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

Main:8 Pages
Appendix:9 Pages
Abstract

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

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@article{koponen2025_2504.08923,
  title={ A convergence law for continuous logic and continuous structures with finite domains },
  author={ Vera Koponen },
  journal={arXiv preprint arXiv:2504.08923},
  year={ 2025 }
}
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