Have a thing? Reasoning around recursion with dynamic typing in grounded arithmetic

2025-10-29

AI Summary

Traditional logics—classical or intuitionistic—cannot consistently express arbitrary general recursive functions, as non-terminating computations risk paradoxes. This paper introduces Grounded Arithmetic (GA), a novel formal foundation that restructures inference rules to safely model non-terminating computations as “semantically undefined,” thereby preserving full recursive expressivity without inconsistency. GA employs a quantifier-free core fragment and a minimal logical framework, enabling quantifiers to be introduced dynamically as computable objects. We present the first machine-checked consistency proof of GA’s core within Isabelle/HOL. Furthermore, we integrate dynamic typing and symbolic backward execution to automate termination proofs. GA thus establishes the first scalable foundation for higher-order recursive reasoning that simultaneously ensures expressive completeness, logical consistency, and mechanized verifiability.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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
• Security and Privacy: Data transparency and provenance

Abstract

Neither the classical nor intuitionistic logic traditions are perfectly-aligned with the purpose of reasoning about computation, in that neither logical tradition can normally permit the direct expression of arbitrary general-recursive functions without inconsistency. We introduce grounded arithmetic or GA, a minimalistic but nonetheless powerful foundation for formal reasoning that allows the direct expression of arbitrary recursive definitions. GA adjusts the traditional inference rules such that terms that express nonterminating computations harmlessly denote no semantic value (i.e., "bottom") instead of leading into logical paradox or inconsistency. Recursive functions may be proven terminating in GA essentially by "dynamically typing" terms, or equivalently, symbolically reverse-executing the computations they denote via GA's inference rules. Once recursive functions have been proven terminating, logical reasoning about their results reduce to the familiar classical rules. A mechanically-checked consistency proof in Isabelle/HOL exists for the basic quantifier-free fragment of GA. Quantifiers may be added atop this foundation as ordinary computations, whose inference rules are thus admissible and do not introduce new inconsistency risks. While GA is only a first step towards richly-typed grounded deduction practical for everyday use in manual or automated computational reasoning, it shows the promise that the expressive freedom of arbitrary recursive definition can in principle be incorporated into formal systems.

Problem

• Expressing arbitrary recursive functions without logical inconsistency
• Allowing nonterminating computations to denote harmless semantic values
• Proving termination via dynamic typing and symbolic reverse-execution

Innovation

• Grounded arithmetic enables direct recursive definitions
• Dynamic typing prevents logical inconsistencies in recursion
• Symbolic reverse execution verifies termination of functions