Bad News on Decision Problems for Patterns

Abstract

Abstract. We study the inclusion problem for pattern languages, which is shown to be undecidable by Jiang et al. (J. Comput. System Sci. 50, 1995). 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.

Expressive Languages for Path Queries over Graph-Structured Data

“…Combining this with a recent result on the undecidability of pattern containment [17], we can prove the following. Theorem 6.1.…”

Existence and Nonexistence of Descriptive Patterns

“…Therefore, quite a number of basic properties of pattern languages, e. g. regarding the usual decision problems for classes of formal languages, are known (cf. the surveys by Mateescu and Salomaa [7] and Salomaa [11] and our recent paper [4]). Furthermore, pattern languages have been a focus of interest of inductive inference from the very beginning, investigating whether it is possible to infer a pattern from the words in its pattern language (see Ng and Shinohara [8]).…”

“…Jiang et al [6], Freydenberger and Reidenbach [4]). A significant part of our subsequent technical considerations, however, can be restricted to terminal-free E-pattern languages, and here the inclusion problem is known to be decidable.…”

Closure properties of pattern languages

“…Hence, the knowledge on pattern languages is still patchy, despite recent progress mainly regarding decision problems (see, e. g., Freydenberger, Reidenbach [5], Fernau, Schmid [3], Fernau et al [4] and Reidenbach, Schmid [13]) and the relation to the Chomsky hierarchy (see Jain et al [6] and Reidenbach, Schmid [14]). …”