Daniel Reidenbach

A non-learnable class of E-pattern languages

We investigate the inferrability of E-pattern languages (also known as extended or erasing pattern languages) from positive data in Golds learning model. As the main result, our analysis yields a negative outcome for the full class of E-pattern languages--and even for the subclass of terminal-free E-pattern languages--if the corresponding terminal alphabet consists of exactly two distinct letters. Furthermore, we present a positive result for a manifest subclass of terminal-free E-pattern languages. We point out that the considered problems are closely related to fundamental questions concerning the nondeterminism of E-pattern languages.

Discontinuities in pattern inference

This paper deals with the inferrability of classes of E-pattern languages-also referred to as extended or erasing pattern languages-from positive data in Golds model of identification in the limit. The first main part of the paper shows that the recently presented negative result on terminal-free E-pattern languages over binary alphabets does not hold for other alphabet sizes, so that the full class of these languages is inferrable from positive data if and only if the corresponding terminal alphabet does not consist of exactly two distinct letters. The second main part yields the insight that the positive result on terminal-free E-pattern languages over alphabets with three or four letters cannot be extended to the class of general E-pattern languages. With regard to larger alphabets, the extensibility remains open. The proof methods developed for these main results do not directly discuss the (non-)existence of appropriate learning strategies, but they deal with structural properties of classes of E-pattern languages, and, in particular, with the problem of finding telltales for these languages. It is shown that the inferrability of classes of E-pattern languages is closely connected to some problems on the ambiguity of morphisms so that the technical contributions of the paper largely consist of combinatorial insights into morphisms in word monoids.

UNAMBIGUOUS MORPHIC IMAGES OF STRINGS

We study a fundamental combinatorial problem on morphisms in free semigroups: With regard to any string α over some alphabet we ask for the existence of a morphism σ such that σ(α) is unambiguous, i.e. there is no morphism τ with τ(i) ≠ σ(i) for some symbol i in α and, nevertheless, τ(α) = σ(α). As a consequence of its elementary nature, this question shows a variety of connections to those topics in discrete mathematics which are based on finite strings and morphisms such as pattern languages, equality sets and, thus, the Post Correspondence Problem. Our studies demonstrate that the existence of unambiguous morphic images essentially depends on the structure of α: We introduce a partition of the set of all finite strings into those that are decomposable (referred to as prolix) in a particular manner and those that are indecomposable (called succinct). This partition, that is also known to be of major importance for the research on pattern languages and on finite fixed points of morphisms, allows to formulate our main result according to which a string α can be mapped by an injective morphism onto an unambiguous image if and only if α is succinct.

Bad news on decision problems for patterns

We study the inclusion problem for pattern languages, which-due to Jiang et al. [T. Jiang, A. Salomaa, K. Salomaa, S. Yu, Decision problems for patterns, Journal of Computer and System Sciences 50 (1995) 53-63]-is known to be undecidable. More precisely, Jiang et al. demonstrate that there is no effective procedure deciding the inclusion for the class of all pattern languages over all alphabets. Most applications of pattern languages, however, consider classes over fixed alphabets, and therefore it is practically more relevant to ask for the existence of alphabet-specific decision procedures. Our first main result states that, for all but very particular cases, this version of the inclusion problem is also undecidable. The second main part of our paper disproves the prevalent conjecture on the inclusion of so-called similar E-pattern languages, and it explains the devastating consequences of this result for the intensive previous research on the most prominent open decision problem for pattern languages, namely the equivalence problem for general E-pattern languages.

Patterns with bounded treewidth

We show that any parameter of patterns that is an upper bound for the treewidth of appropriate encodings of patterns as relational structures, if restricted to a constant, allows the membership problem for pattern languages to be solved in polynomial time. Furthermore, we identify a new such parameter, called the scope coincidence degree.

A polynomial time match test for large classes of extended regular expressions

In the present paper, we study the match test for extended regular expressions. We approach this NP-complete problem by introducing a novel variant of two-way multihead automata, which reveals that the complexity of the match test is determined by a hidden combinatorial property of extended regular expressions, and it shows that a restriction of the corresponding parameter leads to rich classes with a polynomial time match test. For presentational reasons, we use the concept of pattern languages in order to specify extended regular expressions. While this decision, formally, slightly narrows the scope of our results, an extension of our concepts and results to more general notions of extended regular expressions is straightforward.

A Discontinuity in Pattern Inference

This paper examines the learnability of a major subclass of E-pattern languages – also known as erasing or extended pattern languages – in Gold’s learning model: We show that the class of terminal-free E-pattern languages is inferrable from positive data if the corresponding terminal alphabet consists of three or more letters. Consequently, the recently presented negative result for binary alphabets is unique.

On the Learnability of E-pattern Languages over Small Alphabets

This paper deals with two well discussed, but largely open problems on E-pattern languages, also known as extended or erasing pattern languages: primarily, the learnability in Gold’s learning model and, secondarily, the decidability of the equivalence. As the main result, we show that the full class of E-pattern languages is not inferrable from positive data if the corresponding terminal alphabet consists of exactly three or of exactly four letters – an insight that remarkably contrasts with the recent positive finding on the learnability of the subclass of terminal-free E-pattern languages for these alphabets. As a side-effect of our reasoning thereon, we reveal some particular example patterns that disprove a conjecture of Ohlebusch and Ukkonen (Theoretical Computer Science 186, 1997) on the decidability of the equivalence of E-pattern languages.

The unambiguity of segmented morphisms

A segmented morphism σn : Δ* → {a, b}*, n ∈ N, maps each symbol in Δ onto a word which consists of n distinct subwords in ab+a. In the present paper, we examine the impact of n on the unambiguity of σn with respect to any α ∈ Δ+, i. e. the question of whether there does not exist a morphism τ satisfying τ(α) = σn(α) and, for some symbol x in α, τ(x) ≠ σn(x). To this end, we consider the set U(σn) of those α ∈ Δ+ with respect to which σn is unambiguous, and we comprehensively describe its relation to any U(σm), m ≠ n. Our paper thus contributes fundamental (and, in parts, fairly counter-intuitive) results to the recently initiated research on the ambiguity of morphisms.

A Negative Result on Inductive Inference of Extended Pattern Languages

The question of learnability of the class of extended pattern languages is considered to be one of the eldest and outstanding open problems in inductive inference of formal languages. This paper provides an appropriate answer presenting a subclass - the terminal-free extended pattern languages - that is not learnable in the limit. In order to achieve this result we will have to limit the respective alphabet of terminal symbols to exactly two letters.In addition we will focus on the impact of ambiguity of pattern languages on inductive inference of terminal-free extended pattern languages. The conventional view on nondeterminism in patterns inspired by formal language theory is transformed into an approach that meets the requirements of inductive inference. These studies will lead to some useful learnability criteria for classes of terminal-free extended pattern languages.