Asymptotic elimination of partially continuous aggregation functions in directed graphical models

Abstract

For a finite and relational signature σ and finite domain D we consider the set WD of all σ-structures with domain D. On WD a probability distribution is determined by a so-called parametrized probabilistic graphical model, a concept studied in statistical relational artificial intelligence. We also consider a many valued logic, denoted PLA, with truth values in the unit interval for expressing queries. PLA uses aggregation functions, for example the arithmetic mean, geometric mean, maximum and minimum, instead of quantifiers. In this setting we prove that every formula of PLA with only admissible aggregation functions is asymptotically equivalent to a formula without aggregation functions, as the domain size tends to infinity. A corollary of this is a probabilistic convergence law for PLA-formulas with only admissible aggregation functions.