text: 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 quantitative 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 string-to-tree 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 “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 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 weighted automata, an extensively studied nondeterministic model, and this connection results in new insights and new open problems.
  @cite_16: Weighted finite automata are classical nondeterministic finite automata in which the transitions carry weights. These weights may model, for example, the cost involved when executing a transition, the resources or time needed for this, or the probability or reliability of its successful execution. Weights can also be added to classical automata with infinite state sets like pushdown automata, and this extension constitutes the general concept of weighted automata. Since their introduction in the 1960s they have stimulated research in related areas of theoretical computer science, including formal language theory, algebra, logic, and discrete structures. Moreover, weighted automata and weighted context-free grammars have found application in natural-language processing, speech recognition, and digital image compression. This book covers all the main aspects of weighted automata and formal power series methods, ranging from theory to applications. The contributors are the leading experts in their respective areas, and each chapter presents a detailed survey of the state of the art and pointers to future research. The chapters in Part I cover the foundations of the theory of weighted automata, specifically addressing semirings, power series, and fixed point theory. Part II investigates different concepts of weighted recognizability. Part III examines alternative types of weighted automata and various discrete structures other than words. Finally, Part IV deals with applications of weighted automata, including digital image compression, fuzzy languages, model checking, and natural-language processing. Computer scientists and mathematicians will find this book an excellent survey and reference volume, and it will also be a valuable resource for students exploring this exciting research area.
  @cite_13: 
  @cite_12: Weighted automata and transducers are widely used in modern applications in bioinformatics and text, speech, and image processing. This chapter describes several fundamental weighted automata and shortest-distance algorithms including composition, determinization, minimization, and synchronization, as well as single-source and all-pairs shortest distance algorithms over general semirings. It presents the pseudocode of these algorithms, gives an analysis of their running time complexity, and illustrates their use in some simple cases. Many other complex weighted automata and transducer algorithms used in practice can be obtained by combining these core algorithms.
summary: Finite-state have been an active area of research, with numerous articles studying their algebraic and algorithmic properties. See @cite_16 for a comprehensive exposition. An important problem for is that of determinization @cite_12 @cite_13 . A deterministic is defined in the usual sense: no two outgoing transitions from a state share the same input label. It has been shown that there are that do not admit equivalent deterministic . In contrast, the cost register automata that we introduce in this paper are deterministic machines with equivalent expressive power and equally efficient decision problems as weighted automata. We believe that this makes them a more suitable model for expressing weighted computations.