Decision Problems for Patterns

Abstract

We settle an open problem, the inclusion problem for pattern languages 1, 2]. This is the rst known case where inclusion is undecidable for generative devices having a trivially decidable equivalence problem. The study of patterns goes back to the seminal work of Thue 16] and is important also, for instance, in recent work concerning inductive inference and learning. Our results concern both erasing and nonerasing patterns.

A Discontinuity in Pattern Inference

“…We say that the inclusion problem for ePAT ⋆ is decidable if and only if there exists a computable function which, given two arbitrary patterns α, β with L(α), L(β) ∈ ePAT ⋆ , decides whether or not L(α) ⊆ L(β). In [JSSY95] it is shown that the inclusion problem for the full class of E-pattern languages is not decidable. Fortunately, this fact does not hold for terminal-free E-pattern languages.…”

“…Fortunately, this fact does not hold for terminal-free E-pattern languages. As this is of great importance for the following studies, we now cite two respective theorems of [JSSY95]: Fact 1. Let Σ be an alphabet, |Σ| ≥ 2, and α, β ∈ X * two arbitrarily given terminal-free patterns.…”

“…[Fil88] and [JSSY95]) as well -with a rather surprising outcome: We show that it is inferrable from positive data if and only if the terminal alphabet does not consist of two letters. Thus, we present the first class of pattern languages to be known for which different non-trivial alphabets imply different answers to the question of learnability.…”

“…in [JKS + 94] and [JSSY95] (for a survey see [RS97]). These examinations reveal that a definition disallowing the substitution of variables with the empty word -as given by Angluin -leads to a language with particular features being quite different from the one allowing the empty substitution (that has been applied when generating w 3 in our example).…”

“…Thus, we present the first class of pattern languages to be known for which different non-trivial alphabets imply different answers to the question of learnability. Using several theorems in [Ang80] and [JSSY95], our respective reasoning is chiefly combinatorial; therefore it touches on some prominent topics within the research on word monoids and combinatorics of words.…”