Nested Weighted Limit-Average Automata of Bounded Width

TL;DR

This paper introduces a subclass of nested weighted automata with bounded width and limit average, showing they are more expressive than standard weighted automata while maintaining similar decision problem complexities.

Contribution

It defines nested weighted automata with bounded width, proves their increased expressiveness, and analyzes the complexity of key decision problems.

Findings

More expressive than weighted automata, e.g., for average response time.

Decision problems like emptiness and universality have matching complexity to weighted automata.

PSPACE-completeness when the bound is given in unary.

Abstract

While weighted automata provide a natural framework to express quantitative properties, many basic properties like average response time cannot be expressed with weighted automata. Nested weighted automata extend weighted automata and consist of a master automaton and a set of slave automata that are invoked by the master automaton. Nested weighted automata are strictly more expressive than weighted automata (e.g., average response time can be expressed with nested weighted automata), but the basic decision questions have higher complexity (e.g., for deterministic automata, the emptiness question for nested weighted automata is PSPACE-hard, whereas the corresponding complexity for weighted automata is PTIME). We consider a natural subclass of nested weighted automata where at any point at most a bounded number k of slave automata can be active. We focus on automata whose master value function is the limit average. We show that these nested weighted automata with bounded width are strictly more expressive than weighted automata (e.g., average response time with no overlapping requests can be expressed with bound k=1, but not with non-nested weighted automata). We show that the complexity of the basic decision problems (i.e., emptiness and universality) for the subclass with k constant matches the complexity for weighted automata. Moreover, when k is part of the input given in unary we establish PSPACE-completeness.