Tom Head

Fixed languages and the adult languages of ol schemest

The fixed language of an endomorphism of a free monoid is shown to be method that provides a concide description of the basis of the fixed language. A supplementary disscussion shown that the adult language of an OL scheme is free and provides a simple listing procedure for the basis element.

Expanded subalphabets in the theories of languages and semigroups

A new type of code, called an expanded subalphabet, is introduced. It is shown that the following four conditions on a subset L of the free monoid A ∗ over a finite or infinite alphabet A are equivalent: (1) L is the submonoid generated by an expanded subalphabet; (2) L is a retract; (3) L is the fixed language of an endomorphism; (4) L is the stationary language of an endomorphism. Expanded subalphabets are used as a tool for the investigation of fixed languages (=retracts). Special results for the case of a finite alphabet are given and the relationship with the theory of L-systems is indicated.

The decidability of equivalence for deterministic finite transducers

Abstract An algorithm is given which will decide, for two given deterministic finite transducers M and M ′, whether the input-output behaviours of M and M ′ are identical.

Fixed and Stationary ω —Words and ω —Languages

Explicit representations of the ω-words that are fixed (resp., stationary) relative to a function h: A -→ A* are given. A procedure is provided for constructing a concise expression for the fixed (resp., stationary) ω -language of such an h. The equivalence problem for fixed (resp., stationary) ω-languages of functions h & k: A -→ A* is shown to be decidable. The fundamental tool for this latter procedure is the recently developed algorithm of K. Culik II & T. Harju for deciding the ω -sequence equivalence problem.

The adherences of languages as topological spaces

The problem of characterizing the topological spaces that arise as adherences of languages of specified types is raised and pertinent concepts of general topology are reviewed. It is observed that the spaces that arise as adherences of arbitrary languages may be characterized as either: (1) the closed subsets of the Cantor ternary set; (2) the zero-dimensional compact metrizable spaces; or (3) the Stone spaces of the countable Boolean algebras. R.S.Pierces concept of a space of finite type is reviewed and his theorem characterizing the zero-dimensional compact metric spaces of finite type by means of an associated finite structural invariant is reviewed. It is shown that a topological space is homeomorphic with the adherence of a regular language if and only if it is zero-dimensional compact metrizable and of finite type. The structural invariant of the adherence of a regular language is algorithmically constructiole from any automaton recognizing the language. Comparing these invariants provides a procedure for deciding homeomorphism of adherences for regular languages.

Periodic D0L languages

Abstract Periodic D0L systems and languages are defined, and periodicity is shown to be a decidable property of D0L systems. The fundamental tools used are recent results on the representation of stationary ω-words (Head and Lando, 1986) and the decidability of ultimate periodicity of ω-words (Harju and Linna, 1986; Pansiot, 1986). The relation of D0L periodicity to local catenativity and to n -codes is examined.

Commutative semigroups having greatest regular images

Let S be a commutative semigroup. S has a greatest regular image if and only if each of its Archimedean components contains an idempotent. S has a greatest group-with-zero image if and only if S has precisely two Archimedean components and the upper component contains an idempotent. The existence and structure of these images and of greatest group images is related to tensor products.

DOL schemes and the periodicity of string embeddings

Abstract For a DOL scheme ( A , h ), integers i and k with i k are specified for which for every string w in A ∗ , h i ( w ) is a subsequence of h k ( w ). For a finite non-empty set A integers I and K with I K are specified for which for every DOL system ( A , h , w ), h I ( w ) is a subsequence of h K ( w ). These results allow clarification and simplification of earlier results in the literature of DOL languages. Conclusions are drawn concerning the family of finite DOL languages sharing a common scheme and the family of finite DOL languages sharing a common alphabet. The stationary sets of DOL schemes are shown to be finitely generated free monoids.

Adherences of D0L languages

Abstract Procedures for constructing representations of each of the finite number of sequences, i.e., ω-words, that belong to the adherence of an arbitrary D0L language are given. A procedure is then given for deciding whether or not the languages generated by an arbitrary pair of D0L systems have the same adherence. From an arbitrary D0L system simpler systems are constructed which have the same adherence as the original system. Representations of the sequences in the adherence of these simpler systems are then constructed. Such sequences either have the form uv ω for finite strings u and v or they have a form widely discussed by Salomaa: wsh ( s ) h 2 ( s )… h n ( s )… where h ( w ) = ws . The problem of deciding equality of two sequences of the latter type was recently solved by Culik and Harju (1981) and their algorithm is a major tool used here for deciding the adherence equivalence of D0L languages.

Finite DOL languages and codes

Abstract A finite DOL language that does not contain the null string is either a prefix code or a suffix code. When the null string is removed from a finite DOL language that contains the null string a code remains.