On Rational Recursive Sequences

TL;DR

This paper investigates rational recursive sequences, proposing algebraic techniques to analyze their properties and conjecturing equivalence between two classes, with implications for complexity and decidability problems.

Contribution

It proves a variant of the conjecture relating ratrec and simple ratrec sequences using symbolic initial conditions, and explores complexity bounds for related polynomial recursive sequences.

Findings

Proved a variant of the ratrec and simple ratrec class conjecture.

Established PSPACE lower bounds for the zeroness and equivalence problems.

Demonstrated PSPACE-hardness of the Skolem problem for polynomial recursive sequences.

Abstract

We study the class of rational recursive sequences (ratrec) over the rational numbers. A ratrec sequence is defined via a system of sequences using mutually recursive equations of depth 1, where the next values are computed as rational functions of the previous values. An alternative class is that of simple ratrec sequences, where one uses a single recursive equation, however of depth k: the next value is defined as a rational function of k previous values. We conjecture that the classes ratrec and simple ratrec coincide. The main contribution of this paper is a proof of a variant of this conjecture where the initial conditions are treated symbolically, using a formal variable per sequence, while the sequences themselves consist of rational functions over those variables. While the initial conjecture does not follow from this variant, we hope that the introduced algebraic techniques may eventually be helpful in resolving the problem. The class ratrec strictly generalises a well-known class of polynomial recursive sequences (polyrec). These are defined like ratrec, but using polynomial functions instead of rational ones. One can observe that if our conjecture is true and effective, then we can improve the complexities of the zeroness and the equivalence problems for polyrec sequences. Currently, the only known upper bound is Ackermanian, which follows from results on polynomial automata. We complement this observation by proving a PSPACE lower bound for both problems for polyrec. Our lower bound construction also implies that the Skolem problem is PSPACE-hard for the polyrec class.