skip to main content
research-article
Free Access

On the origins of bisimulation and coinduction

Published:26 May 2009Publication History
Skip Abstract Section

Abstract

The origins of bisimulation and bisimilarity are examined, in the three fields where they have been independently discovered: Computer Science, Philosophical Logic (precisely, Modal Logic), Set Theory.

Bisimulation and bisimilarity are coinductive notions, and as such are intimately related to fixed points, in particular greatest fixed points. Therefore also the appearance of coinduction and fixed points is discussed, though in this case only within Computer Science. The paper ends with some historical remarks on the main fixed-point theorems (such as Knaster-Tarski) that underpin the fixed-point theory presented.

References

  1. Aceto, L., Ingulfsduttir, A., Larsen, K. G., and Srba, J. 2007. Reactive Systems: Modelling, Specification and Verification. Cambridge University Press. Google ScholarGoogle ScholarCross RefCross Ref
  2. Aczel, P. 1988. Non-Well-Founded Sets. CSLI lecture notes, no. 14.Google ScholarGoogle Scholar
  3. Aczel, P. 1993. Final universes of processes. In Proceedings of the Mathematical Foundations of Programming Semantcs (MFPS'93), B. et al., Ed. Lecture Notes in Computer Science, vol. 802. Springer, 1--28. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. Alvarez, C., Balcázar, J. L., Gabarró, J., and Santha, M. 1991. Parallel complexity in the design and analysis on conurrent systems. In Proceedings of the Parallel Architectures and Languages Europe, Volume I: Parallel Architectures and Algorithms (PARLE'91). Lecture Notes in Computer Science, vol. 505. Springer, 288--303. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Amadio, R. M. and Cardelli, L. 1993. Subtyping recursive types. ACM Trans. Program. Lang. Syst. 15, 4, 575--631. A preliminary version appeared in POPL '91 (pp. 104--118), and as DEC Systems Research Center Res. rep. number 62, August 1990. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Arden, D. N. 1960. Delayed logic and finite state machines. In Theory of Computing Machine Design. University of Michigan Press, 1--35.Google ScholarGoogle Scholar
  7. Bakker, J. W. D. 1971. Recursive Procedures. Mathematical Centre Tracts 24, Mathematisch Centrum, Amsterdam.Google ScholarGoogle Scholar
  8. Bakker, J. W. D. 1975. The fixed-point approach in semantics: Theory and applications. In Foundations of Computer Science, J. de Bakker, Ed. Mathematical Centre Tracts 63, Mathematisch Centrum, Amsterdam, 3--53.Google ScholarGoogle Scholar
  9. Bakker, J. W. D. and Roever, W. P. D. 1973. A calculus for recursive program schemes. In Proceedings of the IRIA Symposium on on Automata, Languages and Programming 1972, M. Nivat, Ed. North-Holland, 167--196.Google ScholarGoogle Scholar
  10. Balcázar, J. L., Gabarró, J., and Santha, M. 1992. Deciding bisimilarity is P-complete. Formal Asp. Comput. 4, 6A, 638--648.Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. Barwise, J. and Etchemendy, J. 1987. The Liar: An Essay in Truth and Circularity. Oxford University Press. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. Barwise, J., Gandy, R. O., and Moschovakis, Y. N. 1971. The next admissible set. J. Symb. Log. 36, 108--120.Google ScholarGoogle ScholarCross RefCross Ref
  13. Barwise, J. and Moss, L. 1996. Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena. CSLI (Center for the Study of Language and Information). Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Bekič, H. 1969. Definable operations in general algebras and the theory of automata and flowcharts. Unpublished manuscript, IBM Lab. Vienna 1969. Also appeared in Jones {1984}.Google ScholarGoogle Scholar
  15. Benthem, J. V. 1976. Modal correspondence theory. Ph.D. thesis, Mathematish Instituut and Instituut voor Grondslagenonderzoek, University of Amsterdam.Google ScholarGoogle Scholar
  16. Benthem, J. V. 1983. Modal Logic and Classical Logic. Bibliopolis.Google ScholarGoogle Scholar
  17. Benthem, J. V. 1984. Correspondence theory. In Handbook of Philosophical Logic, D. Gabbay and F. Guenthner, Eds. Vol. 2. Reidel, 167--247.Google ScholarGoogle Scholar
  18. Bernays, P. 1954. A system of axiomatic set theory--Part VII. J. Symb. Log. 19, 2, 81--96.Google ScholarGoogle ScholarCross RefCross Ref
  19. Birkhoff, G. 1948. Lattice Theory (revised edition). Vol. 25 of American Mathematical Society Colloquium Publications. American Mathematical Society.Google ScholarGoogle Scholar
  20. Blackburn, P., Rijke, M. D., and Venema, Y. 2001. Modal Logic. Cambridge University Press. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Blikle, A. 1977. A comparative review of some program verification methods. In 6th Symposium on Mathematical Foundations of Computer Science (MFCS'77), J. Gruska, Ed. Lecture Notes in Computer Science, vol. 53. Springer, 17--33.Google ScholarGoogle ScholarCross RefCross Ref
  22. Boffa, M. 1968. Les ensembles extraordinaires. Bull. Société Math. Belgique XX, 3--15.Google ScholarGoogle Scholar
  23. Boffa, M. 1969. Sur la théorie des ensembles sans axiome de fondement. Bull. Société Math. Belgique XXXI, 16--56.Google ScholarGoogle Scholar
  24. Boffa, M. 1972. Forcing et negation de l'axiome de fondement. Académie Royale de Belgique, Mémoires de la classe des sciences, 2e série XL, 7, 1--53.Google ScholarGoogle Scholar
  25. Bourbaki, N. 1950. Sur le théorème de Zorn. Arch. Math. 2, 434--437.Google ScholarGoogle ScholarCross RefCross Ref
  26. Brand, D. June 1978. Algebraic simulation between parallel programs. Res. rep. RC 7206, Yorktown Heights, New Yok, 39 pp.Google ScholarGoogle Scholar
  27. Brandt, M. and Henglein, F. 1997. Coinductive axiomatization of recursive type equality and subtyping. In Proceedings of the 3rd Conference on Typed Lambda Calculi and Applications (TLCA'97), R. Hindley, Ed. Lecture Notes in Computer Science, vol. 1210. Springer, 63--81. Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. Burge, W. H. 1975. Stream processing functions. IBM J. Res. Development 19, 1, 12--25.Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. Cadiou, J. M. 1972. Recursive definitions of partial functions and their computations. Ph.D. thesis, Computer Science Department, Stanford University. Google ScholarGoogle ScholarDigital LibraryDigital Library
  30. Clarke, E. M. 1977. Program invariants as fixed points (preliminary reports). In FOCS. IEEE, 18--29. 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. Google ScholarGoogle ScholarDigital LibraryDigital Library
  31. Coquand, T. 1993. Infinite objects in type theory. In TYPES, H. Barendregt and T. Nipkow, Eds. Lecture Notes in Computer Science, vol. 806. Springer, 62--78. Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. Cousot, P. and Cousot, R. 1979. Constructive versions of Tarski's fixed point theorems. Pacific J. Math. 81, 1, 43--57.Google ScholarGoogle ScholarCross RefCross Ref
  33. de Roever, W. P. 1977. On backtracking and greatest fixpoints. In 4th Colloquium on Automata, Languages and Programming (ICALP), A. Salomaa and M. Steinby, Eds. Lecture Notes in Computer Science, vol. 52. Springer, 412--429. Google ScholarGoogle ScholarDigital LibraryDigital Library
  34. Devidé, V. 1963. On monotonous mappings of complete lattices. Fundam. Math. LIII, 147--154.Google ScholarGoogle Scholar
  35. Ehrenfeucht, A. 1961. An application of games to the completeness problem for formalized theories. Fundam. Math. 49, 129--141.Google ScholarGoogle ScholarCross RefCross Ref
  36. Finsler, P. 1926. Über die Grundlagen der Mengenlehre. I. Math. Zeitschrift 25, 683--713.Google ScholarGoogle ScholarCross RefCross Ref
  37. Floyd, R. W. 1967. Assigning meaning to programs. In Proceedings of the Symposia in Applied Mathematics. Vol. 19. American Mathematical Society, 19--32.Google ScholarGoogle ScholarCross RefCross Ref
  38. Forti, M. and Honsell, F. 1983. Set theory with free construction principles. Annali Scuola Normale Superiore, Pisa, Serie IV X, 3, 493--522.Google ScholarGoogle Scholar
  39. Fraenkel, A. 1922. Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre. Math. Annalen 86, 230--237.Google ScholarGoogle ScholarCross RefCross Ref
  40. Fraïssé, R. 1953. Sur quelques classifications des syst&mgrave;es de relations. Ph.D. thesis, University of Paris. Also in Publications Scientifiques de l'Universite d'Alger, series A 1, 35--182, 1954.Google ScholarGoogle Scholar
  41. Friedman, H. 1973. The consistency of classical set theory relative to a set theory with intuitionistic logic. J. Symb. Log. 38, 315--319.Google ScholarGoogle ScholarCross RefCross Ref
  42. Giarratana, V. Gimona, F. and Montanari, U. 1976. Observability concepts in abstract data type specification. In 5th Symposium on Mathematical Foundations of Computer Science, A. Mazurkievicz, Ed. Lecture Notes in Computer Science, vol. 45. Springer, 576--587.Google ScholarGoogle Scholar
  43. Giménez, E. 1996. Un calcul de constructions infinies et son application a la verification des systemes communicants. Ph.D. thesis, Laboratoire de l'Informatique du Parallélisme, Ecole Normale Supérieure de Lyon.Google ScholarGoogle Scholar
  44. Ginsburg, S. and Rice, H. G. 1962. Two families of languages related to algol. J. ACM 9, 3, 350--371. Google ScholarGoogle ScholarDigital LibraryDigital Library
  45. Ginzburg, A. 1968. Algebraic Theory of Automata. Academic Press.Google ScholarGoogle Scholar
  46. Glabbeek, R. V. 1990. The linear time-branching time spectrum (extended abstract). In First Conference on Concurrency Theory (CONCUR'90), J. C. M. Baeten and J. W. Klop, Eds. Lecture Notes in Computer Science, vol. 458. Springer, 278--297. Google ScholarGoogle ScholarDigital LibraryDigital Library
  47. Glabbeek, R. V. 1993. The linear time—Branching time spectrum II (the semantics of sequential systems with silent moves). In 4th Conference on Concurrency Theory (CONCUR'93), E. Best, Ed. Lecture Notes in Computer Science, vol. 715, 66--81. Google ScholarGoogle ScholarDigital LibraryDigital Library
  48. Goldblatt, R. 1989. Varieties of complex algebras. Ann. Pure Applied Logic 44, 173--242.Google ScholarGoogle ScholarCross RefCross Ref
  49. Gordeev, L. 1982. Constructive models for set theory with extensionality. In The L.E.J. Brouwer Centenary Symposium, A. Troelstra and D. van Dalen, Eds. 123--147.Google ScholarGoogle Scholar
  50. Gourlay, J. S., Rounds, W. C., and Statman, R. 1979. On properties preserved by contraction of concurrent systems. In International Symposium on Semantics of Concurrent Computation, G. Kahn, Ed. Lecture Notes in Computer Science, vol. 70. Springer, 51--65. Google ScholarGoogle ScholarDigital LibraryDigital Library
  51. Heijenoort (Ed.), J. V. 1967. From Frege to Gödel: A Source Book in Mathematical Logic 1879-1931. Harvard University Press.Google ScholarGoogle Scholar
  52. Hennessy, M. and Milner, R. 1980. On observing nondeterminism and concurrency. In Proceedings of the 7th Colloquium Automata, Languages and Programming, J. W. de Bakker and J. van Leeuwen, Eds. Lecture Notes in Computer Science, vol. 85. Springer, 299--309. Google ScholarGoogle ScholarDigital LibraryDigital Library
  53. Hennessy, M. and Milner, R. 1985. Algebraic laws for nondeterminism and concurrency. J. ACM 32, 137--161. Google ScholarGoogle ScholarDigital LibraryDigital Library
  54. Hinnion, R. 1980. Contraction de structures et application à NFU. Comptes Rendus Acad. des Sciences de Paris 290, Sér. A, 677--680.Google ScholarGoogle Scholar
  55. Hinnion, R. 1981. Extensional quotients of structures and applications to the study of the axiom of extensionality. Bull. Société Math. Belgique XXXIII (Fas. II, Sér. B), 173--206.Google ScholarGoogle Scholar
  56. Hinnion, R. 1986. Extensionality in Zermelo-Fraenkel set theory. Zeitschr. Math. Logik und Grundlagen Math. 32, 51--60.Google ScholarGoogle ScholarCross RefCross Ref
  57. Hitchcock, P. and Park, D. 1973. Induction rules and termination proofs. In Proceedings of the IRIA symposium on on Automata, Languages and Programming 1972, M. Nivat, Ed. North-Holland, 225--251.Google ScholarGoogle Scholar
  58. Hoare, T. 1972. Proof of correctness of data representations. Acta Inf. 1, 271--281.Google ScholarGoogle ScholarDigital LibraryDigital Library
  59. Honsell, F. 1981. Modelli della teoria degli insiemi, principi di regolarità e di libera costruzione. Tesi di Laurea, Universita' di Pisa.Google ScholarGoogle Scholar
  60. Huffman, D. 1954. The synthesis of sequential switching circuits. J. Franklin Institute 257, 3--4, 161--190 and 275--303.Google ScholarGoogle ScholarCross RefCross Ref
  61. Immerman, N. 1982. Upper and lower bounds for first order expressibility. J. Comput. Syst. Sci. 25, 1, 76--98.Google ScholarGoogle ScholarCross RefCross Ref
  62. Jacobs, B. and Rutten, J. 1996. A tutorial on (co)algebras and (co)induction. Bull. EATCS 62, 222--259.Google ScholarGoogle Scholar
  63. Jensen, K. 1980. A method to compare the descriptive power of different types of petri nets. In Proceedings of the 9th Mathematical Foundations of Computer Science (MFCS'80), P. Dembinski, Ed. Lecture Notes in Computer Science, vol. 88. Springer, 348--361. Google ScholarGoogle ScholarDigital LibraryDigital Library
  64. Jones, C. B., Ed. 1984. Programming Languages and Their Definition -- Hans Bekic (1936-1982). Lecture Notes in Computer Science, vol. 177. Springer.Google ScholarGoogle Scholar
  65. Jongh, D. D. and Troelstra, A. 1966. On the connection of partially ordered sets with some pseudo-boolean algebras. Indagationes Math. 28, 317--329.Google ScholarGoogle ScholarCross RefCross Ref
  66. Kahn, G. 1974. The semantics of simple language for parallel programming. In IFIP Congress. North-Holland, 471--475.Google ScholarGoogle Scholar
  67. Kanellakis, P. C. and Smolka, S. A. 1990. CCS expressions, finite state processes, and three problems of equivalence. Inf. Comput. 86, 1, 43--68. Google ScholarGoogle ScholarDigital LibraryDigital Library
  68. Kantorovich, L. V. 1939. The method of successive approximations for functional equations. Acta Math. 71, 63--97.Google ScholarGoogle ScholarCross RefCross Ref
  69. Kleene, S. C. 1952. Introduction to Metamathematics. Van Nostrand.Google ScholarGoogle Scholar
  70. Kleene, S. C. 1970. The origin of recursive function theory. In 20th Annual Symposium on Foundations of Computer Science. IEEE, 371--382. Google ScholarGoogle ScholarDigital LibraryDigital Library
  71. Knaster, B. 1928. Un théorèm sur les fonctions d'ensembles. Annals Soc. Pol. Math 6, 133--134.Google ScholarGoogle Scholar
  72. Kwong, Y. S. 1977. On reduction of asynchronous systems. Theor. Comput. Sci. 5, 1, 25--50.Google ScholarGoogle ScholarCross RefCross Ref
  73. Landin, P. 1969. A program-machine symmetric automata theory. Mach. Intell. 5, 99--120.Google ScholarGoogle Scholar
  74. Landin, P. J. 1964. The mechanical evaluation of expressions. The Comput. J. 6, 4, 308--320.Google ScholarGoogle ScholarCross RefCross Ref
  75. Landin, P. J. 1965a. Correspondence between ALGOL 60 and Church's Lambda-notation: Part I. Commun. ACM 8, 2, 89--101. Google ScholarGoogle ScholarDigital LibraryDigital Library
  76. Landin, P. J. 1965b. A correspondence between ALGOL 60 and Church's Lambda-notations: Part II. Commun. ACM 8, 3, 158--167. Google ScholarGoogle ScholarDigital LibraryDigital Library
  77. Lassez, J.-L., Nguyen, V. L., and Sonenberg, L. 1982. Fixed point theorems and semantics: A folk tale. Inf. Process. Lett. 14, 3, 112--116.Google ScholarGoogle ScholarCross RefCross Ref
  78. Manna, Z. 1969. The correctness of programs. J. Comput. Syst. Sci. 3, 2, 119--127.Google ScholarGoogle ScholarDigital LibraryDigital Library
  79. Manna, Z. 1974. Mathematical Theory of Computation. McGraw-Hill. Google ScholarGoogle ScholarDigital LibraryDigital Library
  80. Mazurkiewicz, A. 1973. Proving properties of processes. Tech. rep. 134, Computation Center ol Polish Academy of Sciences, Warsaw. Also in Algorytmy 11, 5--22, 1974.Google ScholarGoogle Scholar
  81. Mazurkiewicz, A. W. 1971. Proving algorithms by tail functions. Inf. Control 18, 3, 220--226.Google ScholarGoogle ScholarCross RefCross Ref
  82. McCarthy, J. 1961. A basis for a mathematical theory of computation. In Proceedings of the Western Joint Computer Conference, Vol. 19. Spartan Books, 225--238. Reprinted, with corrections and an added tenth section, in Computer Programming and Formal Systems, P. Braffort and D. Hirschberg, eds., North-Holland, 1963, pp. 33--70.Google ScholarGoogle Scholar
  83. McCarthy, J. 1963. Towards a mathematical science of computation. In Proceedings of the IFIP Congress 62. North-Holland, 21--28.Google ScholarGoogle Scholar
  84. Meyer, A. R. and Stockmeyer, L. J. 1972. The equivalence problem for regular expressions with squaring requires exponential space. In 13th Annual Symposium on Switching and Automata Theory (FOCS). IEEE, 125--129. Google ScholarGoogle ScholarDigital LibraryDigital Library
  85. Milner, R. 1970. A formal notion of simulation between programs. Memo 14, Computers and Logic Resarch Group, University College of Swansea, U.K.Google ScholarGoogle Scholar
  86. Milner, R. 1971b. Program simulation: An extended formal notion. Memo 17, Computers and Logic Resarch Group, University College of Swansea, U.K.Google ScholarGoogle Scholar
  87. Milner, R. 1980. A Calculus of Communicating Systems. Lecture Notes in Computer Science, vol. 92. Springer. Google ScholarGoogle ScholarDigital LibraryDigital Library
  88. Milner, R. 1989. Communication and Concurrency. Prentice Hall. Google ScholarGoogle ScholarDigital LibraryDigital Library
  89. Milner, R. London, 1971a. An algebraic definition of simulation between programs. In Proceedings of the 2nd International Joint Conferences on Artificial Intelligence. British Computer Society.Google ScholarGoogle ScholarDigital LibraryDigital Library
  90. Milner, R. and Tofte, M. 1991. Co-Induction in relational semantics. Theor. Comput. Sci. 87, 209--220. Also Tech. rep. ECS-LFCS-88-65, University of Edinburgh, 1988.Google ScholarGoogle ScholarDigital LibraryDigital Library
  91. Mirimanoff, D. 1917a. Les antinomies de Russell et de Burali-Forti et le problème fondamental de la théorie des ensembles. L'Enseignement Math. 19, 37--52.Google ScholarGoogle Scholar
  92. Mirimanoff, D. 1917b. Remarques sur la théorie des ensembles et les antinomies cantoriennes I. L'Enseignement Math. 19, 209--217.Google ScholarGoogle Scholar
  93. Mirimanoff, D. 1920. Remarques sur la théorie des ensembles et les antinomies cantoriennes II. L'Enseignement Math. 21, 29--52.Google ScholarGoogle Scholar
  94. Moore, E. 1956. Gedanken experiments on sequential machines. Automata Studies, Ann. Math. Series 34, 129--153.Google ScholarGoogle Scholar
  95. Morris, J. H. Dec. 1968. Lambda-Calculus models of programming languages. Ph.D. thesis, M.I.T., project MAC.Google ScholarGoogle Scholar
  96. Moschovakis, Y. N. 1974. Elementary Induction on Abstract Structures. Studies in Logic and the Foundations of Mathematics, vol. 77. North-Holland, Amsterdam. Google ScholarGoogle ScholarDigital LibraryDigital Library
  97. Nerode, A. 1958. Linear automaton transformations. In Proceedings of the American Mathematical Society. Vol. 9. 541--544.Google ScholarGoogle ScholarCross RefCross Ref
  98. Paige, R. and Tarjan, R. E. 1987. Three partition refinement algorithms. SIAM J. Comput. 16, 6, 973--989. Google ScholarGoogle ScholarDigital LibraryDigital Library
  99. Park, D. 1969. Fixpoint induction and proofs of program properties. In Machine Intelligence 5, B. Meltzer and D. Michie, Eds. Edinburgh University Press, 59--78.Google ScholarGoogle Scholar
  100. Park, D. 1970. The Y-combinator in Scott's lambda-calculus models. Symposium on Theory of Programming, unpublished (A revised version: Res. rep. CS-RR-013, Department of Computer Science, University of Warwick, June 1976.). Google ScholarGoogle ScholarDigital LibraryDigital Library
  101. Park, D. 1979. On the semantics of fair parallelism. In Proceedings of the Confernce on Abstract Software Specifications, Copenhagen Winter School. Lecture Notes in Computer Science. Springer, 504--526. Google ScholarGoogle ScholarDigital LibraryDigital Library
  102. Park, D. 1981a. Concurrency on automata and infinite sequences. In Conference on Theoretical Computer Science, P. Deussen, Ed. Lecture Notes in Computer Science, vol. 104. Springer, 167--183. Google ScholarGoogle ScholarDigital LibraryDigital Library
  103. Park, D. 1981b. A new equivalence notion for communicating systems. In Bull. EATCS, G. Maurer, Ed. Vol. 14. 78--80. Abstract of the talk presented at the 2nd Workshop on the Semantics of Programming Languages. Abstracts collected in the Bulletin by B. Mayoh.Google ScholarGoogle Scholar
  104. Pasini, A. 1974. Some fixed point theorems of the mappings of partially ordered sets. Rendiconti del Seminario Matematico della Università di Padova 51, 167--177.Google ScholarGoogle Scholar
  105. Pous, D. 2007. Complete lattices and up-to techniques. In 5th Asian Symposium on Programming Languages and Systems (APLAS). Lecture Notes in Computer Science, vol. 4807. Springer, 351--366. Google ScholarGoogle ScholarDigital LibraryDigital Library
  106. Reynolds, J. C. 1993. The discoveries of continuations. Lisp Symbol. Comput. 6, 3-4, 233--248. Google ScholarGoogle ScholarDigital LibraryDigital Library
  107. Rogers, H. 1967. Theory of Recursive Functions and Effective Computability. McGraw Hill. Reprinted, MIT Press 1987. Google ScholarGoogle ScholarDigital LibraryDigital Library
  108. Russell, B. 1903. Principles of Mathematics. Cambridge University Press.Google ScholarGoogle Scholar
  109. Russell, B. 1908. Mathematical logic as based on the theory of types. Amer. J. Math. 30, 222--262.Google ScholarGoogle ScholarCross RefCross Ref
  110. Russell, B. and Whitehead, A. N. 1910, 1912, 1913. Principia Mathematica, 3 vols. Cambridge University Press.Google ScholarGoogle Scholar
  111. Rutten, J. and Turi, D. 1992. On the foundation of final semantics: Non-Standard sets, metric spaces, partial orders. In REX Workshop, J. W. de Bakker, W. P. de Roever, and G. Rozenberg, Eds. Lecture Notes in Computer Science, vol. 666. Springer, 477--530. Google ScholarGoogle ScholarDigital LibraryDigital Library
  112. Sangiorgi, D. 1998. On the bisimulation proof method. J. Math. Structures Comput. Sci. 8, 447--479. Google ScholarGoogle ScholarDigital LibraryDigital Library
  113. Sangiorgi, D., Kobayashi, N., and Sumii, E. 2007. Environmental bisimulations for higher-order languages. In 22nd Annual IEEE Symposium on Logic in Computer Science (LICS'07). IEEE Computer Society, 293--302. Google ScholarGoogle ScholarDigital LibraryDigital Library
  114. Sangiorgi, D. and Walker, D. 2001. The π-Calculus: A Theory of Mobile Processes. Cambridge University Press. Google ScholarGoogle ScholarDigital LibraryDigital Library
  115. Scott, D. 1960. A different kind of model for set theory. Unpublished paper, given at the 1960 Stanford Congress of Logic, Methodology and Philosophy of Science.Google ScholarGoogle Scholar
  116. Scott, D. 1972a. Continuous lattices. In Toposes, Algebraic Geometry and Logic, E. Lawvere, Ed. Lecture Notes in Mathematics, vol. 274. Springer, 97--136.Google ScholarGoogle Scholar
  117. Scott, D. 1972b. The lattice of flow diagrams. In Symposium of Semantics of Algorithmic Languages, E. Engeler, Ed. Lecture Notes in Mathematics, vol. 188. Springer, 311--366.Google ScholarGoogle Scholar
  118. Scott, D. 1976. Data types as lattices. SIAM J. Comput. 5, 522--587.Google ScholarGoogle ScholarDigital LibraryDigital Library
  119. Scott, D. December 1969b. Models for the λ-calculus. Manuscript, raft, Oxford.Google ScholarGoogle Scholar
  120. Scott, D. November 1969a. A construction of a model for the λ-calculus. Manuscript, Oxford.Google ScholarGoogle Scholar
  121. Scott, D. October 1969c. A type-theoretical alternative to CUCH, ISWIM, OWHY. Typed script, Oxford. Also appeared as Scott {1993}.Google ScholarGoogle Scholar
  122. Scott, D. and de Bakker, J. 1969. A theory of programs. Handwritten notes. IBM Lab., Vienna, Austria.Google ScholarGoogle Scholar
  123. Scott, D. S. 1993. A type-theoretical alternative to ISWIM, CUCH, OWHY. Theor. Comput. Sci. 121, 1-2, 411--440. Google ScholarGoogle ScholarDigital LibraryDigital Library
  124. Segerberg, K. 1968. Decidability of S4.1. Theoria 34, 7--20.Google ScholarGoogle ScholarCross RefCross Ref
  125. Segerberg, K. 1970. Modal logics with linear alternative relations. Theoria 36, 301--322.Google ScholarGoogle ScholarCross RefCross Ref
  126. Segerberg, K. 1971. An essay in classical modal logic. Filosofiska Studier, Uppsala.Google ScholarGoogle Scholar
  127. Skolem, T. 1923. Einige Bemerkungen zur Axiomatischen Begründung der Mengenlehre. In Proceedings of the 5th Scandinavian Mathematics Congress, 1922. Akademiska Bokhandeln, Helsinki, 217--232. English translation, “Some remarks on axiomatized set theory”, in Heijenoort (Ed.) {1967}, pages 290--301.Google ScholarGoogle Scholar
  128. Specker, E. 1957. Zur axiomatik der Mengenlehre. Z. Math. Logik 3, 3, 173--210.Google ScholarGoogle ScholarCross RefCross Ref
  129. Tarski, A. 1949. A fixpoint theorem for lattices and its applications (preliminary report). Bull. Amer. Math. Soc. 55, 1051--1052 and 1192.Google ScholarGoogle Scholar
  130. Tarski, A. 1955. A lattice-theoretical fixpoint theorem and its applications. Pac. J. Math. 5, 285--309.Google ScholarGoogle ScholarCross RefCross Ref
  131. Thomas, W. 1993. On the Ehrenfeucht-Fraïssé game in theoretical computer science. In TAPSOFT, M.-C. Gaudel and J.-P. Jouannaud, Eds. Lecture Notes in Computer Science, vol. 668. Springer, 559--568. Google ScholarGoogle ScholarDigital LibraryDigital Library
  132. Thomason, S. K. 1972. Semantic analysis of tense logics. J. Symb. Log. 37, 1, 150--158.Google ScholarGoogle ScholarCross RefCross Ref
  133. Turi, D. and Plotkin, G. D. 1997. Towards a mathematical operational semantics. In 12th Annual IEEE Symposium on Logic in Computer Science (LICS'97). IEEE Computer Society Press, 280--291. Google ScholarGoogle ScholarDigital LibraryDigital Library
  134. Zermelo, E. 1908. Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen 65, 261--281. English translation, “Investigations in the foundations of set theory”, in Heijenoort (Ed.) {1967}, 199--215.Google ScholarGoogle ScholarCross RefCross Ref

Index Terms

  1. On the origins of bisimulation and coinduction

      Recommendations

      Comments

      Login options

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

      Sign in

      Full Access

      • Published in

        cover image ACM Transactions on Programming Languages and Systems
        ACM Transactions on Programming Languages and Systems  Volume 31, Issue 4
        May 2009
        181 pages
        ISSN:0164-0925
        EISSN:1558-4593
        DOI:10.1145/1516507
        Issue’s Table of Contents

        Copyright © 2009 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 26 May 2009
        • Accepted: 1 July 2008
        • Revised: 1 June 2008
        • Received: 1 October 2007
        Published in toplas Volume 31, Issue 4

        Permissions

        Request permissions about this article.

        Request Permissions

        Check for updates

        Qualifiers

        • research-article
        • Research
        • Refereed

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader
      About Cookies On This Site

      We use cookies to ensure that we give you the best experience on our website.

      Learn more

      Got it!