Limited Math: Aligning Mathematical Semantics with Finite Computation

2026-01-08

arXiv.org

AI Summary

Classical mathematical semantics relies on infinite precision and unbounded sets, making it ill-suited for precise reasoning about numerical behavior, algebraic properties, and termination in finite computational settings. This work proposes the Limited Math (LM) framework, which— for the first time—takes finiteness of numerical magnitude, precision, and structural complexity as foundational semantic principles. By introducing a unified bound parameter \( M \), a deterministic value-mapping operator, and a semantic boundary separation mechanism, LM constructs a finite-state semantic model. Within its representable domain, the framework aligns with classical arithmetic while explicitly characterizing out-of-bound behaviors and eliminating implicit infinities. This provides a rigorous, analyzable foundation for program semantics, numerical analysis, and termination verification grounded in finite computation.

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

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Classical mathematical models used in the semantics of programming languages and computation rely on idealized abstractions such as infinite-precision real numbers, unbounded sets, and unrestricted computation. In contrast, concrete computation is inherently finite, operating under bounded precision, bounded memory, and explicit resource constraints. This discrepancy complicates semantic reasoning about numerical behavior, algebraic properties, and termination under finite execution. This paper introduces Limited Math (LM), a bounded semantic framework that aligns mathematical reasoning with finite computation. Limited Math makes constraints on numeric magnitude, numeric precision, and structural complexity explicit and foundational. A finite numeric domain parameterized by a single bound \(M\) is equipped with a deterministic value-mapping operator that enforces quantization and explicit boundary behavior. Functions and operators retain their classical mathematical interpretation and are mapped into the bounded domain only at a semantic boundary, separating meaning from bounded evaluation. Within representable bounds, LM coincides with classical arithmetic; when bounds are exceeded, deviations are explicit, deterministic, and analyzable. By additionally bounding set cardinality, LM prevents implicit infinitary behavior from re-entering through structural constructions. As a consequence, computations realized under LM induce finite-state semantic models, providing a principled foundation for reasoning about arithmetic, structure, and execution in finite computational settings.

Problem

• finite computation
• mathematical semantics
• bounded precision
• resource constraints
• semantic reasoning

Innovation

• Limited Math
• bounded semantics
• finite computation
• deterministic quantization
• finite-state models