequent in a very small time; 3. Heuristic version This implementation performs a “two-phase” computation: On a valid sequent with over 120 connectives, the heuristic version succeeds in 460 msec versus 4326 msec of the free variable version. Design of CondLean 2.0

The performances of the three versions are promising. We have tested CondLean free variable version obtaining the following results; we define the sequent degree as the maximum level of nesting of the conditional operator. Sequent degree Time to succeed (ms) Design of CondLean 2.0

One can download the source code and the application CondLean 2.0 at the following address: 2.0

5 Future work

We are working on some extensions of CondLean 2.0 to stronger conditional systems We have found cut-free and terminating calculi for conditional logics CS and CEM (Stalnaker logic): Future work CK+CEM (A  B)  (A   B) CK+CS | f (x, [ A ]) |  1 SystemAxiomSelection function property w  [ A ]  f (w, [ A ])  {w} (A  B)  (A  B)

6 References

[BeckertPosegga96] Bernard Beckert and Joachim Posegga. leanTAP: Lean Tableau-based Deduction. Journal of Automated Reasoning, 15(3), pp [BeckertGorè97] Bernard Beckert and Rajeev Gorè. Free Variable Tableaux for Propositional Modal Logics. Tableaux-97, LNCS 1227, Springer, pp [Gabbay96] Dov. M. Gabbay. Labelled deductive systems (vol. i). Oxford logic guides, Oxford University Press.

References [OlivettiPozzatoSchwind04] Nicola Olivetti, Gian Luca Pozzato and Camilla B. Schwind. A Sequent Calculus and a Theorem Prover for Standard Conditional Logics. Technical Report 81/04, Dipartimento di Informatica, Università degli Studi di Torino, Italy, November [OlivettiPozzato03] Nicola Olivetti and Gian Luca Pozzato. CondLean: A Theorem Prover for Conditional Logics. In Proc. of TABLEAUX 2003 (Automated Reasoning with Analytic Tableaux and Related Methods), volume 2796 of LNAI, Springer, pp

References [OlivettiSchwind01] Nicola Olivetti and Camilla B. Schwind. A Calculus and Complexity Bound for Minimal Conditional Logic. Proc. ICTCS01 - Italian Conference on Theoretical Computer Science, vol. LNCS 2202, pp [Viganò00] Luca Viganò. Labelled Non-classical Logics. Kluwer Academic Publishers, Dordrecht. [Pozzato03] Gian Luca Pozzato. Deduzione Automatica per Logiche Condizionali: Analisi e Sviluppo di un Theorem Prover. Tesi di laurea, Informatica, Università di Torino. In Italian, download at