Symbolic Automata: $\omega$-Regularity Modulo Theories

Abstract

Symbolic automata are finite state automata that support potentially infinite alphabets, such as the set of rational numbers, generally applied to regular expressions/languages over finite words. In symbolic automata (or automata modulo theories), an alphabet is represented by an effective Boolean algebra, supported by a decision procedure for satisfiability. Regular languages over infinite words (so called $\omega$-regular languages) have a rich history paralleling that of regular languages over finite words, with well known applications to model checking via B\"uchi automata and temporal logics. We generalize symbolic automata to support $\omega$-regular languages via symbolic transition terms and symbolic derivatives, bringing together a variety of classic automata and logics in a unified framework that provides all the necessary ingredients to support symbolic model checking modulo $A$, $NBW_A$. In particular, we define: (1) alternating B\"uchi automata modulo $A$, $ABW_A$ as well (non-alternating) non-deterministic B\"uchi automata modulo $A$, $NBW_A$; (2) an alternation elimination algorithm that incrementally constructs an $NBW_A$ from an $ABW_A$, and can also be used for constructing the product of two $NBW_A$'s; (3) a definition of linear temporal logic (LTL) modulo $A$ that generalizes Vardi's construction of alternating B\"uchi automata from LTL, using (2) to go from LTL modulo $A$ to $NBW_A$ via $ABW_A$. Finally, we present a combination of LTL modulo $A$ with extended regular expressions modulo $A$ that generalizes the Property Specification Language (PSL). Our combination allows regex complement, that is not supported in PSL but can be supported naturally by using symbolic transition terms.