Partial Regularization of First-Order Resolution Proofs

Resolution and superposition are common techniques which have seen widespread use with propositional and first-order logic in modern theorem provers. In these cases, resolution proof production is a key feature of such tools; however, the proofs that they produce are not necessarily as concise as possible. For propositional resolution proofs, there are a wide variety of proof compression techniques. There are fewer techniques for compressing first-order resolution proofs generated by automated theorem provers. This paper describes an approach to compressing first-order logic proofs based on lifting proof compression ideas used in propositional logic to first-order logic. One method for propositional proof compression is partial regularization, which removes an inference $\eta$ when it is redundant in the sense that its pivot literal already occurs as the pivot of another inference in every path from $\eta$ to the root of the proof. This paper describes the generalization of the partial-regularization algorithm RecyclePivotsWithIntersection [10] from propositional logic to first-order logic. The generalized algorithm performs partial regularization of resolution proofs containing resolution and factoring inferences with unification. An empirical evaluation of the generalized algorithm and its combinations with the previously lifted GreedyLinearFirstOrderLowerUnits algorithm [12] is also presented

[1]  G. S. Tseitin On the Complexity of Derivation in Propositional Calculus , 1983 .

[2]  Christoph Weidenbach,et al.  SPASS Version 3.5 , 2009, CADE.

[3]  Geoff Sutcliffe,et al.  Proof Generation for Saturating First-Order Theorem Provers , 2014 .

[4]  Roberto Bruttomesso,et al.  An Efficient and Flexible Approach to Resolution Proof Reduction , 2010, Haifa Verification Conference.

[5]  Josef Urban,et al.  Automated Proof Compression by Invention of New Definitions , 2010, LPAR.

[6]  Tomer Libal,et al.  Understanding Resolution Proofs through Herbrand's Theorem , 2013, TABLEAUX.

[7]  Harrie de Swart,et al.  Automated Reasoning with Analytic Tableaux and Related Methods , 2000, Lecture Notes in Computer Science.

[8]  Alexander Leitsch,et al.  Algorithmic introduction of quantified cuts , 2014, Theor. Comput. Sci..

[10]  Bruno Woltzenlogel Paleo,et al.  Atomic Cut Introduction by Resolution: Proof Structuring and Compression , 2010, LPAR.

[11]  Simon Cruanes,et al.  Extending Superposition with Integer Arithmetic, Structural Induction, and Beyond. (Extensions de la Superposition pour l'Arithmétique Linéaire Entière, l'Induction Structurelle, et bien plus encore) , 2015 .

[12]  Carsten Sinz Compressing Propositional Proofs by Common Subproof Extraction , 2007, EUROCAST.

[13]  Hasan Amjad Compressing Propositional Refutations , 2007, Electron. Notes Theor. Comput. Sci..

[14]  Michaël Rusinowitch,et al.  Proving refutational completeness of theorem-proving strategies: the transfinite semantic tree method , 1991, JACM.

[15]  Rüdger Thiele,et al.  Hilbert's Twenty-Fourth Problem , 2002, Journal of Automated Reasoning.

[16]  Victor W. Marek,et al.  Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer , 2016, SAT.

[17]  Stephan Merz,et al.  Exploring and Exploiting Algebraic and Graphical Properties of Resolution , 2010 .

[18]  Joseph Boudou,et al.  Compression of Propositional Resolution Proofs by Lowering Subproofs , 2013, TABLEAUX.

[19]  Giselle Reis,et al.  Importing SMT and Connection proofs as expansion trees , 2015, PxTP@CADE.

[20]  Geoff Sutcliffe The TPTP Problem Library and Associated Infrastructure , 2017, Journal of Automated Reasoning.

[21]  Stephan Schulz,et al.  System Description: E 1.8 , 2013, LPAR.

[22]  Scott Cotton Two Techniques for Minimizing Resolution Proofs , 2010, SAT.

[23]  Tim Miller,et al.  Explanation in Artificial Intelligence: Insights from the Social Sciences , 2017, Artif. Intell..

[24]  J. A. Robinson,et al.  Automatic Deduction with Hyper-Resolution , 1983 .

[25]  Holger Hermanns,et al.  Logic for Programming, Artificial Intelligence, and Reasoning , 2010, Lecture Notes in Computer Science.

[26]  Ofer Strichman,et al.  Linear-Time Reductions of Resolution Proofs , 2008, Haifa Verification Conference.

[27]  Ross A. Overbeek,et al.  Complexity and related enhancements for automated theorem-proving programs , 1976 .

[28]  Armin Biere,et al.  Clausal Proof Compression , 2015, IWIL@LPAR.

[29]  Danna Zhou,et al.  d. , 1934, Microbial pathogenesis.

[30]  Christoph Weidenbach,et al.  Combining Superposition, Sorts and Splitting , 2001, Handbook of Automated Reasoning.

[31]  Bruno Woltzenlogel Paleo,et al.  Towards the Compression of First-Order Resolution Proofs by Lowering Unit Clauses , 2015, CADE.

[32]  Frieder Stolzenburg,et al.  CoRg: Commonsense Reasoning Using a Theorem Prover and Machine Learning , 2018, LuxLogAI.

[33]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[34]  Stephan Merz,et al.  Compression of Propositional Resolution Proofs via Partial Regularization , 2011, CADE.

[35]  Alexander Leitsch,et al.  Herbrand Sequent Extraction , 2008, AISC/MKM/Calculemus.

[36]  Ross A. Overbeek An implementation of hyper-resolution , 1975 .

[37]  Maria Paola Bonacina Automated Reasoning for Explainable Artificial Intelligence , 2017, ARCADE@CADE.

[38]  Andrei Voronkov,et al.  The design and implementation of VAMPIRE , 2002, AI Commun..