Summaries for Context-Free Games

TL;DR

This paper introduces a novel, summary-based algorithm for solving context-free games with automaton-based winning conditions, achieving optimal complexity and significant practical speedups.

Contribution

It presents a new summary-based algorithm for deciding winners in context-free games, utilizing Boolean formulas over transition monoids, with proven optimal complexity and practical efficiency.

Findings

Algorithm has doubly exponential time complexity.

Compatible with recent antichain optimizations.

Preliminary experiments show a three orders of magnitude speedup.

Abstract

We study two-player games played on the infinite graph of sentential forms induced by a context-free grammar (that comes with an ownership partitioning of the non-terminals). The winning condition is inclusion of the derived terminal word in the language of a finite automaton. Our contribution is a new algorithm to decide the winning player and to compute her strategy. It is based on a novel representation of all plays starting in a non-terminal. The representation uses the domain of Boolean formulas over the transition monoid of the target automaton. The elements of the monoid are essentially procedure summaries, and our approach can be seen as the first summary-based algorithm for the synthesis of recursive programs. We show that our algorithm has optimal (doubly exponential) time complexity, that it is compatible with recent antichain optimizations, and that it admits a lazy evaluation strategy. Our preliminary experiments indeed show encouraging results, indicating a speed up of three orders of magnitude over a competitor.