Abstract
Let A be a finite alphabet and A ∗ the free monoid generated by A. A language is any subset of A ∗ . Assume that all the languages of the form {a}, where a is either the empty word or a letter in A, are given. Close this basic family of languages under Boolean operations; let B (0) be the resulting Boolean algebra of languages. Next, close B (0) under concatenation and then close the resulting family under Boolean operations. Call this new Boolean algebra B (1), etc. The sequence B (0), B (1),…, B k ,… of Boolean algebras is called the dot-depth hierarchy. The union of all these Boolean algebras is the family A of star-free or aperiodic languages which is the same as the family of noncounting regular languages. Over an alphabet of one letter the hierarchy is finite; in fact, B (2) = B (1). We show in this paper that the hierarchy is infinite for any alphabet with two or more letters.
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