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

Abstract

We consider continuous relational structures with finite domain [n] := \1, , 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 _R^ic : [0, 1] [0, ). We also consider a probability distribution on the set W_n of continuous structures with domain [n] which is such that for every relation symbol R, identity constraint ic, and tuple a satisfying ic, the distribution of the value of R(a) is given by _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 CLA has no free variable and I [0, 1] is an interval, then there is [0, 1] such that, as n tends to infinity, the probability that the value of is in I tends to .

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