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.

Ambiguity in Graphs and Expressions

Publisher Summary The theory of regular languages and finite automata was developed in the early 1950s and is one of the oldest branches of theoretical computer science. Regular languages constitute the best known family of formal languages, and finite automata constitute the best known family of abstract machine models. The concepts of regular languages and finite automata appear frequently in theoretical computer science and have several important applications. There is a vast literature on these subjects. Despite the fact that many researchers have worked in this field, there remain several difficult open problems. The chapter discusses six of these problems. These problems are of fundamental importance and considerable difficulty. Most of them are intimately involved with the fundamental property of finite automata, namely finiteness. In a monograph published in 1971, McNaughton and Papert included a collection of open problems concerning regular languages. Their list is headed by the star height problem and until now, no progress has been made on such an intriguing question. The bounds on star height apply only to languages whose syntactic monoids are groups. In that case, the corresponding semiautomata are permutation semiautomata.

Open problems about regular languages

The state complexity of a regular language is the number of states in the minimal deterministic automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity n of languages in that class. We prove that nn-1 is a tight upper bound on the complexity of right ideals and prefix-closed languages, and that there exist left ideals and suffix-closed languages of syntactic complexity nn-1 + n - 1, and two-sided ideals and factor-closed languages of syntactic complexity nn-2 + (n - 2)2n-2 + 1.

Syntactic complexity of ideal and closed languages

A regular event W is a star event if there exists another event V such that W = V*. In that case, V is called a root of W. It is shown that every star event has a unique minimum root, which is contained in every other root. An algorithm for finding the minimum root of a regular event is presented, and the root is shown to be regular. The results have applications to languages, codes, canonical forms for regular expressions, simplification of expressions, decomposition of sequential machines, and semigroup theory.

Roots of Star Events

Descriptional complexity is the study of the conciseness of the various models representing formal languages. The state complexity of a regular language is the size, measured by the number of states of the smallest, either deterministic or nondeterministic, finite automaton that recognises it. Operational state complexity is the study of the state complexity of operations over languages. In this survey, we review the state complexities of individual regularity preserving language operations on regular and some subregular languages. Then we revisit the state complexities of the combination of individual operations. We also review methods of estimation and approximation of state complexity of more complex combined operations.

A Survey on Operational State Complexity

New classes of events and finite automata which generalize the noninitial definite class are introduced. These events, called “ultimate definite” (u.d.), “reverse u.d.” and “symmetric definite,” are investigated as to algebraic and closure properties. Effective decision procedures whereby it can be decided whether a given finite automata defines such an event are given and unique canonical representations for these events are derived.

Ultimate-Definite and Symmetric-Definite Events and Automata

—Finding simple or "canonical representations" for regular events is one of the important problems concerning finite automata. The present paper is an attempt to find concatenative canonical decompositions for regular events. Five different decomposition types of regular events are defined, imustrated by examples, and their properties investigated. The main tool in our investigations is the notion of a decomposition set; it is a generalization of the notion of a decomposition state, introduced by the authors in a previous paper.[7]Let A =(S, M, s 0 , F) be a finite automaton; a subset S⊂S is a decomposition set if A goes through a state of S whenever it accepts a tape. In order to determine whether a given subset S⊂S is a decomposition set one has to check only tapes whose length is not greater than |S|-|S|-1, where |S| and |S| are the number of states in S and S, respectively. Thus, one can deternine all decomposition sets of A. The knowledge of the decomposition sets of A enables one to determine whether and in what form T(A), the set of tapes accepted by A, is decomposable.

On Concatenative Decompositions of Regular Events

Using a technique based on "state splitting" it is shown that every n-state Markov system is decomposable into a whirl interconnection of n−1 two-state Markov systems.

Whirl Decomposition of Stochastic Systems

R70-45 On Decompositions of Regular Events

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Papers

Ultimate-Definite and Symmetric-Definite Events and Automata

On Concatenative Decompositions of Regular Events

Whirl Decomposition of Stochastic Systems

R70-45 On Decompositions of Regular Events