Semantic preserving translations

Abstract

Let X1, X2 be derivation systems (free x-categories) generated by context free grammars. Let Xo be a translation category with x-functors fi: Xo --~ X, i = l, 2." Let T be an ~*-theory, a generalization of algebraic theories. Let It: X~ -~ T be algebraic interpretations of the derivations systems, giving the semantics of derivation systems. The translation category Xo is shown to preserve the common semantics through the translation if there is a natural transformation from the functor ]'2 °/2 to the functor fl o 11. This is used to show that certain elementary conditions on well-behaved generalized 2 sequential machine maps (g2sm maps) result in semantics preservation by the g2sm maps.1. Introduction. Consider two formal languages, L1 and L 2. If one does not consider the grammars for these languages, then a translation from L 1 to L2 is a relation from L1 to L 2. However, this notion gives no insight as to why two sentences, ~o 1 e LI and to 2 E L2, are or are not related by the translation. If we accept the idea that each syntactic structure of a sentence determines exactly one meaning, then it is correct to consider a translation as mapping each syntactic structure of sentences in L~ to one or more syntactic structures of various sentences in L 2. If one further admits the possibility that not all syntactic structures of the first language system are translatable, we see that a translation can be viewed as a relation from the syntactic structures of L 1 to the syntactic structures of L 2. This is the general nature of the translations we will consider. Such syntactically based translations of formal languages, frequently called syntax directed translations, have been studied recently by Thatcher [21], A h o and Ullman [l, 2], Lewis and Stearns [13], and in an algebraic setting by Shepard [17]. F o r a rather complete bibliography, see [21].Translations must be effective to be of interest in current formal studies of natural and p r o g r a m m i n g languages. A natural method of obtaining the

On the Correctness of Semantic-Syntax-Directed Translations

“…[] CLAIM 2. If Z satisfies the conditions of Theorem 6 except for condition (2), then it is undecidable whether rz is a total semanttc-preserving translation. If Z sattsfies the condttions of Theorem 6 except for condition (3), then tt is undecidable whether *z is a total semantic-preservmg translation.…”

“…Denote the root node oft2 in t by c, and the root node of t~ in t' by d. Thus A(d) ffi c. Since~rt~ ~~r;~ for q = 1, 2, we have 7rt~ (c, a) --~r~(d, a), q = 1, 2, We now prove that ~rt ~ ~r~,. Clearly conditions (1), (2), and (4) of Definition ! i are met.…”

“…This definition is reasonable in a "semantacs-free" environment. WRhin the last decade, however, several formal semantic models have appeared [2,7,14,16]. These developments prompt a change in the definition of"language" to include meaning.…”

Transductions of dags and trees

“…In [7], Benson has studied such transformations in a general algebraic framework. For example, it is interesting to simplify the syntactic rules used to describe given objects.…”