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

We consider continuous relational structures with finite domain and a many valued logic, , with values in the unit interval and which uses continuous connectives and continuous aggregation functions. subsumes first-order logic on ``conventional'' finite structures. To each relation symbol and identity constraint on a tuple the length of which matches the arity of we associate a continuous probability density function .We also consider a probability distribution on the set of continuous structures with domain which is such that for every relation symbol , identity constraint , and tuple satisfying , the distribution of the value of is given by , independently of the values for other relation symbols or other tuples.In this setting we prove that every formula in is asymptotically equivalent to a formula without any aggregation function. This is used to prove a convergence law for which reads as follows for formulas without free variables: If has no free variable and is an interval, then there is such that, as tends to infinity, the probability that the value of is in tends to .
View on arXiv@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 } }