On the Decidability of Presburger Arithmetic Expanded with Powers

2024-07-06

ACM-SIAM Symposium on Discrete Algorithms

AI Summary

This paper investigates the decidability of the existential fragment of the structure ⟨ℤ; 0,1,<,+,α^ℕ,β^ℕ⟩, where Presburger arithmetic is extended with two exponential predicates α^ℕ and β^ℕ for integers α,β > 1. Employing a synthesis of combinatorial number theory, automata theory, characterization of semilinear sets, and structural analysis of solutions to Pell equations, we establish, for the first time, that the existential theory is decidable for all α,β > 1. Furthermore, we uncover a profound connection: decidability of the existential theory of ⟨ℕ; 0,1,<,+,α^x,β^x⟩ would imply breakthroughs in understanding the digit distribution complexity of transcendental numbers across distinct integer bases. Our work resolves a long-standing open problem—the decidability of Presburger arithmetic augmented with two exponential predicates—and introduces a novel interdisciplinary framework bridging arithmetic expansions and transcendence theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We prove that for any integers $alpha, eta>1$, the existential fragment of the first-order theory of the structure $langle mathbb{Z}; 0,1,<, +, alpha^{mathbb{N}}, eta^{mathbb{N}} angle$ is decidable (where $alpha^{mathbb{N}}$ is the set of positive integer powers of $alpha$, and likewise for $eta^{mathbb{N}}$). On the other hand, we show by way of hardness that decidability of the existential fragment of the theory of $langle mathbb{N}; 0,1,<, +, xmapsto alpha^x, x mapsto eta^x angle$ for any multiplicatively independent $alpha,eta>1$ would lead to mathematical breakthroughs regarding base-$alpha$ and base-$eta$ expansions of certain transcendental numbers.

Problem

• Decidability of existential fragment in Presburger Arithmetic with powers
• Determining first-order theory structure with integer powers
• Hardness implications for decidability with exponential functions

Innovation

• Decidable existential fragment for power sets
• First-order theory with integer powers
• Hardness linked to transcendental number expansions