research-article

Public Access

Expressive Completeness of Separation Logic with Two Variables and No Separating Conjunction

Authors:

Stephane Demri,

Morgan DetersAuthors Info & Claims

ACM Transactions on Computational Logic (TOCL), Volume 17, Issue 2

Article No.: 12, Pages 1 - 44

https://doi.org/10.1145/2835490

Published: 07 January 2016 Publication History

Abstract

Separation logic is used as an assertion language for Hoare-style proof systems about programs with pointers, and there is an ongoing quest for understanding its complexity and expressive power. Herein, we show that first-order separation logic with one record field restricted to two variables and the separating implication (no separating conjunction) is as expressive as weak second-order logic, substantially sharpening a previous result. Capturing weak second-order logic with such a restricted form of separation logic requires substantial updates to known proof techniques. We develop these and, as a by-product, identify the smallest fragment of separation logic known to be undecidable: first-order separation logic with one record field, two variables, and no separating conjunction. Because we forbid ourselves the use of many syntactic resources, this underscores even further the power of separating implication on concrete heaps.

Supplementary Material

a12-demri-apndx.pdf (demri.zip)

Supplemental movie, appendix, image and software files for, Expressive Completeness of Separation Logic with Two Variables and No Separating Conjunction

Download
148.92 KB

References

[1]

T. Antonopoulos. 2010. Expressive Power of Query Languages. Ph.D. Dissertation. University of Cambridge.

[2]

T. Antonopoulos and A. Dawar. 2009. Separating graph logic from MSO. In FOSSACS’09 (LNCS), Vol. 5504. Springer, 63--77.

Digital Library

[3]

T. Antonopoulos, N. Gorogiannis, C. Haase, M. Kanovich, and J. Ouaknine. 2014. Foundations for decision problems in separation logic with general inductive predicates. In FOSSACS’14 (LNCS), Vol. 8412. Springer, 411--425.

[4]

K. Apt. 1981. Ten years of Hoare’s logic. ACM Transactions on Programming Languages and Systems 3, 4 (1981), 431--483.

Digital Library

[5]

K. Bansal, A. Reynolds, T. King, C. Barrett, and Th. Wies. 2015. Deciding local theory extensions via E-matching. In CAV’15 (LNCS), Vol. 9207. Springer, 87--105.

[6]

M. Bojanczyk, A. Muscholl, Th. Schwentick, and L. Segoufin. 2009. Two-variable logic on data trees and XML reasoning. Journal of the Association for Computing Machinery 56, 3 (2009).

Digital Library

[7]

E. Börger, E. Grädel, and Y. Gurevich. 1997. The Classical Decision Problem. Springer.

[8]

M. Bozga, R. Iosif, and S. Perarnau. 2010. Quantitative separation logic and programs with lists. Journal of Automated Reasoning 45, 2 (2010), 131--156.

Digital Library

[9]

D. Bresolin, D. Della Monica, V. Goranko, A. Montanari, and G. Sciavicco. 2010. Metric propositional neighborhood logics: Expressiveness, decidability, and undecidability. In ECAI’10 (Frontiers in Artificial Intelligence and Applications), Vol. 215. IOS Press, 695--700.

Digital Library

[10]

R. Brochenin. 2013. Separation Logic: Expressiveness, Complexity, Temporal Extension. Ph.D. Dissertation. LSV, ENS Cachan.

[11]

R. Brochenin, S. Demri, and E. Lozes. 2012. On the almighty wand. Information and Computation 211 (2012), 106--137.

Digital Library

[12]

J. Brotherston and M. Kanovich. 2014. Undecidability of propositional separation logic and Its neighbours. Journal of the Association for Computing Machinery 61, 2 (2014).

Digital Library

[13]

C. Calcagno, D. Distefano, P. O’Hearn, and H. Yang. 2011. Compositional shape analysis by means of Bi-abduction. Journal of the Association for Computing Machinery 58, 6 (2011), 26.

Digital Library

[14]

C. Calcagno, P. O’Hearn, and H. Yang. 2001. Computability and complexity results for a spatial assertion language for data structures. In FSTTCS’01 (LNCS), Vol. 2245. Springer, 108--119.

Digital Library

[15]

S. Chakraborty. 2012. Reasoning about heap manipulating programs using automata techniques. In Modern Applications of Automata Theory, D. D’Souza and P. Shankar (Eds.). IISc Research Monographs Series, Vol. 2. World Scientific, Chapter 7, 193--228.

[16]

B. Cook, C. Haase, J. Ouaknine, M. Parkinson, and J. Worrell. 2011. Tractable reasoning in a fragment of separation logic. In CONCUR’11 (LNCS), Vol. 6901. Springer, 235--249.

Digital Library

[17]

A. Dawar, Ph. Gardner, and G. Ghelli. 2007. Expressiveness and complexity of graph logic. Information and Computation 205, 3 (2007), 263--310.

Digital Library

[18]

S. Demri and M. Deters. 2014. Expressive completeness of separation logic with two variables and no separating conjunction. In CSL-LICS’14. ACM, 37.

Digital Library

[19]

S. Demri and M. Deters. 2015. Two-variable separation logic and its inner circle. ACM Transactions on Computational Logic 16, 2 (2015), 15.

Digital Library

[20]

S. Demri, D. Galmiche, D. Larchey-Wendling, and D. Mery. 2014. Separation logic with one quantified variable. In CSR’14 (LNCS), Vol. 8476. Springer, 125--138.

[21]

K. Etessami, M. Vardi, and Th. Wilke. 1997. First-order logic with two variables and unary temporal logics. In LICS’97. IEEE, 228--235.

Digital Library

[22]

D. Gabbay. 1981. Expressive functional completeness in tense logic. In Aspects of Philosophical Logic. Reidel, 91--117.

[23]

D. Gabbay, I. Hodkinson, and M. Reynolds. 1994. Temporal Logic - Mathematical Foundations and Computational Aspects, Vol. 1. Oxford University Press.

Digital Library

[24]

D. Galmiche and D. Méry. 2010. Tableaux and resource graphs for separation logic. Journal of Logic and Computation 20, 1 (2010), 189--231.

Digital Library

[25]

E. Grädel, Ph. Kolaitis, and M. Vardi. 1997. On the decision problem for two-variable first-order logic. Bulletin of Symbolic Logic 3, 1 (1997), 53--69.

[26]

E. Grädel, M. Otto, and E. Rosen. 1999. Undecidability results on two-variable logics. Archives of Mathematical Logic 38, 4--5 (1999), 313--354.

[27]

C. Haase, S. Ishtiaq, J. Ouaknine, and M. Parkinson. 2013. SeLoger: A tool for graph-based reasoning in separation logic. In CAV’13 (LNCS), Vol. 8044. Springer, 790--795.

Digital Library

[28]

Z. Hou, R. Clouston, R. Goré, and A. Tiu. 2014. Proof search for propositional abstract separation logics via labelled sequents. In POPL’14. ACM, 465--476.

Digital Library

[29]

Z. Hou, R. Goré, and A. Tiu. 2015. Automated theorem proving for assertions in separation logic with all connectives. In CADE’15 (LNCS), Vol. 9195. Springer, 501--516.

[30]

N. Immerman, A. Rabinovich, Th. Reps, M. Sagiv, and G. Yorsh. 2004. The boundary between decidability and undecidability for transitive-closure logics. In CSL’04 (LNCS), Vol. 3210. Springer, 160--174.

[31]

R. Iosif, A. Rogalewicz, and J. Simacek. 2013. The tree width of separation logic with recursive definitions. In CADE’13 (LNCS), Vol. 7898. Springer, 21--38.

Digital Library

[32]

D. Janin and I. Walukiewicz. 1996. On the expressive completeness of the propositional mu-calculus with respect to monadic second order logic. In CONCUR’96 (LNCS), Vol. 1119. 263--277.

Digital Library

[33]

H. Kamp. 1968. Tense Logic and the Theory of Linear Order. Ph.D. Dissertation. UCLA.

[34]

V. Kuncak and M. Rinard. 2004. On Spatial Conjunction as Second-order Logic. Technical Report MIT--CSAIL--TR--2004--067. MIT CSAIL.

[35]

M. Lange. 2007. Linear time logics around PSL: Complexity, expressiveness, and a little bit of succinctness. In CONCUR’07 (LNCS), Vol. 4703. Springer, 90--104.

Digital Library

[36]

D. Larchey-Wendling and D. Galmiche. 2013. Nondeterministic phase semantics and the undecidability of boolean BI. ACM Transactions on Computational Logic 14, 1 (2013).

Digital Library

[37]

W. Lee and S. Park. 2014. A proof system for separation logic with magic wand. In POPL’14. ACM, 477--490.

Digital Library

[38]

E. Lozes. 2012. Separation logic: Expressiveness and copyless message-passing. ENS Cachan. (2012). Habilitation thesis.

[39]

C. Lutz, U. Sattler, and F. Wolter. 2001. Modal logic and the two-variable fragment. In CSL’01 (LNCS), Vol. 2142. Springer, 247--261.

Digital Library

[40]

J. Marcinkowski. 2006. On the expressive power of graph logic. In CSL’06 (LNCS), Vol. 4207. Springer, 486--500.

Digital Library

[41]

M. Marx and M. de Rijke. 2005. Semantic characterizations of navigational XPath. SIGMOD Record 34, 2 (2005), 41--46.

Digital Library

[42]

J. Navarro Pérez and A. Rybalchenko. 2013. Separation logic modulo theories. In APLAS’13 (LNCS), Vol. 8301. 90--106.

Digital Library

[43]

R. Piskac, Th. Wies, and D. Zufferey. 2013. Automating separation logic using SMT. In CAV’13 (LNCS), Vol. 8044. Springer, 773--789.

Digital Library

[44]

R. Piskac, Th. Wies, and D. Zufferey. 2014. GRASShopper: Complete heap verification with mixed specifications. In TACAS’14 (LNCS), Vol. 8413. Springer, 124--139.

[45]

A. Rabinovich. 2014. A Proof of Kamp’s theorem. LMCS 10, 1 (2014).

[46]

J. C. Reynolds. 2002. Separation logic: A logic for shared mutable data structures. In LICS’02. IEEE, 55--74.

Digital Library

[47]

M. Schwerhoff and A. Summers. 2015. Lightweight support for magic wands in an automatic verifier. In ECOOP’15. Leibniz-Zentrum für Informatik, LIPICS, 999--1023.

[48]

M. Sighireanu and D. Cok. 2014. Report on SL-COMP 2014. Journal of Satisfiability, Boolean Modeling and Computation (2014). To appear.

[49]

A. V. Sreejith. 2013. Regular Quantifiers in Logic. Ph.D. Dissertation. The Institute of Mathematical Sciences, Chennai.

[50]

H. Straubing. 1994. Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser.

Digital Library

[51]

A. Thakur, J. Breck, and Th. Reps. 2014. Satisfiability modulo abstraction for separation logic with linked lists. In SPIN’14. ACM, 58--67.

Digital Library

[52]

B. Trakhtenbrot. 1963. Impossibility of an algorithm for the decision problem in finite classes. AMS Translations, Series 2 23 (1963), 1--5.

[53]

V. Vafeiadis and M. Parkinson. 2007. A marriage of rely/guarantee and separation logic. In CONCUR’07 (LNCS), Vol. 4703. Springer, 256--271.

Digital Library

[54]

M. Y. Vardi. 1988. A temporal fixpoint calculus. In 15th Annual ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages, San Diego. ACM, 250--259.

Digital Library

[55]

Y. Venema. 1991. A modal logic for chopping intervals. Journal of Logic and Computation 1, 4 (1991), 453--476.

[56]

Ph. Weis. 2011. Expressiveness and Succinctness of First-order Logic on Finite Words. Ph.D. Dissertation. University of Massachussetts.

Digital Library

[57]

P. Wolper. 1983. Temporal logic can be more expressive. Information and Computation 56 (1983), 72--99.

Cited By

de Boer FHiep Hde Gouw S(2023)The Logic of Separation Logic: Models and ProofsAutomated Reasoning with Analytic Tableaux and Related Methods10.1007/978-3-031-43513-3_22(407-426)Online publication date: 18-Sep-2023
https://dl.acm.org/doi/10.1007/978-3-031-43513-3_22
Jin ZZhang BCao TCao YWang H(2022)Reasoning about block-based cloud storage systems via separation logicTheoretical Computer Science10.1016/j.tcs.2022.09.015936:C(43-76)Online publication date: 10-Nov-2022
https://dl.acm.org/doi/10.1016/j.tcs.2022.09.015
Haase DGrädel EWilke R(2022)Separation logic and logics with team semanticsAnnals of Pure and Applied Logic10.1016/j.apal.2021.103063173:10(103063)Online publication date: Dec-2022
https://doi.org/10.1016/j.apal.2021.103063
Show More Cited By

Index Terms

Expressive Completeness of Separation Logic with Two Variables and No Separating Conjunction
1. Theory of computation
  1. Logic
  2. Semantics and reasoning
    1. Program reasoning

Recommendations

Parametric completeness for separation theories
POPL '14

In this paper, we close the logical gap between provability in the logic BBI, which is the propositional basis for separation logic, and validity in an intended class of separation models, as employed in applications of separation logic such as program ...
Expressive completeness of separation logic with two variables and no separating conjunction
CSL-LICS '14: Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

We show that first-order separation logic with one record field restricted to two variables and the separating implication (no separating conjunction) is as expressive as weak second-order logic, substantially sharpening a previous result. Capturing ...
Two-Variable Separation Logic and Its Inner Circle

Separation logic is a well-known assertion language for Hoare-style proof systems. We show that first-order separation logic with a unique record field restricted to two quantified variables and no program variables is undecidable. This is among the ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Computational Logic

ACM Transactions on Computational Logic Volume 17, Issue 2

March 2016

266 pages

ISSN:1529-3785

EISSN:1557-945X

DOI:10.1145/2851089

Editor:
Orna Kupferman
School of Computer Science and Engineering, Hebrew University, Israel

Issue’s Table of Contents

Copyright © 2016 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 07 January 2016

Accepted: 01 October 2015

Revised: 01 September 2015

Received: 01 April 2015

Published in TOCL Volume 17, Issue 2

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Funding Sources

Air Force Office of Scientific Research
EU Seventh Framework Programme
National Science Foundation

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

8
Total Citations
View Citations
367
Total Downloads

Downloads (Last 12 months)111
Downloads (Last 6 weeks)10

Reflects downloads up to 27 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

de Boer FHiep Hde Gouw S(2023)The Logic of Separation Logic: Models and ProofsAutomated Reasoning with Analytic Tableaux and Related Methods10.1007/978-3-031-43513-3_22(407-426)Online publication date: 18-Sep-2023
https://dl.acm.org/doi/10.1007/978-3-031-43513-3_22
Jin ZZhang BCao TCao YWang H(2022)Reasoning about block-based cloud storage systems via separation logicTheoretical Computer Science10.1016/j.tcs.2022.09.015936:C(43-76)Online publication date: 10-Nov-2022
https://dl.acm.org/doi/10.1016/j.tcs.2022.09.015
Haase DGrädel EWilke R(2022)Separation logic and logics with team semanticsAnnals of Pure and Applied Logic10.1016/j.apal.2021.103063173:10(103063)Online publication date: Dec-2022
https://doi.org/10.1016/j.apal.2021.103063
Demri SLozes EMansutti A(2021)The Effects of Adding Reachability Predicates in Quantifier-Free Separation LogicACM Transactions on Computational Logic10.1145/344826922:2(1-56)Online publication date: 21-Jun-2021
https://dl.acm.org/doi/10.1145/3448269
Echenim MIosif RPeltier N(2020)The Bernays-Schönfinkel-Ramsey Class of Separation Logic with Uninterpreted PredicatesACM Transactions on Computational Logic10.1145/338080921:3(1-46)Online publication date: 2-Mar-2020
https://dl.acm.org/doi/10.1145/3380809
Demri SFervari R(2019)The power of modal separation logicsJournal of Logic and Computation10.1093/logcom/exz01929:8(1139-1184)Online publication date: 19-Dec-2019
https://doi.org/10.1093/logcom/exz019
Hóu ZClouston RGoré RTiu A(2018)Modular Labelled Sequent Calculi for Abstract Separation LogicsACM Transactions on Computational Logic10.1145/319738319:2(1-35)Online publication date: 28-Apr-2018
https://dl.acm.org/doi/10.1145/3197383
Demri SLozes ÉMansutti A(2018)The Effects of Adding Reachability Predicates in Propositional Separation LogicFoundations of Software Science and Computation Structures10.1007/978-3-319-89366-2_26(476-493)Online publication date: 14-Apr-2018
https://doi.org/10.1007/978-3-319-89366-2_26

View Options

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

Media

Figures

Other

Tables

View Issue’s Table of Contents