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

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 μRic : [0, 1] → [0, ∞). We also consider a probability distribution on the set Wn of continuous structures with domain [n] which is such that for every relation symbol R, identity constraint ic, and tuple bara satisfying ic, the distribution of the value of R(bara) is given by μRic, 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 ∈ CLA has no free variable and I subseteq [0, 1] is an interval, then there is α ∈ [0, 1] such that, as n tends to infinity, the probability that the value of varphi is in I tends to α.