We introduce the Calculus of Delayed Reduction (cdr), that expresses that redexes can only be contracted when brought to the right position in a term, and will show that Call by Name or Value (cbn, cbv) reduction for the λ -calculus can be modelled through reduction in cdr, and that the cbn fragment of the -calculus can model reduction in cdr. cdr is a Call by Push Value calculus (cbpv) in that it separates terms in computations and values, with their corresponding types. Some simulation results were already achieved by others for cbpv, but only up to equality for cbv; for cbn past results are rather weak.

We present $\cal L$, an extension of Parigot's $\lambda\mu$-calculus by adding negation as a type constructor, together with syntactic constructs that represent negation introduction and elimination. We will define a notion of reduction that extends $\lambda\mu$'s reduction system with two new reduction rules, and show that the system satisfies subject reduction. Using Aczel's generalisation of Tait and Martin-L\"of's notion of parallel reduction, we show that this extended reduction is confluent. Although the notion of type assignment has its limitations with respect to representation of proofs in natural deduction with implication and negation, we will show that all propositions that can be shown in there have a witness in $\cal L$. Using Girard's approach of reducibility candidates, we show that all typeable terms are strongly normalisable, and conclude the paper by showing that type assignment for $\cal L$ enjoys the principal typing property.

We present λtry, an extension of the λ-calculus with named exception handling, via try, throw and catch, and present a basic notion of type assignment expressing recoverable exception handling and show that it is sound. We define an interpretation for λtry to Parigot's λμ-calculus, and show that reduction (both lazy and call by value) is preserved by the interpretation. We will show that also types assignable in the basic system are preserved by the interpretation.

We introduce an intersection type system for the lambda-mu calculus that is invariant under subject reduction and expansion. The system is obtained by describing Streicher and Reus's denotational model of continuations in the category of omega-algebraic lattices via Abramsky's domain-logic approach. This provides at the same time an interpretation of the type system and a proof of the completeness of the system with respect to the continuation models by means of a filter model construction. We then define a restriction of our system, such that a lambda-mu term is typeable if and only if it is strongly normalising. We also show that Parigot's typing of lambda-mu terms with classically valid propositional formulas can be translated into the restricted system, which then provides an alternative proof of strong normalisability for the typed lambda-mu calculus.

We show characterisation results for normalisation, head-normalisation, and strong normalisation for λ μ using intersection types. We reach these results for a strict notion of type assignment for λ μ that is the natural restriction of the domain-based system of van Bakel et al. (2011) for λ μ by limiting the type inclusion relation to just intersection elimination. We show that this system respects β μ-equality, by showing both soundness and completeness results. We then define a notion of reduction on derivations that corresponds to cut-elimination, and show that this is strongly normalisable. We use this strong normalisation result to show an approximation result, and through that a characterisation of head-normalisation. Using the approximation result, we show that there is a very strong relation between the system of van Bakel et al. (2011) and ours. We then introduce a notion of type assignment that eliminates ω as an assignable type, and show, using the strong normalisation result for derivation reduction, that all terms typeable in this system are strongly normalisable as well, and show that all strongly normalisable terms are typeable. We conclude by adding type variables to our system, and show that system essentially is that of van Bakel (2010b).

With the eye on defining a type-based semantics, this paper defines intersection and union type assignment for the sequent calculus 𝒳, a substitution-free language that enjoys the Curry-Howard correspondence with respect to the implicative fragment

We apply the principles of the intersection type discipline to the study of class-based object oriented programs and; our work follows from a similar approach (in the context of Abadi and Cardelli's ς-object calculus) taken by van Bakel and de'Liguoro. We define an extension of Featherweight Java, pFJ, and present a predicate system which we show to be sound and expressive. We also show that our system provides a semantic underpinning for the object oriented paradigm by generalising the concept of approximant from the Lambda Calculus and demonstrating an approximation result: all expressions to which we can assign a predicate have an approximant that satisfies the same predicate. Crucial to this result is the notion of predicate language, which associates a family of predicates with a class.

$\X$ is an untyped continuation-style formal language with a typed subset that provides a Curry–Howard isomorphism for a sequent calculus for implicative classical logic. $\X$ can also be viewed as a language for describing nets by composition of basic components connected by wires. These features make ${\X}$ an expressive platform on which many different (applicative) programming paradigms can be mapped. In this paper we will present the syntax and reduction rules for $\X$ ; in order to demonstrate its expressive power, we will show how elaborate calculi can be embedded, such as the λ-calculus, Bloo and Rose's calculus of explicit substitutions λx, Parigot's λμ and Curien and Herbelin's $\lmmt$ . ${\X}$ was first presented in Lengrand (2003), where it was called the λξ-calculus. It can be seen as the pure untyped computational content of the reduction system for the implicative classical sequent calculus of Urban (2000).

A notion of type assignment on Curryfied Term Rewriting Systems is introduced that uses Intersection Types of Rank 2, and in which all function symbols are assumed to have a type. Type assignment will consist of specifying derivation rules that describe how types can be assigned to terms, using the types of function symbols. Using a modified unification procedure, for each term the principal pair (of basis and type) will be defined in the following sense: from these all admissible pairs can be generated by chains of operations on pairs, consisting of the operations substitution, copying, and weakening. In general, given an arbitrary typeable GTRS, the subject reduction property does not hold. Using the principal type for the left-hand side of a rewrite rule, a sufficient and decidable condition will be formulated that typeable rewrite rules should satisfy in order to obtain this property.

We study the strict type assignment system, a restriction of the intersection type discipline, and prove that it has the principal type property. We define, for a term M, the principal pair (of basis and type). We specify three operations on pairs, and prove that all pairs deducible for M can be obtained from the principal one by these operations, and that these map deducible pairs to deducible pairs.