Checking Whether an Automaton Is Monotonic Is NP-complete

An automaton is monotonic if its states can be arranged in a linear order that is preserved by the action of every letter. We prove that the problem of deciding whether a given automaton is monotonic is NP-complete. The same result is obtained for oriented automata, whose states can be arranged in a cyclic order. Moreover, both problems remain hard under the restriction to binary input alphabets.Comment: 13 pages, 4 figures. CIAA 2015. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-22360-5_2

Synchronization Problems in Automata without Non-trivial Cycles

We study the computational complexity of various problems related to synchronization of weakly acyclic automata, a subclass of widely studied aperiodic automata. We provide upper and lower bounds on the length of a shortest word synchronizing a weakly acyclic automaton or, more generally, a subset of its states, and show that the problem of approximating this length is hard. We investigate the complexity of finding a synchronizing set of states of maximum size. We also show inapproximability of the problem of computing the rank of a subset of states in a binary weakly acyclic automaton and prove that several problems related to recognizing a synchronizing subset of states in such automata are NP-complete.Comment: Extended and corrected version, including arXiv:1608.00889. Conference version was published at CIAA 2017, LNCS vol. 10329, pages 188-200, 201

Preimage problems for deterministic finite automata

Given a subset of states $S$ of a deterministic finite automaton and a word $w$, the preimage is the subset of all states mapped to a state in $S$ by the action of $w$. We study three natural problems concerning words giving certain preimages. The first problem is whether, for a given subset, there exists a word \emph{extending} the subset (giving a larger preimage). The second problem is whether there exists a \emph{totally extending} word (giving the whole set of states as a preimage)---equivalently, whether there exists an \emph{avoiding} word for the complementary subset. The third problem is whether there exists a \emph{resizing} word. We also consider variants where the length of the word is upper bounded, where the size of the given subset is restricted, and where the automaton is strongly connected, synchronizing, or binary. We conclude with a summary of the complexities in all combinations of the cases

Revisiting Reachability in Timed Automata

We revisit a fundamental result in real-time verification, namely that the binary reachability relation between configurations of a given timed automaton is definable in linear arithmetic over the integers and reals. In this paper we give a new and simpler proof of this result, building on the well-known reachability analysis of timed automata involving difference bound matrices. Using this new proof, we give an exponential-space procedure for model checking the reachability fragment of the logic parametric TCTL. Finally we show that the latter problem is NEXPTIME-hard

Constraint LTL Satisfiability Checking without Automata

This paper introduces a novel technique to decide the satisfiability of formulae written in the language of Linear Temporal Logic with Both future and past operators and atomic formulae belonging to constraint system D (CLTLB(D) for short). The technique is based on the concept of bounded satisfiability, and hinges on an encoding of CLTLB(D) formulae into QF-EUD, the theory of quantifier-free equality and uninterpreted functions combined with D. Similarly to standard LTL, where bounded model-checking and SAT-solvers can be used as an alternative to automata-theoretic approaches to model-checking, our approach allows users to solve the satisfiability problem for CLTLB(D) formulae through SMT-solving techniques, rather than by checking the emptiness of the language of a suitable automaton A_{\phi}. The technique is effective, and it has been implemented in our Zot formal verification tool.Comment: 39 page

Pure Nash Equilibria in Concurrent Deterministic Games

We study pure-strategy Nash equilibria in multi-player concurrent deterministic games, for a variety of preference relations. We provide a novel construction, called the suspect game, which transforms a multi-player concurrent game into a two-player turn-based game which turns Nash equilibria into winning strategies (for some objective that depends on the preference relations of the players in the original game). We use that transformation to design algorithms for computing Nash equilibria in finite games, which in most cases have optimal worst-case complexity, for large classes of preference relations. This includes the purely qualitative framework, where each player has a single omega-regular objective that she wants to satisfy, but also the larger class of semi-quantitative objectives, where each player has several omega-regular objectives equipped with a preorder (for instance, a player may want to satisfy all her objectives, or to maximise the number of objectives that she achieves.)Comment: 72 page

Timed pushdown automata revisited

This paper contains two results on timed extensions of pushdown automata (PDA). As our first result we prove that the model of dense-timed PDA of Abdulla et al. collapses: it is expressively equivalent to dense-timed PDA with timeless stack. Motivated by this result, we advocate the framework of first-order definable PDA, a specialization of PDA in sets with atoms, as the right setting to define and investigate timed extensions of PDA. The general model obtained in this way is Turing complete. As our second result we prove NEXPTIME upper complexity bound for the non-emptiness problem for an expressive subclass. As a byproduct, we obtain a tight EXPTIME complexity bound for a more restrictive subclass of PDA with timeless stack, thus subsuming the complexity bound known for dense-timed PDA.Comment: full technical report of LICS'15 pape

Complexity of problems concerning reset words for some partial cases of automata

A word w is called a reset word for a deterministic finite automaton A if it maps all states of A to one state. A word w is called a compressing to M states for a deterministic finite automaton A if it maps all states of A to at most M states. We consider several subclasses of automata: aperiodic, D-trivial, monotonic, partially monotonic automata and automata with a zero state. For these subclasses we study the computational complexity of the following problems. Does there exist a reset word for a given automaton? Does there exist a reset word of given length for a given automaton? What is the length of the shortest reset word for a given automaton? Moreover, we consider complexity of the same problems for compressing words