The Mathematical Theory of Context-Free Languages
The Mathematical Theory of Context-Free Languages
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This paper presents the results of recent investigations into the question of when the context-free (CF) languages of an algebra are all recognizable. First, it is shown (Theorem 2.1.3) that phrase structure (PS) languages are the same as the CF languages in any (finitely generated) generic algebra. Mezei & Wright (MW) previously generalized the equality of CF and recognizable languages in generic monadic algebras to arbitrary generic algebras. Thus it now follows that the PS, CF and recognizable languages are the same in any generic algebra. This result has the following consequence: Let &rgr; be a congruence on a generic algebra A with the property that there exists a PS grammar G on A such that the language generated by G from any element x of A is the &rgr; congruence class of x. Then in A/&rgr;, the CF languages (which are precisely the homomorphic images of the CF languages in A) are the same as the recognizable languages (which are precisely the inverse homomorphic images of subsets of finite algebras). Any algebra with this latter property is called regular. The following is a more powerful sufficient condition for the regularity of an algebra: Suppose A and B are regular algebras and that f1 and f2 are homomorphisms from A to B. Let R be a recognizable language in A and define &rgr; B × B by &rgr; &equil; {(x,y) ¦ ( @@@@ z &egr; R) (f1 (z) &equil; x &Lgr; f2 (z) &equil; y }. Then if &rgr; is a congruence on B, B/&rgr; is regular. Any relation &rgr; (not necessarily a congruence) defined in the above way is called an algebraic transduction. It is shown that the notion of algebraic transduction generalizes the notion of binary transduction as presented for generic monadic algebras by Elgot and Mezei (EM). Some proofs have been omitted below, especially when they are standard or straightforward. Details can be found in [S].