History-deterministic Parikh Automata

Parikh automata extend finite automata by counters that can be tested for membership in a semilinear set, but only at the end of a run. Thereby, they preserve many of the desirable properties of finite automata. Deterministic Parikh automata are strictly weaker than nondeterministic ones, but enjoy better closure and algorithmic properties. This state of affairs motivates the study of intermediate forms of nondeterminism. Here, we investigate history-deterministic Parikh automata, i.e., automata whose nondeterminism can be resolved on the fly. This restricted form of nondeterminism is well-suited for applications which classically call for determinism, e.g., solving games and composition. We show that history-deterministic Parikh automata are strictly more expressive than deterministic ones, incomparable to unambiguous ones, and enjoy almost all of the closure and some of the algorithmic properties of deterministic automata.Comment: arXiv admin note: text overlap with arXiv:2207.0769

Parikh One-Counter Automata

Counting abilities in finite automata are traditionally provided by two orthogonal extensions: adding a single counter that can be tested for zeroness at any point, or adding ?-valued counters that are tested for equality only at the end of runs. In this paper, finite automata extended with both types of counters are introduced. They are called Parikh One-Counter Automata (POCA): the "Parikh" part referring to the evaluation of counters at the end of runs, and the "One-Counter" part to the single counter that can be tested during runs. Their expressiveness, in the deterministic and nondeterministic variants, is investigated; it is shown in particular that there are deterministic POCA languages that cannot be expressed without nondeterminism in the original models. The natural decision problems are also studied; strikingly, most of them are no harder than in the original models. A parametric version of nonemptiness is also considered

Remarks on Parikh-recognizable omega-languages

Several variants of Parikh automata on infinite words were recently introduced by Guha et al. [FSTTCS, 2022]. We show that one of these variants coincides with blind counter machine as introduced by Fernau and Stiebe [Fundamenta Informaticae, 2008]. Fernau and Stiebe showed that every $\omega$-language recognized by a blind counter machine is of the form $\bigcup_iU_iV_i^\omega$ for Parikh recognizable languages $U_i, V_i$, but blind counter machines fall short of characterizing this class of $\omega$-languages. They posed as an open problem to find a suitable automata-based characterization. We introduce several additional variants of Parikh automata on infinite words that yield automata characterizations of classes of $\omega$-language of the form $\bigcup_iU_iV_i^\omega$ for all combinations of languages $U_i, V_i$ being regular or Parikh-recognizable. When both $U_i$ and $V_i$ are regular, this coincides with B\"uchi's classical theorem. We study the effect of $\varepsilon$-transitions in all variants of Parikh automata and show that almost all of them admit $\varepsilon$-elimination. Finally we study the classical decision problems with applications to model checking.Comment: arXiv admin note: text overlap with arXiv:2302.04087, arXiv:2301.0896

Parikh Automata over Infinite Words

Parikh automata extend finite automata by counters that can be tested for membership in a semilinear set, but only at the end of a run, thereby preserving many of the desirable algorithmic properties of finite automata. Here, we study the extension of the classical framework onto infinite inputs: We introduce reachability, safety, B\"uchi, and co-B\"uchi Parikh automata on infinite words and study expressiveness, closure properties, and the complexity of verification problems. We show that almost all classes of automata have pairwise incomparable expressiveness, both in the deterministic and the nondeterministic case; a result that sharply contrasts with the well-known hierarchy in the $\omega$-regular setting. Furthermore, emptiness is shown decidable for Parikh automata with reachability or B\"uchi acceptance, but undecidable for safety and co-B\"uchi acceptance. Most importantly, we show decidability of model checking with specifications given by deterministic Parikh automata with safety or co-B\"uchi acceptance, but also undecidability for all other types of automata. Finally, solving games is undecidable for all types

Bounded Parikh Automata

The Parikh finite word automaton model (PA) was introduced and studied by Klaedtke and Ruess in 2003. Here, by means of related models, it is shown that the bounded languages recognized by PA are the same as those recognized by deterministic PA. Moreover, this class of languages is the class of bounded languages whose set of iterations is semilinear.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Two-Way Parikh Automata

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 a 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

Real-time and Probabilistic Temporal Logics: An Overview

Over the last two decades, there has been an extensive study on logical formalisms for specifying and verifying real-time systems. Temporal logics have been an important research subject within this direction. Although numerous logics have been introduced for the formal specification of real-time and complex systems, an up to date comprehensive analysis of these logics does not exist in the literature. In this paper we analyse real-time and probabilistic temporal logics which have been widely used in this field. We extrapolate the notions of decidability, axiomatizability, expressiveness, model checking, etc. for each logic analysed. We also provide a comparison of features of the temporal logics discussed