Skip to main content
Springer Nature Link
Log in

Separation Logic with Linearly Compositional Inductive Predicates and Set Data Constraints

  • Conference paper
  • First Online:
SOFSEM 2019: Theory and Practice of Computer Science (SOFSEM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11376))

Abstract

We identify difference-bound set constraints (DBS), an analogy of difference-bound arithmetic constraints for sets. DBS can express not only set constraints but also arithmetic constraints over set elements. We integrate DBS into separation logic with linearly compositional inductive predicates, obtaining a logic thereof where set data constraints of linear data structures can be specified. We show that the satisfiability of this logic is decidable. A crucial step of the decision procedure is to compute the transitive closure of DBS-definable set relations, to capture which we propose an extension of quantified set constraints with Presburger Arithmetic (RQSPA). The satisfiability of RQSPA is then shown to be decidable by harnessing advanced automata-theoretic techniques.

Partially supported by the NSFC grants (No. 61472474, 61572478, 61872340), UK EPSRC grant (EP/P00430X/1), and the INRIA-CAS joint research project VIP.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    This shall be usually referred to as “transitive closure of \(\mathcal {DBS}\)” to avoid clumsiness.

  2. 2.

    The operators < and > can be seen as abbreviations, for instance, \(x < y\) is equivalent to \(x \le y -1\), which will be used later on as well.

  3. 3.

    An unrestricted extension of quantified set constraints with Presburger Arithmetic is undecidable, as shown in [6].

References

  1. Bouajjani, A., Drăgoi, C., Enea, C., Sighireanu, M.: Accurate invariant checking for programs manipulating lists and arrays with infinite data. In: Chakraborty, S., Mukund, M. (eds.) ATVA 2012. LNCS, vol. 7561, pp. 167–182. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-33386-6_14

    Chapter  MATH  Google Scholar 

  2. Bozga, M., Gîrlea, C., Iosif, R.: Iterating octagons. In: TACAS, pp. 337–351 (2009)

    Google Scholar 

  3. Bozga, M., Iosif, R., Konecný, F.: Fast acceleration of ultimately periodic relations. In: CAV, pp. 227–242 (2010)

    Google Scholar 

  4. Bozga, M., Iosif, R., Lakhnech, Y.: Flat parametric counter automata. Fundam. Inf. 91(2), 275–303 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Büchi, R.J.: Weak Second-Order arithmetic and finite automata. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 6(1–6), 66–92 (1960)

    Article  MathSciNet  Google Scholar 

  6. Cantone, D., Cutello, V., Schwartz, J.T.: Decision problems for tarski and presburger arithmetics extended with sets. In: Börger, E., Kleine Büning, H., Richter, M.M., Schönfeld, W. (eds.) CSL 1990. LNCS, vol. 533, pp. 95–109. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-54487-9_54

    Chapter  Google Scholar 

  7. Chin, W.-N., David, C., Nguyen, H.H., Qin, S.: Automated verification of shape, size and bag properties via user-defined predicates in separation logic. Sci. Comput. Program. 77(9), 1006–1036 (2012)

    Article  Google Scholar 

  8. Chu, D.-H., Jaffar, J., Trinh, M.-T.: Automatic induction proofs of data-structures in imperative programs. In: PLDI, pp. 457–466 (2015)

    Google Scholar 

  9. Comon, H., Jurski, Y.: Multiple counters automata, safety analysis and presburger arithmetic. In: Hu, A.J., Vardi, M.Y. (eds.) CAV 1998. LNCS, vol. 1427, pp. 268–279. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0028751

    Chapter  Google Scholar 

  10. Cristiá, M., Rossi, G.: A decision procedure for restricted intensional sets. In: de Moura, L. (ed.) CADE 2017. LNCS (LNAI), vol. 10395, pp. 185–201. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63046-5_12

    Chapter  Google Scholar 

  11. Elgot, C.C.: Decision problems of finite automata design and related arithmetics. Trans. Am. Math. Soc. 98(1), 21–51 (1961)

    Article  MathSciNet  Google Scholar 

  12. Enea, C., Lengál, O., Sighireanu, M., Vojnar, T.: Compositional entailment checking for a fragment of separation logic. In: APLAS, pp. 314–333 (2014)

    Google Scholar 

  13. Enea, C., Sighireanu, M., Wu, Z.: On automated lemma generation for separation logic with inductive definitions. In: Finkbeiner, B., Pu, G., Zhang, L. (eds.) ATVA 2015. LNCS, vol. 9364, pp. 80–96. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24953-7_7

    Chapter  MATH  Google Scholar 

  14. Gao, C., Chen, T., Wu, Z.: Separation logic with linearly compositional inductive predicates and set data constraints (full version). http://arxiv.org/abs/1811.00699

  15. Gu, X., Chen, T., Wu, Z.: A complete decision procedure for linearly compositional separation logic with data constraints. In: Olivetti, N., Tiwari, A. (eds.) IJCAR 2016. LNCS (LNAI), vol. 9706, pp. 532–549. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40229-1_36

    Chapter  Google Scholar 

  16. Halpern, J.Y.: Presburger arithmetic with unary predicates is \({\varPi }_{1}^{1}\)-complete. J. Symb. Logic 56(2), 637–642 (1991)

    Article  MathSciNet  Google Scholar 

  17. Horbach, M., Voigt, M., Weidenbach, C.: On the combination of the Bernays–Schönfinkel–Ramsey fragment with simple linear integer arithmetic. In: de Moura, L. (ed.) CADE 2017. LNCS (LNAI), vol. 10395, pp. 77–94. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63046-5_6

    Chapter  Google Scholar 

  18. Klaedtke, F., Rueß, H.: Monadic second-order logics with cardinalities. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 681–696. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-45061-0_54

    Chapter  Google Scholar 

  19. Konečný, F.: PTIME computation of transitive closures of octagonal relations. In: Chechik, M., Raskin, J.-F. (eds.) TACAS 2016. LNCS, vol. 9636, pp. 645–661. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49674-9_42

    Chapter  Google Scholar 

  20. Kuncak, V., Piskac, R., Suter, P.: Ordered sets in the calculus of data structures. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 34–48. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15205-4_5

    Chapter  Google Scholar 

  21. Le, Q.L., Sun, J., Chin, W.-N.: Satisfiability modulo heap-based programs. In: Chaudhuri, S., Farzan, A. (eds.) CAV 2016. LNCS, vol. 9779, pp. 382–404. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-41528-4_21

    Chapter  Google Scholar 

  22. Madhusudan, P., Parlato, G., Qiu, X.: Decidable logics combining heap structures and data. In: POPL 2011, pp. 611–622. ACM (2011)

    Google Scholar 

  23. Madhusudan, P., Qiu, X., Stefanescu, A.: Recursive proofs for inductive tree data-structures. In: POPL, pp. 123–136 (2012)

    Google Scholar 

  24. 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). https://doi.org/10.1007/3-540-44978-7_10

    Chapter  Google Scholar 

  25. O’Hearn, P., Reynolds, J., Yang, H.: Local reasoning about programs that alter data structures. In: Fribourg, L. (ed.) CSL 2001. LNCS, vol. 2142, pp. 1–19. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44802-0_1

    Chapter  Google Scholar 

  26. Piskac, R., Wies, T., Zufferey, D.: Automating separation logic using SMT. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 773–789. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39799-8_54

    Chapter  Google Scholar 

  27. Piskac, R., Wies, T., Zufferey, D.: Automating separation logic with trees and data. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 711–728. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08867-9_47

    Chapter  Google Scholar 

  28. Reynolds, J.C.: Separation logic: a logic for shared mutable data structures. In: LICS, pp. 55–74 (2002)

    Google Scholar 

  29. Seidl, H., Schwentick, T., Muscholl, A., Habermehl, P.: Counting in trees for free. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1136–1149. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-27836-8_94

    Chapter  Google Scholar 

  30. Suter, P., Dotta, M., Kuncak, V.: Decision procedures for algebraic data types with abstractions. In: POPL 2010, pp. 199–210. ACM (2010)

    Google Scholar 

  31. Tatsuta, M., Le, Q.L., Chin, W.-N.: Decision procedure for separation logic with inductive definitions and presburger arithmetic. In: Igarashi, A. (ed.) APLAS 2016. LNCS, vol. 10017, pp. 423–443. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-47958-3_22

    Chapter  MATH  Google Scholar 

  32. Voigt, M.: The Bernays–Schönfinkel–Ramsey fragment with bounded difference constraints over the reals is decidable. In: Dixon, C., Finger, M. (eds.) FroCoS 2017. LNCS (LNAI), vol. 10483, pp. 244–261. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66167-4_14

    Chapter  Google Scholar 

  33. Wies, T., Piskac, R., Kuncak, V.: Combining theories with shared set operations. In: Ghilardi, S., Sebastiani, R. (eds.) FroCoS 2009. LNCS (LNAI), vol. 5749, pp. 366–382. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04222-5_23

    Chapter  MATH  Google Scholar 

  34. Xu, Z., Chen, T., Wu, Z.: Satisfiability of compositional separation logic with tree predicates and data constraints. In: de Moura, L. (ed.) CADE 2017. LNCS (LNAI), vol. 10395, pp. 509–527. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63046-5_31

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China

    Chong Gao & Zhilin Wu

  2. University of Chinese Academy of Sciences, Beijing, China

    Chong Gao

  3. Department of Computer Science and Information Systems, Birkbeck, University of London, London, UK

    Taolue Chen

Authors
  1. Chong Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Taolue Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhilin Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhilin Wu .

Editor information

Editors and Affiliations

  1. University of Genoa, Genoa, Italy

    Barbara Catania

  2. Comenius University, Bratislava, Slovakia

    Rastislav Královič

  3. Poznań University of Technology, Poznań, Poland

    Jerzy Nawrocki

  4. Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Gao, C., Chen, T., Wu, Z. (2019). Separation Logic with Linearly Compositional Inductive Predicates and Set Data Constraints. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds) SOFSEM 2019: Theory and Practice of Computer Science. SOFSEM 2019. Lecture Notes in Computer Science(), vol 11376. Springer, Cham. https://doi.org/10.1007/978-3-030-10801-4_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-10801-4_17

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10800-7

  • Online ISBN: 978-3-030-10801-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics