Abstract
This paper considers some known abstract domains for affine-relation analysis (ARA), along with several variants, and studies how they relate to each other. We show that the abstract domains of Müller-Olm/Seidl (MOS) and King/Søndergaard (KS) are, in general, incomparable, but give sound interconversion methods. We also show that the methods of King and Søndergaard can be applied without bit-blasting—while still using a bit-precise concrete semantics.
Supported, in part, by NSF under grants CCF-{0810053, 0904371}, by ONR under grants N00014-{09-1-0510, 10-M-0251}, by ARL under grant W911NF-09-1-0413, and by AFRL under grants FA9550-09-1-0279 and FA8650-10-C-7088. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors, and do not necessarily reflect the views of the sponsoring agencies.
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References
Bach, E.: Linear algebra modulo n. Unpublished manuscript (December 1992)
Bouajjani, A., Esparza, J., Touili, T.: A generic approach to the static analysis of concurrent programs with procedures. In: POPL (2003)
Dutertre, B., de Moura, L.: Yices: An SMT solver (2006), http://yices.csl.sri.com/
Elder, M., Lim, J., Sharma, T., Andersen, T., Reps, T.: Abstract domains of affine relations. Tech. Rep. TR-1691, Comp. Sci. Dept., Univ. of Wisconsin, Madison, WI (June 2011)
Gulwani, S., Necula, G.: Discovering affine equalities using random interpretation. In: POPL (2003)
Gulwani, S., Necula, G.: Precise interprocedural analysis using random interpretation. In: POPL (2005)
Hafner, J., McCurley, K.: Asymptotically fast triangularization of matrices over rings. SIAM J. Comput. 20(6) (1991)
Howell, J.: Spans in the module (ℤ m )s. Linear and Multilinear Algebra 19 (1986)
Karr, M.: Affine relationship among variables of a program. Acta Inf. 6, 133–151 (1976)
Kidd, N., Lal, A., Reps, T.: WALi: The Weighted Automaton Library (2007), http://www.cs.wisc.edu/wpis/wpds/download.php
King, A., Søndergaard, H.: Inferring congruence equations using SAT. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 281–293. Springer, Heidelberg (2008)
King, A., Søndergaard, H.: Automatic abstraction for congruences. In: Barthe, G., Hermenegildo, M. (eds.) VMCAI 2010. LNCS, vol. 5944, pp. 197–213. Springer, Heidelberg (2010)
Knoop, J., Steffen, B.: The interprocedural coincidence theorem. In: Pfahler, P., Kastens, U. (eds.) CC 1992. LNCS, vol. 641. Springer, Heidelberg (1992)
Lal, A., Reps, T.: Improving pushdown system model checking. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 343–357. Springer, Heidelberg (2006)
Lal, A., Reps, T., Balakrishnan, G.: Extended weighted pushdown systems. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 434–448. Springer, Heidelberg (2005)
Lim, J., Reps, T.: A system for generating static analyzers for machine instructions. In: Hendren, L. (ed.) CC 2008. LNCS, vol. 4959, pp. 36–52. Springer, Heidelberg (2008)
Meyer, C.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2000)
Müller-Olm, M., Seidl, H.: Precise interprocedural analysis through linear algebra. In: POPL (2004)
Müller-Olm, M., Seidl, H.: Analysis of modular arithmetic. In: Sagiv, M. (ed.) ESOP 2005. LNCS, vol. 3444, pp. 46–60. Springer, Heidelberg (2005)
Müller-Olm, M., Seidl, H.: Personal communication (April 2005)
Müller-Olm, M., Seidl, H.: Analysis of modular arithmetic. TOPLAS 29(5) (2007)
Reps, T., Sagiv, M., Yorsh, G.: Symbolic implementation of the best transformer. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 252–266. Springer, Heidelberg (2004)
Reps, T., Schwoon, S., Jha, S., Melski, D.: Weighted pushdown systems and their application to interprocedural dataflow analysis. SCP 58(1-2) (October 2005)
Sharir, M., Pnueli, A.: Two approaches to interprocedural data flow analysis. In: Program Flow Analysis: Theory and Applications. Prentice-Hall, Englewood Cliffs (1981)
Storjohann, A.: Algorithms for Matrix Canonical Forms. PhD thesis, ETH Zurich, Zurich, Switzerland, Diss. ETH No. 13922 (2000)
Warren Jr., H.: Hacker’s Delight. Addison-Wesley, Reading (2003)
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Elder, M., Lim, J., Sharma, T., Andersen, T., Reps, T. (2011). Abstract Domains of Affine Relations. In: Yahav, E. (eds) Static Analysis. SAS 2011. Lecture Notes in Computer Science, vol 6887. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23702-7_17
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