Related papers: Statman's Hierarchy Theorem

We introduce basic notions in category theory to type theorists, including comprehension categories, categories with attributes, contextual categories, type categories, and categories with families along with additional discussions that are…

Intersection types have been originally developed as an extension of simple types, but they can also be used for refining simple types. In this survey we concentrate on the latter option; more precisely, on the use of intersection types for…

We investigate the class of models of a general dependent theory. We continue math.LO/0702292 in particular investigating so called "decomposition of types"; thesis is that what holds for stable theory and for Th(Q,<) hold for dependent…

In a constructive setting, no concrete formulation of ordinal numbers can simultaneously have all the properties one might be interested in; for example, being able to calculate limits of sequences is constructively incompatible with…

We study the coherence and conservativity of extensions of dependent type theories by additional strict equalities. By considering notions of congruences and quotients of models of type theory, we reconstruct Hofmann's proof of the…

By reformulating a learning process of a set system L as a game between Teacher (presenter of data) and Learner (updater of the abstract independent set), we define the order type dim L of L to be the order type of the game tree. The theory…

Since Val Tannen's pioneer work on the combination of simply-typed lambda-calculus and first-order rewriting (LICS'88), many authors have contributed to this subject by extending it to richer typed lambda-calculi and rewriting paradigms,…

Szpilrajn's Lemma entails that each partial order extends to a linear order. Dushnik and Miller use Szpilrajn's Lemma to show that each partial order has a relizer. Since then, many authors utilize Szpilrajn's Theorem and the Well-ordering…

Hahn's embedding theorem asserts that linearly ordered abelian groups embed in some lexicographic product of real groups. Hahn's theorem is generalized to a class of residuated semigroups in this paper, namely, to odd involutive commutative…

Type and effect systems are a tool to analyse statically the behaviour of programs with effects. We present a proof based on the so called reducibility candidates that a suitable stratification of the type and effect system entails the…

Performing $n$ steps of $\beta$-reduction to a given term in the $\lambda$-calculus can lead to an increase in the size of the resulting term that is exponential in $n$. The same is true for the possible depth increase of terms along a…

We present a general and user-extensible equality checking algorithm that is applicable to a large class of type theories. The algorithm has a type-directed phase for applying extensionality rules and a normalization phase based on…

Originating in Girard's Linear logic, Ehrhard and Regnier's Taylor expansion of $\lambda$-terms has been broadly used as a tool to approximate the terms of several variants of the $\lambda$-calculus. Many results arise from a Commutation…

Simple type theory is formulated for use with the generic theorem prover Isabelle. This requires explicit type inference rules. There are function, product, and subset types, which may be empty. Descriptions (the eta-operator) introduce the…

A forcing extension may create new isomorphisms between two models of a first order theory. Certain model theoretic constraints on the theory and other constraints on the forcing can prevent this pathology. A countable first order theory is…

Twenty years after its introduction by Ehrhard and Regnier, differentiation in $\lambda$-calculus and in linear logic is now a celebrated tool. In particular, it allows to establish a Taylor expansion formula for various $\lambda$-calculi,…

We develop a dependent type theory that is based purely on inductive and coinductive types, and the corresponding recursion and corecursion principles. This results in a type theory with a small set of rules, while still being fairly…

We study the combinatorial equivalence of separable elements in types $A$ and $B$. A bijection is constructed from the set of separable permutations in the symmetric group $S_{n+1}$ to the set of separable signed permutations in the…

A definable type of a first-order theory is the same as a section (retraction) of the simplicial path space (decalage) of its space of types viewed as a simplicial topological space; as is well-known, in the category of simplicial sets such…

We prove that the problem of deciding the consequence relation of the full Lambek calculus with weakening is complete for the class HAck of hyper-Ackermannian problems (i.e., level F_{\omega}^{\omega} of the ordinal-indexed hierarchy of…