Closure Properties of Pattern Languages

Pattern languages are a well-established class of languages that is particularly popular in algorithmic learning theory, but very little is known about their closure properties. In the present paper we establish a large number of closure properties of the terminal-free pattern languages, and we characterise when the union of two terminal-free pattern languages is again a terminal-free pattern language. We demonstrate that the equivalent question for general pattern languages is characterised differently, and that it is linked to some of the most prominent open problems for pattern languages. We also provide fundamental insights into a well-known construction of E-pattern languages as unions of NE-pattern languages, and vice versa. © 2014 Springer International Publishing Switzerland

Developments from enquiries into the learnability of the pattern languages from positive data

AbstractThe pattern languages are languages that are generated from patterns, and were first proposed by Angluin as a non-trivial class that is inferable from positive data [D. Angluin, Finding patterns common to a set of strings, Journal of Computer and System Sciences 21 (1980) 46–62; D. Angluin, Inductive inference of formal languages from positive data, Information and Control 45 (1980) 117–135]. In this paper we chronologize some results that developed from the investigations on the inferability of the pattern languages from positive data

Revisiting Shinohara's algorithm for computing descriptive patterns

A pattern α is a word consisting of constants and variables and it describes the pattern language L(α) of all words that can be obtained by uniformly replacing the variables with constant words. In 1982, Shinohara presents an algorithm that computes a pattern that is descriptive for a finite set S of words, i.e., its pattern language contains S in the closest possible way among all pattern languages. We generalise Shinohara’s algorithm to subclasses of patterns and characterise those subclasses for which it is applicable. Furthermore, within this set of pattern classes, we characterise those for which Shinohara’s algorithm has a polynomial running time (under the assumption P 6= N P). Moreover, we also investigate the complexity of the consistency problem of patterns, i.e., finding a pattern that separates two given finite sets of words

Closure properties of pattern languages

Pattern languages are a well-established class of languages, but very little is known about their closure properties. In the present paper we establish a large number of closure properties of the terminal-free pattern languages, and we characterise when the union of two terminal-free pattern languages is again a terminal-free pattern language. We demonstrate that the equivalent question for general pattern languages is characterised differently, and that it is linked to some of the most prominent open problems for pattern languages. We also provide fundamental insights into a well-known construction of E-pattern languages as unions of NE-pattern languages, and vice versa