4 research outputs found

    Inferring descriptive generalisations of formal languages

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    In the present paper, we introduce a variant of Gold-style learners that is not required to infer precise descriptions of the languages in a class, but that must find descriptive patterns, i.e., optimal generalisations within a class of pattern languages. Our first main result characterises those indexed families of recursive languages that can be inferred by such learners, and we demonstrate that this characterisation shows enlightening connections to Angluin’s corresponding result for exact inference. Using a notion of descriptiveness that is restricted to the natural subclass of terminal-free E-pattern languages, we introduce a generic inference strategy, and our second main result characterises those classes of languages that can be generalised by this strategy. This characterisation demonstrates that there are major classes of languages that can be generalised in our model, but not be inferred by a normal Gold-style learner. Our corresponding technical considerations lead to deep insights of intrinsic interest into combinatorial and algorithmic properties of pattern languages

    Inferring descriptive generalisations of formal languages

    Get PDF
    In the present paper, we introduce a variant of Gold-style learners that is not required to infer precise descriptions of the languages in a class, but that must nd descriptive patterns, i. e., optimal generalisations within a class of pattern languages. Our rst main result characterises those indexed families of recursive languages that can be inferred by such learners, and we demonstrate that this characterisation shows enlightening connections to Angluin's corresponding result for exact inference. Furthermore, this result reveals that our model can be interpreted as an instance of a natural extension of Gold's model of language identi cation in the limit. Using a notion of descriptiveness that is restricted to the natural subclass of terminal-free E-pattern languages, we introduce a generic inference strategy, and our second main result characterises those classes of languages that can be generalised by this strategy. This characterisation demonstrates that there are major classes of languages that can be generalised in our model, but not be inferred by a normal Gold-style learner. Our corresponding technical considerations lead to insights of intrinsic interest into combinatorial and algorithmic properties of pattern languages

    Inklusion von Patternsprachen und verwandte Probleme

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    A pattern is a word that consists of variables and terminal symbols. The pattern language that is generated by a pattern A is the set of all terminal words that can be obtained from A by uniform replacement of variables with terminal words. For example, the pattern A = a x y a x (where x and y are variables, and the letter a is a terminal symbol) generates the set of all words that have some word a x both as prefix and suffix (where these two occurrences of a x do not overlap). Due to their simple definition, pattern languages have various connections to a wide range of other areas in theoretical computer science and mathematics. Among these areas are combinatorics on words, logic, and the theory of free semigroups. On the other hand, many of the canonical questions in formal language theory are surprisingly difficult. The present thesis discusses various aspects of the inclusion problem of pattern languages. It can be divide in two parts. The first one examines the decidability of pattern languages with a limited number of variables and fixed terminal alphabets. In addition to this, the minimizability of regular expressions with repetition operators is studied. The second part deals with descriptive patterns, the smallest generalizations of arbitrary languages through pattern languages ("smallest" with respect to the inclusion relation). Main questions are the existence and the discoverability of descriptive patterns for arbitrary languages.Ein Pattern ist ein Wort aus Variablen und Terminalsymbolen. Die von einem Pattern A erzeugte Patternsprache ist die Menge aller Terminalwörter, die durch eine uniforme Ersetzung der Variablen in A durch Terminalwörter erzeugt werden können. So beschreibt das Pattern A = a x y a x (wobei x und y Variablen sind und a ein Terminal ist) die Menge aller Wörter, die ein Wort der Form a x sowohl als Präfix, als auch als Suffix haben (ohne dass sich diese beiden Vorkommen von a x überlappen). Wegen ihrer einfachen Definition besitzen Patternsprachen eine Vielzahl von Verbindungen zu verschiedenen anderen Gebieten der theoretischen Informatik und Mathematik, unter anderem zur Wortkombinatorik, Logik und der Theorie freier Halbgruppen. Andererseits führen viele der üblichen sprachtheoretischen Fragestellungen bei Patternsprachen zu kombinatorischen Problemen von überraschender Schwierigkeit. Die vorliegende Dissertation widmet sich verschiedenen Aspekten des Inklusionsproblems von Patternsprachen und kann in zwei Teile unterteilt werden. Der erste Teil untersucht die Entscheidbarkeit des Inklusionsproblems für Sprachen, die von Pattern mit beschränkter Variablenzahl über Terminalalphabeten von beschränkter Größe erzeugt werden. Darüber hinaus werden verschiedene Aspekte der Minimierbarkeit von regulären Ausdrücken mit Rückreferenzen betrachtet. Der zweite Teil der Dissertation handelt von deskriptiven Pattern; d.h. denjenigen Pattern, die die (hinsichtlich der Inklusion) kleinsten Verallgemeinerungen einer gegebenen Sprache erzeugen. Hauptfragen sind hierbei die Existenz und die Auffindbarkeit deskriptiver Pattern für beliebige Sprachen
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