A Type Assignment System for the Game Semantics

The aim of this paper is to give an alternative description of the game semantics (). The main idea is to mimic, in the game semantics setting, a duality existing in domain theory. In domain theory, the denotational semantics of a language can be given in two forms: a term can be interpreted by some point in a particular domain, or by a set of properties. The Stone-duality theorems, for domain theory , tell us that these two alternative descriptions are equivalent. In this setting, the properties of a term are normally called “types” and a set of rules, that allow to derive the properties satisfied by a term, is called “type assignment system”. In the literature type assignment systems are mainly used in the semantics of lambda calculi (). But, in principle, they can be used as an alternative description of any kind of denotational semantics, based on domain theory. Type assignment systems provide a concrete finitary way for defining the semantics of a term. In this paper we introduce a finitary description of λ-terms in a particular game model. We think that type assignment systems are useful in the context of game semantics since they provide a more concrete and intuitive account of the interpretation of terms given by game models. In fact, classically game semantics is given using a categorical definition. To derive the concrete definition from the categorical one can be a tiring task.