This chapter is divided into two major sections. Section B.1 reviews the classes of finite state machines used in electronics design and their equivalence relationships. Although this material is strongly related to automata theory — or actually part of it — no attempt is made to cover the theory since there are excellent and comprehensive textbooks on the subject. Rather, the emphasis is on a number of mathematical facts relevant to hardware design that are not normally found in such references. Section B.2 then looks at finite state machines more from an implementation point of view, yet without committing one to any specific technology.

Abstract automata

Automata theory is a mathematical discipline concerned with fundamental issues of discrete computation such as formal languages and grammars, parsing, decidability, and computability. The underlying formal models are crude abstractions that essentially simplify computing equipment to transducers that, while changing from state to state, convert a given input string into some output string. Most issues relevant to digital design such as hardware architecture, computer arithmetics, parasitic states, state encoding, transient effects, delays, synchronization, etc. are neglected, which raises the question

“Why study the abstract subject of automata theory in the context of electronics design?”

The motivation is threefold:

Functional specification. Describing what a digital system has to do is not always easy. Automata theory often helps to specify the relationship between a circuit's inputs and outputs in a more formal way, especially for control- and protocol-oriented tasks.