Deciding quantifier-free presburger formulas using parameterized solution bounds

Logical Methods in Computer Science

Volumn 1, Issue 2, 2005, Pages

  Seshia, Sanjit A ^a   Bryant, Randal E ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b Carnegie Mellon University   (United States)

Author keywords

Boolean satisfiability; Decision procedures; Difference (separation) constraints; Finite instantiation; Integer linear programming; Presburger arithmetic

Indexed keywords

DIGITAL ARITHMETIC; FORMAL LOGIC; INTEGER PROGRAMMING; VERIFICATION;

BOOLEAN SATISFIABILITY; DECISION PROCEDURE; DIFFERENCE (SEPARATION) CONSTRAINTS; FINITE INSTANTIATION; INTEGER LINEAR PROGRAMMING; PRESBURGER ARITHMETIC;

BOOLEAN ALGEBRA;

EID: 84944459228     PISSN: None     EISSN: 18605974     Source Type: Journal    
DOI: 10.2168/LMCS-1(2:6)2005     Document Type: Article

References (47)

1. W. Ackermann. Solvable Cases of the Decision Problem. North-Holland, Amsterdam, 1954.
2. Joel Brenner and Larry Cummings. The Hadamard maximum determinant problem. American Mathematical Monthly, 79:626-630, June-July 1972.
3. R. Brinkmann and R. Drechsler. RTL-datapath verification using integer linear programming. In Proceedings of the IEEE VLSI Design Conference, pages 741-746, 2002.
4. C. Barrett, D. L. Dill, and A. Stump. Checking satisfiability of first-order formulas by incremental translation to SAT. In E. Brinksma and K. G. Larsen, editors, Proc. 14th Intl. Conference on Computer-Aided Verification (CAV’02), LNCS 2404, pages 236-249. Springer-Verlag, July 2002.
5. I. Borosh, M. Flahive, D. Rubin, and L. B. Treybig. A sharp bound for solutions of linear Diophantine equations. Proceedings of the American Mathematical Society, 105(4):844-846, April 1989.
6. I. Borosh, M. Flahive, and L. B. Treybig. Small solutions of linear Diophantine equations. Dis-crete Mathematics, 58:215-220, 1986.
7. S. Berezin, V. Ganesh, and D. L. Dill. An online proof-producing decision procedure for mixedinteger linear arithmetic. In Proc. Tools and Algorithms for the Construction and Analysis of Systems (TACAS’03), LNCS 2619, pages 521-536, April 2003.
8. R. E. Bryant, S. K. Lahiri, and S. A. Seshia. Modeling and verifying systems using a logic of counter arithmetic with lambda expressions and uninterpreted functions. In E. Brinksma and K. G. Larsen, editors, Proc. Computer-Aided Verification (CAV’02), LNCS 2404, pages 78-92, July 2002.
9. I. Borosh and L. B. Treybig. Bounds on positive integral solutions of linear Diophantine equations. Proceedings of the American Mathematical Society, 55(2):299-304, March 1976.
10. I. Borosh and L. B. Treybig. A sharp bound on positive solutions of linear Diophantine equations. SIAM Journal on Matrix Analysis and Applications, 13(2):454-458, April 1992.
11. Sagar Chaki, Edmund M. Clarke, Alex Groce, Somesh Jha, and Helmut Veith. Modular verification of software components in C. In Proc. 25th International Conference on Software Engineering (ICSE), pages 385-395, 2003.
12. Vijay Chandru. Variable elimination in linear constraints. The Computer Journal, 36(5):463-472, August 1993.
13. CPLEX Optimization Tool. Available from ILOG. http://www.ilog.com/products/cplex/.
14. CVC-Lite: Cooperating Validity Checker. Available at http://verify.stanford.edu/CVCL/.
15. Leonardo de Moura, Harald Rueß, and Maria Sorea. Lazy theorem proving for bounded model checking over infinite domains. In Proc. 18th International Conference on Automated Deduction (CADE), pages 438-455, 2002.
16. David Detlefs, Greg Nelson, and James B. Saxe. Simplify: A theorem prover for program checking. Technical Report HPL-2003-148, HP Laboratories Palo Alto, 2003.
17. M. J. Fischer and M. O. Rabin. Super-exponential complexity of Presburger arithmetic. Pro-ceedings of SIAM-AMS, 7:27-41, 1974.
18. V. Ganesh, S. Berezin, and D. L. Dill. Deciding Presburger arithmetic by model checking and comparisons with other methods. In Formal Methods in Computer-Aided Design (FMCAD ’02), LNCS 2517, pages 171-186. Springer-Verlag, November 2002.
19. E. Goldberg and Y. Novikov. BerkMin: A fast and robust SAT solver. In Design Automation and Test in Europe (DATE) 2002, pages 142-149, 2002.
20. Thomas A. Henzinger, Ranjit Jhala, Rupak Majumdar, and Gregoire Sutre. Lazy abstraction. In Proc. 29th ACM Symposium on Principles of Programming Languages, pages 58-70, 2002.
21. ICS: Integrated Canonizer and Solver. Available at http://www.icansolve.com.
22. R. Kannan and C. L. Monma. On the computational complexity of integer programming problems. In Optimisation and Operations Research, volume 157 of Lecture Notes in Economics and Mathematical Systems, pages 161-172. Springer-Verlag, 1978.
23. Daniel Kroening, Jo¨el Ouaknine, Sanjit A. Seshia, and Ofer Strichman. Abstraction-based satisfiability solving of Presburger arithmetic. In Proc. 16th International Conference on Computer-Aided Verification (CAV), pages 308-320, July 2004.
24. LASH Toolset. Available at http://www. montefiore.ulg.ac.be/~boigelot/research/lash.
25. Stephen McCamant and Michael D. Ernst. Predicting problems caused by component upgrades. In Proceedings of the 11th ACM SIGSOFT Symposium on Foundations of Software Engineering (FSE), pages 287-296, 2003.
26. M. Moskewicz, C. Madigan, Y. Zhao, L. Zhang, and S. Malik. Chaff: Engineering an efficient SAT solver. In 38th Design Automation Conference (DAC ’01), pages 530-535, June 2001.
27. Moscow ML. Available at http://www.dina.dk/~sestoft/mosml.html.
28. G. Nelson and D. C. Oppen. Simplification by cooperating decision procedures. ACM Transac-tions on Programming Languages and Systems, 1(2):245-257, 1979.
29. Christos H. Papadimitriou. On the complexity of integer programming. Journal of the ACM, 28(4):765-768, 1981.
30. Vaughan Pratt. Two easy theories whose combination is hard. Technical report, Massachusetts Institute of Technology, 1977. Cambridge, MA.
31. M. Preßburger. Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. Comptes-rendus du Premier Congrès des Mathématiciens des Pays Slaves, 395:92-101, 1929.
32. A. Pnueli, Y. Rodeh, O. Shtrichman, and M. Siegel. Deciding equality formulas by small-domain instantiations. In N. Halbwachs and D. Peled, editors, Computer-Aided Verification, volume 1633 of Lecture Notes in Computer Science, pages 455-469. Springer-Verlag, July 1999.
33. Christos H. Papadimitriou and Kenneth Steiglitz. Combinatorial Optimization: Algorithms and Complexity, chapter 13. Prentice-Hall, 1982.
34. William Pugh. The omega test: A fast and practical integer programming algorithm for dependence analysis. In Supercomputing, pages 4-13, 1991.
35. Alexander Schrijver. Theory of Linear and Integer Programming. John Wiley and Sons, 1986.
36. Sanjit A. Seshia. Adaptive Eager Boolean Encoding for Arithmetic Reasoning in Verification. PhD thesis, Carnegie Mellon University, 2005.
37. R. E. Shostak. Deciding combinations of theories. Journal of the ACM, 31(1):1-12, 1984.
38. Natarajan Shankar and Harald Rueß. Combining Shostak theories. In Sophie Tison, editor, Proc. Rewriting Techniques and Applications, LNCS 2378, pages 1-18. Springer-Verlag, July 2002.
39. O. Strichman, S. A. Seshia, and R. E. Bryant. Deciding separation formulas with SAT. In E. Brinksma and K. G. Larsen, editors, Proc. 14th Intl. Conference on Computer-Aided Verification (CAV’02), LNCS 2404, pages 209-222. Springer-Verlag, July 2002.
40. Sanjit A. Seshia, K. Subramani, and Randal E. Bryant. On solving Boolean combinations of generalized 2SAT constraints. Technical Report CMU-CS-04-179, Carnegie Mellon University, 2004.
41. O. Strichman. On solving Presburger and linear arithmetic with SAT. In Formal Methods in Computer-Aided Design (FMCAD ’02), LNCS 2517, pages 160-170. Springer-Verlag, November 2002.
42. UCLID Verification System. Available at http://www.cs.cmu.edu/_uclid.
43. J. von zur Gathen and M. Sieveking. A bound on solutions of linear integer equalities and inequalities. Proceedings of the American Mathematical Society, 72(1):155-158, October 1978.
44. Pierre Wolper and Bernard Boigelot. An automata-theoretic approach to Presburger arithmetic constraints. In Proc. Static Analysis Symposium, LNCS 983, pages 21-32, September 1995.
45. Wisconsin Safety Analyzer Project. http://www.cs.wisc.edu/wisa.
46. Steven A. Wolfman and Daniel S. Weld. The LPSAT engine and its application to resource planning. In Proceedings of the International Joint Conference in Artificial Intelligence (IJCAI), pages 310-317, 1999.
47. zChaff Boolean Satisfiability Solver. Available at http://ee.princeton.edu/~chaff/zchaff.php.