Robert McNaughton

Infinite games played on finite graphs

Abstract The concept of an infinite game played on a finite graph is perhaps novel in the context of an rather extensive recent literature in which infinite games are generally played on an infinite game tree. We claim two advantages for our model, which is admittedly more restrictive. First, our games have a more apparent resemblance to ordinary parlor games in spite of their infinite duration. Second, by distinguishing those nodes of the graph that determine the winning and losing of the game (winning-condition nodes), we are able to offer a complexity analysis that is useful in computer science applications.

Parenthesis Grammars

A decision procedure is given which determines whether the languages defined by two parenthesis grammars are equal.

Church-Rosser Thue systems and formal languages

Since about 1971, much research has been done on Thue systems that have properties that ensure viable and efficient computation. The strongest of these is the Church-Rosser property, which states that two equivalent strings can each be brought to a unique canonical form by a sequence of length-reducing rules. In this paper three ways in which formal languages can be defined by Thue systems with this property are studied, and some general results about the three families of languages so determined are studied.

The loop complexity of pure-group events

An algorithm is presented for finding the loop complexity (sometimes known as “star height≓) of certain regular events. The algorithm works for those regular events whose syntactic monoid (i.e., semigroup) is a group. The existence of an algorithm that works for all regular events remains an open question.

A polynomial time algorithm for the local testability problem of deterministic finite automata

The local testability problem of deterministic finite automata is investigated. A locally testable language is a language with the property that, for some nonnegative integer k, whether or not a word w is in the language depends on (1) the prefix and suffix of w of length k, and (2) the set of substrings of w length k+1, without regard to the order in which these substrings occur. The local testability problem is, given a deterministic finite automation, to decide whether or not it accepts a locally testable language. The authors present an O(n/sup 2/) time algorithm for the local testability problem based on two simple properties that characterize locally testable automata. >

The Burnside problem for semigroups

Abstract We show that the strong Burnside problem has an affirmative answer for semigroups of finite dimensional matrices over a field. As a corollary of this result and the proof of a theorem of Procesi, it follows that a torsion semigroup embeddable in the multiplicative semigroup of an algebra over a field satisfying a polynomial identity is locally finite. We prove, more generally, that a torsion semigroup of matrices over a skew field all of whose subgroups are locally finite is locally finite.

Computing the Order of a Locally Testable Automaton

A locally testable language is a language with the property that, for some positive integer

Semi-Thue Systems with an Inhibitor

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The undecidability of the preperfectness of thue systems

, whether or not a string

An Insertion into the Chomsky Hierarchy

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