Automata vs Linear-Programming Discounted-Sum Inclusion

Abstract

The problem of quantitative inclusion formalizes the goal of comparing quantitative dimensions between systems such as worst-case execution time, resource consumption, and the like. Such systems are typically represented by formalisms such as weighted logics or weighted automata. Despite its significance in analyzing the quality of computing systems, the study of quantitative inclusion has mostly been conducted from a theoretical standpoint. In this work, we conduct the first empirical study of quantitative inclusion for discounted-sum weighted automata (DS-inclusion, in short). Currently, two contrasting approaches for DS-inclusion exist: the linear-programming based DetLP and the purely automata-theoretic BCV. Theoretical complexity of DetLP is exponential in time and space while of BCV is PSPACE-complete. All practical implementations of BCV, however, are also exponential in time and space. Hence, it is not clear which of the two algorithms renders a superior implementation. In this work we present the first implementations of these algorithms, and perform extensive experimentation to compare between the two approaches. Our empirical analysis shows how the two approaches complement each other. This is a nuanced picture that is much richer than the one obtained from the theoretical study alone.

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

“…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. A key ingredient of Büchi language-inclusion is Büchi complementation [36].…”

“…3.2 makes use of newly obtained safety/co-safety properties of DS-comparator to present the first deterministic constructions for DS-comparators. These deterministic construction are compact in the sense that they match their non-deterministic counterparts in number of states [11]. Section 3.3 evaluates the complexity of quantitative inclusion with regular safety/co-safety comparators, and observes that its complexity is lower than the complexity for quantitative inclusion with regular comparators.…”

“…Prior work [11] shows that we can define DS-comparison language with upper bound μ, discount-factor d > 1, and relation R to accept infinite and bounded weight-sequence C over {−μ, . .…”

“…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.…”

“…These algorithms use DS-comparator automata (DS-comparator, in short) as their main technique, and are purely automata-theoretic. While these algorithms outperform other existing approaches for DS-inclusion in runtime [15,17], even these do not scale well on weighted-automata with more than few hundreds of states [11]. This work contributes novel techniques and algorithms for DSinclusion to address the scalability challenge of DS-inclusion An in-depth examination of the DS-comparator based algorithm exposes their scalability bottleneck.…”