Symbolic Automata: $\omega$-Regularity Modulo Theories

TL;DR

This paper extends symbolic automata to infinite words ($$-regular languages) using symbolic derivatives and transition terms, enabling symbolic model checking with a unified framework for automata and logics over infinite alphabets.

Contribution

It introduces a generalized framework for symbolic automata over infinite words, including new automata types, an alternation elimination algorithm, and a logic extension supporting regex complement.

Findings

Defined alternating and non-deterministic Bbuchi automata modulo theories

Developed an algorithm to convert alternating automata to non-deterministic automata

Extended LTL and PSL with symbolic automata features, supporting regex complement

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.