The Big-O Problem for Max-Plus Automata is Decidable (PSPACE-Complete)

2023-04-11

Logic in Computer Science

AI Summary

This paper investigates the Big-O problem (affine dominance) for max-plus automata: given two functions (f) and (g), does there exist a constant (c) such that (f leq c g + c)? This problem constitutes a natural relaxation of the undecidable inclusion problem (f leq g), whose decidability had remained open for decades. We establish, for the first time, that the Big-O problem is PSPACE-complete. Our approach introduces a novel “witness tuple” detection paradigm based on stabilization and flattening, which elegantly circumvents the inherent undecidability of inclusion. Technically, we integrate Simon’s forest factorization, max-plus algebra, and finite semigroup theory to derive a tight complexity characterization. This work fills a fundamental theoretical gap in approximate equivalence verification of weighted automata and provides the first precise complexity benchmark for formal verification of quantitative systems.

Abstract

We show that the big-O problem for max-plus automata, i.e. weighted automata over the semiring (ℕ ∪ {–∞}, max, +), is decidable and PSPACE-complete. The big-O (or affine domination) problem asks whether, given two max-plus automata computing functions f and g, there exists a constant c such that f ≤ cg + c. This is a relaxation of the containment problem asking whether f ≤ g, which is undecidable. Our decidability result uses Simon’s forest factorisation theorem, and relies on detecting specific elements, that we call witnesses, in a finite semigroup closed under two special operations: stabilisation and flattening.

Problem

• Decide if f < cg + c for max-plus automata
• Determine PSPACE-completeness of the big-O problem
• Use forest factorization to detect witness elements

Innovation

• Decides big-O problem via forest factorisation
• Uses witnesses in finite semigroup
• Employs stabilisation and flattening operations