Affiliations: Department of Computer Science, University of California, Santa Barbara, CA 93106, USA. ibarra@cs.ucsb.edu | Helsinki Institute for Information Technology (HIIT), Department of Information and Computer Science, Aalto University, P. O. Box 15400, FI-00076, Aalto, Finland. shinnosuke.seki@aalto.fi
Note: [] Supported in part by NSF Grant CCF-1117708. Address for correspondence: Department of Computer Science, University of California, Santa Barbara, CA 93106, USA
Note: [] Supported in part by Academy of Finland, Postdoctoral Researcher Grant No. 13266670/T30606.
Abstract: 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.
