Article

Formalising the π-calculus using nominal logic

Authors:

Jesper Bengtson,

Joachim ParrowAuthors Info & Claims

FOSSACS'07: Proceedings of the 10th international conference on Foundations of software science and computational structures

Pages 63 - 77

Published: 24 March 2007 Publication History

Abstract

We formalise the pi-calculus using the nominal datatype package, a package based on ideas from the nominal logic by Pitts et al., and demonstrate an implementation in Isabelle/HOL. The purpose is to derive powerful induction rules for the semantics in order to conduct machine checkable proofs, closely following the intuitive arguments found in manual proofs. In this way we have covered many of the standard theorems of bisimulation equivalence and congruence, both late and early, and both strong and weak in a unison manner. We thus provide one of the most extensive formalisations of a process calculus ever done inside a theorem prover.

A significant gain in our formulation is that agents are identified up to alpha-equivalence, thereby greatly reducing the arguments about bound names. This is a normal strategy for manual proofs about the pi-calculus, but that kind of hand waving has previously been difficult to incorporate smoothly in an interactive theorem prover. We show how the nominal logic formalism and its support in Isabelle accomplishes this and thus significantly reduces the tedium of conducting completely formal proofs. This improves on previous work using weak higher order abstract syntax since we do not need extra assumptions to filter out exotic terms and can keep all arguments within a familiar first-order logic.

References

[1]

Brian E. Aydemir, Aaron Bohannon, Matthew Fairbairn, Nathan J. Foster, Benjamin C. Pierce, Peter Sewell, Dimitrios Vytiniotis, Geoffrey Washburn, Stephanie Weirich, and Steve Zdancewic. Mechanized metatheory for the masses: The POPLmark challenge. In International Conference on Theorem Proving in Higher Order Logics (TPHOLs), August 2005.

Digital Library

[2]

Jesper Bengtson. Generic implementations of process calculi in Isabelle. In The 16th Nordic Workshop on Programming Theory (NWPT'04), pages 74-78, 2004.

[3]

M. J. Gabbay. A theory of inductive definitions with α-equivalence, PhD thesis, University of Cambridge, 2000.

[4]

M. J. Gabbay. The π-calculus in FM. In Fairouz Kamareddine, editor, Thirty-five years of Automath. Kluwer, 2003.

[5]

M. J. Gabbay and A. M. Pitts. A new approach to abstract syntax with variable binding. Formal Aspects of Computing, 13:341-363, 2001.

Digital Library

[6]

Furio Honsell, Marino Miculan, and Ivan Scagnetto. π-calculus in (co)inductive type theory. Theoretical Computer Science, 253(2):239-285, 2001.

Digital Library

[7]

Thomas F. Melham. A mechanized theory of the pi-calculus in HOL. Nordic Journal of Computing, 1(1):50-76, 1994.

Digital Library

[8]

R. Milner. A Calculus of Communicating Systems. Number 92 in LNCS. Springer-Verlag, 1980.

Digital Library

[9]

Robin Milner, Joachim Parrow, and David Walker. A calculus of mobile processes, I/II. Inf. Comput., 100(1):1-77, 1992.

Digital Library

[10]

Otmane Aït Mohamed. Mechanizing a pi-calculus equivalence in HOL. In Proceedings of the 8th International Workshop on Higher Order Logic Theorem Proving and Its Applications, pages 1-16, London, UK, 1995. Springer-Verlag.

Digital Library

[11]

T. Nipkow, L. C. Paulson, and M. Wenzel. Isabelle/HOL: a proof assistant for higher-order logic. Springer-Verlag, 2002.

Digital Library

[12]

Joachim Parrow. An introduction to the pi-calculus. In Handbook of Process Algebra, pages 479-543. Elsevier, 2001.

[13]

A. M. Pitts. Nominal logic, a first order theory of names and binding. Information and Computation, 186:165-193, 2003.

Digital Library

[14]

A. M. Pitts. Alpha-structural recursion and induction. Journal of the ACM, 53:459-506, 2006.

Digital Library

[15]

Christine Röckl and Daniel Hirschkoff. A fully adequate shallow embedding of the π-calculus in Isabelle/HOL with mechanized syntax analysis. J. Funct. Program., 13(2):415-451, 2003.

Digital Library

[16]

C. Urban and S. Berghoffer. A recursion combinator for nominal datatypes implemented in Isabelle/HOL, Accepted to IJCAR 2006.

Digital Library

[17]

C. Urban, A. M. Pitts, and M. J. Gabbay. Nominal unification. Theoretical Computer Science, 323:473-497, 2004.

Digital Library

[18]

Christian Urban and Michael Norrish. A formal treatment of the barendregt variable convention in rule inductions. In MERLIN '05: Proceedings of the 3rd ACM SIGPLAN workshop on Mechanized reasoning about languages with variable binding , pages 25-32, New York, NY, USA, 2005. ACM Press.

Digital Library

[19]

Christian Urban and Christine Tasson. Nominal techniques in Isabelle/HOL. In CADE, pages 38-53, 2005.

Digital Library

[20]

Björn Victor and Faron Moller. The Mobility Workbench -- a tool for the π-calculus. In David Dill, editor, CAV'94: Computer Aided Verification, volume 818 of Lecture Notes in Computer Science, pages 428-440. Springer-Verlag, 1994.

Digital Library

[21]

Markus Wenzel. Isar - a generic interpretative approach to readable formal proof documents. In Theorem Proving in Higher Order Logics, pages 167-184, 1999.

Digital Library

Cited By

Gabbay MLitak TPetrişan D(2011)Stone duality for nominal Boolean algebras with ИProceedings of the 4th international conference on Algebra and coalgebra in computer science10.5555/2040096.2040112(192-207)Online publication date: 30-Aug-2011
https://dl.acm.org/doi/10.5555/2040096.2040112
Tiu AMiller D(2010)Proof search specifications of bisimulation and modal logics for the π-calculusACM Transactions on Computational Logic10.1145/1656242.165624811:2(1-35)Online publication date: 22-Jan-2010
https://dl.acm.org/doi/10.1145/1656242.1656248
Doczkal CSchwinghammer JCheney JFelty A(2009)Formalizing a strong normalization proof for Moggi's computational metalanguageProceedings of the Fourth International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice10.1145/1577824.1577834(57-63)Online publication date: 2-Aug-2009
https://dl.acm.org/doi/10.1145/1577824.1577834
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Formalising the π-calculus using nominal logic

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cover image Guide Proceedings

FOSSACS'07: Proceedings of the 10th international conference on Foundations of software science and computational structures

March 2007

378 pages

Editor:
Helmut Seidl
TU München,Institut für Informatik, München, Germany

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Fundacao para a Ciencia e Tecnologia
TAP Air Portugal
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CISCO

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 24 March 2007

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Cited By

Gabbay MLitak TPetrişan D(2011)Stone duality for nominal Boolean algebras with ИProceedings of the 4th international conference on Algebra and coalgebra in computer science10.5555/2040096.2040112(192-207)Online publication date: 30-Aug-2011
https://dl.acm.org/doi/10.5555/2040096.2040112
Tiu AMiller D(2010)Proof search specifications of bisimulation and modal logics for the π-calculusACM Transactions on Computational Logic10.1145/1656242.165624811:2(1-35)Online publication date: 22-Jan-2010
https://dl.acm.org/doi/10.1145/1656242.1656248
Doczkal CSchwinghammer JCheney JFelty A(2009)Formalizing a strong normalization proof for Moggi's computational metalanguageProceedings of the Fourth International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice10.1145/1577824.1577834(57-63)Online publication date: 2-Aug-2009
https://dl.acm.org/doi/10.1145/1577824.1577834
Johansson MParrow JVictor BBengtson J(2008)Extended pi-CalculiProceedings of the 35th international colloquium on Automata, Languages and Programming, Part II10.1007/978-3-540-70583-3_8(87-98)Online publication date: 7-Jul-2008
https://dl.acm.org/doi/10.1007/978-3-540-70583-3_8
Kahsai TMiculan M(2008)Implementing Spi Calculus Using Nominal TechniquesProceedings of the 4th conference on Computability in Europe: Logic and Theory of Algorithms10.1007/978-3-540-69407-6_33(294-305)Online publication date: 15-Jun-2008
https://dl.acm.org/doi/10.1007/978-3-540-69407-6_33
Urban CBerghofer SNorrish M(2007)Barendregt's Variable Convention in Rule InductionsProceedings of the 21st international conference on Automated Deduction: Automated Deduction10.1007/978-3-540-73595-3_4(35-50)Online publication date: 17-Jul-2007
https://dl.acm.org/doi/10.1007/978-3-540-73595-3_4

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