a ke tableau system for c1 - dainfadolfo/publications/2009/slides_cleaips... · a ke tableau system...

A KE tableau system for C1

Adolfo Neto (DAINF-UTFPR), Celso A. A. Kaestner (DAINF-UTFPR)and Marcelo Finger (IME-USP)

August 24th, 2009

Original title: A KE Tableau System for the Paraconsistent Logic C1 and an Efficient Implementation of a TheoremProver for C1

Presented at the “CLE/AIPS - Science, Truth and Consistency” meeting, in tribute to the 80 years of ProfessorNewton Carneiro Affonso da Costa: http://www.cle.unicamp.br/cle-aips-event/.

Adolfo Neto, Celso Kaestner and Marcelo Finger: A KE tableau system for C1, 1

http://www.cle.unicamp.br/cle-aips-event/

Outline

� Introduction

� C1, a paraconsistent logic

� The KE System for C1

� ImplementationA Strategy for the C1 KE SystemProblem Families to Evaluate C1 ProversEvaluation

� Conclusion


Introduction

Introduction

I Inconsistency is a phenomena that appears naturally in theworld.

I “Ex contradictione sequitur quod libet” principle (principle ofexplosion).

I Paraconsistent logics are logics in which theories can beinconsistent but nontrivial.

I Paraconsistent logics have several applications, such as inrobot control and medicine.

I Our objective: a theorem prover for C1


C1, a paraconsistent logic

� Introduction




� Conclusion




I C1 is a paraconsistent logic

I It is part of the hierarchy of logics Cn, 1 ≤ n < ω .

I C1 is of historical importance because it was one of the firstparaconsistent logics to be presented [da Costa, 1963]




I A consistency operator (◦) is introduced.

I The intended meaning of ◦A is “A is consistent”.

I The consistency connective “◦” is not a primitive connective,but an abbreviation:

◦A def= ¬(A ∧ ¬A).



C1’s Axiomatization (1/2)

Axiom schemas:

(Ax1) α→ (β → α)

(Ax2) (α→ β)→ ((α→ (β → γ))→ (α→ γ))

(Ax3) α→ (β → (α ∧ β))

(Ax4) (α ∧ β)→ α

(Ax5) (α ∧ β)→ β

(Ax6) α→ (α ∨ β)

(Ax7) β → (α ∨ β)

(Ax8) (α→ γ)→ ((β → γ)→ ((α ∨ β)→ γ))

(Ax10) α ∨ ¬α(Ax11) ¬¬α→ α



C1’s Axiomatization (2/2)

Axiom schemas:

(bc1) ◦α→ (α→ (¬α→ β))

(ca1) (◦α ∧ ◦β)→ ◦(α ∧ β)

(ca2) (◦α ∧ ◦β)→ ◦(α ∨ β)

(ca3) (◦α ∧ ◦β)→ ◦(α→ β)

Inference rule:

(MP)α, α→ β

β



From C1 to Classical Logic

To obtain classical propositional logic (CPL) from C1:

I remove ◦-axioms: (bc1), (ca1), (ca2) and (ca3)

I add the explosion law :

(exp) α→ (¬α→ β)



C1’s Valuation

I v(α1 ∧ α2) = 1 ⇐⇒ v(α1) = 1 and v(α2) = 1;

I v(α1 ∨ α2) = 1 ⇐⇒ v(α1) = 1 or v(α2) = 1;

I v(α1 → α2) = 1 ⇐⇒ v(α1) = 0 or v(α2) = 1;

I v(¬α) = 0 =⇒ v(α) = 1;

I v(¬¬α) = 1 =⇒ v(α) = 1;

I v(◦α) = 1 =⇒ v(α) = 0 or v(¬α) = 0.

I v(◦(α� β)) = 0 =⇒ v(◦α) = 0 or v(◦β) = 0, for� ∈ {∧,∨,→};



Proof systems for C1

I Axiomatic systems

I Tableau systems



Tableau systems for C1

I [Carnielli and Lima-Marques, 1992]

I [Buchsbaum and Pequeno, 1993]

I [Carnielli et al., 2007]

I [D’Ottaviano and de Castro, 2006]

None of them is KE-based

I All of them have many branching rules

I Branching is computationally expensive



Theorem provers for C1

To the best of our knowledge, before this work only two theoremprovers for C1 have been implemented:

I [Carnielli and Lima-Marques, 1992]: a prover based on ananalytic tableaux system for C1 (but its source code is lost)

I [Buchsbaum and Pequeno, 1993]: a prover based on adifferent analytic tableaux system for C1 whose source code iswritten in an old dialect of LISP: muLisp.

• It is not very easy to port this source code to a newer LISPdialect

• muLisp is very limited (it runs only on Windows with limitedaccess to main memory)


The KE System for C1

� Introduction




� Conclusion



C1 KE System

I The KE inference system is a tableau method[D’Agostino and Mondadori, 1994]

I Improves on efficiency over Smullyan’s Analytic Tableaux

I KE has only one branching rule, corresponding to the CutRule in the sequent calculus

I KE-proofs may (or may not) be analytic



C1 KE System - Evolution

I Classical KE system [D’Agostino and Mondadori, 1994]

I Non-KE tableaux systems for Logics of Formal Inconsistency,a class of paraconsistent logics that includes mbC, mCi andC1([Carnielli et al., 2007], preprint available in 2005)

I KE System for mbC [Neto and Finger, 2006a]

I KE System for mCi [Neto and Finger, 2007]

I KE System for C1


Rules

FA→ BTAFB

(F→)TA→ B

TATB

(T→1)TA→ B

FBFA

(T→2)

TA ∧ BTATB

(T∧)FA ∧ BTAFB

(F∧1)FA ∧ BTBFA

(F∧2)

FA ∨ BFAFB

(F∨)TA ∨ BFATB

(T∨1)TA ∨ BFBTA

(T∨2)

T A F A(PB)

F¬ATA

(F¬)T¬¬ATA

(T¬¬)

T ◦ AT¬AFA

(T ◦ ¬)F ◦ (A� B)

T ◦ AF ◦ B

(F ◦ �1)F ◦ (A� B)

T ◦ BF ◦ A

(F ◦ �2)


C1 KE System

Results:

I The KE System for C1 is sound, complete and decidable.


Implementation

� Introduction




� Conclusion


The implemented rules

FA→ BTAFB

(F→)TA→ B

TATB

(T→1)TA→ B

FBFA

(T→2)

TA ∧ BTATB

(T∧)FA ∧ BTAFB

(F∧1)FA ∧ BTBFA

(F∧2)

FA ∨ BFAFB

(F∨)TA ∨ BFATB

(T∨1)TA ∨ BFBTA

(T∨2)

T A F A(PB)

F¬ATA

(F¬)T¬¬ATA

(T¬¬)

T ◦ AT¬AFA

(T ◦ ¬)

T¬(A� B)TA� BT ◦ AF ◦ B

(T¬�1)

T¬(A� B)TA� BT ◦ BF ◦ A

(T¬�2)

Implementation

What’s the difference?

F ◦ (A� B)T ◦ A

F ◦ B(F ◦ �1) −→

T¬(A� B)T (A� B)T ◦ A

F ◦ B

(T¬�1)

Where F ◦ (A� B)def= F¬((A� B) ∧ ¬(A� B))

Using (F¬): T (A� B) ∧ ¬(A� B)Using (T∧): T (A� B) and T¬(A� B)

The same process for 2 rules: (F ◦ �1) and (F ◦ �2).


Implementation

Why did we do this?

1. Because the premises are simpler in the implemented rules(thus it is easier to match the pattern)

2. Because in the original rules, F¬(∗) is the main pattern forthree rules: (F¬), (F ◦ �1) and (F ◦ �2). We want that aformula, once “analysed” (used as main premise), should notbe “analysed” again.

3. There is a price to pay: we have to deal with three premiserules. . .


Implementation

The procedure

1. Apply all one-premise rules

2. Apply all two-premise rules

3. Apply all three-premise rules

4. Apply PB motivated by (F∧1), (T∨1) and (T→1).

5. Apply PB motivated by (T ◦ ¬).

6. Apply PB motivated by (T¬�1)


Implementation

Analytic Branching in KE

PB (principle of bivalence) is only applied in a branch Θ

T A F A(PB)

if Θ already contains the main premiss of a 2/3-premiss rule. E.g.TA→ B.

Analytic PB respects the subformula property


Implementation

C1 KE Simple Strategy:

1. Apply all possible linear rules

2. If the current branch closes, pick an open branch (if there issome) and go back to the first step. If there is no remainingopen branch, the procedure ends and the result is that thetableau is declared closed;

3. If the current branch is linearly saturated, but not closed, thestrategy tries to apply the PB rule.


Implementation


I Apply analytic PB

I If the strategy can apply the PB rule, then the (new) rightbranch is put in the stack of open branches and the leftbranch becomes the current branch.

I If the strategy cannot apply the PB rule, then the procedurefinishes by declaring the tableau open.


Implementation


I This strategy was implemented in KEMS[Neto and Finger, 2006b]

I It was able to prove instances from all families of problemsdesigned


Implementation

Problems Families for Evaluating Theorem Provers

I A problem is a sequent that can be given as input for atheorem prover.

I A problem family is a set of problems that we know, byconstruction, whether they are valid, satisfiable orunsatisfiable.

I We have developed nine families of difficult problems (two ofthem specially for C1) that can be used to evaluate theoremprovers for paraconsistent logics.

I The motivation for developing and presenting these problemfamilies before the actual prover for C1 was implemented wasto use them as tests for the C1 KE system as well as for theimplemented sofwtare.


Implementation

Fifth Family

Φ5n is:

◦A1,

n∧i=1

(Ai ),n∧

i=1

[An+1 → ((Ai∨Bi )→ (◦Ai+1))], (n∧

i=1

◦Ai )→ ¬An+1

` ¬¬¬An+1


Implementation

Fifth Family

Φ53 is:

T ◦ A1

T A1 ∧ A2 ∧ A3

T [A4 → ((A1 ∨ B1)→ (◦A2))]∧ [A4 → ((A2 ∨ B2)→ (◦A3))]∧ [A4 → ((A3 ∨ B3)→ (◦A4))]

T ((◦A1) ∧ (◦A2) ∧ (◦A3))→ ¬A4

F ¬¬¬A4


Implementation

Sixth Family

Φ6n is:

n∧i=1

(Bi ),n∧

i=1

(◦Ci ),n∧

i=1

((Ai ∨ Bi )→ (◦Ai+1)), (n∧

i=1

Ci )→ (D ∧ ¬C1)

` [n∨

i=1

(◦(Ai+1 → Ci ))] ∨ D


Implementation

Sixth Family

Φ63 is:

T B1 ∧ B2 ∧ B3

T ◦ C1 ∧ ◦C2 ∧ ◦C3

T ((A1 ∨ B1)→ (◦A2)) ∧ ((A2 ∨ B2)→ (◦A3)) ∧ ((A3 ∨ B3)→ (◦A4))T (C1 ∧ C2 ∧ C3)→ (D ∧ ¬C1)F [◦(A2 → C1)] ∨ [◦(A3 → C2)] ∨ [◦(A4 → C3)] ∨ D


Implementation

Demo

A demonstration


Implementation

Running the problems


Conclusion

� Introduction




� Conclusion


Conclusion

Conclusion

I We have presented a sound, complete and decidableKE system for Da Costa’s C1 calculus for paraconsistent logic.

I Our system has less branching rules than other tableausystems for C1 described in the literature

I We described a strategy for this KE system that wasimplemented in KEMS.


Conclusion

Future work

I Implement at least another strategy in KEMS

I Compare the results obtained by our strategies amongthemselves as well as with Arthur Buchsbaum’s prover usingour problem families


Conclusion

More. . .

How to access the full text of the paper that gave origin to thispresentation:I Search for “adolfo neto publications”

• Worked on Google, Bing, Cuil (first link returned on2009-08-23)

I The name of the paper presented at LSFA 2009 is “Towardsan efficient prover for the C1 paraconsistent logic”[Neto et al., 2009]

I A paper covering more details of the implementation will soonbe available


Conclusion

Thanks for your attentionAny Questions?


References

Buchsbaum, A. and Pequeno, T. (1993).A reasoning method for a paraconsistent logic.Studia Logica, 52(2):281–289.

Carnielli, W., Coniglio, M. E., and Marcos, J. (2007).Handbook of the Philosophical Logic, volume 14, chapterLogics of Formal Inconsistency, pages 15–107.Springer-Verlag, second edition.

Carnielli, W. A. and Lima-Marques, M. (1992).Reasoning under inconsistent knowledge.Journal of Applied Non-Classical Logics, 2(1).

da Costa, N. C. A. (1963).Sistemas Formais Inconsistentes.Rio de Janeiro, NEPE.Reprinted by Editora da UFPR, Curitiba, 1993.

D’Agostino, M. and Mondadori, M. (1994).


References

The taming of the cut: Classical refutations with analytic cut.Journal of Logic and Computation, pages 285–319.

D’Ottaviano, I. M. L. and de Castro, M. A. (2006).Analytical tableaux for da costa’s hierarchy of paraconsistentlogics.Electronic Notes in Theoretical Computer Science, 143:27 –44.Proceedings of the 12th Workshop on Logic, Language,Information and Computation (WoLLIC 2005).

Neto, A. and Finger, M. (2006a).Effective Prover for Minimal Inconsistency Logic.In Artificial Intelligence in Theory and Practice, IFIP, pages465–474. Springer Verlag.Available at http://www.springerlink.com/content/b80728w7m6885765.Last accessed, November 2006.


http://www.springerlink.com/content/b80728w7m6885765

http://www.springerlink.com/content/b80728w7m6885765

References

Neto, A. and Finger, M. (2006b).KEMS - A KE-based Multi-Strategy Tableau Prover.http://www.dainf.ct.utfpr.edu.br/~adolfo/KEMS. Lastaccessed, April 2009.

Neto, A. and Finger, M. (2007).A KE tableau for a logic of formal inconsistency.In Proceedings of TABLEAUX’07 position papers andWorkshop on Agents, Logic and Theorem Proving. TechnicalReport (LSIS.RR.2007.002) of the LSIS/Universite PaulCezanne, Marseille, France.

Neto, A., Kaestner, C. A. A., and Finger, M. (2009).Towards an efficient prover for the C1 paraconsistent logic.In Proceeding of LSFA 2009.http://www.dainf.ct.utfpr.edu.br/~adolfo/publications/


http://www.dainf.ct.utfpr.edu.br/~adolfo/KEMS

http://www.dainf.ct.utfpr.edu.br/~adolfo/publications/2009/NetoKaestnerFinger_LSFA09_preprint.pdf



References

2009/NetoKaestnerFinger_LSFA09_preprint.pdf. Lastaccessed, August 2009.




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