Automata-Based Quantitative Reasoning

Abstract

Existing solution approaches for problems in formal quantitative analysis suffer from two challenges that adversely impact their theoretical understanding and large-scale applicability. These are the lack of generalizability , and separation-of-techniques. Lack of generalizability refers to the issue that solution approaches are often specialized to the underlying cost model that evaluates the quantitative property. Different cost models deploy such disparate algorithms that there is no transfer of knowledge from one cost model to another. Separation-of-techniques refers to the inherent dichotomy in solving problems in quantitative analysis. Most algorithms comprise of two phases: A structural phase , which reasons about the structure of the quantitative system(s) using techniques from automata or graphs; and a numerical phase , which reasons about the quantitative dimension/cost model using numerical methods. These techniques are incompatible with one another, forcing the phases to be performed sequentially, thereby impacting scalability. The article presents a novel framework that addresses the aforementioned challenges. The introduced framework, called comparator automata or comparators in short, builds on automata-theoretic foundations to generalize across a variety of cost models. The crux of comparators is that they enable automata-based methods in the numerical phase, hence eradicating the dependence on numerical methods. In doing so, comparators are able to integrate the structural and numerical phases. On the theoretical front, we demonstrate that comparator-based solutions have the advantage of generalizable results, and yield complexity-theoretic improvements over a range of problems in quantitative analysis. On the practical front, we demonstrate through empirical analysis that comparator-based solutions render more efficient, scalable, and robust performance, and demonstrate broader applicability than traditional methods for quantitative reasoning.