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2-Pointer Logic

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Taming the Infinities of Concurrency

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14660))


For reasoning about properties of pointers, we consider conjunctions of equalities and dis-equalities between terms built up from address constants by addition of offsets and dereferencing. We call the resulting class of formulas 2-pointer logic. We introduce a quantitative version of congruence closure to provide polynomial time algorithms for deciding satisfiability as well as implication between formulas. By generalizing quantitative congruence closure to quantitative finite automata, we succeed in constructing canonical normal forms so that checking of equivalence between conjunctions reduces to syntactic equality.

We apply our techniques to realize abstract transformers for dedicated forms of assignments via pointers, in particular, indefinite, definite and locally invertible assignments. Quantitative finite automata here allow us to restrict formulas to properties expressible by some subterm-closed subset of terms only.

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  1. Arceri, V., Olliaro, M., Cortesi, A., Ferrara, P.: Relational string abstract domains. In: Finkbeiner, B., Wies, T. (eds.) VMCAI 2022. LNCS, vol. 13182, pp. 20–42. Springer, Cham (2022).

    Chapter  Google Scholar 

  2. Chang, B.E., Rival, X.: Modular construction of shape-numeric analyzers. In: Banerjee, A., Danvy, O., Doh, K., Hatcliff, J. (eds.) Semantics, Abstract Interpretation, and Reasoning about Programs: Essays Dedicated to David A. Schmidt on the Occasion of his Sixtieth Birthday, Manhattan, Kansas, USA, 19–20th September 2013, EPTCS, vol. 129, pp. 161–185 (2013).

  3. Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among variables of a program. In: Aho, A.V., Zilles, S.N., Szymanski, T.G. (eds.) Conference Record of the Fifth Annual ACM Symposium on Principles of Programming Languages, Tucson, Arizona, USA, January 1978, pp. 84–96. ACM Press (1978).

  4. Giet, J., Ridoux, F., Rival, X.: A product of shape and sequence abstractions. In: Hermenegildo, M.V., Morales, J.F. (eds.) SAS 2023. LNCS, vol. 14284, pp. 310–342. Springer, Cham (2023).

    Chapter  Google Scholar 

  5. Gotsman, A., Berdine, J., Cook, B., Sagiv, M.: Thread-modular shape analysis. In: PLDI 2007, pp. 266–277. ACM (2007).

  6. Gulwani, S., Tiwari, A., Necula, G.C.: Join algorithms for the theory of uninterpreted functions. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 311–323. Springer, Heidelberg (2004).

    Chapter  Google Scholar 

  7. Illous, H., Lemerre, M., Rival, X.: A relational shape abstract domain. Formal Methods Syst. Des. 57(3), 343–400 (2021).

    Article  Google Scholar 

  8. Korovin, K., Kovács, L., Reger, G., Schoisswohl, J., Voronkov, A.: ALASCA: reasoning in quantified linear arithmetic. In: Sankaranarayanan, S., Sharygina, N. (eds.) TACAS 2023. LNCS, vol. 13993, pp. 647–665. Springer, Cham (2023).

    Chapter  Google Scholar 

  9. Kroening, D., Strichman, O.: Decision Procedures - An Algorithmic Point of View. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2008). ISBN 978-3-540-74104-6

  10. Miné, A.: A new numerical abstract domain based on difference-bound matrices. In: Danvy, O., Filinski, A. (eds.) PADO 2001. LNCS, vol. 2053, pp. 155–172. Springer, Heidelberg (2001).

    Chapter  Google Scholar 

  11. Miné, A.: The octagon abstract domain. In: WCRE 2001, p. 310. IEEE Computer Society (2001).

  12. Miné, A.: Field-sensitive value analysis of embedded C programs with union types and pointer arithmetics. In: Irwin, M.J., Bosschere, K.D. (eds.) Proceedings of the 2006 ACM SIGPLAN/SIGBED Conference on Languages, Compilers, and Tools for Embedded Systems (LCTES 2006), Ottawa, Ontario, Canada, 14–16 June 2006, pp. 54–63. ACM (2006).

  13. Miné, A.: The octagon abstract domain. Higher Order Symbol. Comput. 19(1), 31–100 (2006). ISSN 1388-3690

  14. Müller-Olm, M., Seidl, H.: A note on Karr’s algorithm. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1016–1028. Springer, Heidelberg (2004).

    Chapter  Google Scholar 

  15. Müller-Olm, M., Seidl, H.: Precise interprocedural analysis through linear algebra. In: Jones, N.D., Leroy, X. (eds.) Proceedings of the 31st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2004, Venice, Italy, 14–16 January 2004, pp. 330–341. ACM (2004).

  16. Müller-Olm, M., Seidl, H.: Upper adjoints for fast inter-procedural variable equalities. In: Drossopoulou, S. (ed.) ESOP 2008. LNCS, vol. 4960, pp. 178–192. Springer, Heidelberg (2008).

    Chapter  Google Scholar 

  17. Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Trans. Program. Lang. Syst. 1(2), 245–257 (1979). ISSN 0164-0925

  18. Nelson, G., Oppen, D.C.: Fast decision procedures based on congruence closure. J. ACM 27(2), 356–364 (1980). ISSN 0004-5411

  19. O’Hearn, P.W.: Resources, concurrency, and local reasoning. Theoret. Comput. Sci. 375(1), 271–307 (2007).

    Article  MathSciNet  Google Scholar 

  20. O’Hearn, P.W.: Separation logic. Commun. ACM 62(2), 86–95 (2019).

    Article  Google Scholar 

  21. Reps, T.W., Sagiv, M., Wilhelm, R.: Shape analysis and applications. In: Srikant, Y.N., Shankar, P. (eds.) The Compiler Design Handbook: Optimizations and Machine Code Generation, 2nd edn, p. 12. CRC Press (2007)

    Google Scholar 

  22. Rueß, H., Shankar, N.: Deconstructing Shostak. In: Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science, Boston, Massachusetts, USA, 16–19 June 2001, pp. 19–28. IEEE Computer Society (2001).

  23. Sagiv, S., Reps, T.W., Wilhelm, R.: Parametric shape analysis via 3-valued logic. ACM Trans. Program. Lang. Syst. 24(3), 217–298 (2002).

    Article  Google Scholar 

  24. Seidl, H., Erhard, J., Tilscher, S., Schwarz, M.: Non-numerical weakly relational domains (2024).

  25. Seidl, H., Vojdani, V., Vene, V.: A smooth combination of linear and Herbrand equalities for polynomial time must-alias analysis. In: Cavalcanti, A., Dams, D.R. (eds.) FM 2009. LNCS, vol. 5850, pp. 644–659. Springer, Heidelberg (2009).

    Chapter  Google Scholar 

  26. Shostak, R.E.: Deciding combinations of theories. J. ACM 31(1), 1–12 (1984).

    Article  MathSciNet  Google Scholar 

  27. Singh, G., Püschel, M., Vechev, M.: Fast polyhedra abstract domain. In: Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages, POPL 2017, New York, NY, USA, pp. 46–59. Association for Computing Machinery (2017). ISBN 9781450346603

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This work was supported in part by the Shota Rustaveli National Science Foundation of Georgia under the project FR-21-7973 and Deutsche Forschungsgemeinschaft (DFG) - 378803395/2428 ConVeY 2.

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Correspondence to Helmut Seidl .

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Seidl, H., Erhard, J., Schwarz, M., Tilscher, S. (2024). 2-Pointer Logic. In: Kiefer, S., Křetínský, J., Kučera, A. (eds) Taming the Infinities of Concurrency. Lecture Notes in Computer Science, vol 14660. Springer, Cham.

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