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Partial Commutation and Traces

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Abstract

Parallelism and concurrency are fundamental concepts in computer science. Specification and verification of concurrent programs are of first importance. It concerns our daily life whether software written for distributed systems behaves correctly.

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References

  1. IJsbrand Jan Aalbersberg and Hendrik Jan Hoogeboom. Characterizations of the decidability of some problems for regular trace languages. Mathematical Systems Theory, 22:1–19, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  2. IJsbrand Jan Aalbersberg and Emo Welzl. Trace languages defined by regular string languages. R.A.I.R.O. - Informatique Théorique et Applications, 20:103–119, 1986.

    MathSciNet  MATH  Google Scholar 

  3. Anatolij V. Anisimov and Donald E. Knuth. Inhomogeneous sorting. International Journal of Computer and Information Sciences,8:255–260, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  4. Jean Berstel. Transductions and context-free languages. Teubner Studienbücher, Stuttgart, 1979.

    MATH  Google Scholar 

  5. Jean Berstel and Dominique Perrin. Theory of Codes. Pure and Applied Mathematics; 117. Academic Press, Orlando, Florida, 1985.

    Google Scholar 

  6. Alberto Bertoni, Giancarlo Mauri, and Nicoletta Sabadini. A hierarchy of regular trace languages and some combinatorial applications. In A. Ballester, D. Cardus, and E. Trillas, editors, Proceedings of the 2nd World Conf. on Mathematics at the Service of Man, Las Palmas (Canary Island) Spain,pages 146–153. Universidad Politecnica de Las Palmas, 1982.

    Google Scholar 

  7. Alberto Bertoni, Giancarlo Mauri, and Nicoletta Sabadini. Concurrency and commutativity. Technical report, Instituto di Cibernetica, Università di Milano, 1983. Presented at the Workshop on Petri nets, Varenna (Italy), 1982.

    Google Scholar 

  8. Alberto Bertoni, Giancarlo Mauri, and Nicoletta Sabadini. Unambiguous regular trace languages. In G. Demetrovics, J. Katona and A. Salomaa, editors, Proceedings of the Coll. on Algebra, Combinatorics and Logic in Computer Science, volume 42 of Colloquia Mathematica Soc. J. Bolyai, pages 113–123. North Holland, Amsterdam, 1985.

    Google Scholar 

  9. Eike Best and Raymond Devillers. Sequential and concurrent behaviour in Petri net theory. Theoretical Computer Science, 55:87–136, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  10. Paola Bonizzoni, Giancarlo Mauri, and Giovanni Pighizzini. About infinite traces. In V. Diekert, editor, Proceedings of the ASMICS workshop Free Partially Commutative Monoids, Kochel am See, Oktober 1989,Report TUMI9002, Technical University of Munich, pages 1–10, 1990.

    Google Scholar 

  11. Véronique Bruyère and Clelia De Felice. Trace codings. In E. W. Mayr and C. Puech, editors, Proceedings of the 12th Annual Symposium on Theoretical Aspects of Computer Science (STACS’95), 1995, Lecture Notes in Computer Science 900, pages 373–384, Springer-Verlag, Berlin, 1995.

    Google Scholar 

  12. Véronique Bruyère, Clelia De Felice, and Giovanna Guaiana. Coding with traces. In P. Enjalbert, E. W. Mayr, and K. W. Wagner, editors, Proceedings of the 11th Annual Symposium on Theoretical Aspects of Computer Science (STACS’94), 1994, Lecture Notes in Computer Science 775, pages 353–364, Springer-Verlag, Berlin, 1994.

    Google Scholar 

  13. Pierre Cartier and Dominique Foata. Problèmes combinatoires de commutation et réarrangements. Lecture Notes in Mathematics 85, Springer-Verlag, Berlin, 1969.

    Google Scholar 

  14. Marek Chrobak and Wojciech Rytter. Unique decipherability for partially commutative alphabets. Fundamenta Informaticae,X:323–336, 1987.

    MathSciNet  Google Scholar 

  15. Mireille Clerbout. Commutations Partielles et Familles de Langages. Thèse, Université des Sciences et Technologies de Lille (France), 1984.

    Google Scholar 

  16. Mireille Clerbout and Michel Latteux. Partial commutations and faithful rational transductions. Theoretical Computer Science, 34:241–254, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  17. Mireille Clerbout and Michel Latteux. Semi-Commutations. Information and Computation, 73:59–74, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  18. Mireille Clerbout, Michel Latteux, and Yves Roos. Decomposition of partial commutations. In M. S. Paterson, editor, Proceedings of the 17th International Colloquium on Automata, Languages and Programming (ICALP’90), Warwick (England) 1990, Lecture Notes in Computer Science 443, pages 501–511, Springer-Verlag, Berlin, 1990.

    Google Scholar 

  19. Robert Cori and Yves Métivier. Recognizable subsets of some partially abelian monoids. Theoretical Computer Science,35:179–189, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  20. Robert Cori, Yves Métivier, and Wieslaw Zielonka. Asynchronous mappings and asynchronous cellular automata. Information and Computation,106:159–202, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  21. Robert Cori and Dominique Perrin. Automates et commutations partielles. R.A.I.R.O. - Informatique Théorique et Applications, 19:21–32, 1985.

    MathSciNet  MATH  Google Scholar 

  22. Bruno Courcelle. The monadic second-order logic of graphs X: linear orderings. Theoretical Computer Science,1996. to appear.

    Google Scholar 

  23. Volker Diekert. Transitive orientations, Möbius functions and complete semiThue systems for free partially commutative monoids. In T. Lepistö et al., editors, Proceedings of the 15th International Colloquium on Automata, Languages and Programming (ICALP’88), Tampere (Finland) 1988,Lecture Notes in Computer Science 317, pages 176–187, Springer-Verlag, Berlin, 1988.

    Google Scholar 

  24. Volker Diekert. Combinatorics on Traces. Lecture Notes in Computer Science 454. Springer-Verlag, Berlin, 1990.

    Book  MATH  Google Scholar 

  25. Volker Diekert. Research topics in the theory of free partially commutative monoids. Bulletin of the European Association for Theoretical Computer Science (EATCS), 40:479–491, 1990.

    Google Scholar 

  26. Volker Diekert. Möbius functions and confluent semi-commutations. Theoretical Computer Science, 108:25–43, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  27. Volker Diekert. On the concatenation of infinite traces. Theoretical Computer Science, 113:35–54, 1993. Special issue STACS’91.

    Article  MathSciNet  MATH  Google Scholar 

  28. Volker Diekert. A partial trace semantics for Petri nets. Theoretical Computer Science, 134:87–105, 1994. Special issue of ICWLC 92, Kyoto (Japan).

    Article  MathSciNet  MATH  Google Scholar 

  29. Volker Diekert and Paul Gastin. A domain for concurrent termination: A generalization of Mazurkiewicz traces. In Z. Fülöp and F. Gécseg, editors, Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (ICALP’95), Szeged (Hungary) 1995, Lecture Notes in Computer Science 944, pages 15–26. Springer-Verlag, Berlin, 1995.

    Google Scholar 

  30. Volker Diekert, Paul Gastin, and Antoine Petit. Rational and recognizable complex trace languages. Information and Computation, 116:134–153, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  31. Volker Diekert and Anca Muscholl. Deterministic asynchronous automata for infinite traces. Acta Informatica, 31:379–397, 1994. A preliminary version was presented at STACS’93, Lecture Notes in Computer Science 665, 1993.

    Google Scholar 

  32. Volker Diekert and Anca Muscholl. A note on Métivier’s construction of asynchronous automata for triangulated graphs. Fundamenta Informaticae,to appear 1996. Special issue on Formal Language Theory.

    Google Scholar 

  33. Volker Diekert, Anca Muscholl, and Klaus Reinhardt. On codings of traces. In E. W. Mayr and C. Puech, editors, Proceedings of the 12th Annual Symposium on Theoretical Aspects of Computer Science (STACS’95), 1995, Lecture Notes in Computer Science 900, pages 385–396, Springer-Verlag, Berlin, 1995.

    Google Scholar 

  34. Volker Diekert and Grzegorz Rozenberg, editors. The Book of Traces. World Scientific, Singapore, 1995.

    Google Scholar 

  35. Manfred Droste. Concurrent automata and domains. International Journal of Foundations of Computer Science, 3:389–418, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  36. Manfred Droste. A Kleene Theorem for recognizable languages over con-currency monoids. In S. Abiteboul and E. Shamir, editors, Proceedings of the 21st International Colloquium on Automata, Languages and Programming (ICALP’94), Jerusalem (Israel) 1994, Lecture Notes in Computer Science 820, pages 388–398, Springer-Verlag, Berlin, 1994.

    Google Scholar 

  37. Manfred Droste. Aperiodic languages over concurrency monoids. 1995. submitted.

    Google Scholar 

  38. Manfred Droste. Recognizable languages in concurrency monoids. Theoretical Computer Science, 150:77–109, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  39. Manfred Droste and Dietrich Kuske. Logical definability of recognizable and aperiodic languages in concurrency monoids. In Proceedings of the CSL’9, Paderborn, Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1996.

    Google Scholar 

  40. Christine Duboc. Commutations dans les monoïdes libres: un cadre théorique pour l’étude du parallelisme Thèse, Faculté des Sciences de l’Université de Rouen, 1986.

    Google Scholar 

  41. Werner Ebinger. Charakterisierung von Sprachklassen unendlicher Spuren durch Logiken. Dissertation, Institut für Informatik, Universität Stuttgart, 1994.

    Google Scholar 

  42. Werner Ebinger and Anca Muscholl. Logical definability on infinite traces. Theoretical Computer Science,154:67–84, 1996. A preliminary version appeared in Proceedings of the 20th International Colloquium on Automata, Languages and Programming (ICALP’93), Lund (Sweden) 1993, Lecture Notes in Computer Science 700, Springer-Verlag, Berlin, 1993.

    Google Scholar 

  43. Samuel Eilenberg and Marcel Paul Schützenberger. Rational sets in commutative monoids. Journal of Algebra, 13:173–191, 1969.

    Article  MathSciNet  MATH  Google Scholar 

  44. Marie-Paule Flé and Gérard Roucairol. On serializability of iterated transactions. In Proceedings of the 15th ACM SIGACT-SIGOPS Symp. on Princ. of Distrib. Comp., Ottawa (1982), pages 194–200, 1982.

    Google Scholar 

  45. Marie-Paule Flé and Gérard Roucairol. Fair serializability of iterated transactions using fifo-nets. In G. Rozenberg, editor, Advances in Petri Nets, Lecture Notes in Computer Science 188, pages 154–168. Springer-Verlag, Berlin, 1985.

    Google Scholar 

  46. Michel Fliess. Matrices de Hankel. J. Math. Pures et Appl., 53:197–224, 1974.

    MathSciNet  MATH  Google Scholar 

  47. Paul Gastin. Infinite traces. In I. Guessarian, editor, Proceedings of the Spring School of Theoretical Computer Science on Semantics of Systems of Concurrent Processes, Lecture Notes in Computer Science 469, pages 277–308, Springer-Verlag, Berlin, 1990.

    Google Scholar 

  48. Paul Gastin. Recognizable and rational trace languages of finite and infinite traces. In C. Choffrut et al., editors, Proceedings of the 8th Annual Symposium on Theoretical Aspects of Computer Science (STACS’91), Hamburg 1991, Lecture Notes in Computer Science 480, pages 89–104, Springer-Verlag, Berlin, 1991.

    Google Scholar 

  49. Paul Gastin, Edward Ochmariski, Antoine Petit, and Brigitte Rozoy. Decidability of the Star problem in A* × {b}*. Information Processing Letters, 44:65–71, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  50. Paul Gastin and Antoine Petit. Asynchronous automata for infinite traces. In W. Kuich, editor, Proceedings of the 19th International Colloquium on Automata, Languages and Programming (ICALP’92), Vienna (Austria) 1992, Lecture Notes in Computer Science 623, pages 583–594, Springer-Verlag, Berlin, 1992.

    Google Scholar 

  51. Paul Gastin, Antoine Petit, and Wieslaw Zielonka. An extension of Kleene’s and Ochmanski’s theorems to infinite traces. Theoretical Computer Science, 125:167–204, 1994. A preliminary version was presented at ICALP’91, Lecture Notes in Computer Science 510, Springer-Verlag, Berlin, 1991.

    Google Scholar 

  52. Paul Gastin and Brigitte Rozoy. The poset of infinitary traces. Theoretical Computer Science, 120:101–121, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  53. Alan Gibbons and Wojciech Rytter. On the decidability of some problems about rational subsets of free partially commutative monoids. Theoretical Computer Science, 48:329–337, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  54. Gerhard Gierz, Karl Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael W. Mislove, and Dana S. Scott. A Compendium of Continuous Lattices. Springer-Verlag, Berlin, 1980.

    Book  MATH  Google Scholar 

  55. Seymour Ginsburg and Edwin H. Spanier. Semigroups, Presburger formulas and languages. Pacific Journal of Mathematics,16(2):285–296, 1966.

    MathSciNet  MATH  Google Scholar 

  56. Martin C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980.

    MATH  Google Scholar 

  57. Sheila A. Greibach. The undecidability of the ambiguity problem for minimal linear grammars. Information and Control,6:119–125, 1963.

    Article  MathSciNet  MATH  Google Scholar 

  58. Giovanna Guaiana, Antonio Restivo, and Sergio Salemi. Star-free trace languages. Theoretical Computer Science, 97:301–311, 1992. A preliminary version was presented at STACS’91, Lecture Notes in Computer Science 480, Springer-Verlag, Berlin, 1991.

    Google Scholar 

  59. Kosaburo Hashiguchi. Limitedness theorem on finite automata with distance functions. Journal of Computer and System Sciences, 24:233–244, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  60. Kosaburo Hashiguchi. Recognizable closures and submonoids of free partially commutative monoids. Theoretical Computer Science, 86:233–241, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  61. Hendrik Jan Hoogeboom and Anca Muscholl. The code problem for traces - improving the boundaries. Submitted for publication, 1995.

    Google Scholar 

  62. Günter Hotz and Volker Claus. Automatentheorie und Formale Sprachen, Band III. Bibliographisches Institut, Mannheim, 1972.

    Google Scholar 

  63. Oscar H. Ibarra. Reversal-bounded multicounter machines and their decision problems. Journal of the Association of Computing Machinery, 25(1):116–133, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  64. Robert M. Keller. Parallel program schemata and maximal parallelism I. Fundamental results. Journal of the Association of Computing Machinery, 20(3):514–537, 1973.

    Article  MATH  Google Scholar 

  65. Nils Klarlund, Madhavan Mukund, and Millind Sohoni. Determinizing asynchronous automata. In S. Abiteboul and E. Shamir, editors, Proceedings of the 21st International Colloquium on Automata Languages and Programming (ICALP’94), Jerusalem (Israel) 1994, Lecture Notes in Computer Science 820, pages 130–141, Springer-Verlag, Berlin, 1994.

    Google Scholar 

  66. Marta Z. Kwiatkowska. A metric for traces. Information Processing Letters, 35:129–135, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  67. Leslie Lamport. Time, clocks, and the ordering of events in a distributed system. Communications of the Association of Computing Machinery,21:558–564, 1978.

    Article  MATH  Google Scholar 

  68. Gennadií Semjonovich Makanin. The problem of solvability of equations in free semigroups. Math. USSR Izvestiya, 21:483–546, 1983.

    Article  Google Scholar 

  69. Antoni Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

    Google Scholar 

  70. Antoni Mazurkiewicz. Trace theory. In W. Brauer et al., editors, Petri Nets, Applications and Relationship to other Models of Concurrency, Lecture Notes in Computer Science 255, pages 279–324, Springer-Verlag, Berlin, 1987.

    Google Scholar 

  71. Yves Métivier. On recognizable subsets of free partially commutative monoids. In L. Kott, editor, Proceedings of the 13th International Colloquium on Automata, Languages and Programming (ICALP’86), Rennes (France) 1986, Lecture Notes in Computer Science 226, pages 254–264, Springer-Verlag, Berlin, 1986.

    Google Scholar 

  72. Yves Métivier. Une condition suffisante de reconnaissabilité dans un monoïde partiellement commutatif. R.A.I.R.O. - Informatique Théorique et Applications, 20:121–127, 1986.

    MATH  Google Scholar 

  73. Yves Métivier. An algorithm for computing asynchronous automata in the case of acyclic non-commutation graph. In Th. Ottmann, editor, Proceedings of the 14th International Colloquium on Automata, Languages and Programming (ICALP’87), Karlsruhe (FRG) 1987, Lecture Notes in Computer Science 267, pages 226–236, Springer-Verlag, Berlin, 1987.

    Google Scholar 

  74. Yves Métivier. Contribution à l’étude des monoïdes de commutations. Thèse d’état, 1987. Université Bordeaux I.

    Google Scholar 

  75. Yves Métivier and Edward Ochmaiiski. On lexicographic semi-commutations. Information Processing Letters,26:55–59, 1987/88.

    Article  MathSciNet  MATH  Google Scholar 

  76. Yves Métivier, Gwénaël Richomme, and Pierre-André Wacrenier. Computing the closure of sets of words under partial commutations. In Z. Fülöp and F. Gécseg, editors, Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (ICALP’95), Szeged (Hungary) 1995, Lecture Notes in Computer Science 944, pages 75–86, Springer-Verlag, Berlin, 1995.

    Google Scholar 

  77. Anca Muscholl. On the complementation of Büchi asynchronous cellular automata. In S. Abiteboul and E. Shamir, editors, Proceedings of the 21st International Colloquium on Automata, Languages and Programming (ICALP’94), Jerusalem (Israel) 1994, Lecture Notes in Computer Science 820, pages 142–153, Springer-Verlag, Berlin, 1994.

    Google Scholar 

  78. Anca Muscholl. Über die Erkennbarkeit unendlicher Spuren. Teubner, Stuttgart, 1996.

    Google Scholar 

  79. Anca Muscholl and Holger Petersen. A note on the commutative closure of star-free languages. Information Processing Letters,57:71–74, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  80. Edward Ochmanski. Regular behaviour of concurrent systems. Bulletin of the European Association for Theoretical Computer Science (EATCS), 27:56–67, Oct 1985.

    Google Scholar 

  81. Edward Ochmanski. On morphisms of trace monoids. In R. Cori and M. Wirsing, editors, Proceedings of the 5th Annual Symposium on Theoretical Aspects of Computer Science (STACS’88), Lecture Notes in Computer Science 294, pages 346–355, Springer-Verlag, Berlin, 1988.

    Google Scholar 

  82. Friedrich Otto. Finite canonical rewriting systems for congruences generated by concurrency relations. Mathematical Systems Theory,20:253–260, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  83. Prakas Panangaden and Eugene W. Stark. Computations, residuals and the power of indeterminacy. In T. Lepistö et al., editors, Proceedings of the 15th International Colloquium on Automata, Languages and Programming (ICALP’88), Tampere (Finland) 1988, Lecture Notes in Computer Science 317, pages 439–454, Springer-Verlag, Berlin, 1988.

    Google Scholar 

  84. Dominique Perrin. Words over a partially commutative alphabet. In A. Apostolico, editor, Combinatorial Algorithms on Words, volume F12 of NATO-ASI Series, pages 329–340. Springer-Verlag, Berlin, 1986.

    Google Scholar 

  85. Giovanni Pighizzini. Asynchronous automata versus asynchronous cellular automata. Theoretical Computer Science,132:179–207, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  86. Vaughan R. Pratt. Modelling concurrency with partial orders. International Journal of Parallel Programming, 15(1):33–71, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  87. Gwénaël Richomme. Some trace monoids where both the Star Problem and the Finite Power Property Problem are decidable. In I. Privara et al., editors, Proceedings of the 19th Symposium on Mathematical Foundations of Computer Science (MFCS’94), Kosice (Slovakia) 1994, Lecture Notes in Computer Science 841, pages 577–586, Springer-Verlag, Berlin, 1994.

    Google Scholar 

  88. Jacques Sakarovitch. On regular trace languages. Theoretical Computer Science, 52:59–75, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  89. Jacques Sakarovitch. The “last” decision problem for rational trace languages. In I. Simon, editor, Proceedings of the 1st Latin American Symposium on Theoretical Informatics (LATIN’92), Lecture Notes in Computer Science 583, pages 460–473, Springer-Verlag, Berlin, 1992. Springer.

    Google Scholar 

  90. Wolfgang Thomas. Classifying regular events in symbolic logic. Journal of Computer and System Sciences,25:360–376, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  91. Wolfgang Thomas. Automata on infinite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, chapter 4, pages 133–191. Elsevier Science Publishers, Amsterdam, 1990.

    Google Scholar 

  92. Wolfgang Thomas. On logical definability of trace languages. In V. Diekert, editor, Proceedings of a workshop of the ESPRIT Basic Research Action No 3166: Algebraic and Syntactic Methods in Computer Science (ASMICS), Kochel am See, Bavaria, FRG (1989), Report TUM-19002, Technical University of Munich, pages 172–182, 1990.

    Google Scholar 

  93. Wieslaw Zielonka. Notes on finite asynchronous automata. R.A.I.R.O. — Informatique Théorique et Applications, 21:99–135, 1987.

    MathSciNet  MATH  Google Scholar 

  94. Wieslaw Zielonka. Safe executions of recognizable trace languages by asynchronous automata. In A. R. Mayer et al., editors, Proceedings of the Symposium on Logical Foundations of Computer Science, Logic at Botik ‘89, Pereslavl-Zalessky (USSR) 1989, Lecture Notes in Computer Science 363, pages 278–289, Springer-Verlag, Berlin, 1989.

    Google Scholar 

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Diekert, V., Métivier, Y. (1997). Partial Commutation and Traces. In: Rozenberg, G., Salomaa, A. (eds) Handbook of Formal Languages. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59126-6_8

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