On the relative asymptotic expressivity of inference frameworks

We consider logics with truth values in the unit interval [0,1]. Such logics are used to define queries and to define probability distributions. In this context the notion of almost sure equivalence of formulas is generalized to the notion of asymptotic equivalence. We prove two new results about the asymptotic equivalence of formulas where each result has a convergence law as a corollary. These results as well as several older results can be formulated as results about the relative asymptotic expressivity of inference frameworks. An inference framework F is a class of pairs (mathbbP, L), where mathbbP = (mathbbPn : n = 1, 2, 3, ldots), mathbbPn are probability distributions on the set Wn of all σ-structures with domain 1, ldots, n (where σ is a first-order signature) and L is a logic with truth values in the unit interval [0, 1]. An inference framework F' is asymptotically at least as expressive as an inference framework F if for every (mathbbP, L) ∈ F there is (mathbbP', L') ∈ F' such that mathbbP is asymptotically total variation equivalent to mathbbP' and for every varphi(barx) ∈ L there is varphi'(barx) ∈ L' such that varphi'(barx) is asymptotically equivalent to varphi(barx) with respect to mathbbP. This relation is a preorder. If, in addition, F is at least as expressive as F' then we say that F and F' are asymptotically equally expressive. Our third contribution is to systematize the new results of this paper and several previous results in order to get a preorder on a number of inference systems that are of relevance in the context of machine learning and artificial intelligence.