We present two mutual encodings, respectively of the Calculus of Inductive Constructions in Zermelo-Frrnkel set theory and the opposite way. More precisely, we actually construct two families of en-codings, relating the number of universes in the type theory with the number of inaccessible cardinals in the set theory. The main result is that both hierarchies of logical formalisms interleave w.r...

We show that strictly positive inductive types, constructed from polynomial functors, constant exponentiation and arbitrarily nested inductive types exist in any Martin-Löf category (extensive locally cartesian closed category with W-types) by exploiting our work on container types. This generalises a result by Dybjer (1997) who showed that non-nested strictly positive inductive types can be re...

Data Types without the Types Philip Wadler (Edinburgh University, Scotland [email protected]) Dedicated to David Turner on the occasion of his 70’th birthday Abstract: The data abstraction mechanism of Miranda may be adapted to a dynamically typed programming language by applying ideas from gradual typing.

We consider Martin-Löf’s well-ordering type constructor in the context of an impredicative type theory. We show that the well-ordering types can represent various inductive types faithfully in the presence of the filling-up equality rules or η-rules. We also discuss various properties of the filling-up rules. ! " ! # $ % &" '( ) '* + '' + ,. '' " ''/0 % &..'1 " '2 34",5 6 ,5/0 7 % ! 0 " ! # &" ...

Behavioural type systems ensure more than the usual safety guarantees of static analysis. They are based on the idea of “types-as-processes”, providing dedicated type algebras for particular properties, ranging from protocol compatibility to race-freedom, lock-freedom, or even responsiveness. Two successful, although rather different, approaches, are session types and process types. The former ...

Semantic substitution errors (e.g., saying "arm" when "leg" is intended) are among the most common types of errors occurring during spontaneous speech. It has been shown that grammatical gender of German target nouns is preserved in the errors (E. Mane, 1999). In 3 experiments, the authors explored different accounts of the grammatical gender preservation effect in German. In all experiments, s...

We prove that, if a geodesic metric space has Markov type 2 with constant 1, then it is an Alexandrov space of nonnegative curvature. The same technique provides a lower bound of the Markov type 2 constant of a space containing a tripod or a branching point.

In this paper, we firstly review the neutrosophic set, and then construct two new concepts called neutrosophic implication of type 1 and of type 2 for neutrosophic sets. Furthermore, some of their basic properties and some results associated with the two neutrosophic implications are proven.