Title:

Algorithmic verification problems in automatatheoretic settings

Problems in formal verification are often stated in terms of finite automata and extensions thereof. In this thesis we investigate several such algorithmic problems. In the first part of the thesis we develop a theory of completeness thresholds in Bounded Model Checking. A completeness threshold for a given model M and a specification φ is a bound k such that, if no counterexample to φ of length k or less can be found in M, then M in fact satisfies φ. We settle a problem of Kroening et al. [KOS^{+}11] in the affirmative, by showing that the linearity problem for both regular and ωregular specifications (provided as finite automata and Buchi automata respectively) is PSPACEcomplete. Moreover, we establish the following dichotomies: for regular specifications, completeness thresholds are either linear or exponential, whereas for ωregular specifications, completeness thresholds are either linear or at least quadratic in the recurrence diameter of the model under consideration. Given a formula in a temporal logic such as LTL or MTL, a fundamental problem underpinning automatabased model checking is the complexity of evaluating the formula on a given finite word. For LTL, the complexity of this task was recently shown to be in NC [KF09]. In the second part of the thesis we present an NC algorithm for MTL, a quantitative (or metric) extension of LTL, and give an AC^{1} algorithm for UTL, the unary fragment of LTL. We then establish a connection between LTL path checking and planar circuits which, among others, implies that the complexity of LTL path checking depends on the Boolean connectives allowed: adding Boolean exclusive or yields a temporal logic with Pcomplete pathchecking problem. In the third part of the thesis we study the decidability of the reachability problem for parametric timed automata. The problem was introduced over 20 years ago by Alur, Henzinger, and Vardi [AHV93]. It is known that for three or more parametric clocks the problem is undecidable. We translate the problem to reachability questions in certain extensions of parametric onecounter machines. By further reducing to satisfiability in Presburger arithmetic with divisibility, we obtain decidability results for several classes of parametric onecounter machines. As a corollary, we show that, in the case of a single parametric clock (with arbitrarily many nonparametric clocks) the reachability problem is NEXPcomplete, improving the nonelementary decision procedure of Alur et al. The case of two parametric clocks is open. Here, we show that the reachability is decidable in this case of automata with a single parameter.
