Abstract
We consider commutative string rewriting systems (Vector Addition Systems, Petri nets), i.e., string rewriting systems in which all pairs of letters commute. We are interested in reachability: given a rewriting system R and words v and w, can v be rewritten to w by applying rules from R? A famous result states that reachability is decidable for commutative string rewriting systems. We show that reachability is decidable for a union of two such systems as well. We obtain, as a special case, that if h:U→S and g:U→T are homomorphisms of commutative monoids, then their pushout has a decidable word problem. Finally, we show that, given commutative monoids U, S and T satisfying S ∩ T = U, it is decidable whether there exists a monoid M such that \(S\cup T\subseteq M\); we also show that the problem remains decidable if we require M to be commutative, too.
First author supported by the EC Research Training Network Games, second author by EC project Sensoria (No. 016004).
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References
Mayr, E.W.: An algorithm for the general Petri net reachability problem. SIAM J. Comp. 13(3), 441–459 (1984)
Kosaraju, S.R.: Decidability of reachability in vector addition systems (preliminary version). In: STOC’82, ACM, pp. 267–281. ACM Press, New York (1982)
Sapir, M.V.: Algorithmic problems for amalgams of finite semigroups. J. Algebra 229, 514–531 (2000)
Hoffman, P.: Unions of equational monadic theories. In: Pfenning, F. (ed.) RTA 2006. LNCS, vol. 4098, pp. 81–95. Springer, Heidelberg (2006)
Howie, J.M.: Fundamentals of Semigroup Theory. London Math. Soc. Monogr (N.S.), vol. 12. Oxford Univ. Press, Oxford (1996)
Howie, J.M.: Embedding theorems with amalgamation for semigroups. Proc. London Math. 12(3), 511–534 (1962)
Howie, J.M.: Epimorphisms and amalgamations: A survey of recent progress. Coll. Math. Soc. J. Bolyai 39, 63–82 (1981)
Hall, T.E.: Representation extension and amalgamation for semigroups. Quart. J. Math. Oxford 29(2), 309–334 (1978)
Birget, J.C., Margolis, S., Meakin, J.: On the word problem for tensor products and amalgams of monoids. Intnl. J. Alg. Comp. 9, 271–294 (1999)
Taiclin, M.A.: Algorithmic problems for commutative semigroups. Soviet Math. Dokl. 9(1), 201–204 (1968)
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Bojańczyk, M., Hoffman, P. (2007). Reachability in Unions of Commutative Rewriting Systems Is Decidable. In: Thomas, W., Weil, P. (eds) STACS 2007. STACS 2007. Lecture Notes in Computer Science, vol 4393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70918-3_53
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DOI: https://doi.org/10.1007/978-3-540-70918-3_53
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