Word Automata Research Articles

From complementation to certification

In the automata-theoretic approach to model checking we check the emptiness of the product of a system S with an automaton A ¬ ψ for the complemented specification. This gives rise to two automata-theoretic problems: complementation of word automata, which is used in order to generate A ¬ ψ , and the emptiness problem, to which model checking is reduced. Both problems have numerous other applications, and have been extensively studied for nondeterministic Büchi word automata (NBW). Nondeterministic generalized Büchi word automata (NGBW) have become popular in specification and verification and are now used in applications traditionally assigned to NBW. This is due to their richer acceptance condition, which leads to automata with fewer states and a simpler underlying structure. In this paper we analyze runs of NGBW and use the analysis in order to describe a new complementation construction and a symbolic emptiness algorithm for NGBW. The complementation construction exponentially improves the best known construction for NGBW and is easy to implement. The emptiness algorithm is almost identical to a known variant of the Emerson–Lei algorithm, and our contribution is the strong relation we draw between the complementation construction and the emptiness algorithm—both naturally follow from the analysis of the runs, which easily implies their correctness. This relation leads to a new certified model-checking procedure, where a positive answer to the model-checking query is accompanied by a certificate whose correctness can be checked by methods independent of the model checker. Unlike certificates generated in previous works on certified model checking, our analysis enables us to generate a certificate that can be checked automatically and symbolically.

Relating word and tree automata

In the automata-theoretic approach to verification, we translate specifications to automata. Complexity considerations motivate the distinction between different types of automata. Already in the 60s, it was known that deterministic Büchi word automata are less expressive than nondeterministic Büchi word automata. The proof is easy and can be stated in a few lines. In the late 60s, Rabin proved that Büchi tree automata are less expressive than Rabin tree automata. This proof is much harder. In this work we relate the expressiveness gap between deterministic and nondeterministic Büchi word automata and the expressiveness gap between Büchi and Rabin tree automata. We consider tree automata that recognize derived languages. For a word language L , the derived language of L , denoted L △ , is the set of all trees all of whose paths are in L . Since often we want to specify that all the computations of the program satisfy some property, the interest in derived languages is clear. Our main result shows that L is recognizable by a nondeterministic Büchi word automaton but not by a deterministic Büchi word automaton iff L △ is recognizable by a Rabin tree automaton and not by a Büchi tree automaton. Our result provides a simple explanation for the expressiveness gap between Büchi and Rabin tree automata. Since the gap between deterministic and nondeterministic Büchi word automata is well understood, our result also provides a characterization of derived languages that can be recognized by Büchi tree automata. Finally, it also provides an exponential determinization of Büchi tree automata that recognize derived languages.

Decision procedures for inductive Boolean functions based on alternating automata

We show how alternating automata provide decision procedures for the equality of inductively defined Boolean functions and present applications to reasoning about parameterized families of circuits. We use alternating word automata to formalize families of linearly structured circuits and alternating tree automata to formalize families of tree structured circuits. We provide complexity bounds for deciding the equality of function (or circuit) families and show how our decision procedures can be implemented using BDDs. In comparison to previous work, our approach is simpler, has better complexity bounds, and, in the case of tree-structured families, is more general.