[Paper Review] Advanced Automata Minimization

Lorenzo Clemente, Richard Mayr|arXiv (Cornell University)|Oct 24, 2012

semigroups and automata theory20 references4 citations

TL;DR

This paper presents an advanced automata minimization algorithm for nondeterministic Büchi word automata that uses novel transition pruning based on combined forward and backward trace inclusions, approximated via lookahead simulations. The method achieves near-quadratic average-case complexity and significantly outperforms prior techniques, enabling scalable solving of PSPACE-complete problems like language inclusion and equivalence on much larger automata instances (1–3 orders of magnitude larger).

ABSTRACT

We present an efficient algorithm to reduce the size of nondeterministic Buchi word automata, while retaining their language. Additionally, we describe methods to solve PSPACE-complete automata problems like universality, equivalence and inclusion for much larger instances (1-3 orders of magnitude) than before. This can be used to scale up applications of automata in formal verification tools and decision procedures for logical theories. The algorithm is based on new transition pruning techniques. These use criteria based on combinations of backward and forward trace inclusions. Since these relations are themselves PSPACE-complete, we describe methods to compute good approximations of them in polynomial time. Extensive experiments show that the average-case complexity of our algorithm scales quadratically. The size reduction of the automata depends very much on the class of instances, but our algorithm consistently outperforms all previous techniques by a wide margin. We tested our algorithm on Buchi automata derived from LTL-formulae, many classes of random automata and automata derived from mutual exclusion protocols, and compared its performance to the well-known automata tool GOAL.

Motivation & Objective

• Address the scalability limitations of PSPACE-complete automata problems such as universality, equivalence, and inclusion checking in formal verification.
• Develop a practical minimization technique that reduces Büchi automata size without altering their language, improving efficiency for downstream verification tasks.
• Overcome the limitations of traditional simulation-based minimization by introducing refined trace inclusion approximations using lookahead simulations.
• Enable the handling of much larger automata instances—1 to 3 orders of magnitude larger—than previously feasible with existing tools like GOAL.
• Provide a robust, parallelizable transition pruning mechanism based on hybrid forward and backward trace inclusion criteria to maximize reduction while preserving language.

Proposed method

• Introduce transition pruning based on combinations of forward and backward trace inclusions, ensuring language preservation during reduction.
• Develop lookahead simulations as polynomial-time approximations of full trace inclusions, which are themselves PSPACE-complete.
• Use parallel pruning of transitions by pre-computing simulation and trace inclusion relations, avoiding repeated recomputation after each removal.
• Apply a fixpoint iteration process that repeatedly applies pruning until no further reductions are possible, ensuring maximal compression.
• Integrate the minimization pipeline into a three-step language inclusion checking algorithm: (1) fast simulation checks, (2) lookahead simulation, and (3) fallback to Piterman’s construction.
• Optimize performance by tuning lookahead depth (e.g., k=12) and using least-squares fitting to model average-case complexity as a function of automaton size.

Experimental results

Research questions

• RQ1 Can we design a minimization algorithm that achieves significant size reduction for Büchi automata while maintaining polynomial-time complexity?
• RQ2 To what extent can lookahead simulations approximate full trace inclusions in practice, and how does this affect minimization quality?
• RQ3 Does the proposed method enable solving PSPACE-complete automata problems (e.g., language inclusion) on larger instances than previous approaches?
• RQ4 How does the average-case complexity of the algorithm scale with automaton size, and can it achieve near-quadratic performance despite multiple simulation computations?
• RQ5 Can the method be applied effectively across diverse automaton classes, including LTL-derived, random, and protocol-based automata?

Key findings

• The Heavy-12 variant of the algorithm achieves near-quadratic average-case complexity, with fitted exponents of 2.05–2.39 across different density classes, indicating strong scalability.
• For automata with transition density td=2.0, the method reduced minimization time to an average of 258 seconds on 1000-state automata, solving all 300 language inclusion instances.
• At td=1.6, the algorithm solved 98% of 300 language inclusion instances (294/300), compared to only 13 solved by GOAL and fair simulation under the same timeout.
• The method consistently outperformed GOAL and fair simulation across all tested densities, solving up to 143 included instances at td=2.0, while GOAL solved only one.
• The average transition reduction rate was 65% for td=1.4, dropping to 2% at td=2.0 for n=1000, indicating strong performance on dense but non-trivial instances.
• The lookahead depth of 12 proved sufficient across all tested sizes (n=50 to n=1000), showing that larger automata do not require increased lookahead for effective minimization.