The Target Discounted-Sum Problem

TL;DR

This work introduces the Target Discounted-Sum (TDS) problem, establishing the finite version as decidable while proving hardness for the infinite variant and connecting it to β-expansions, discounted-sum automata, piecewise affine maps, and generalized Cantor sets. It develops a β-expansion-centric framework, including the 0-1 reduction TDS_01, to derive decidability results (notably for λ ≥ 1/2 and λ = 1/n) and PSPACE bounds for eventual-periodic representations. The study extends to generalized forms GTDS/CGTDS and CTDS^F, yielding PSPACE decidability for eventual-periodic solutions and enabling applications to open problems in discounted-sum automata (exact-value, universality, inclusion). It further exposes deep connections to dynamical systems and fractal geometry, showing reductions to and from piecewise affine maps and middle-k Cantor sets. Overall, the paper advances understanding of discounting phenomena across multiple mathematical and computational domains and provides concrete algorithmic implications for automata with discounted sums.

Abstract

The target discounted-sum problem is the following: Given a rational discount factor $0<λ<1$ and three rational values $a,b$, and $t$, does there exist a finite or an infinite sequence $w \in \{a,b\}^*$ or $w \in \{a,b\}^ω$, such that $\sum_{i=0}^{|w|} w(i) λ^i$ 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: $β$-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 $λ\geq \frac{1}{2}$ or $λ=\frac{1}{n}$, 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.

Figures (2)

• Figure 1: The automata of the example for the proof of Theorem \ref{['thm:GTDSEventuallyPeriodic']}: The safety automaton ${\cal A}$ for the exploration of the relevant gaps (stopping the exploration for too big/small gaps and for gaps with odd numerators), the Büchi automaton ${\cal B}$ expressing the $\omega$-regular constraint, and the Büchi automaton ${\cal C}$ for the intersection of ${\cal A}$ and ${\cal B}$.
• Figure 2: The discounted-sum automaton, with a discount factor $\lambda$, whose $<$-universality problem is equivalent to a TDS$_{01}$ instance with a discount factor $\lambda$ and a target value $t$.

Theorems & Definitions (37)

• Definition 1: TDS, TDS$^F$
• Definition 2: TDS$_{01}$
• Theorem 3
• Definition 4: GTDS
• Definition 5: CTDS
• Theorem 6
• Proposition 7
• Theorem 8
• Example 9
• Lemma 9