Abstract. Soft linear logic ([Lafont02]) is a subsystem of linear logic characterizing the class PTIME. We introduce Soft lambda-calculus as a calculus typable in the intuitionistic and affine variant of this logic. We prove that the (untyped) terms of this calculus are reducible in polynomial time. We then extend the type system of Soft logic with recursive types. This allows us to consider non-standard types for representing lists. Using these datatypes we examine the concrete expressiveness of Soft lambda-calculus with the example of the insertion sort algorithm. 1

Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: self-application is meaningful, but not necessarily total. It has turned out that theories with self-application provide a natural setting for studying notions of abstract computability, especially from a proof-theoretic perspective.

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We provide a natural characterization of the type two Mehlhorn-CookUrquhart basic feasible functionals as the provably total type two functionals of our (classical) applicative theory PT introduced in [27], thus providing a proof of a result claimed in the conclusion of [27].

This is a survey paper on various weak systems of Feferman’s explicit mathematics and their proof theory. The strength of the systems considered in measured in terms of their provably terminating operations typically belonging to some natural classes of computational time or space complexity. Keywords: Proof theory, Feferman’s explicit mathematics, applicative theories, higher types, types and names, partial truth, feasible operations 1

A typed lambda calculus with recursion in all finite types is defined such that the first order terms exactly characterize the parallel complexity class NC. This is achieved by use of the appropriate forms of recursion (concatenation recursion and logarithmic recursion), a ramified type structure and imposing of a linearity constraint.

Abstract. We formalize SℓPCF, namely a programming language which is able to represent linear function between coherence spaces. We give an interpretation of SℓPCF into the model of linear coherence spaces and we show that such semantics is fully abstract with respect to it. SℓPCF is not syntactically linear, namely its programs can contain the same variable more than once. Last, we address the universality problem. 1

A purely syntactical proof is given that all functions definable in a certain affine linear typed -calculus with iteration in all types are polynomial time computable. The proof also gives explicit polynomial bounds that can easily be calculated. 1 Summary In [6] Hofmann presented a linear type system for nonsize -increasing polynomial time computation allowing unrestricted recursion for inductive datatypes. The proof that all definable functions of type N ( N are polynomial time computable essentially used semantic concepts, such as the set-theoretic interpretation of terms. We present a different proof of the same result for a slightly modified version of the term system, which uses syntactical arguments only. However, this paper is more than a new proof of an already known result, as the method choosen has several benefits: ffl A reduction relation is defined on the term system such that the term system is closed under reduction. Therefore calculations can be done within the syst...