Enhancing Symbolic Manipulation through Pairing Primitive Recursive String Functions and Interplay with Generalized Pairing PRSF

Abstract

In earlier studies, the notion of generalized primitive recursive string functions has been presented, and their connections with abstract paring-based primitive recursive string functions have been investigated. Our study is centered around establishing a fundamental theorem that states a connection between these two distinct sorts of functions. The theorem specifically establishes that the universal definition of every generalized pairing primitive recursive string function is contingent upon its correspondence with a conventional(abstract) pairing primitive recursive string function. This article introduces innovative concept of Pairing Primitive Recursive String Functions (P-PRSF) for manipulating and interacting with word pairs. Based on the principles of primitive recursion and pairing functions, P-PRSF enables the extraction, transformation, and combination of word components. The proposed theorems validate the effectiveness of P-PRSF in capturing relationships within word pairs. Moreover, the interplay between P-PRSF and Generalized Pairing PRSF (GP-PRSF) extends the concept to involve more intricate interactions.