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Semilinear Sets and Counter Machines : a Brief Survey

100%

Ibarra O. H. , Seki S.

Fundamenta Informaticae

2015

tom Vol. 138, nr 1/2

61--76

Semilinear sets are one of the most important concepts in theoretical computer science, as illustrated by the fact that the set of nonnegative integer solutions to any system of Diophantine equations is semilinear. Parikh’s theorem enables us to represent any semilinear set as a pushdown automaton (PDA).We summarize recent results on the descriptional complexity of conversions among different representations of a semilinear set: as a vector set (conventional), a finite automaton (FA), a PDA, etc.. We also discuss semilinearity-preserving operations like union, intersection, and complement. We use Parikh’s theorem to enlarge the class of finite-state machines that can represent semilinear sets. In particular, we give a simpler proof of a known result that characterizes semilinear sets in terms of machines with reversal-bounded counters. We then investigate the power of such a machine with only one counter in the context of a long-standing conjecture about repetition on words.

Visibly Pushdown Automata and Transducers with Counters

100%

Ibarra O. H.

Fundamenta Informaticae

2016

tom Vol. 148, nr 3/4

291--308

We generalize the models of visibly pushdown automata (VPDAs) and visibly pushdown transducers (VPDTs) by equipping them with reversal-bounded counters. We show that some of the results for VPDAs and VPDTs (e.g., closure under intersection and decidability of emptiness for VPDA languages) carry over to the generalized models, but other results (e.g., determinization and closure under complementation) do not carry over, in general. We define a model that combines the desirable features of a VPDA and reversal-bounded counters, called 2- phase VPCM, and show that the deterministic and nondeterministic versions are equivalent and that the family of languages they define is closed under Boolean operations and has decidable emptiness, infiniteness, disjointness, containment, and equivalence problems. We also investigate the finite-ambiguity and finite-valuedness problems concerning these devices.