# Simulations in Rank-Based B\"uchi Automata Complementation (Technical Report)

## TL;DR

This paper introduces simulation-based techniques to optimize the size of automata produced by rank-based B"uchi automata complementation, making the process more efficient in practice.

## Contribution

It proposes novel methods using simulation relations to reduce automaton size in rank-based complementation, improving practical efficiency.

## Key findings

- Techniques significantly reduce automaton size in experiments.
- Methods effectively ignore non-contributing macrostates.
- Saturation with simulation-smaller states decreases macrostate count.

## Abstract

Complementation of B\"uchi automata is an essential technique used in some approaches for termination analysis of programs. The long search for an optimal complementation construction climaxed with the work of Schewe, who proposed a worst-case optimal rank-based procedure that generates complements of a size matching the theoretical lower bound of $(0.76n)^n$, modulo a polynomial factor of $O(n^2)$. Although worst-case optimal, the procedure in many cases produces automata that are unnecessarily large. In this paper, we propose several ways of how to use the direct and delayed simulation relations to reduce the size of the automaton obtained in the rank-based complementation procedure. Our techniques are based on either (i) ignoring macrostates that cannot be used for accepting a word in the complement or (ii) saturating macrostates with simulation-smaller states, in order to decrease their total number. We experimentally showed that our techniques can indeed considerably decrease the size of the output of the complementation.