Dipartimento di Informatica - Università degli studi di Torino

Dipartimento di Informatica - Università degli studi di Torino

CondLean 3.0: improving CondLean

for stronger 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

• CondLean 3.0: a SICStus Prolog implementation of sequent calculi SeqS

• Conclusions, future work and references

Outline

11

Conditional logicsConditional 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:

< 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.

• 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)

(A1 A2 … An) B

(C A1 C A2 … C An) (C B)

Conditional logics - System CK

With some properties of the selection function, we have the following extensions:

System Axiom Selection function property

Conditional logics

Extensions of CK

CK+ID A A f (x, [ A ]) [ A ]

CK+MP (A B) (A B) w [ A ] w f (w, [ A ])

CK+CS (A B) (A B) w [ A ] f (w, [ A ]) {w}

CK+CEM (A B) (A B) | f (w, [ A ]) | 1

22

Sequent Calculi SeqSSequent Calculi SeqS

• In [OlivettiSchwind01], [OlivettiPozzatoSchwind05] sequent calculi for conditional logics, called SeqS, are introduced.

• SeqS consider CK and extensions CK+{ID, MP, CS, CEM} and all the combinations of them, except for those combining both CEM and MP

• 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 labelled formulas; we have two kinds of formulas:


world formulas, like x: A;x yA transition formulas, like .

• A transition formula represents that y f (x, [A]).

• A world formula x: A represents that the formula A is true in the world x.

x yA

SeqCK:


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


Rules for the extensions of CK:

33

How to obtain a How to obtain a decision proceduredecision procedure

• SeqS calculi have the following rules:

, x: A B (L)

How to obtain a decision procedure

, x: A B , x yA , x: A B, y: B

(CEM), x y

A

, ,x yA (, )[y,z/u]x yAx zA

• In backward proof search, the above rules add a formula in the premise (i.e. they copy their principal formula in their premises)

• In order to obtain a decision procedure, it is essential to control the application of these rules.


• In [OlivettiPozzatoSchwind05 : submitted] it is shown that:

1. one needs to apply (L) at most once on the same formula x: A B by using the same transition

2. one needs to apply (L) by using only when x=y or

3. the same restrictions on the applications of (CEM)


x y Cx y

A

x yA

• SeqCK and SeqID are complete even if we reformulate as follows:


(L)

, x: A B (L)

x yA , y: B

x y,C

x yC

x y,C

44

DDesign of esign of

CondLean 3.0CondLean 3.0

• CondLean 3.0 is a Prolog implementation of SeqS calculi; it is written in SICStus Prolog and it is inspired by leanTAP, introduced 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 3.0

• CondLean 3.0 vs CondLean:

1. CondLean is a t.p. for CK and its extensions MP, ID, and MP+ID, whereas CondLean 3.0 includes extensions CS and CEM and all combinations of ID, MP, CS, and CEM, except those combining both CEM and MP

2. CondLean implements sequent calculi with explicit contractions, whereas CondLean 3.0 implements SeqS as in [OlivettiPozzatoSchwind05], where the crucial rule (L) is invertible


• The sequent calculi are implemented by the predicate

prove(Cond, 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

- Labels is the list of labels introduced in that branch

- Cond is a list of pairs [F, Used], where F is a conditional formula and Used the list of transitions already used to apply (L) on F



•Each clause of predicate prove implements one axiom or rule of SeqS.

• 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)

Example 2: clause implementing (R):

prove(Cond,[LitSigma,TransSigma,ComplexSigma],

, x: A B

, , y: B(R) x yA


select([X,A => B],ComplexDelta,ResComplexDelta),!,createLabels(Y,Labels),put([Y,B], LitDelta, ResComplexDelta, NewLitDelta, NewComplexDelta),prove(Cond,[LitSigma, [[X,A,Y]|TransSigma], ComplexSigma],[NewLitDelta,TransDelta, NewComplexDelta],[Y|Labels]).

[LitDelta,TransDelta,ComplexDelta],Labels):

Example 3: clause implementing (L):



member([X,A => B],ComplexSigma),select([[X,A => B],Used],Cond,TempCond),

put([Y,B], LitSigma, ComplexSigma, NewLitSigma, NewComplexSigma),…


, x: A B (L)

, x: A B , x yA , x: A B, y: B

member([X,C,Y],TransSigma),\+member([X,C,Y],Used),!,

Example 3: clause implementing (L):



…prove([[[X,A=>B],[[X,C,Y]|Used]]|TempCond], [LitSigma,TransSigma,ComplexSigma], [LitDelta,[X,A,Y]|TransDelta],ComplexDelta],Labels),


, x: A B (L)

, x: A B , x yA , x: A B, y: B

prove([[[X,A=>B],[[X,C,Y]|Used]]|TempCond], [NewLitSigma,TransSigma,NewComplexSigma], [LitDelta,TransDelta,ComplexDelta],Labels).


• For systems allowing (CEM) another parameter Tr is added to the predicate prove:

prove(Tr, Cond, Sigma, Delta, Labels)

• It is a list of pairs [T,Used] where T is a transition formula and Used the list of transitions already used to apply (CEM) on T

• The application of (CEM) is restricted as in the case of (L)

• We present three different implmentations for our theorem provers:

1. Constant labels version (for all the systems)

2. Free-variables version

3. Heuristic version


(only for SeqCK and SeqID)}

1. Constant labels version

• This version makes use of Prolog constants to represent SeqS’s labels, introdouced by the (R) rule.

• In SeqCK and SeqID…


• 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)

x yA , y: B


• 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 3.0 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].


, x: A B

,(L)

x VA , V: B

Each free variable will be then istantiated by Prolog’s pattern matching either to apply the (EQ) rule, or to close a branch with an axiom.

Free variable


• 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.


3. Heuristic version

• This implementation performs a “two-phase” 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;2. In case of failure of phase 1, the free variable version is called to complete the computation.• On a valid sequent with over 120 connectives, the heuristic version succeeds in 460 msec versus 4326 msec of the free variable version.

• The performances of the three versions are promising.

• We have tested CondLean 3.0 - free variable version – for SeqCK 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)

2 6 9 11 15

5 500 650 1000 2000


• One can download the source code and the application CondLean 3.0 at the following address:

www.di.unito.it/~pozzato/CondLean 3.0

55

CConclusion andonclusion and

FFuture workuture work

• To the best of our knowldege, CondLean 3.0 is the first theorem prover for CK and extensions with ID, MP, CEM, and CS.

•We are working on extending CondLean to other conditional systems (AC, CV, …)

• We intend to develop free variable and heuristic versions for systems with MP, CS, and CEM

Conclusions and Future work

66

RReferenceseferences

References

[BeckertPosegga96] Bernard Beckert and Joachim Posegga. leanTAP: Lean Tableau-based Deduction. Journal of Automated Reasoning, 15(3), pp. 339-358.

[BeckertGorè97] Bernard Beckert and Rajeev Gorè. Free Variable Tableaux for Propositional Modal Logics. Tableaux-97, LNCS 1227, Springer, pp. 91-106.

[Gabbay96] Dov. M. Gabbay. Labelled deductive systems (vol. i). Oxford logic guides, Oxford University Press.

References

[OlivettiPozzatoSchwind05] Nicola Olivetti, Gian Luca Pozzato and Camilla B. Schwind. A Sequent Calculus and a Theorem Prover for Standard Conditional Logics: Extended version. Technical Report 87/05, Dipartimento di Informatica, Università degli Studi di Torino, Italy, 2005.

[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. 264-270.

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. 384-404.

[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 athttp://www.di.unito.it/~pozzato/tesiPozzato.html

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