We present a Myhill-Nerode style characterization for languages recognized by one-clock deterministic timed automata ( $$1$$ -DTA). Although there is only one clock, distinct automata may reset it differently along the same word. This adds a significant challenge in the search for a canonical automaton. Our characterization is based on a new perspective of $$1$$ -DTAs in terms of “half-integral” words that they accept, along with the reset information encoded by them We apply our results to develop $$\mathsf {L^*}$$ style algorithms that learn the canonical $$1$$ -DTA.

The well-known Nerode equivalence for finite words plays a fundamental role in our understanding of the class of regular languages. The equivalence leads to the Myhill-Nerode theorem and a canonical automaton, which in turn, is the basis of several automata learning algorithms. A Nerode-like equivalence has been studied for various classes of timed languages. In this work, we focus on timed automata with integer resets. This class is known to have good automata-theoretic properties and is also useful for practical modeling. Our main contribution is a Nerode-style equivalence for this class that depends on a constant K. We show that the equivalence leads to a Myhill-Nerode theorem and a canonical one-clock integer-reset timed automaton with maximum constant K. Based on the canonical form, we develop an Angluin-style active learning algorithm whose query complexity is polynomial in the size of the canonical form.

Abstract We define novel algorithms for the inclusion problem between two visibly pushdown languages of infinite words, an EXPTime -complete problem. Our algorithms search for counterexamples to inclusion in the form of ultimately periodic words i.e. words of the form $$uv^{\omega }$$ u v ω where $$u$$ u and $$v$$ v are finite words. They are parameterized by a pair of quasiorders telling which ultimately periodic words need not be tested as counterexamples to inclusion without compromising completeness. The pair of quasiorders enables distinct reasoning for prefixes and periods of ultimately periodic words thereby allowing to discard even more words compared to using the same quasiorder for both. We put forward two families of quasiorders: the state-based quasiorders based on automata and the syntactic quasiorders based on languages. We also implemented our algorithm and conducted an empirical evaluation on benchmarks from software verification.

Abstract We propose a novel algorithm to decide the language inclusion between (nondeterministic) Büchi automata, a PSpace-complete problem. Our approach, like others before, leverage a notion of quasiorder to prune the search for a counterexample by discarding candidates which are subsumed by others for the quasiorder. Discarded candidates are guaranteed to not compromise the completeness of the algorithm. The novelty of our work lies in the quasiorder used to discard candidates. We introduce FORQs (family of right quasiorders) that we obtain by adapting the notion of family of right congruences put forward by Maler and Staiger in 1993. We define a FORQ-based inclusion algorithm which we prove correct and instantiate it for a specific FORQ, called the structural FORQ, induced by the Büchi automaton to the right of the inclusion sign. The resulting implementation, called Forklift, scales up better than the state-of-the-art on a variety of benchmarks including benchmarks from program verification and theorem proving for word combinatorics. Artifact: https://doi.org/10.5281/zenodo.6552870

We introduce an algorithmic framework to decide whether inclusion holds between languages of infinite words over a finite alphabet. Our approach falls within the class of Ramsey-based methods and relies on a least fixpoint characterization of ω-languages leveraging ultimately periodic infinite words of type uv^ω, with u a finite prefix and v a finite period of an infinite word. We put forward an inclusion checking algorithm between Büchi automata, called BAInc, designed as a complete abstract interpretation using a pair of well-quasiorders on finite words. BAInc is quite simple: it consists of two least fixpoint computations (one for prefixes and the other for periods) manipulating finite sets (of pairs) of states compared by set inclusion, so that language inclusion holds when the sets (of pairs) of states of the fixpoints satisfy some basic conditions. We implemented BAInc in a tool called BAIT that we experimentally evaluated against the state-of-the-art. We gathered, in addition to existing benchmarks, a large number of new case studies stemming from program verification and word combinatorics, thereby significantly expanding both the scope and size of the available benchmark set. Our experimental results show that BAIT advances the state-of-the-art on an overwhelming majority of these benchmarks. Finally, we demonstrate the generality of our algorithmic framework by instantiating it to the inclusion problem of Büchi pushdown automata into Büchi automata.