Thomas Streicher

Classical logic, continuation semantics and abstract machines

One of the goals of this paper is to demonstrate that denotational semantics is useful for operational issues like implementation of functional languages by abstract machines. This is exemplified in a tutorial way by studying the case of extensional untyped call-by-name λ-calculus with Felleisens control operator C. We derive the transition rules for an abstract machine from a continuation semantics which appears as a generalization of the ¬¬-translation known from logic. The resulting abstract machine appears as an extension of Krivines machine implementing head reduction. Though the result, namely Krivines machine, is well known our method of deriving it from continuation semantics is new and applicable to other languages (as e.g. call-by-value variants). Further new results are that Scotts D∞-models are all instances of continuation models. Moreover, we extend our continuation semantics to Parigots λμ-calculus from which we derive an extension of Krivines machine for λμ-calculus. The relation between continuation semantics and the abstract machines is made precise by proving computational adequacy results employing an elegant method introduced by Pitts.

Games semantics for linear logic

An attempt is made to relate various notions of duality used in mathematics with the denotational semantics of linear logic. The author proposes a naive semantics for linear logic that, in a certain sense, generalizes various notions such as finite-dimensional vector spaces, topological spaces, and J.-Y. Girards (1987) coherence spaces. A game consists of a set of vectors (or strategies), a set of forms (or co-strategies) and an evaluation bracket. This is enough to interpret the connectives of full propositional linear logic, including exponentials.<<ETX>>

The groupoid model refutes uniqueness of identity proofs

We give a model of intensional Martin-Lof type theory based on groupoids and fibrations of groupoids in which identity types may contain two distinct elements which are not even prepositionally equal. This shows that the principle of uniqueness of identity proofs is not derivable in the syntax.<<ETX>>

Domain-theoretic Foundations of Functional Programming

PCF and Its Operational Semantics The Scott Model of PCF Computational Adequacy Milners Context Lemma The Full Abstraction Problem Logical Relations Some Structural Properties of the D Solutions of Recursive Domain Equations Characterisation of Fully Abstract Models Sequential Domains as a Model of PCF The Model of PCF in S is Fully Abstract Computability in Domains.

General synthetic domain theory – a logical approach

Synthetic domain theory (SDT) is a version of Domain Theory where ‘all functions are continuous’. Following the original suggestion of Dana Scott, several approaches to SDT have been developed that are logical or categorical, axiomatic or model-oriented in character and that are either specialised towards Scott domains or aim at providing a general theory axiomatising the structure common to the various notions of domains studied so far.In Reus and Streicher (1993), Reus (1995) and Reus (1998), we have developed a logical and axiomatic version of SDT, which is special in the sense that it captures the essence of Domain Theory a la Scott but rules out, for example, Stable Domain Theory, as it requires order on function spaces to be pointwise. In this article we will give a logical and axiomatic account of a general SDT with the aim of grasping the structure common to all notions of domains.As in loc.cit., the underlying logic is a sufficiently expressive version of constructive type theory. We start with a few basic axioms giving rise to a core theory on top of which we study various notions of predomains (such as, for example, complete and well-complete S-spaces (Longley and Simpson 1997)), define the appropriate notion of domain and verify the usual induction principles of domain theory.Although each domain carries a logically definable ‘specialization order’, we avoid order-theoretic notions as much as possible in the formulation of axioms and theorems. The reason is that the order on function spaces cannot be required to be pointwise, as this would rule out the model of stable domains a la Berry.The consequent use of logical language – understood as the internal language of some categorical model of type theory – avoids the irritating coexistence of the internal and the external view pervading purely categorical approaches. Therefore, the paper is aimed at providing an elementary introduction to synthetic domain theory, albeit requiring some knowledge of basic type theory.

Impredicativity entails untypedness

In standard realizability one works with respect to an untyped universe of realizers called a partial combinatory algebra (pca). It is well known that a pca [Ascr ] gives rise to a categorical model of impredicative type theory via the category Asm ([Ascr ]) of assemblies over [Ascr ] or the realizability topos over [Ascr ]. Recently, John Longley introduced a typed version of pcas (Longley 1999b). The above mentioned construction of categorical models extends to the typed case. However, in general these are no longer impredicative. We show that for a typed pca [Tscr ] the ensuing models are impredicative if and only if [Tscr ] has a universal type U . Such a type U can be endowed with the structure of an untyped pca such that U and [Tscr ] induce equivalent realizability models: in other words, a typed pca [Tscr ] with a universal type is essentially untyped. Thus, a posteriori it turns out that nothing is lost by restricting to (untyped) pcas as far as realizability models of impredicative type theories are concerned. For instance, we show that for a typed pca [Tscr ] the fibred category of discrete families in Asm ([Tscr ]) is small if and only if [Tscr ] has a universal type. As the category of ¬¬-separated objects of the modified realizability topos is equivalent to Asm ([Tscr ]) for an appropriate typed pca [Tscr ] without a universal type, it follows that the discrete families in the subcategory of ¬¬-separated objects of the modified realizability topos do not provide a model of polymorphic λ-calculus.

Shoenfield is Gödel after Krivine

We show that Shoenfields functional interpretation of Peano arithmetic can be factorized as a negative translation due to J. L. Krivine followed by Godels Dialectica interpretation. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On the non-sequential nature of the interval-domain model of real-number computation

We show that real-number computations in the interval-domain environment are ‘inherently parallel’ in a precise mathematical sense. We do this by reducing computations of the weak parallel-or operation on the Sierpinski domain to computations of the addition operation on the interval domain.

Reduction-free normalisation for a polymorphic system

We give a semantical proof that every term of a combinator version of system F has a normal form. As the argument is entirely formalisable in an impredicative constructive type theory a reduction-free normalisation algorithm can be extracted from this. The proof is presented as the construction of a model of the calculus inside a category of presheaves. Its definition is given entirely in terms of the internal language.

About hoare logics for higher-order store

We present a Hoare logic for a simple imperative while-language with stored commands, ie. stored parameterless procedures. Stores that may contain procedures are called higher-order. Soundness of our logic is established by using denotational rather than operational semantics. The former is employed to elegantly account for an inherent difficulty of higher-order store, namely that assertions necessarily describe recursive predicates on a recursive domain. In order to obtain proof rules for mutually recursive procedures, assertions have to explicitly refer to the code of the procedures.