Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2025-01-03T13:25:29.674Z Has data issue: false hasContentIssue false

1 - Origins of bisimulation and coinduction

Published online by Cambridge University Press:  05 November 2011

By
Davide Sangiorgi
Edited by
Davide Sangiorgi  and
Jan Rutten
Davide Sangiorgi
Affiliation:
Università di Bologna
Davide Sangiorgi
Affiliation:
University of Bologna, Italy
Jan Rutten
Affiliation:
Stichting Centrum voor Wiskunde en Informatica (CWI), Amsterdam
Get access

Summary

Introduction

In this chapter,we look at the origins of bisimulation.We showthat bisimulation has been discovered not only in computer science, but also – and roughly at the same time – in other fields: philosophical logic (more precisely, modal logic), and set theory. In each field, we discuss the main steps that led to the discovery, and introduce the people who made these steps possible.

In computer science, philosophical logic, and set theory, bisimulation has been derived through refinements of notions of morphism between algebraic structures. Roughly, morphisms are maps (i.e. functions) that are ‘structurepreserving’. The notion is therefore fundamental in all mathematical theories in which the objects of study have some kind of structure, or algebra. The most basic forms of morphism are the homomorphisms. These essentially give us a way of embedding a structure (the source) into another one (the target), so that all the relations in the source are present in the target. The converse, however, need not be true; for this, stronger notions of morphism are needed. One such notion is isomorphism, which is, however, extremely strong – isomorphic structures must be essentially the same, i.e. ‘algebraically identical’. It is a quest for notions in between homomorphism and isomorphism that led to the discovery of bisimulation.

The kind of structures studied in computer science, philosophical logic, and set theory were forms of rooted directed graphs.

Type
Chapter
Information
Advanced Topics in Bisimulation and Coinduction , pp. 1 - 37
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[Acz88] P., Aczel. Non-Well-Founded Sets. CSLI Lecture Notes, no. 14, 1988.
[AIS12] L., Aceto, A., Ingolfsdottir, and J., Srba. The algorithmics of bisimilarity. Chapter 3 of this volume.
[Ard60] D.N., Arden. Delayed logic and finite state machines. In Theory of Computing Machine Design, pages 1–35. University of Michigan Press, 1960.
[Bak71] J.W., Bakker. Recursive Procedures. Mathematical Centre Tracts 24, Mathematisch Centrum, Amsterdam, 1971.
[Bak75] J.W., Bakker. The fixed-point approach in semantics: theory and applications. In J.W., Bakker, editor, Foundations of Computer Science, pages 3–53. Mathematical Centre Tracts 63, Mathematisch Centrum, Amsterdam, 1975.
[BE87] J., Barwise and J., Etchemendy. The Liar: an Essay in Truth and Circularity. Oxford University Press, 1987.
[Bek69] H., Bekič. Definable operations in general algebras and the theory of automata and flowcharts. Unpublished Manuscript, IBM Lab. Vienna 1969. Also appeared in [Jon84].
[Ben76] J., Benthem. Modal correspondence theory. PhD thesis, Mathematish Instituut & Instituut voor Grondslagenonderzoek, University of Amsterdam, 1976.
[Ben83] J., Benthem. Modal Logic and Classical Logic. Bibliopolis, 1983.
[Ben84] J., Benthem. Correspondence theory. In D.M., Gabbay and F., Guenthner, editors, Handbook of Philosophical Logic, volume 2, pages 167–247. Reidel, 1984.
[Ber54] P., Bernays. A system of axiomatic set theory–Part VII. Journal of Symbolic Logic, 19(2):81–96, 1954.Google Scholar
[BGM71] J., Barwise, R.O., Gandy, and Y.N., Moschovakis. The next admissible set. Journal of Symbolic Logic, 36:108–120, 1971.Google Scholar
[Bir48] G., Birkhoff. Lattice Theory (Revised Edition). Volume 25 of American Mathematical Society Colloquium Publications. American Mathematical Society, 1948.
[Bli77] A., Blikle. A comparative review of some program verification methods. In Jozef, Gruska, editor, 6th Symposium on Mathematical Foundations of Computer Science (MFCS'77), volume 53 of Lecture Notes in Computer Science, pages 17–33. Springer, 1977.
[BM96] J., Barwise and L., Moss. Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena. CSLI (Center for the Study of Language and Information), 1996.
[Bof68] M., Boffa. Les ensembles extraordinaires. Bulletin de la Société Mathématique de Belgique, XX:3–15, 1968.Google Scholar
[Bof69] M., Boffa. Sur la théorie des ensembles sans axiome de fondement. Bulletin de la Société Mathématique de Belgique, XXXI:16–56, 1969.Google Scholar
[Bof72] M., Boffa. Forcing et negation de l'axiome de fondement. Académie Royale de Belgique, Mémoires de la Classe des Sciences, 2e série, XL(7):1–53, 1972.Google Scholar
[Bou50] N., Bourbaki. Sur le théorème de Zorn. Archiv der Mathematik, 2:434–437, 1950.Google Scholar
[BR73] J.W., Bakker and W.P., Roever. A calculus for recursive program schemes. In M., Nivat, editor, Proceedings of the IRIA Symposium on Automata, Languages and Programming, Paris, France, July, 1972, pages 167–196. North-Holland, 1973.
[Bra78] D., Brand. Algebraic simulation between parallel programs. Research Report RC 7206, Yorktown Heights, NY, 39 pp., June 1978.
[BRV01] P., Blackburn, M., Rijke, and Y., Venema. Modal Logic. Cambridge University Press, 2001.
[Buc94] P., Buchholz. Markovian process algebra: composition and equivalence. In U., Herzog and M., Rettelbach, editors, Proceedings of the 2nd Workshop on Process Algebras and Performance Modelling, pages 11–30. Arbeitsberichte des IMMD, Band 27, Nr. 4, 1994.
[Bur75] W.H., Burge. Stream processing functions. IBM Journal of Research and Development, 19(1):12–25, 1975.Google Scholar
[Cad72] J.M., Cadiou. Recursive definitions of partial functions and their computations. PhD thesis, Computer Science Department, Stanford University, 1972.
[CC79] P., Cousot and R., Cousot. Constructive versions of Tarski's fixed point theorems. Pacific Journal of Mathematics, 81(1):43–57, 1979.Google Scholar
[Cla77] E.M., Clarke. Program invariants as fixed points (preliminary reports). In FOCS, pages 18–29. IEEE, 1977. Final version in Computing, 21(4):273–294, 1979. Based on Clarke's PhD thesis, Completeness and Incompleteness Theorems for Hoare-like Axiom Systems, Cornell University, 1976.
[Dev63] V., Devidé. On monotonous mappings of complete lattices. Fundamenta Mathematicae, LIII:147–154, 1963.Google Scholar
[dR77] W.P., Roever. On backtracking and greatest fixpoints. In Arto, Salomaa and Magnus, Steinby, editors, Fourth Colloquium on Automata, Languages and Programming (ICALP), volume 52 of Lecture Notes in Computer Science, pages 412–429. Springer, 1977.
[Ehr61] A., Ehrenfeucht. An application of games to the completeness problem for formalized theories. Fundamenta Mathematicae, 49:129–141, 1961.Google Scholar
[FH83] M., Forti and F., Honsell. Set theory with free construction principles. Annali Scuola Normale Superiore, Pisa, Serie IV, X(3):493–522, 1983.Google Scholar
[Fin26] P., Finsler. Über die Grundlagen der Mengenlehre. I. Math. Zeitschrift, 25:683–713, 1926.Google Scholar
[Flo67] R.W., Floyd. Assigning meaning to programs. In Proceeding of Symposia in Applied Mathematics, volume 19, pages 19–32. American Mathematical Society, 1967.
[Fra22] A., Fraenkel. Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre. Mathematische Annalen, 86:230–237, 1922.Google Scholar
[Fra53] R., Fraïssé. Sur quelques classifications des systèmes de relations. PhD thesis, University of Paris, 1953. Also in Publications Scientifiques de l'Université d'Alger, series A, 1, 35–182, 1954.
[Fri73] H., Friedman. The consistency of classical set theory relative to a set theory with intuitionistic logic. Journal of Symbolic Logic, 38:315–319, 1973.Google Scholar
[Gin68] A., Ginzburg. Algebraic Theory of Automata. Academic Press, 1968.
[Gla90] R.J., Glabbeek. The linear time-branching time spectrum (extended abstract). In J.C.M., Baeten and J.W., Klop, editors, First Conference on Concurrency Theory (CONCUR'90), volume 458 of Lecture Notes in Computer Science, pages 278–297. Springer, 1990.
[Gla93] R.J., Glabbeek. The linear time–branching time spectrum II (the semantics of sequential systems with silent moves). In E., Best, editor, Fourth Conference on Concurrency Theory (CONCUR'93), volume 715, pages 66–81. Springer, 1993.
[GM76] F., Giarratana, V., Gimona and U., Montanari. Observability concepts in abstract data type specification. In A., Mazurkievicz, editor, 5th Symposium on Mathematical Foundations of Computer Science, volume 45 of Lecture Notes in Computer Science, pages 576–587. Springer, 1976.
[Gol89] R., Goldblatt. Varieties of complex algebras. Annals of Pure and Applied Logic, 44:173–242, 1989.Google Scholar
[Gor82] L., Gordeev. Constructive models for set theory with extensionality. In A.S., Troelstra and D., Dalen, editors, The L.E.J. Brouwer Centenary Symposium, pages 123–147, 1982.
[GR62] S., Ginsburg and H., Gordon Rice. Two families of languages related to algol. Journal of the ACM, 9(3):350–371, 1962.Google Scholar
[GRS79] J.S., Gourlay, W.C., Rounds, and R., Statman. On properties preserved by contraction of concurrent systems. In G., Kahn, editor, International Symposium on Semantics of Concurrent Computation, volume 70 of Lecture Notes in Computer Science, pages 51–65. Springer, 1979.
[HE67] J., Heijenoort, editor. From Frege to Gödel: A Source Book in Mathematical Logic 1879–1931. Harvard University Press, 1967.
[Hin80] R., Hinnion. Contraction de structures et application à NFU. Comptes Rendus Académie des Sciences de Paris, Sér. A 290:677–680, 1980.Google Scholar
[Hin81] R., Hinnion. Extensional quotients of structures and applications to the study of the axiom of extensionality. Bulletin de la Société Mathématique de Belgique, XXXIII (Fas. II, Sér. B):173–206, 1981.Google Scholar
[Hin86] R., Hinnion. Extensionality in Zermelo–Fraenkel set theoryZeitschrift für Mathematische Logik und Grundlagen Mathematik, 32:51–60, 1986.
[HM80] M., Hennessy and R., Milner. On observing nondeterminism and concurrency. In J.W., Bakker and J., Leeuwen, editors, Proc. 7th Colloquium Automata, Languages and Programming, volume 85 of Lecture Notes in Computer Science, pages 299–309. Springer, 1980.
[HM85] M., Hennessy and R., Milner. Algebraic laws for nondeterminism and concurrency. Journal of the ACM, 32:137–161, 1985.Google Scholar
[Hon81] F., Honsell. Modelli della teoria degli insiemi, principi di regolarità e di libera costruzione. Tesi di Laurea, Università di Pisa, 1981.
[HP73] P., Hitchcock and D., Park. Induction rules and termination proofs. In M., Nivat, editor, Proceedings of the IRIA Symposium on Automata, Languages and Programming, Paris, France, July, 1972, pages 225–251. North-Holland, 1973.
[Huf54] D.A., Huffman. The synthesis of sequential switching circuits. Journal of the Franklin Institute (Mar. 1954) and (Apr. 1954), 257(3–4):161–190 and 275–303, 1954.Google Scholar
[Imm82] N., Immerman. Upper and lower bounds for first order expressibility. Journal of Computer and System Science, 25(1):76–98, 1982.Google Scholar
[Jen80] K., Jensen. A method to compare the descriptive power of different types of petri nets. In P., Dembinski, editor, Proc. 9th Mathematical Foundations of Computer Science 1980 (MFCS'80), Rydzyna, Poland, September 1980, volume 88 of Lecture Notes in Computer Science, pages 348–361. Springer, 1980.
[Jon84] C.B., Jones, editor. Programming Languages and Their Definition – Hans Bekic (1936–1982), volume 177 of Lecture Notes in Computer Science. Springer, 1984.
[JT66] D.H.J., Jongh and A.S., Troelstra. On the connection of partially ordered sets with some pseudo-Boolean algebras. Indagationes Mathematicae, 28:317–329, 1966.Google Scholar
[Kah74] G., Kahn. The semantics of simple language for parallel programming. In IFIP Congress, pages 471–475. North-Holland, 1974.
[Kan39] L.V., Kantorovich. The method of successive approximations for functional equations. Acta Mathematica, 71:63–97, 1939.Google Scholar
[Kle52] S.C., Kleene. Introduction to Metamathematics. Van Nostrand, 1952.
[Kle70] S.C., Kleene. The origin of recursive function theory. In 20th Annual Symposium on Foundations of Computer Science, pages 371–382. IEEE, 1970.
[Kna28] B., Knaster. Un théorème sur les fonctions d'ensembles. Annales de la Société Polonaise de Mathématique, 6:133–134, 1928.Google Scholar
[KS60] J., Kemeny and J.L., Snell. Finite Markov Chains. Van Nostrand, 1960.
[Kwo77] Y.S., Kwong. On reduction of asynchronous systems. Theoretical Computer Science, 5(1):25–50, 1977.Google Scholar
[Lan64] P.J., Landin. The mechanical evaluation of expressions. The Computer Journal, 6(4):308–320, 1964.Google Scholar
[Lan65a] P.J., Landin. Correspondence between ALGOL 60 and Church's Lambdanotation: Part I. Communications of the ACM, 8(2):89–101, 1965.Google Scholar
[Lan65b] P.J., Landin. A correspondence between ALGOL 60 and Church's Lambdanotations: Part II. Communications of the ACM, 8(3):158–167, 1965.Google Scholar
[Lan69] P., Landin. A program-machine symmetric automata theory. Machine Intelligence, 5:99–120, 1969.Google Scholar
[LNS82] J.-L., Lassez, V.L., Nguyen, and L., Sonenberg. Fixed point theorems and semantics: A folk tale. Information Processing Letters, 14(3):112–116, 1982.Google Scholar
[LS91] K.G., Larsen and A., Skou. Bisimulation through probabilistic testing. Information and Computation, 94(1):1–28, 1991. Preliminary version in POPL'89, 344–352, 1989.Google Scholar
[Man69] Z., Manna. The correctness of programs. Journal of Computer and System Sciences, 3(2):119–127, 1969.Google Scholar
[Man74] Z., Manna. Mathematical Theory of Computation. McGraw-Hill, 1974.
[Maz71] A.W., Mazurkiewicz. Proving algorithms by tail functions. Information and Control, 18(3):220–226, 1971.Google Scholar
[Maz73] A., Mazurkiewicz. Proving properties of processes. Technical report 134, Computation Center of Polish Academy of Sciences,Warsaw, 1973. Also in Algorytmy, 11, 5–22, 1974.Google Scholar
[McC61] J., McCarthy. A basis for a mathematical theory of computation. In Proceedings of the Western Joint Computer Conference, volume 19, pages 225–238. Spartan Books, 1961. Reprinted, with corrections and an added tenth section, in Computer Programming and Formal Systems, P. Braffort and D. Hirschberg, editors, pages 33–70, North-Holland, 1963.
[McC63] J., McCarthy. Towards a mathematical science of computation. In Proceedings of IFIP Congress 62, pages 21–28. North-Holland, 1963.
[Mil70] R., Milner. A formal notion of simulation between programs. Memo 14, Computers and Logic Research Group, University College of Swansea, UK, 1970.
[Mil71a] R., Milner. Program simulation: an extended formal notion. Memo 17, Computers and Logic Research Group, University College of Swansea, UK, 1971.
[Mil71b] R., Milner. An algebraic definition of simulation between programs. In Proceedings of the 2nd International Joint Conferences on Artificial Intelligence. British Computer Society, London, 1971.
[Mil80] R., Milner. A Calculus of Communicating Systems, volume 92 of Lecture Notes in Computer Science. Springer, 1980.
[Mil89] R., Milner. Communication and Concurrency. Prentice Hall, 1989.
[Mir17a] D., Mirimanoff. Les antinomies de Russell et de Burali-Forti et le problème fondamental de la théorie des ensembles. L'Enseignement Mathématique, 19:37–52, 1917.Google Scholar
[Mir17b] D., Mirimanoff. Remarques sur la théorie des ensembles et les antinomies cantoriennes I. L'Enseignement Mathématique, 19:209–217, 1917.Google Scholar
[Mir20] D., Mirimanoff. Remarques sur la théorie des ensembles et les antinomies cantoriennes II. L'Enseignement Mathématique, 21:29–52, 1920.Google Scholar
[Moo56] E.F., Moore. Gedanken experiments on sequential machines. Automata Studies, Annals of Mathematics Series, 34:129–153, 1956.Google Scholar
[Mor68] J.H., Morris. Lambda-calculus models of programming languages. PhD thesis, MIT, project MAC, Dec. 1968.
[Mos74] Y.N., Moschovakis. Elementary Induction on Abstract Structures, volume 77 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1974.
[Ner58] A., Nerode. Linear automaton transformations. In Proceedings of the American Mathematical Society, 9, 541–544, 1958.Google Scholar
[Par69] D., Park. Fixpoint induction and proofs of program properties. In B., Meltzer and D., Michie, editors, Machine Intelligence 5, pages 59–78. Edinburgh University Press, 1969.
[Par70] D., Park. The Y-combinator in Scott's lambda-calculus models. Symposium on Theory of Programming, University of Warwick, unpublished (A revised version: Research Report CS-RR-013, Department of Computer Science, University of Warwick, June 1976), 1970.
[Par79] D., Park. On the semantics of fair parallelism. In Proceedings of Abstract Software Specifications, Copenhagen Winter School, Lecture Notes in Computer Science, pages 504–526. Springer, 1979.
[Par81a] D., Park. Concurrency on automata and infinite sequences. In P., Deussen, editor, Conference on Theoretical Computer Science, volume 104 of Lecture Notes in Computer Science, pages 167–183. Springer, 1981.
[Par81b] D., Park. A new equivalence notion for communicating systems. In G., Maurer, editor, Bulletin EATCS, volume 14, pages 78–80, 1981. Abstract of the talk presented at the Second Workshop on the Semantics of Programming Languages, Bad Honnef, March 16–20 1981. Abstracts collected in the Bulletin by B. Mayoh.Google Scholar
[Pas74] A., Pasini. Some fixed point theorems of the mappings of partially ordered sets. Rendiconti del Seminario Matematico della Università di Padova, 51:167–177, 1974.Google Scholar
[Rey93] J.C., Reynolds. The discoveries of continuations. Lisp and Symbolic Computation, 6(3–4):233–248, 1993.Google Scholar
[Rog67] H., Rogers. Theory of Recursive Functions and Effective Computability. McGraw Hill, 1967. Reprinted, MIT Press, 1987.
[Rus03] B., Russell. Principles of Mathematics. Cambridge University Press, 1903.
[Rus08] B., Russell. Mathematical logic as based on the theory of types. American Journal of Mathematics, 30:222–262, 1908. Also in [HE67], pages 153–168.Google Scholar
[RW13] B., Russell and A.N., Whitehead. Principia Mathematica, 3 vols. Cambridge University Press, 1910, 1912, 1913.
[San09] D., Sangiorgi. On the origins of bisimulation and coinduction. ACM Transactions on Programming Languages and Systems, 31(4), 2009.Google Scholar
[San12] D., Sangiorgi. An Introduction to Bisimulation and Coinduction. Cambridge University Press, 2012.
[Sco60] D., Scott. A different kind of model for set theory. Unpublished paper, given at the 1960 Stanford Congress of Logic, Methodology and Philosophy of Science, 1960.Google Scholar
[Sco69a] D., Scott. Models for the λ-calculus. Manuscript, draft, Oxford, December 1969.
[Sco69b] D., Scott. A construction of a model for the λ-calculus. Manuscript, Oxford, November 1969.
[Sco69c] D., Scott. A type-theoretical alternative to CUCH, ISWIM, OWHY. Typed script, Oxford. Also appeared as [Sco93], October 1969.
[Sco72a] D., Scott. Continuous lattices. In E., Lawvere, editor, Toposes, Algebraic Geometry and Logic, volume 274 of Lecture Notes in Mathematics, pages 97–136. Springer, 1972.
[Sco72b] D., Scott. The lattice of flow diagrams. In E., Engeler, editor, Symposium of Semantics of Algorithmic Languages, volume 188 of Lecture Notes in Mathematics, pages 311–366. Springer, 1972.
[Sco76] D., Scott. Data types as lattices. SIAM Journal on Computing, 5:522–587, 1976. A manuscript with the same title was written in 1972.Google Scholar
[Sco93] D.S., Scott. A type-theoretical alternative to ISWIM, CUCH, OWHY. Theoretical Computer Science, 121(1&2):411–440, 1993.Google Scholar
[SdB69] D., Scott and J.W., Bakker. A theory of programs. Handwritten notes. IBM Lab., Vienna, Austria, 1969.
[Seg68] K., Segerberg. Decidability of S4.1. Theoria, 34:7–20, 1968.Google Scholar
[Seg70] K., Segerberg. Modal logics with linear alternative relations. Theoria, 36:301–322, 1970.Google Scholar
[Seg71] K., Segerberg. An essay in classical modal logic. Filosofiska Studier, Uppsala, 1971.
[Sko23] T., Skolem. Einige Bemerkungen zur Axiomatischen Begründung der Mengenlehre. In Proceedings of the 5th Scandinavian Mathematics Congress, Helsinki, 1922, pages 217–232. Akademiska Bokhandeln, Helsinki, 1923. English translation, ‘Some remarks on axiomatized set theory’, in [HE67], pages 290–301.
[Spe57] E., Specker. Zur Axiomatik der Mengenlehre. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 3(3):173–210, 1957.Google Scholar
[Sti12] C., Stirling. Bisimulation and Logic. Chapter 4 of this volume.
[Tar49] A., Tarski. A fixpoint theorem for lattices and its applications (preliminary report). Bulletin of the American Mathematical Society, 55:1051–1052 and 1192, 1949.Google Scholar
[Tar55] A., Tarski. A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics, 5:285–309, 1955.Google Scholar
[Tho93] W., Thomas. On the Ehrenfeucht–Fraïssé game in theoretical computer science. In M.-C., Gaudel and J.-P., Jouannaud, editors, TAPSOFT, volume 668 of Lecture Notes in Computer Science, pages 559–568. Springer, 1993.
[Zer08] E., Zermelo. Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen, 65:261–281, 1908. English translation, ‘Investigations in the foundations of set theory’, in [HE67], pages 199–215.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×