Inherently Reversible Grammars, Logic Programming And Computability

This paper a t tempts to clarify two distinct notions of "reversibility": (i) Uniformity of implementation of parsing and generation, and (it) reversibility as an inherent (or intrinsic) property of grammars. On the one hand, we explain why grammars specified as definite programs (or the various related "unification grammars") lead to uniformity of implementation. On the other hand, we define different intrinsic reversibility properties for such g rammars the most important being finite reversibility, which says that both parsing and generation are finitely enumerable (see t e x t ) and give examples and counter-examples of grammars which possess or do not possess these intrinsic properties. We also show that , under a certain "moderation" condition on linguistic description, finite enumerability of parsing is equivalent to finite enumerability of generation. 1 I n t r o d u c t i o n From the linguist's point of view, a grammar is a formal device which defines a recursively enumerable set of well-formed linguistic structures, each having, among other aspects, a phonological content (or, when dealing with writ ten text, a string content) and a semantic content. Such a device is completely neutral as regards its uses for parsing (recovering semantic content from string content) or generation (recovering string content from semantic content). From the computat ional linguist's point of view, on the other hand, the problem is how to implement such a grammar both as a parsing program and as a generation program, in such a way that these programs exactly reflect the content of the grammar. This we will call the reversibility problem. Let us assume, for specificity, tha t the grammar has been presented as a definite program (a Prolog program)J Then the reversibility problem has a simple solution: use a complete interpreter for definite programs--for instance a top-down interpreter having a breadth-first search procedure2--and directly use the grammar as the program both for parsing and for generation. In the parsing mode, for any given string x, the program will enumerate all semantics Yl,Y2, . . . assigned to it by the grammar, and similarly, in the generation mode, for any given semantics y, the program will enumerate all semantics xl , x 2 , . . , assigned to it by the grammar. This is a striking property of definite programs: they are reversible in the sense that they naturally lead to uniformity of implementation of the parsing and generation modes (see §4). So the reversibility problem is solved, and we can spend the next few years skimming through Fodor's (not Jerry 's) guides in travel bookstores? Not quite. First, the s tandard depth-first interpreter for definite programs is an incomplete one, and this problem must be circumvented in some way. Second, and more crucially, even when using a complete interpreter, parsing (and similarly generation) does not in general terminate: the program may well enumerate Yl,Y2, . . . ad infinitum. This is even true if, in fact, there are only a finite number of solutions Yl ,Y2, . . . , Yk, or even, in the extreme case, no solution at all: the program may not be "aware" that it has at some point already exhausted all the solutions that it will eventually 1We could have made s o m e o t h e r choice, for instance s o m e unification grammar formalism. The advantage of using definite programs in t h e p r e s e n t discussion is that they embody the whole unification paradigm in its purest form, that unification of terms is conceptually simpler (and less p r o n e to misunderstandings) than unification of DAGs, and that the denotational and operational semantics of definite programs have been thoroughly studied. 2See e.g. [7, p. 59] and section §2.1.3. See also [19] in this volume for a related approach.