Efficient reduction of nondeterministic automata with application to language inclusion testing

TL;DR

This paper introduces efficient algorithms for reducing nondeterministic automata sizes, enabling the handling of larger automata in verification tasks and improving the scalability of automata-based decision procedures.

Contribution

The paper presents novel reduction techniques combining pruning, saturation, and quotienting, along with lookahead simulations as efficient approximations for trace inclusion relations.

Findings

Algorithms scale slightly above quadratically in time complexity.

Automata size reduction exceeds previous techniques significantly.

Methods enable handling automata with over 1000 states in verification tasks.

Abstract

We present efficient algorithms to reduce the size of nondeterministic B\"uchi word automata (NBA) and nondeterministic finite word automata (NFA), while retaining their languages. Additionally, we describe methods to solve PSPACE-complete automata problems like language universality, equivalence, and inclusion for much larger instances than was previously possible ($\ge 1000$ states instead of 10-100). This can be used to scale up applications of automata in formal verification tools and decision procedures for logical theories. The algorithms are based on new techniques for removing transitions (pruning) and adding transitions (saturation), as well as extensions of classic quotienting of the state space. These techniques use criteria based on combinations of backward and forward trace inclusions and simulation relations. Since trace inclusion relations are themselves PSPACE-complete, we introduce lookahead simulations as good polynomial time computable approximations thereof. Extensive experiments show that the average-case time complexity of our algorithms scales slightly above quadratically. (The space complexity is worst-case quadratic.) The size reduction of the automata depends very much on the class of instances, but our algorithm consistently reduces the size far more than all previous techniques. We tested our algorithms on NBA derived from LTL-formulae, NBA derived from mutual exclusion protocols and many classes of random NBA and NFA, and compared their performance to the well-known automata tool GOAL.