Abstract
This paper explores the spatial domain of sets of inequalities where each inequality contains at most two variables — a domain that is richer than intervals and more tractable than general polyhedra. We present a complete suite of efficient domain operations for linear systems with two variables per inequality with unrestricted coefficients. We exploit a tactic in which a system of inequalities with at most two variables per inequality is decomposed into a series of projections — one for each two dimensional plane. The decomposition enables all domain operations required for abstract interpretation to be expressed in terms of the two dimensional case. The resulting operations are efficient and include a novel planar convex hull algorithm. Empirical evidence suggests that widening can be applied effectively, ensuring tractability.
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References
R. Bagnara. Data-Flow Analysis for Constrant Logic-Based Languages. PhD thesis, Dipartimento di Informatica, Università di Pisa, 1997.
V. Balasundaram and K. Kennedy. A Technique for Summarizing Data Access and its Use in Parallelism Enhancing Transformations. In Programming Language Design and Implementation, pages 41–53. ACM Press, 1989.
F. Bancilhon and R. Ramakrishnan. An Amateur’s Introduction to Recursive Query Processing Strategies. In International Conference on Management of Data, pages 16–52. ACM Press, 1986.
F. Benoy and A. King. Inferring Argument Size Relationships with CLP(ℝ). In Logic Program Synthesis and Transformation (Selected Papers), volume 1207 of Lecture Notes in Computer Science, pages 204–223. Springer-Verlag, 1997.
J. L. Bentley, H. T. Kung, M. Schkolnick, and C. D. Thompson. On the Average Number of Maxima in a Set of Vectors. Journal of the ACM, 25:536–543, 1978.
N. V. Chernikova. Algorithm for Discovering the Set of All Solutions of a Linear Programming Problem. USSR Computational Mathematics and Mathematical Physics, 8(6):282–293, 1968.
P. Cousot and R. Cousot. Comparing the Galois Connection and Widening/Narrowing Approaches to Abstract Interpretation. In Programming Language Implementation and Logic Programming, volume 631 of Lecture Notes in Computer Science, pages 269–295. Springer-Verlag, 1992.
P. Cousot and N. Halbwachs. Automatic Discovery of Linear Restraints among Variables of a Program. In Principles of Programming Languages, pages 84–97. ACM Press, 1978.
E. Davis. Constraint Propagation with Interval Labels. Artificial Intelligence, 32(3):281–331, 1987.
S. D. Feit. A Fast Algorithm for the Two-Variable Integer Programming Problem. Journal of the ACM, 31(1):99–113, 1984.
R. L. Graham. An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set. Information Processing Letters, 1(4):132–133, 1972.
P. Granger. Static Analysis of Linear Congruence Equalities among Variables of a Program. In International Joint Conference on the Theory and Practice of Software Development, volume 493 of Lecture Notes in Computer Science, pages 169–192. Springer-Verlag, 1991.
W. H. Harrison. Compiler Analysis of the Value Ranges for Variables. IEEE Transactions on Software Engineering, SE-3(3), 1977.
W. Harvey. Computing Two-Dimensional Integer Hulls. SIAM Journal on Computing, 28(6):2285–2299, 1999.
W. Harvey and P. J. Stuckey. A Unit Two Variable per Inequality Integer Constraint Solver for Constraint Logic Programming. Australian Computer Science Communications, 19(1):102–111, 1997.
D. S. Hochbaum and J. Naor. Simple and Fast Algorithms for Linear and Integer Programs with Two Variables per Inequality. SIAM Journal on Computing, 23(6):1179–1192, 1994.
J. M. Howe and A. King. Specialising Finite Domain Programs using Polyhedra. In Logic Programming, Synthesis and Transformation (Selected Papers), volume 1817 of Lecture Notes in Computer Science, pages 118–135. Springer-Verlag, 1999.
J. Jaffar, M. J. Maher, P. J. Stuckey, and R. H. C. Yap. Beyond Finite Domains. In International Workshop on Principles and Practice of Constraint Programming, volume 874 of Lecture Notes in Computer Science, pages 86–94. Springer-Verlag, 1994.
M. Karr. Affine Relationships Among Variables of a Program. Acta Informatica, 6:133–151, 1976.
V. Klee and G. J. Minty. How Good is the Simplex Algorithm? In Inequalities-III. Academic Press, New York and London, 1972.
J. C. Lagarias. The Computational Complexity of Simultaneous Diophantine Approximation Problems. SIAM Journal on Computing, 14(1):196–209, 1985.
K. G. Larsen, J. Pearson, C. Weise, and W. Yi. Clock Difference Diagrams. Nordic Journal of Computing, 6(3):271–298, 1999.
H. Le Verge. A Note on Chernikova’s Algorithm. Technical Report 1662, Institut de Recherche en Informatique, Campus Universitaire de Beaulieu, France, 1992.
N. Lindenstrauss and Y. Sagiv. Automatic Termination Analysis of Logic Programs. In International Conference on Logic Programming, pages 63–77. MIT Press, 1997.
A. Miné. A New Numerical Abstract Domain Based on Difference-Bound Matrices. In Programs as Data Objects, volume 2053 of Lecture Notes in Computer Science, pages 155–172. Springer, 2001.
A. Miné. The Octagon Abstract Domain. In Eighth Working Conference on Reverse Engineering, pages 310–319. IEEE Computer Society, 2001.
A. Miné. A Few Graph-Based Relational Numerical Abstract Domains. In Ninth International Static Analysis Symposium, volume 2477 of Lecture Notes in Computer Science, pages 117–132. Springer-Verlag, 2002.
J. Møller, J. Lichtenberg, H. R. Andersen, and H. Hulgaard. Difference Decision Diagrams. In Conference of the European Association for Computer Science Logic, volume 1683 of Lecture Notes in Computer Science, pages 111–125. Springer-Verlag, 1999.
C. G. Nelson. An n log(n) Algorithm for the Two-Variable-Per-Constraint Linear Programming Satisfiability Problem. Technical Report STAN-CS-78-689, Stanford University, Department of Computer Science, 1978.
V. R. Pratt. Two Easy Theories Whose Combination is Hard, September 1977. http://boole.stanford.edu/pub/sefnp.pdf.
H. Raynaud. Sur L’enveloppe Convexe des Nuages de Points Aléatoires dans ℝn. Journal of Applied Probability, 7(1):35–48, 1970.
R. Seidel. Convex Hull Computations. In J. E. Goodman and J. O’Rourke, editors, Handbook of Discrete and Computational Geometry, pages 361–376. CRC Press, 1997.
R. Shaham, H. Kolodner, and M. Sagiv. Automatic Removal of Array Memory Leaks in Java. In Compiler Construction, volume 1781 of Lecture Notes in Computer Science, pages 50–66. Springer, 2000.
R. Shostak. Deciding Linear Inequalities by Computing Loop Residues. Journal of the ACM, 28(4): 769–779, 1981.
Z. Su and D. Wagner. Efficient Algorithms for General Classes of Integer Range Constraints, July 2001. http://www.cs.berkeley.edu/~zhendong/.
D. Wagner, J. S. Foster, E. A. Brewer, and A. Aiken. A First Step Towards Detection of Buffer Overrun Vulnerabilities. In Network and Distributed System Security Symposium. Internet Society, 2000.
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Simon, A., King, A., Howe, J.M. (2003). Two Variables per Linear Inequality as an Abstract Domain. In: Leuschel, M. (eds) Logic Based Program Synthesis and Transformation. LOPSTR 2002. Lecture Notes in Computer Science, vol 2664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45013-0_7
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DOI: https://doi.org/10.1007/3-540-45013-0_7
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