Mathematical and Computational Models of Transformational Grammar

INTRODUCTION In this paper we compare three models of transformational grammar: the mathematical model of Ginsburg and Partee (1969) as applied by Salomaa (1971), the mathematical model of Peters and Ritchie (1971 and forthcoming), and the computer model of Friedman et al. (1971). All of these are, of course, based on the work of Chomsky as presented in Aspects of the Theory of Syntax (1965). We were led to this comparison by the observation that the computer model is weaker in three important ways: search depth is not unbounded, structures matching variables cannot be compared, and structures matching variables cannot be moved. All of these are important to the explanatory adequacy of transformational grammar. Both mathematical models allow the first, they each allow some form of the second, one of them allows the third. We were interested in the mathematical consequences of our restrictions. The comparison will be carried out by reformulating in the computer system the most interesting proofs to date of the ability of transformational grammars to generate any recursively enumerable set. These are Salomaa's proof that the Ginsburg-Partee model can generate any recursively enumerable (r.e.) set from a regular base, and the Peters-Ritchie proof that any r.e. set can be obtained from a minimal linear base. Although modifications are required, it is, as we shall show, possible to obtain these results within the weaker computer model. Thus, every recursively enumerable language is generated by a transformational grammar with limited search depth, without equality comparisons of variables, and without moving structures corresponding to variables. The comparison reinforces the observation that transformational grammars can be excessively powerful in terms of generative capacity while at the same time lacking features necessary for explanatory adequacy.