Regular Functions, Cost Register Automata, and Generalized Min-Cost Problems

TL;DR

This paper introduces a deterministic model called cost register automata for analyzing quantitative string properties, extending classical automata with operations like addition and min, and explores their theoretical properties and applications.

Contribution

It defines a new class of regular functions based on string-to-tree transducers, connecting deterministic cost automata with weighted automata, and studies their decision problems and expressiveness.

Findings

Classical shortest-path and discounted cost algorithms can be adapted for these functions.

Cost register automata with min and increment are equivalent to weighted automata.

The model provides a foundation for analyzing quantitative properties of finite-state systems.

Abstract

Motivated by the successful application of the theory of regular languages to formal verification of finite-state systems, there is a renewed interest in developing a theory of analyzable functions from strings to numerical values that can provide a foundation for analyzing {\em quantitative} properties of finite-state systems. In this paper, we propose a deterministic model for associating costs with strings that is parameterized by operations of interest (such as addition, scaling, and $\min$), a notion of {\em regularity} that provides a yardstick to measure expressiveness, and study decision problems and theoretical properties of resulting classes of cost functions. Our definition of regularity relies on the theory of string-to-tree transducers, and allows associating costs with events that are conditional upon regular properties of future events. Our model of {\em cost register automata} allows computation of regular functions using multiple "write-only" registers whose values can be combined using the allowed set of operations. We show that classical shortest-path algorithms as well as algorithms designed for computing {\em discounted costs}, can be adopted for solving the min-cost problems for the more general classes of functions specified in our model. Cost register automata with $\min$ and increment give a deterministic model that is equivalent to {\em weighted automata}, an extensively studied nondeterministic model, and this connection results in new insights and new open problems.