Review: Seymour Ginsburg, Edwin H. Spanier, Semigroups, Presburger Formulas, and Languages

. Spot is a C ++ 17 library for LTL and ω -automata manip-ulation, with command-line utilities, and Python bindings. This paper summarizes its evolution over the past six years, since the release of Spot 2.0, which was the first version to support ω -automata with arbitrary acceptance conditions, and the last version presented at a conference. Since then, Spot has been extended with several features such as acceptance transformations, alternating automata, games, LTL synthesis, and more. We also shed some lights on the data-structure used to store automata.

From Spot 2.0 to Spot 2.10: What's New?

We investigate the connection between properties of formal languages and properties of their generating series, with a focus on the class of holonomic power series. We first prove a strong version of a conjecture by Castiglione and Massazza: weakly-unambiguous Parikh automata are equivalent to unambiguous two-way reversal bounded counter machines, and their multivariate generating series are holonomic. We then show that the converse is not true: we construct a language whose generating series is algebraic (thus holonomic), but which is inherently weakly-ambiguous as a Parikh automata language. Finally, we prove an effective decidability result for the inclusion problem for weakly-unambiguous Parikh automata, and provide an upper-bound on its complexity.

https://drops.dagstuhl.de/opus/volltexte/2020/12521/pdf/LIPIcs-ICALP-2020-114.pdf/

Weakly-Unambiguous Parikh Automata and Their Link to Holonomic Series

HyperLTL is a temporal logic that can express hyperproperties, i.e., properties that relate multiple execution traces of a system. Such properties are becoming increasingly important and naturally occur, e.g., in information-flow control, robustness, mutation testing, path planning, and causality checking. Thus far, complete model checking tools for HyperLTL have been limited to alternation-free formulas, i.e., formulas that use only universal or only existential trace quantification. Properties involving quantifier alternations could only be handled in an incomplete way, i.e., the verification might fail even though the property holds. In this paper, we present AutoHyper, an explicit-state automata-based model checker that supports full HyperLTL and is complete for properties with arbitrary quantifier alternations. We show that language inclusion checks can be integrated into HyperLTL verification, which allows AutoHyper to benefit from a range of existing inclusion-checking tools. We evaluate AutoHyper on a broad set of benchmarks drawn from different areas in the literature and compare it with existing (incomplete) methods for HyperLTL verification.

AutoHyper: Explicit-State Model Checking for HyperLTL

Parikh automata extend automata with counters whose values can only be tested at the end of the computation, with respect to membership into a semi-linear set. Parikh automata have found several applications, for instance in transducer theory, as they enjoy decidable emptiness problem. In this paper, we study two-way Parikh automata. We show that emptiness becomes undecidable in the non-deterministic case. However, it is PSpace-C when the number of visits to any input position is bounded and the semi-linear set is given as an existential Presburger formula. We also give tight complexity bounds for the inclusion, equivalence and universality problems. Finally, we characterise precisely the complexity of those problems when the semi-linear constraint is given by an arbitrary Presburger formula.

Two-way Parikh Automata

We introduce a logic to express structural properties of automata with string inputs and, possibly, outputs in some monoid. In this logic, the set of predicates talking about the output values is parametric, and we provide sufficient conditions on the predicates under which the model-checking problem is decidable. We then consider three particular automata models (finite automata, transducers and automata weighted by integers – sum-automata –) and instantiate the generic logic for each of them. We give tight complexity results for the three logics and the model-checking problem, depending on whether the formula is fixed or not. We study the expressiveness of our logics by expressing classical structural patterns characterising for instance finite ambiguity and polynomial ambiguity in the case of finite automata, determinisability and finite-valuedness in the case of transducers and sum-automata. Consequently to our complexity results, we directly obtain that these classical properties can be decided in PTime.

A Pattern Logic for Automata with Outputs

Two-Way Parikh Automata

We introduce a logic to express structural properties of automata with string inputs and, possibly, outputs in some monoid. In this logic, the set of predicates talking about the output values is p...

. We propose a novel algorithm to decide the language inclusion between (nondeterministic) B¨uchi 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 B¨uchi 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.

Nicolas Mazzocchi

Papers

A Pattern Logic for Automata with Outputs

Two-way Parikh Automata

Two-Way Parikh Automata

A Pattern Logic for Automata with Outputs

FORQ-based Language Inclusion Formal Testing