Ramified Recurrence and Computational Complexity I: Word Recurrence and Poly-time

Abstract

We exhibit a notion of recurrence over free algebras ramified into tiers, based on unfolding an implicit circularity of the recurrence schema; recurrence in two tiers was introduced independently in [4, 13, 24].) We show that the functions generated by ramified recurrence over a word algebra \(\mathbb{A}\) are obtained already using only two tiers, and are exactly the functions computable by a register machine over \(\mathbb{A}\) in time polynomial in the length of the input. For the algebra \(\mathbb{W}\) of words over a binary alphabet, these are exactly the poly-time functions in the usual sense, as shown in [4]. When the medium of computation is the algebra ℕ of unary numerals, then the numeric functions generated are exactly the ones computable in linear space, as shown independently in [2, 10, 21].