A regular expression is called unambiguous if every tape in the event can be generated from the expression in one way only. The flow-graph technique for constructing an expression is shown to preserve ambiguities of the graph, and thus, if the graph is that of a deterministic automaton, the expression is unambiguous. A procedure for generating a nondeterministic automaton which preserves the ambiguities of the given regular expression is described. Finally, a procedure for testing whether a given expression is ambiguous is given.

We treat the problem of encoding the outputs from a controllable rate source whose output statistics can be accurately described as a stationary finite-state Markov process. The class of encoding schemes considered is restricted to those in which each output from the source is encoded as a separate, uniquely decodable codeword. We present a uniquely decodable scheme in this class which minimizes the ensemble average length of the transmitted message corresponding to any number of outputs from the source, i.e., a compact encoding scheme. The code used will vary with time as a function of the entire past history of the source. The future encoding of the outputs from a sample source with any given past history will be compact also. Although the efficiency of the encoding scheme can be improved by enlarging the class of schemes considered, e.g., by applying the technique presented herein to higher order extensions of the source, this topic is not pursued.

Procedures are given to convert any regular expression into a state diagram description and neural net realization of a machine that recognizes the regular set of sequences described by the given expression. It is established that any regular expression with a finite number of connectives describes a regular set of sequences that can be recognized by a finite state machine. All the procedures given are guaranteed to terminate in a finite number of steps, and a generalized computer program can be written to handle the entire conversion. An incidental result of the theory is the design of multiple output sequential machines. The potential usefulness of regular expressions and a long neglected form of a state diagram are demonstrated.