Model-Theoretic Expressivity Analysis

Abstract

In the preceding chapter the problem of comparing languages was considered from a behavioral perspective. In this chapter we develop an alternative, model-theoretic approach.

In this approach we compare the expressiveness of probabilistic-logic (pl-) languages by considering the models that can be characterized in a language. Roughly speaking, one language L′ is at least as expressive as another language L, if every model definable in L also is definable in L′. Results obtained in the model-theoretic approach can be somewhat stronger than results obtained in the behavioral approach in that equivalence of models entails equivalent behavior with respect to any possible type of inference tasks. On the other hand, the model-theoretic approach is somewhat less flexible than the behavioral approach, because only languages can be compared that define comparable types of models. A comparison between Bayesian Logic Programs (defining probability distributions on possible worlds) and Stochastic Logic Programs (defining probability distributions over derivations), therefore, is already quite challenging in a model-theoretic approach, as it requires first to define a unifying semantic framework. In this chapter, therefore, we focus on pl-languages that exhibit stronger semantic similarities (Bayesian Logic Programs (BLPs) [6], Probabilistic Relational Models (PRMs) [1], Multi-Entity Bayesian Networks [7], Markov Logic Networks (MLNs) [12], Relational Bayesian Networks (RBNs) [4]), and first establish a unifying semantics for these languages. However, the framework we propose is flexible to enough (with a slightly bigger effort) to also accommodate languages like Stochastic Logic Programs [9] or Prism [3].