 
Summary: Regular Functions, Cost Register Automata, and Generalized
MinCost Problems
Rajeev Alur Loris D'Antoni Jyotirmoy V. Deshmukh Mukund Ragothaman
Yifei Yuan
February 23, 2012
Abstract
Motivated by the successful application of the theory of regular languages to formal verifi
cation of finitestate 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 quan
titative properties of finitestate 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 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 stringtotree transducers, and allows associating
costs with events that are conditional upon regular properties of future events. Our model of
cost register automata allows computation of regular functions using multiple "writeonly" reg
isters whose values can be combined using the allowed set of operations. We show that classical
shortestpath algorithms as well as algorithms designed for computing discounted costs, can be
adopted for solving the mincost problems for the more general classes of functions specified
in our model. Cost register automata with min and increment give a deterministic model that
