Abstract
This paper extends automata-theoretic techniques to unbounded parallel behaviour, as seen for instance in Petri nets. Languages are defined to be sets of (labelled) series-parallel posets – or, equivalently, sets of terms in an algebra with two product operations: sequential and parallel. In an earlier paper, we restricted ourselves to languages of posets having bounded width and introduced a notion of branching automaton. In this paper, we drop the restriction to bounded width. We define rational expressions, a natural generalization of the usual ones over words, and prove a Kleene theorem connecting them to regular languages (accepted by finite branching automata). We also show that recognizable languages (inverse images by a morphism into a finite algebra) are strictly weaker.
Part of this work was done while the second author was visiting the Institute of Mathematical Sciences, in Chennai.
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References
Bloom, S., Ésik, Z.: Free shuffle algebras in language varieties. TCS 163, 55–98 (1996)
Boudol, G.: Notes on algebraic calculi of processes. In: Apt, K.R. (ed.) Logics and models of concurrent systems. NATO ASI Series F13, pp. 261–305 (1985)
Büchi, J.R.: Finite automata, their algebras and grammars: Towards a theory of formal expressions. Siefkes, D. (ed.) Springer, Heidelberg (1989)
Courcelle, B.: Graph rewriting: an algebraic and logical approach. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B. Elsevier, Amsterdam (1990)
Courcelle, B.: The monadic second-order logic of graphs V: on closing the gap between definability and recognizability. Theoret. Comp. Sci. 80, 153–202 (1991)
Diekert, V., Rozenberg, G.: The book of traces. World Scientific, Singapore (1995)
Garg, V.K., Ragunath, M.T.: Concurrent regular expressions and their relationship to Petri nets. Theoret. Comp. Sci. 96, 285–304 (1992)
Gécseg, F., Steinby, M.: Tree automata, Akadémiai Kiadó, Budapest (1984)
Gischer, J.L.: The equational theory of pomsets. TCS 61, 199–224 (1988)
Grabowski, J.: On partial languages. Fund. Inform. IV, 427–498 (1981)
Lodaya, K., Weil, P.: Series-parallel posets: algebra, automata and languages. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 555–565. Springer, Heidelberg (1998)
Lodaya, K., Weil, P.: Series-parallel languages and the bounded width property, IMSc Tech Rep 98/07/36 (1998), http://www.imsc.ernet.in/~kamal/splbwp.ps.gz
Petri, C.A.: Fundamentals of a theory of asynchronous information flow. In: Proc. IFIP (Amsterdam 1962), pp. 386–390. North-Holland, Amsterdam (1963)
Pratt, V.: Modelling concurrency with partial orders. IJPP 15(1), 33–71 (1986)
Reisig, W.: Petri nets, an introduction. Springer, Heidelberg (1985)
Thatcher, J.W., Wright, J.B.: Generalized finite automata with an application to a decision problem of second order logic. Math. Syst. Theory 2, 57–82 (1968)
Valdes, J., Tarjan, R.E., Lawler, E.L.: The recognition of series-parallel digraphs. SIAM J. Comput. 11, 298–313 (1981)
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Lodaya, K., Weil, P. (1998). A Kleene Iteration for Parallelism. In: Arvind, V., Ramanujam, S. (eds) Foundations of Software Technology and Theoretical Computer Science. FSTTCS 1998. Lecture Notes in Computer Science, vol 1530. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-49382-2_33
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DOI: https://doi.org/10.1007/978-3-540-49382-2_33
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