Synchronous Regular Relations and Morphological Analysis

We list the major properties of some important classes of subrational relations, mostly to make them easily accessible to computational linguists. We then argue that there are good linguistic reasons for using no class smaller than the class of synchronous regular relations for morphological analysis, and good mathematical reasons for using no class which is larger. 1 Below the Rational Relations We need not stress the importance of finite state transducers and of rational relations for computational linguistics (see Johnson [1972],Koskenniemi [1983],Kaplan and Kay [1994],Beesley and Karttunen [2003]). So we rather start with stressing the importance of sub-rational relations, that is, classes of relations properly contained in the rational relations. As is well-known in the community, rational relations are not closed under intersection. Furthermore, the equivalence and inclusion problems for rational relations are undecidable. So there are a number of arguments for not using rational relations, but rather some weaker class with more favorable decision properties. The question is: if we want to go below the rational relations, which class should we choose? In the literature, we often find the so called sequential relations; these however are quite restricted and will not be considered here. We rather focus on three classes, the strictly synchronous, the kbounded (e.g. Roark and Sproat [2007]), and the synchronous regular relations, which are ordered by inclusion. We present their main closure properties, which are partly already known. For some reason the important class of synchronous regular relations, which has attracted a lot of attention in various fields of mathematics,1 has to our knowledge not gained very much attention in the field of computational linguistics.2 We argue here that 1. there are good linguistic reasons for using no class smaller than the class of synchronous regular relations; and 2. we do not know of any linguistics evidence in morphology to use the more powerful rational relations instead of synchronous regular relations. 2 Closure Properties and Decision Problems We will consider the main closure properties for classes of relations. Union and intersection of two relations R1, R2 are defined in the obvious set-theoretic fashion. The complement of a relation R is defined wrt. two alphabets Σ, T , where R ⊆ Σ∗ × T ∗, and R := (Σ∗ × T ∗)−R. The inversion of a word a1...an ∈ Σ∗ is defined as (a1...an) i := an...a1. For a relation R ⊆ Σ∗ × T ∗, we put Ri := {(wi, vi) : (w, v) ∈ R}. Given two relations R1, R2, we define their composition R1 ◦ R2 := {(x, z) : (x, y) ∈ R1, (y, z) ∈ R2}. Given Two relations R1 ⊆ Σ1 × T ∗ 1 , R2 ⊆ Σ2 × T ∗ 2 , we define their concatenation R1 · R2 := {(w1w2, v1v2) : (w1, v1) ∈ R1, (w2, v2) ∈ R2}. In general, we We just mention Frougny and Sakarovitch [1993], and the research on automatic structures, see Rubin [2008] We have to mention that scholars working on the finite-state manipulation platform Vaucanson have made some efforts in using synchronous regular relations, see Lesaint [2008]