A Brief History of Singlefold Diophantine Definitions

Consider an (m + 1)-ary relation R over the set N of natural numbers. Does there exist an arithmetical formula ZΘ(a0, . . . , am, x1, . . . , xK), not involving universal quantifiers, negation, or implication, such that the representation and univocity conditions, viz., (Formula Presented) are met by each tuple (Formula Presented). A priori, the answer may depend on the richness of the language of arithmetic: Even if solely addition and multiplication operators (along with the equality relator and with positive integer constants) are adopted as primitive symbols of the arithmetical signature, the graph R of any primitive recursive function is representable; but can representability be reconciled with univocity without calling into play one extra operator designating either the dyadic operation [b, n]↠ b n or just the monadic function n ↠ b n associated with a fixed integer b > 1? As a preparatory step toward a hoped-for positive answer to this question, one may consider replacing the exponentiation operator by a dyadic relator designating an exponential-growth relation (a notion made explicit by Julia Bowman Robinson in 1952). We will discuss the said univocity, aka ‘single-fold-ness’, issue-first raised by Yuri V. Matiyasevich in 1974-, framing it in historical context. © 2023 Copyright for this paper by its authors