Periodicity forcing words

The Dual Post Correspondence Problem asks, for a given word α, if there exists a non-periodic morphism g and an arbitrary morphism h such that g(α) = h(α). Thus α satisfies the Dual PCP if and only if it belongs to a non-trivial equality set. Words which do not satisfy the Dual PCP are called periodicity forcing, and are important to the study of word equations, equality sets and ambiguity of morphisms. In this paper, a 'prime' subset of periodicity forcing words is presented. It is shown that when combined with a particular type of morphism it generates exactly the full set of periodicity forcing words. Furthermore, it is shown that there exist examples of periodicity forcing words which contain any given factor/prefix/suffix. Finally, an alternative class of mechanisms for generating periodicity forcing words is developed, resulting in a class of examples which contrast those known already

On the dual post correspondence problem

The Dual Post Correspondence Problem asks whether, for a given word α, there exists a pair of distinct morphisms σ, τ, one of which needs to be non-periodic, such that σ(α) = τ(α) is satisfied. This problem is important for the research on equality sets, which are a vital concept in the theory of computation, as it helps to identify words that are in trivial equality sets only. Little is known about the Dual PCP for words α over larger than binary alphabets. In the present paper, we address this question in a way that simplifies the usual method, which means that we can reduce the intricacy of the word equations involved in dealing with the Dual PCP. Our approach yields large sets of words for which there exists a solution to the Dual PCP as well as examples of words over arbitrary alphabets for which such a solution does not exist

On the dual post correspondence problem

The Dual Post Correspondence Problem asks whether, for a given word α, there exists a pair of distinct morphisms σ,τ, one of which needs to be non-periodic, such that σ(α) = τ(α) is satisfied. This problem is important for the research on equality sets, which are a vital concept in the theory of computation, as it helps to identify words that are in trivial equality sets only. Little is known about the Dual PCP for words α over larger than binary alphabets, especially for so-called ratio-primitive examples. In the present paper, we address this question in a way that simplifies the usual method, which means that we can reduce the intricacy of the word equations involved in dealing with the Dual PCP. Our approach yields large sets of words for which there exists a solution to the Dual PCP as well as examples of words over arbitrary alphabets for which such a solution does not exist

Periodicity forcing words

The Dual Post Correspondence Problem asks, for a given word , if there exists a non-periodic morphism g and an arbitrary morphism h such that g( ) = h( ). Thus satis es the Dual PCP if and only if it belongs to a non-trivial equality set. Words which do not satisfy the Dual PCP are called periodicity forcing, and are important to the study of word equations, equality sets and ambiguity of morphisms. In this paper, a `prime' subset of periodicity forcing words is presented. It is shown that when combined with a particular type of morphism it generates exactly the full set of periodicity forcing words. Furthermore, it is shown that there exist examples of periodicity forcing words which contain any given factor/pre x/su x. Finally, an alternative class of mechanisms for generating periodicity forcing words is developed, resulting in a class of examples which contrast those known already

Periodicity Forcing Words

The Dual Post Correspondence Problem asks, for a given word α, if there exists a non-periodic morphism g and an arbitrary morphism h such that g(α) = h(α). Thus α satisfies the Dual PCP if and only if it belongs to a non-trivial equality set. Words which do not satisfy the Dual PCP are called periodicity forcing, and are important to the study of word equations, equality sets and ambiguity of morphisms. In this paper, a 'prime' subset of periodicity forcing words is presented. It is shown that when combined with a particular type of morphism it generates exactly the full set of periodicity forcing words. Furthermore, it is shown that there exist examples of periodicity forcing words which contain any given factor/prefix/suffix. Finally, an alternative class of mechanisms for generating periodicity forcing words is developed, resulting in a class of examples which contrast those known already

On the Dual Post Correspondence Problem

The Dual Post Correspondence Problem asks whether, for a given word α, there exists a pair of distinct morphisms σ,τ, one of which needs to be non-periodic, such that σ(α) = τ(α) is satisfied. This problem is important for the research on equality sets, which are a vital concept in the theory of computation, as it helps to identify words that are in trivial equality sets only. Little is known about the Dual PCP for words α over larger than binary alphabets, especially for so-called ratio-primitive examples. In the present paper, we address this question in a way that simplifies the usual method, which means that we can reduce the intricacy of the word equations involved in dealing with the Dual PCP. Our approach yields large sets of words for which there exists a solution to the Dual PCP as well as examples of words over arbitrary alphabets for which such a solution does not exist

On restricting the ambiguity in morphic images of words

For alphabets Delta_1, Delta_2, a morphism g : Delta_1* to Delta_2* is ambiguous with respect to a word u in Delta_1* if there exists a second morphism h : Delta_1* to Delta_2* such that g(u) = h(u) and g not= h. Otherwise g is unambiguous. Hence unambiguous morphisms are those whose structure is fully preserved in their morphic images. A concept so far considered in the free monoid, the first part of this thesis considers natural extensions of ambiguity of morphisms to free groups. It is shown that, while the most straightforward generalization of ambiguity to a free monoid results in a trivial situation, that all morphisms are (always) ambiguous, there exist meaningful extensions of (un)ambiguity which are non-trivial - most notably the concepts of (un)ambiguity up to inner automorphism and up to automorphism. A characterization is given of words in a free group for which there exists an injective morphism which is unambiguous up to inner automorphism in terms of fixed points of morphisms, replicating an existing result for words in the free monoid. A conjecture is presented, which if correct, is sufficient to show an equivalent characterization for unambiguity up to automorphism. A rather counterintuitive statement is also established, that for some words, the only unambiguous (up to automorphism) morphisms are non-injective (or even periodic). The second part of the thesis addresses words for which all non-periodic morphisms are unambiguous. In the free monoid, these take the form of periodicity forcing words. It is shown using morphisms that there exist ratio-primitive periodicity forcing words over arbitrary alphabets, and furthermore that it is possible to establish large and varied classes in this way. It is observed that the set of periodicity forcing words is spanned by chains of words, where each word is a morphic image of its predecessor. It is shown that the chains terminate in exactly one direction, meaning not all periodicity forcing words may be reached as the (non-trivial) morphic image of another. Such words are called prime periodicity forcing words, and some alternative methods for finding them are given. The free-group equivalent to periodicity forcing words - a special class of C-test words - is also considered, as well as the ambiguity of terminal-preserving morphisms with respect to words containing terminal symbols, or constants. Moreover, some applications to pattern languages and group pattern languages are discussed