Modular Mix-and-Match Complementation of B\"{u}chi Automata (Technical Report)

TL;DR

This paper introduces a modular algorithm for complementing nondeterministic B"uchi automata that leverages the structure of strongly connected components to improve efficiency and produce more compact automaton complements.

Contribution

It presents a flexible framework for BA complementation that combines multiple procedures, exploiting component structures for better performance and bounds.

Findings

Achieved an exponential improvement with an $O(4n)$ upper bound for elevator automata.

Implemented a prototype demonstrating the framework's effectiveness.

Experimental results show the approach complements existing methods and enhances BA inclusion checking.

Abstract

Complementation of nondeterministic B\"uchi automata (BAs) is an important problem in automata theory with numerous applications in formal verification, such as termination analysis of programs, model checking, or in decision procedures of some logics. We build on ideas from a recent work on BA determinization by Li et al. and propose a new modular algorithm for BA complementation. Our algorithm allows to combine several BA complementation procedures together, with one procedure for a subset of the BA's strongly connected components (SCCs). In this way, one can exploit the structure of particular SCCs (such as when they are inherently weak or deterministic) and use more efficient specialized algorithms, regardless of the structure of the whole BA. We give a general framework into which partial complementation procedures can be plugged in, and its instantiation with several algorithms. The framework can, in general, produce a complement with an Emerson-Lei acceptance condition, which can often be more compact. Using the algorithm, we were able to establish an exponentially better new upper bound of $O(4n)$ for complementation of the recently introduced class of elevator automata. We implemented the algorithm in a prototype and performed a comprehensive set of experiments on a large set of benchmarks, showing that our framework complements well the state of the art and that it can serve as a basis for future efficient BA complementation and inclusion checking algorithms.