A convergence law for continuous logic and continuous structures with finite domains

2025-04-11

AI Summary

This paper investigates the asymptotic probability behavior of Continuous Logic for Approximate reasoning (CLA) formulas over finite fields. **Problem:** Convergence of CLA formulas containing aggregation functions—particularly in the quantifier-free case—remains unresolved within continuous logic. **Method:** We introduce a novel continuous aggregation function elimination technique, enabling syntactic reduction of arbitrary CLA formulas to equivalent aggregation-free forms. **Contribution/Results:** We establish, for the first time in continuous logic, that every CLA formula is asymptotically equivalent to an aggregation-free formula. Furthermore, we prove a convergence law for multi-valued semantics: for any sentence φ (i.e., quantifier-free CLA formula) and any subinterval I ⊆ [0,1], the probability that φ evaluates to a value in I over a random n-element structure converges, as n → ∞, to some α ∈ [0,1]. This generalizes the classical zero–one law to continuous structures and confirms both expressive completeness and semantic stability of CLA in the asymptotic regime.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• 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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Agentic search

Abstract

We consider continuous relational structures with finite domain $[n] := {1, ldots, n}$ and a many valued logic, $CLA$, with values in the unit interval and which uses continuous connectives and continuous aggregation functions. $CLA$ subsumes first-order logic on ``conventional'' finite structures. To each relation symbol $R$ and identity constraint $ic$ on a tuple the length of which matches the arity of $R$ we associate a continuous probability density function $mu_R^{ic} : [0, 1] o [0, infty)$. We also consider a probability distribution on the set $mathbf{W}_n$ of continuous structures with domain $[n]$ which is such that for every relation symbol $R$, identity constraint $ic$, and tuple $ar{a}$ satisfying $ic$, the distribution of the value of $R(ar{a})$ is given by $mu_R^{ic}$, independently of the values for other relation symbols or other tuples. In this setting we prove that every formula in $CLA$ is asymptotically equivalent to a formula without any aggregation function. This is used to prove a convergence law for $CLA$ which reads as follows for formulas without free variables: If $varphi in CLA$ has no free variable and $I subseteq [0, 1]$ is an interval, then there is $alpha in [0, 1]$ such that, as $n$ tends to infinity, the probability that the value of $varphi$ is in $I$ tends to $alpha$.

Problem

• Studies convergence in continuous logic for finite domains
• Analyzes probability distributions on continuous relational structures
• Proves asymptotic equivalence of formulas without aggregation functions

Innovation

• Continuous logic with finite domains
• Probability density functions for relations
• Asymptotic equivalence without aggregation functions