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.

翻译：编程语言与计算语义中使用的经典数学模型依赖于理想化的抽象概念，如无限精度实数、无界集合和无限制计算。相比之下，具体计算本质上是有限的，在有限精度、有限内存和显式资源约束下运行。这种差异使得关于数值行为、代数性质以及有限执行下的终止性的语义推理变得复杂。本文提出有限数学（LM），一种将数学推理与有限计算对齐的有界语义框架。有限数学将数值大小、数值精度和结构复杂度的约束显式化并作为基础。一个由单一界 \(M\) 参数化的有限数值域配备了一个确定性的值映射算子，该算子强制执行量化和显式的边界行为。函数和算子保留其经典数学解释，仅在语义边界处映射到有界域，从而将含义与有界求值分离。在可表示的界内，LM 与经典算术一致；当超出界时，偏差是显式的、确定性的且可分析的。通过额外限制集合基数，LM 防止隐式的无限行为通过结构构造重新进入。因此，在 LM 下实现的计算会导出有限状态语义模型，为在有限计算环境中推理算术、结构和执行提供了原则性的基础。