Tackling the Complexity of Context-Free Representations in Example-Based Machine Translation

Machine Translation (MT) is seen as a mapping from a source language string into a target language string via an internal representation. I restrict the internal representations to derivation trees that can be generated from context-free grammars (CFG). It is shown that there are more than n! possible representations for an input string of length 1 < n < 20. Given this untractable complexity of internal CFrepresentations, all MT systems which rely on such representations implement a strategy or heuristic to reduce the search space. The paper discusses the Generalized Exemplar-Based MT-system which selects among the possible representations a subset at runtime. An analysis of the computational e ort is provided. 1 Complexity of CF-Representation I assume a universal context-free grammar (CFG) G that produces all possible representations for a given string s. I will examine the number of di erent representations that can be derived from the input string s given the universal grammar G. In the next section, I shall discuss the generalized examplarbased MT-systems and analyse its way to tackle the problem of representational complexity. I shall show that this approach has the potential to generate a universal grammar. A string of length 1 (i.e. the string consists of one word only) may either t a grammar rule in G or else it remains unrecognized. Only in the former case the sting can be reduced and an internal representation can be generated. To examine the number of di erent derivation trees that can be generated from a string of length n > 1, I shall assume that the context-free grammar G is su ciently large such that the whole input string can be reduced into one node only. I will henceforth denote the set of possible representations for a string of length n as R(n) and the cardinality of that set as R (n). The ith element in the set R(n) is denoted by R(n). If a special string s is under consideration I shall write R(s), R(s) and R(s) with the same meaning. A string s of length 1 has thus exactly Figure 1: Four possible representations of the string ab of length 2 that a CFG can generate. Each derivation tree Tj may be generated by an appropriate grammar Gj shown in the lower part of the picture. Ti Tii Tiii Tiv X X X X j j j j (a b) (X b) (a X ) (X X ) j j j j a b a b