The Target Discounted-Sum Problem

TL;DR

This paper investigates the computational complexity of the target discounted-sum problem, providing solutions for the finite case, partial results for the infinite case, and applications to automata theory.

Contribution

It solves the finite version of the target discounted-sum problem and links the infinite version to various mathematical and computational open problems.

Findings

Finite version of the problem is decidable.

Infinite version is linked to open problems in mathematics and computer science.

Partial solutions provided for specific cases of the infinite problem.

Abstract

The target discounted-sum problem is the following: Given a rational discount factor $0<\lambda<1$ and three rational values $a,b$, and $t$, does there exist a finite or an infinite sequence $w \in \{a,b\}^*$ or $w \in \{a,b\}^\omega$, such that $\sum_{i=0}^{|w|} w(i) \lambda^i$ equals $t$? The problem turns out to relate to many fields of mathematics and computer science, and its decidability question is surprisingly hard to solve. We solve the finite version of the problem, and show the hardness of the infinite version, linking it to various areas and open problems in mathematics and computer science: $\beta$-expansions, discounted-sum automata, piecewise affine maps, and generalizations of the Cantor set. We provide some partial results to the infinite version, among which are solutions to its restriction to eventually-periodic sequences and to the cases that $\lambda \geq \frac{1}{2}$ or $\lambda=\frac{1}{n}$, for every $n\in \mathbb{N}$. We use our results for solving some open problems on discounted-sum automata, among which are the exact-value, universality and inclusion problems for functional automata.