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Abstract: Hierarchical graph definitions allow a modular description of structures using modules for the specification of repeated substructures. Beside this modularity, hierarchical graph definitions allow us to specify structures of exponential size using polynomial size descriptions. In many cases, this succinctness increases the computational complexity of decision problems when input structures are defined hierarchically. In this paper, the model-checking problem for first-order logic (FO), monadic second-order logic (MSO), and second-order logic (SO) on hierarchically defined input structures is investigated. It is shown that in general these model-checking problems are exponentially harder than their non-hierarchical counterparts, where the input structures are given explicitly. As a consequence, several new complete problems for the levels of the polynomial time hierarchy and the exponential time hierarchy are obtained. Based on classical results of Gaifman and Courcelle, two restrictions on the structure of hierarchical graph definitions that lead to more efficient model-checking algorithms are presented. [Copyright &y& Elsevier]

Abstract: This paper investigates the word problem for inverse monoids generated by a set Γ subject to relations of the form e = f, where e and f are both idempotents in the free inverse monoid generated by Γ. It is shown that for every fixed monoid of this form the word problem can be solved both in linear time on a RAM as well as in deterministic logarithmic space, which solves an open problem of Margolis and Meakin. For the uniform word problem, where the presentation is part of the input, EXPTIME-completeness is shown. For the Cayley-graphs of these monoids, it is shown that the first-order theory with regular path predicates is decidable. Regular path predicates allow to state that there is a path from a node x to a node y that is labeled with a word from some regular language. As a corollary, the decidability of the generalized word problem is deduced. [Copyright &y& Elsevier]

We consider a compressed form of the word problem for finitely presented monoids, where the input consists of two compressed representations of words over the generators of a monoid M, and we ask whether these two words represent the same monoid element of M. Words are compressed using straight-line programs, i.e., context-free grammars that generate exactly one word. For several classes of finitely presented monoids we obtain completeness results for complexity classes in the range from P to EXPSPACE. As a by-product of our results on compressed word problems we obtain a fixed deterministic context-free language with a PSPACE-complete compressed membership problem. The existence of such a language was open so far. Finally, we will investigate the complexity of the compressed membership problem for various circuit complexity classes. [ABSTRACT FROM AUTHOR]

Several complexity and decidability results for automatic monoids are shown: (i) there exists an automatic monoid with a P-complete word problem, (ii) there exists an automatic monoid such that the first-order theory of the corresponding Cayley-graph is not elementary decidable, and (iii) there exists an automatic monoid such that reachability in the corresponding Cayley-graph is undecidable. Moreover, it is shown that for every hyperbolic group the word problem belongs to LOGCFL, which improves a result of Cai [8]. [ABSTRACT FROM AUTHOR]

We study the notion of safe realizability for high-level message sequence charts (HMSCs) (Proceedings of the 28th International Colloquium on Automata, Languages and Programming (ICALP 2001), Crete (Greece), Lecture Notes in Computer Science, Vol. 2076, Springer, Berlin, 2001, pp. 797–808). We show that safe realizability is EXPSPACE-complete for bounded HMSCs but undecidable for the class of all HMSCs. This solves two open problems from Alur et al. Moreover we prove that safe realizability is also EXPSPACE-complete for the larger class of globally-cooperative HMSCs. [Copyright &y& Elsevier]