Abstract. We present a normalization-by-evaluation (NbE) algorithm for System F ω with βη-equality, the simplest impredicative type theory with computation on the type level. Values are kept abstract and requirements on values are kept to a minimum, allowing many different implementations of the algorithm. The algorithm is verified through a general model construction using typed applicative structures, called type and object structures. Both soundness and completeness of NbE are conceived as an instance of a single fundamental theorem.

In recent years, logical frameworks and tile logic have been separately proposed by our research groups, respectively in Udine and in Pisa, as suitable metalanguages with higher-order features for encoding and studying nominal calculi. This paper discusses the main features of the two approaches, tracing di#erences and analogies on the basis of two case studies: late #-calculus and lazy simply typed #-calculus.

We analyze a partial type checking algorithm for the inconsistent domain-free pure type system Type:Type (λ∗). We show that the algorithm is sound and partially complete using a coinductive specification of algorithmic equality. This entails that the algorithm will only diverge due to the presence of diverging computations, in particular it will terminate for all typeable terms. Keywords:

this document that certain implications are not based on a well stated formal theory but require a certain amount of hand-waving.

We present the first verified normalization-by-evaluation algorithm for System F ω, the simplest impredicative type theory with computation on the type level. Types appear in three shapes: As syntactical types, as type values which direct the reification process, and as semantical types, i.e., sets of total values. The three shapes are captured by the new concept of a type structure, and the fundamental theorem now states that an induced structure is a type substructure. This work is an attempt at an algebraic treatment of type theory based on typed applicative structures rather than categories. 1