The Taming of the (X)OR

Many key verification problems such as boundedmodel-checking, circuit verification and logical cryptanalysis are formalized with combined clausal and affine logic (i.e. clauses with xor as the connective) and cannot be efficiently (if at all) solved by using CNF-only provers. We present a decision procedure to efficiently decide such problems. The Gauss-DPLL procedure is a tight integration in a unifying framework of a Gauss-Elimination procedure (for affine logic) and a Davis-Putnam-Logeman-Loveland procedure (for usual clause logic). The key idea, which distinguishes our approach from others, is the full interaction bewteen the two parts which makes it possible to maximize (deterministic) simplification rules by passing around newly created unit or binary clauses in either of these parts. We show the correcteness and the termination of Gauss-DPLL under very liberal assumptions.

[1]  Fabio Massacci,et al.  Using Walk-SAT and Rel-Sat for Cryptographic Key Search , 1999, IJCAI.

[2]  James M. Crawford,et al.  Experimental Results on the Crossover Point in Random 3-SAT , 1996, Artif. Intell..

[3]  Hans van Maaren,et al.  A two phase algorithm for solving a class of hard satissfiability problems , 1998 .

[4]  Bart Selman,et al.  Pushing the Envelope: Planning, Propositional Logic and Stochastic Search , 1996, AAAI/IAAI, Vol. 2.

[5]  John M. Wilson.  Compact normal forms in propositional logic and integer programming formulations , 1990, Comput. Oper. Res..

[6]  David S. Johnson,et al.  Cliques, Coloring, and Satisfiability , 1996 .

[7]  Rolf Drechsler,et al.  Manipulation Algorithms for K*BMDs , 1997, TACAS.

[8]  Peter Baumgartner,et al.  Abductive Coreference by Model Construction , 1999 .

[9]  Chu Min Li,et al.  A Constraint-Based Approach to Narrow Search Trees for Satisfiability , 1999, Inf. Process. Lett..

[10]  Richard C. T. Lee,et al.  Symbolic logic and mechanical theorem proving , 1973, Computer science classics.

[11]  Hans van Maaren,et al.  Recognition of Tractable Satisfiability Problems through Balanced Polynomial Representations , 2000, Discret. Appl. Math..

[12]  Hantao Zhang,et al.  SATO: An Efficient Propositional Prover , 1997, CADE.

[13]  Ralph Howard,et al.  Data encryption standard , 1987 .

[14]  Frieder Stolzenburg,et al.  Loop-Detection in Hyper-Tableaux by Powerful Model Generation , 1999, J. Univers. Comput. Sci..

[15]  Rolf Drechsler,et al.  On the Relation between BDDs and FDDs , 1995, Inf. Comput..

[16]  Enrico Giunchiglia,et al.  Act, and the Rest Will Follow: Exploiting Determinism in Planning as Satisfiability , 1998, AAAI/IAAI.

[17]  Chu Min Li,et al.  Integrating Equivalency Reasoning into Davis-Putnam Procedure , 2000, AAAI/IAAI.

[18]  Fabio Massacci.  Simplification: A General Constraint Propagation Technique for Propositional and Modal Tableaux , 1998, TABLEAUX.

[19]  Randal E. Bryant,et al.  Efficient implementation of a BDD package , 1991, DAC '90.

[20]  Roberto J. Bayardo,et al.  Using CSP Look-Back Techniques to Solve Real-World SAT Instances , 1997, AAAI/IAAI.

[21]  Ulrich Furbach,et al.  Nonmonotonic Reasoning: Towards Efficient Calculi and Implementations , 2001, Handbook of Automated Reasoning.

[22]  Bart Selman,et al.  Ten Challenges in Propositional Reasoning and Search , 1997, IJCAI.

[23]  Andy Schürr,et al.  GXL: toward a standard exchange format , 2000, Proceedings Seventh Working Conference on Reverse Engineering.

[24]  Jens Woch,et al.  Implementation of a Schema-TAG-Parser , 1999 .

[25]  Ullrich Hustadt,et al.  Simplification and Backjumping in Modal Tableau , 1998, TABLEAUX.

[26]  Masahiro Fujita,et al.  Symbolic model checking using SAT procedures instead of BDDs , 1999, DAC '99.

[27]  Jürgen Ebert,et al.  A Formalization of SOCCA , 1999 .

[28]  Andreas Winter,et al.  Querying as an enabling technology in software reengineering , 1999, Proceedings of the Third European Conference on Software Maintenance and Reengineering (Cat. No. PR00090).

[29]  Jürgen Ebert,et al.  GraX-an interchange format for reengineering tools , 1999, Sixth Working Conference on Reverse Engineering (Cat. No.PR00303).

[30]  Oliver Obst,et al.  Spatial Agents Implemented in a Logical Expressible Language , 1999, RoboCup.

[31]  Fabio Massacci,et al.  Logical Cryptanalysis as a SAT Problem ? Encoding and Analysis of the U.S. Data Encryption Standard , 2000 .

[32]  Joseph Douglas Horton,et al.  Merge Path Improvements for Minimal Model Hyper Tableaux , 1999, TABLEAUX.

[33]  Norbert Eisinger,et al.  A Confluent Connection Calculus , 2000, Intellectics and Computational Logic.

[34]  Victor W. Marek,et al.  The Logic Programming Paradigm , 1999, Artificial Intelligence.

[35]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[36]  Jürgen Dix,et al.  Transformation-based bottom-up computation of the well-founded model , 1996, Theory and Practice of Logic Programming.

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