Limited Math: Aligning Mathematical Semantics with Finite Computation

TL;DR

Limited Math (LM) provides a finite, bounded semantic framework that aligns classical mathematical reasoning with the realities of finite computation, explicitly handling numeric and structural constraints.

Contribution

This paper introduces Limited Math, a novel bounded semantic framework explicitly modeling finite numeric and structural constraints in computation.

Findings

LM coincides with classical arithmetic within bounds

Deviations beyond bounds are explicit and analyzable

LM induces finite-state semantic models for reasoning

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.