Equivalences and Transformations Systems-applications to Recursive Program Schemes and Grammars* of Regular

This work presents a unified theory of recursive program schemes, context-free grammars, grammars on arbitrary algebraic structures and, in fact, recursive definitions of all kind by means of regular systems. The equivalences of regular systems associated with either all their solutions or their least solutions (in all domains of appropriate type satisfying a set of algebraic laws expressed by equations) are systematically investigated and characterized (in some cases) in terms of system transformations by folding, unfolding and rewriting according to the equational algebraic laws. Grammars are better characterized in terms of polynomial systems which are regular systems involving the operation of set union, and the same questions are raised for them. We also examine conditions insuring the uniqueness of the solution of a regular or of a polynomial system. This theory applies to grammars of many kinds which generate trees, graphs, etc. We formulate some classical transformations of context-free grammars in terms of correct transformations which only use folding, unfolding and algebraic laws and we immediately obtain their c o r r e c t n e s s .