Comparator Automata in Quantitative Verification

Abstract

The notion of comparison between system runs is fundamental in formal verification. This concept is implicitly present in the verification of qualitative systems, and is more pronounced in the verification of quantitative systems. In this work, we identify a novel mode of comparison in quantitative systems: the online comparison of the aggregate values of two sequences of quantitative weights. This notion is embodied by comparator automata (comparators, in short), a new class of automata that read two infinite sequences of weights synchronously and relate their aggregate values.We show that aggregate functions that can be represented with Büchi automaton result in comparators that are finite-state and accept by the Büchi condition as well. Such ωregular comparators further lead to generic algorithms for a number of well-studied problems, including the quantitative inclusion and winning strategies in quantitative graph games with incomplete information, as well as related non-decision problems, such as obtaining a finite representation of all counterexamples in the quantitative inclusion problem.We study comparators for two aggregate functions: discounted-sum and limit-average. We prove that the discounted-sum comparator is ω-regular iff the discount-factor is an integer. Not every aggregate function, however, has an ω-regular comparator. Specifically, we show that the language of sequence-pairs for which limit-average aggregates exist is neither ω-regular nor ω-context-free. Given this result, we introduce the notion of prefixaverage as a relaxation of limit-average aggregation, and show that it admits ω-context-free comparators.Does an ω-regular comparator for an aggregate function and a relation imply that the aggregate function is also ω-regular? Furthermore, we show that ω-regular comparators lead to generic algorithms for a number of well-studied problems including the quantitative inclusion problem, and in showing existence of winning strategies in incomplete-information quantitative games. Our algorithm yields PSPACE-completeness of quantitative inclusion when the ω-regular comparator is provided. The same algorithm extends to obtaining finite-state representations of counterexample words in inclusion.Next, we show that the discounted-sum aggregation function admits an ω-regular comparator for all relations R iff the discount-factor d > 1 is an integer. We use this result to prove that discounted-sum aggregate function for discount-factor d > 1 is ω-regular iff d is an integer. Furthermore, we use properties of ω-regular comparators to conclude that the discounted-sum inclusion is PSPACE-complete, hence resolving the complexity gap. Finally, we investigate the limit-average comparator. Since limit-average is only defined for sequences in which the average of prefixes converge, limit-average comparison is not welldefined. We show that even a Büchi pushdown automaton cannot separate sequences for which limit-average exists from those for which it does not. Hence, we introduce the novel notion of prefix-avera...

Safety and Co-safety Comparator Automata for Discounted-Sum Inclusion

“…Comparator Automaton: For a finite-set of integers Σ, an aggregate function f : Z ω → R, and equality or inequality relation R ∈ {<, >, ≤, ≥, =, =}, the comparison language for f with relation R is a language of infinite words over the alphabet Σ × Σ that accepts a pair (A, B) iff f (A) R f (B) holds. A comparator automaton (comparator, in short) for aggregate function f and relation R is an automaton that accepts the comparison language for f with R [12]. A comparator is said to be regular if its automaton is a Büchi automaton.…”

“…Comparison language or comparator of a quantitative inclusion problem refer to the comparison language or comparator of the associated aggregate function. [12]. This work investigates algorithmic approaches to DS-inclusion with integer discount-factors.…”

“…The second approach is purely-automata theoretic [12]. This approach uses regular DS-comparator to reduce DS-inclusion to language inclusion between nondeterministic Büchi automata [11,12].…”

“…The second approach is purely-automata theoretic [12]. This approach uses regular DS-comparator to reduce DS-inclusion to language inclusion between nondeterministic Büchi automata [11,12]. While the purely automata-theoretic approach scales better than the hybrid approach in runtime [11], its scalability suffers from fundamental algorithmic limitations of Büchi language inclusion.…”

“…In theory, DS-inclusion is PSPACE-complete [12]. Recent algorithmic approaches have tapped into language-theoretic properties of discounted-sum aggregate function [12,18] to design practical algorithms for DS-inclusion [11,12]. These algorithms use DS-comparator automata (DS-comparator, in short) as their main technique, and are purely automata-theoretic.…”