Monads for composing categorical and probabilistic meanings

In this paper we present a mathematical model for the interpretation of probabilistic meanings in a formal semantic framework. Our model is inspired by work in computer science for the definition of a semantics for probabilistic computations. The model is an instance of a more general approach we are exploring to give a constructive, and so computable, interpretation to a large number of semantic phenomena (Giorgolo and Asudeh 2012a; 2012b). The question of combining categorical meaning representations with probabilistic ones is particularly important for natural language processing, where one often needs to combine information coming from diverse sources to assign an interpretation to a sentence. For instance we may want to combine the output of a stochastic classifier that assigns a certain probability to the assigned class with the output of a database lookup, which is usually a categorical response. Our model allows us to mix probabilistic meaning representations with purely symbolic ones and to propagate the probabilities from constituents to larger expressions in a well understood way. In our model, meanings will be associated with a probability measure. For example an expression denoting an individual will refer to a specific individual with a certain probability and to others with different probability. In this way we can easily and compactly represent and keep track of ambiguous reference. Similarly if a predicate is applied to an argument the resulting proposition may be considered true with a certain probability p and false with probability 1− p. The main advantages of our model are the following: