Quantitative Language Automata

TL;DR

This paper introduces quantitative language automata (QLAs) that assign values to infinite words and sets of words, extending automata theory to quantify hyperproperties and analyze their computational properties.

Contribution

The paper defines QLAs combining quantitative automata with language aggregators, and studies their decision problems and complexity for various classes.

Findings

QLAs can capture hyperproperties and their quantitative generalizations.

Decidability and complexity results are provided for evaluation, nonemptiness, and universality problems.

Analysis includes omega-regular languages and Markov chain distributions.

Abstract

A quantitative word automaton (QWA) defines a function from infinite words to values. For example, every infinite run of a limit-average QWA A obtains a mean payoff, and every word w is assigned the maximal mean payoff obtained by nondeterministic runs of A over w. We introduce quantitative language automata (QLAs) that define functions from language generators (i.e., implementations) to values, where a language generator can be nonprobabilistic, defining a set of infinite words, or probabilistic, defining a probability measure over infinite words. A QLA consists of a QWA and a language aggregator. For example, given a QWA A, the infimum aggregator maps each language L to the greatest lower bound assigned by A to any word in L. For boolean value sets, QWAs capture trace properties, and QLAs capture hyperproperties. For more general value sets, QLAs serve as a specification language for a generalization of hyperproperties, called quantitative hyperproperties. A nonprobabilistic (resp. probabilistic) quantitative hyperproperty assigns a value to each set (resp. distribution) G of traces, e.g., the minimal (resp. expected) average response time exhibited by the traces in G (resp. by traces sampled according to G). We give several examples of quantitative hyperproperties and investigate three paradigmatic problems for QLAs: evaluation, nonemptiness, and universality. In the evaluation problem, given a QLA AA and an implementation G, we ask for the value that AA assigns to G. In the nonemptiness (resp. universality) problem, given a QLA AA, a threshold k, and a comparison in {>, >=} we ask whether AA assigns a value meeting the threshold to some (resp. every) language. We provide a comprehensive picture of decidability and complexity for these problems for QLAs with common aggregators as well as their restrictions to omega-regular languages and distributions generated by finite Markov chains.