On the active and full use of memory in right-boundary grammars and push-down automata

Abstract

A coordinated pair system ( cp system for short) consists of a pair of grammars, the first of which is right-linear ( rl) and the second is right-boundary ( rb). A right-boundary grammar is like a right-linear grammar except that one does not distinguish between terminal and nonterminal symbols—still, the rewriting is applied to the last symbol of a string only (and erasing productions are allowed). A rewriting in a cp system consists of a pair of rewritings: one in the first and one in the second grammar—such a rewriting is possible if the pair of productions involved is in the finite set of rewrites given with the system. Is is easily seen that cp systems correspond very closely to (are another formulation of) push-down automata: the right-linear component models the input and the finite state control while the rb component models the push-down store. An rb grammar G transforms (rewrites) strings which are stored in a one-way (potentially infinite) tape. If one observes during a derivation δ the use of a fixed nth cell of the tape and one notes the symbol stored there, each time that (the contents of) the cell is rewritten, then one gets the n-active record of δ; the set of all n-active records for all successful derivations δ forms the n-active language of G, denoted ACT n ( G). It is proved that, for each rb grammar G and each n ϵ N +, ACT n ( G) is regular and moreover, for each M ⊆ N +, ∪ nϵM ACT n ( G) is regular. Another way to register the use of memory during a derivation δ is to record the contents of (a fixed) nth cell during all consecutive steps of δ—in this way one gets the n-full record of δ. The set of all n-full records for all successful derivations δ forms the n-full record language of G, denoted FR n ( G). It is proved that, as in the case of active records, ∪ nϵM FR n ( G) does not have to be regular even if M = N + (actually, one can get arbitrarily complex languages in this way). Then we provide a representation theorem allowing one to represent a cp system by an rb grammar and using this theorem we transfer the above results on the use of memory to cp systems.