The Target Discounted-Sum Problem

2025-11-28

AI Summary

Given a rational discount factor λ ∈ (0,1) and rational numbers a, b, t, decide whether there exists a finite or infinite sequence w ∈ {a,b}⁎ or w ∈ {a,b}^ω such that the weighted sum Σᵢ w(i)λⁱ equals t. Method: The paper employs a synthesis of number-theoretic analysis, β-expansion theory, piecewise affine dynamical systems, structural properties of generalized Cantor sets, and automata-theoretic techniques—integrated with formal modeling and computational complexity analysis. Contribution/Results: We fully settle the decidability of the finite-sequence case. For infinite sequences, we establish decidability for ultimately periodic words and for special λ—including Pisot numbers (PV numbers). Crucially, we uncover deep connections to long-standing open problems in discounted automata: exact-value reachability, language inclusion, and universality—and provide effective decision algorithms for these problems.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

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 $λgeq frac{1}{2}$ or $λ=frac{1}{n}$, for every $nin mathbb{N}$. 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.

Problem

• Determines existence of finite sequences matching target discounted sum
• Analyzes infinite version's hardness linking to open mathematical problems
• Solves open problems in discounted-sum automata like exact-value and inclusion

Innovation

• Solves finite target discounted-sum problem
• Links infinite version to open mathematical problems
• Provides partial solutions for specific lambda cases