Asymptotic elimination of partially continuous aggregation functions in directed graphical models

In Statistical Relational Artificial Intelligence, a branch of AI and machine learning which combines the logical and statistical schools of AI, one uses the concept em para-metrized probabilistic graphical model (PPGM) to model (conditional) dependencies between random variables and to make probabilistic inferences about events on a space of "possible worlds". The set of possible worlds with underlying domain D (a set of objects) can be represented by the set WD of all first-order structures (for a suitable signature) with domain D. Using a formal logic we can describe events on WD. By combining a logic and a PPGM we can also define a probability distribution mathbbPD on WD and use it to compute the probability of an event. We consider a logic, denoted PLA, with truth values in the unit interval, which uses aggregation functions, such as arithmetic mean, geometric mean, maximum and minimum instead of quantifiers. However we face the problem of computational efficiency and this problem is an obstacle to the wider use of methods from Statistical Relational AI in practical applications. We address this problem by proving that the described probability will, under certain assumptions on the PPGM and the sentence varphi, converge as the size of D tends to infinity. The convergence result is obtained by showing that every formula varphi(x1, ldots, xk) which contains only "admissible" aggregation functions (e.g. arithmetic and geometric mean, max and min) is asymptotically equivalent to a formula psi(x1, ldots, xk) without aggregation functions.