Game semantics for dependent types

We present a model of dependent type theory (DTT) with Π-, 1-, Σ- and intensional Id -types, which is based on a slight variation of the (call-by-name) category of AJM-games and history-free winning well-bracketed strategies. The model satisfies Streicher's criteria of intensionality and refutes function extensionality. The principle of uniqueness of identity proofs is satisfied. We show it contains a submodel as a full subcategory which gives a faithful interpretation of DTT with Π-, 1-, Σ- and intensional Id -types and, additionally, finite inductive type families. This smaller model is fully (and faithfully) complete with respect to the syntax at the type hierarchy built without Id -types, as well as at the more general class of types where we allow for one strictly positive occurrence of an Id -type. Definability for the full type hierarchy with Id -types remains to be investigated.

Authors

• Matthijs Vákár
• Radha Jagadeesan
• Samson Abramsky

Keywords

• Game semantics
• Dependent type theory
• Intensionality