Chistikov, D; Haase, C; (2016) The taming of the semi-linear set. In: Chatzigiannakis, I and Mitzenmacher, M and Rabani, Y and Sangiorgi, D, (eds.) 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). (pp. 128:1-128:13). Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik: Dagstuhl, Germany.

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Abstract

Semi-linear sets, which are rational subsets of the monoid (Z^d,+), have numerous applications in theoretical computer science. Although semi-linear sets are usually given implicitly, by formulas in Presburger arithmetic or by other means, the effect of Boolean operations on semi-linear sets in terms of the size of description has primarily been studied for explicit representations. In this paper, we develop a framework suitable for implicitly presented semi-linear sets, in which the size of a semi-linear set is characterized by its norm—the maximal magnitude of a generator. We put together a toolbox of operations and decompositions for semi-linear sets which gives bounds in terms of the norm (as opposed to just the bit-size of the description), a unified presentation, and simplified proofs. This toolbox, in particular, provides exponentially better bounds for the complement and set-theoretic difference. We also obtain bounds on unambiguous decompositions and, as an application of the toolbox, settle the complexity of the equivalence problem for exponent-sensitive commutative grammars.

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