Weight can also be added to the classical automata with an infinite set of states such as pushdown automata, and this extension is a general concept of weighted automata. In addition, weighted automata and weighted context-free grammar has found applications in natural language processing, speech recognition, and digital image compression. This book covers all major aspects of weighted automata and formal power series methods, ranging from theory to application. The chapters in Part I covers the basics of the theory of weighted automata, special deals semirings, power series, and fixed point theory. Finally, Part IV relates to weighted automata applications, 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 volumes, and also will be a valuable resource for students to explore the research area of interest.