Equal Rights for the Cut: Computable Non-analytic Cuts in Cut-based Proofs

This work studies the structure of proofs containing non-analytic cuts in the cut-based system, a sequent inference system in which the cut rule is not eliminable and the only branching rule is the cut. Such sequent system is invertible, leading to the KE-tableau decision method. We study the structure of such proofs, proving the existence of a normal form for them in the form of a comb-tree proof.We then concentrate on the problem of efficiently computing non-analytic cuts. For that, we study the generalisation of techniques present in many modern theorem provers, namely the techniques of conflict-driven formula learning.

[1]  Joseph R. Shoenfield,et al.  Mathematical logic , 1967 .

[2]  Harald Ganzinger,et al.  Resolution Theorem Proving , 2001, Handbook of Automated Reasoning.

[3]  Donald Loveland.  A Machine Program for Theorem-Provingt , 2000 .

[4]  S. Semmes,et al.  A Graphic Apology for Symmetry and Implicitness , 2000 .

[5]  Daniel P. Miranker,et al.  A Complexity Analysis of Space-Bounded Learning Algorithms for the Constraint Satisfaction Problem , 1996, AAAI/IAAI, Vol. 1.

[6]  William A. Howard,et al.  The formulae-as-types notion of construction , 1969 .

[7]  Dana,et al.  JSL volume 88 issue 4 Cover and Front matter , 1983, The Journal of Symbolic Logic.

[8]  Donald W. Loveland,et al.  A machine program for theorem-proving , 2011, CACM.

[9]  Sharad Malik,et al.  Chaff: engineering an efficient SAT solver , 2001, Proceedings of the 38th Design Automation Conference (IEEE Cat. No.01CH37232).

[10]  Per Martin-Löf,et al.  Constructive mathematics and computer programming , 1984 .

[11]  Fabio Massacci,et al.  Single Step Tableaux for Modal Logics , 2000, Journal of Automated Reasoning.

[12]  Sharad Malik,et al.  Efficient conflict driven learning in a Boolean satisfiability solver , 2001, IEEE/ACM International Conference on Computer Aided Design. ICCAD 2001. IEEE/ACM Digest of Technical Papers (Cat. No.01CH37281).

[13]  Dov M. Gabbay,et al.  Cut and Pay , 2006, J. Log. Lang. Inf..

[14]  Marcello D'Agostino,et al.  The Taming of the Cut. Classical Refutations with Analytic Cut , 1994, J. Log. Comput..

[15]  Marcello D'Agostino,et al.  Are tableaux an improvement on truth-tables? , 1992, J. Log. Lang. Inf..

[16]  R. Smullyan.  First-Order Logic , 1968 .

[17]  Marcello D'Agostino,et al.  Investigations into the complexity of some propositional calculi , 1990 .

[18]  J. P. Marques,et al.  GRASP : A Search Algorithm for Propositional Satisfiability , 1999 .

[19]  Rina Dechter,et al.  Enhancement Schemes for Constraint Processing: Backjumping, Learning, and Cutset Decomposition , 1990, Artif. Intell..

[20]  Marcello D'Agostino,et al.  Tableau Methods for Classical Propositional Logic , 1999 .

[21]  S. Griffis.  EDITOR , 1997, Journal of Navigation.

[22]  M. Fitting.  Proof Methods for Modal and Intuitionistic Logics , 1983 .

[23]  George Boolos,et al.  Don't eliminate cut , 1984, J. Philos. Log..

[24]  Cesare Tinelli,et al.  Abstract DPLL and Abstract DPLL Modulo Theories , 2005, LPAR.

[25]  Raymond M. Smullyan,et al.  Analytic cut , 1969, Journal of Symbolic Logic (JSL).

[26]  Eugene Goldberg,et al.  BerkMin: A Fast and Robust Sat-Solver , 2002 .