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

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 $\mu_R^{ic} : [0, 1] \to [0, \infty)$. 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 $\bar{a}$ satisfying $ic$, the distribution of the value of $R(\bar{a})$ 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 $\alpha$.