The Intersection Problem for Finite Monoids

We investigate the intersection problem for finite monoids, which asks for a given set of regular languages, represented by recognizing morphisms to finite monoids from a variety V, whether there exists a word contained in their intersection. Our main result is that the problem is PSPACE-complete if V is contained in DS and NP-complete if V is non-trivial and contained in DO. Our NP-algorithm for the case that V is contained in DO uses novel methods, based on compression techniques and combinatorial properties of DO. We also show that the problem is log-space reducible to the intersection problem for deterministic finite automata (DFA) and that a variant of the problem is log-space reducible to the membership problem for transformation monoids. In light of these reductions, our hardness results can be seen as a generalization of both a classical result by Kozen and a theorem by Beaudry, McKenzie and Therien.Comment: Extended version of a paper accepted to STACS 201

Deciding FO-definability of regular languages

We prove that, similarly to known PSpace-completeness of recognising FO(<)-definability of the language L(A) of a DFA A, deciding bothFO(<,equiv)- and FO(<,MOD)-definability (corresponding to circuit complexity in AC0 and ACC0) are PSpace-complete. We obtain these results by first showing that known algebraic characterisations of FO-definability of L(A) can be captured by `localisable' properties of the transition monoid of A. Using our criterion, we then generalise the known proof of PSpace-hardness of FO(<)-definability, and establish the upper bounds not only for arbitrary DFAs but also for 2NFAs

Deciding FO-rewritability of ontology-mediated queries in linear temporal logic

Our concern is the problem of determining the data complexity of answering an ontology-mediated query (OMQ) given in linear temporal logic LTL over (Z,<) and deciding whether it is rewritable to an FO(<)-query, possibly with extra predicates. First, we observe that, in line with the circuit complexity and FO-definability of regular languages, OMQ answering in AC0, ACC0 and NC1 coincides with FO(<,\equiv)-rewritability using unary predicates x \equiv 0 mod n), FO(<,MOD)-rewritability, and FO(RPR)-rewritability using relational primitive recursion, respectively. We then show that deciding FO(<)-, \FO(<,\equiv)- and FO(<,MOD)-rewritability of LTL OMQs is ExpSpace-complete, and that these problems become PSpace-complete for OMQs with a linear Horn ontology and an atomic query, and also a positive query in the cases of FO(<)- and FO(<,\equiv)-rewritability. Further, we consider FO(<)-rewritability of OMQs with a binary-clause ontology and identify OMQ classes, for which deciding it is PSpace-, Pi_2^p- and coNP-complete

Regular expression star-freeness is PSPACE-complete

It is proved that the problem of deciding if a regular expression denotes a star-free language is PSPACE-complete. The paper also includes a new proof of the PSPACE-completeness of the finite automaton aperiodicity problem

Deciding FO-rewritability of Regular Languages and Ontology-Mediated Queries in Linear Temporal Logic

Our concern is the problem of determining the data complexity of answering an ontology-mediated query (OMQ) formulated in linear temporal logic LTL over (Z,<) and deciding whether it is rewritable to an FO(<)-query, possibly with some extra predicates. First, we observe that, in line with the circuit complexity and FO-definability of regular languages, OMQ answering in AC0, ACC0 and NC1 coincides with FO(<,≡)-rewritability using unary predicates x ≡ 0 (mod n), FO(<,MOD)-rewritability, and FO(RPR)-rewritability using relational primitive recursion, respectively. We prove that, similarly to known PSᴘᴀᴄᴇ-completeness of recognising FO(<)-definability of regular languages, deciding FO(<,≡)- and FO(<,MOD)-definability is also PSᴘᴀᴄᴇ-complete (unless ACC0 = NC1). We then use this result to show that deciding FO(<)-, FO(<,≡)- and FO(<,MOD)-rewritability of LTL OMQs is ExᴘSᴘᴀᴄᴇ-complete, and that these problems become PSᴘᴀᴄᴇ-complete for OMQs with a linear Horn ontology and an atomic query, and also a positive query in the cases of FO(<)- and FO(<,≡)-rewritability. Further, we consider FO(<)-rewritability of OMQs with a binary-clause ontology and identify OMQ classes, for which deciding it is PSᴘᴀᴄᴇ-, Π2p- and coNP-complete

28th International Symposium on Temporal Representation and Reasoning (TIME 2021)

The 28th International Symposium on Temporal Representation and Reasoning (TIME 2021) was planned to take place in Klagenfurt, Austria, but had to move to an online conference due to the insecurities and restrictions caused by the pandemic. Since its frst edition in 1994, TIME Symposium is quite unique in the panorama of the scientifc conferences as its main goal is to bring together researchers from distinct research areas involving the management and representation of temporal data as well as the reasoning about temporal aspects of information. Moreover, TIME Symposium aims to bridge theoretical and applied research, as well as to serve as an interdisciplinary forum for exchange among researchers from the areas of artifcial intelligence, database management, logic and verifcation, and beyond