A proof of strong normalisation using domain theory

Ulrich Berger presented a powerful proof of strong normalisation using domains, in particular it simplifies significantly Tait's proof of strong normalisation of Spector's bar recursion. The main contribution of this paper is to show that, using ideas from intersection types and Martin-Lof's domain interpretation of type theory one can in turn simplify further U. Berger's argument. We build a domain model for an untyped programming language where U. Berger has an interpretation only for typed terms or alternatively has an interpretation for untyped terms but need an extra condition to deduce strong normalisation. As a main application, we show that Martin-L\"{o}f dependent type theory extended with a program for Spector double negation shift.Comment: 16 page

Strong normalisation for applied lambda calculi

We consider the untyped lambda calculus with constructors and recursively defined constants. We construct a domain-theoretic model such that any term not denoting bottom is strongly normalising provided all its `stratified approximations' are. From this we derive a general normalisation theorem for applied typed lambda-calculi: If all constants have a total value, then all typeable terms are strongly normalising. We apply this result to extensions of G\"odel's system T and system F extended by various forms of bar recursion for which strong normalisation was hitherto unknown.Comment: 14 pages, paper acceptet at electronic journal LMC

On Isomorphism of "Functional" Intersection and Union Types

Type isomorphism is useful for retrieving library components, since a function in a library can have a type different from, but isomorphic to, the one expected by the user. Moreover type isomorphism gives for free the coercion required to include the function in the user program with the right type. The present paper faces the problem of type isomorphism in a system with intersection and union types. In the presence of intersection and union, isomorphism is not a congruence and cannot be characterised in an equational way. A characterisation can still be given, quite complicated by the interference between functional and non functional types. This drawback is faced in the paper by interpreting each atomic type as the set of functions mapping any argument into the interpretation of the type itself. This choice has been suggested by the initial projection of Scott's inverse limit lambda-model. The main result of this paper is a condition assuring type isomorphism, based on an isomorphism preserving reduction.Comment: In Proceedings ITRS 2014, arXiv:1503.0437

Kripke Models for Classical Logic

We introduce a notion of Kripke model for classical logic for which we constructively prove soundness and cut-free completeness. We discuss the novelty of the notion and its potential applications

Extending holomorphic maps from Stein manifolds into affine toric varieties

A complex manifold $Y$ is said to have the interpolation property if a holomorphic map to $Y$ from a subvariety $S$ of a reduced Stein space $X$ has a holomorphic extension to $X$ if it has a continuous extension. Taking $S$ to be a contractible submanifold of $X=\mathbb{C}^n$ gives an ostensibly much weaker property called the convex interpolation property. By a deep theorem of Forstneri\v{c}, the two properties are equivalent. They (and about a dozen other nontrivially equivalent properties) define the class of Oka manifolds. This paper is the first attempt to develop Oka theory for singular targets. The targets that we study are affine toric varieties, not necessarily normal. We prove that every affine toric variety satisfies a weakening of the interpolation property that is much stronger than the convex interpolation property, but the full interpolation property fails for most affine toric varieties, even for a source as simple as the product of two annuli embedded in $\mathbb{C}^4$.Comment: 14 pages, v2 and v3: minor corrections and clarifications. To appear in Proceedings of the AM

Spikes in quantum trajectories

A quantum system subjected to a strong continuous monitoring undergoes quantum jumps. This very well known fact hides a neglected subtlety: sharp scale-invariant fluctuations invariably decorate the jump process even in the limit where the measurement rate is very large. This article is devoted to the quantitative study of these remaining fluctuations, which we call spikes, and to a discussion of their physical status. We start by introducing a classical model where the origin of these fluctuations is more intuitive and then jump to the quantum realm where their existence is less intuitive. We compute the exact distribution of the spikes for a continuously monitored qubit. We conclude by discussing their physical and operational relevance.Comment: 8 pages, 8 figure

Cyclic Datatypes modulo Bisimulation based on Second-Order Algebraic Theories

Cyclic data structures, such as cyclic lists, in functional programming are tricky to handle because of their cyclicity. This paper presents an investigation of categorical, algebraic, and computational foundations of cyclic datatypes. Our framework of cyclic datatypes is based on second-order algebraic theories of Fiore et al., which give a uniform setting for syntax, types, and computation rules for describing and reasoning about cyclic datatypes. We extract the "fold" computation rules from the categorical semantics based on iteration categories of Bloom and Esik. Thereby, the rules are correct by construction. We prove strong normalisation using the General Schema criterion for second-order computation rules. Rather than the fixed point law, we particularly choose Bekic law for computation, which is a key to obtaining strong normalisation. We also prove the property of "Church-Rosser modulo bisimulation" for the computation rules. Combining these results, we have a remarkable decidability result of the equational theory of cyclic data and fold.Comment: 38 page