On computational interpretations of the modal logic S4. I. Cut elimination

A language of constructions for minimal logic is the $\lambda$-calculus, where cut-elimination is encoded as $\beta$-reduction. We examine corresponding languages for the minimal version of the modal logic S4, with notions of reduction that encodes cut-elimination for the corresponding sequent system. It turns out that a natural interpretation of the latter constructions is a $\lambda$-calculus extended by an idealized version of Lisp\u27s \verb/eval/ and \verb/quote/ constructs. In this first part, we analyze how cut-elimination works in the standard sequent system for minimal S4, and where problems arise. Bierman and De Paiva\u27s proposal is a natural language of constructions for this logic, but their calculus lacks a few rules that are essential to eliminate all cuts. The ${\lambda_{\rm S4}}$-calculus, namelyBierman and De Paiva\u27s proposal extended with all needed rules, is confluent. There is a polynomial-time algorithm to compute principal typings of given terms, or answer that the given terms are not typable. The typed ${\lambda_{\rm S4}}$-calculus terminates, and normal forms are exactly constructions for cut-free proofs. Finally, modulo some notion \sqeq of equivalence, there is a natural Curry-Howard style isomorphism between typed ${\lambda_{\rm S4}}$-terms and natural deduction proofs in minimal S4. However, the ${\lambda_{\rm S4}}$-calculus has a non-operational flavor, in that the extra rules include explicit garbage collection, contraction and exchange rules. We shall propose another language of constructions to repair this in Part II

A few notes on formal balls

Using the notion of formal ball, we present a few easy, new results in the theory of quasi-metric spaces. With no specific order: every continuous Yoneda-complete quasi-metric space is sober and convergence Choquet-complete hence Baire in its d-Scott topology; for standard quasi-metric spaces, algebraicity is equivalent to having enough center points; on a standard quasi-metric space, every lower semicontinuous R+-valued function is the supremum of a chain of Lipschitz Yoneda-continuous maps; the continuous Yoneda-complete quasi-metric spaces are exactly the retracts of algebraic Yoneda-complete quasi-metric spaces; every continuous Yoneda-complete quasi-metric space has a so-called quasi-ideal model, generalizing a construction due to K. Martin. The point is that all those results reduce to domain-theoretic constructions on posets of formal balls

Two characterizations of topological spaces with no infinite discrete subspace

We give two characteristic properties of topological spaces with no infinite discrete subspaces. The first one was obtained recently by the first author. The full result extends well-known characterizations of posets with no infinite antichain.Comment: 10 pages, no figures Replace second author by first author in the abstract and in the first line after Theorem

The Directed Homotopy Hypothesis

The homotopy hypothesis was originally stated by Grothendieck: topological spaces should be "equivalent" to (weak) infinite-groupoids, which give algebraic representatives of homotopy types. Much later, several authors developed geometrizations of computational models, e.g., for rewriting, distributed systems, (homotopy) type theory etc. But an essential feature in the work set up in concurrency theory, is that time should be considered irreversible, giving rise to the field of directed algebraic topology. Following the path proposed by Porter, we state here a directed homotopy hypothesis: Grandis\u27 directed topological spaces should be "equivalent" to a weak form of topologically enriched categories, still very close to (infinite,1)-categories. We develop, as in ordinary algebraic topology, a directed homotopy equivalence and a weak equivalence, and show invariance of a form of directed homology

Bisimulations and Unfolding in P-Accessible Categorical Models

In this paper, we propose a categorical framework for bisimulations and unfoldings that unifies the classical approach from Joyal and al. via open maps and unfoldings. This is based on a notion of categories accessible with respect to a subcategory of path shapes, i.e., for which one can define a nice notion of trees as glueing of paths. We prove that transitions systems and pre sheaf models are a particular case of our framework. We also prove that in our framework, several characterizations of bisimulation coincide, in particular an "operational one" akin to the standard definition in transition systems. Also, accessibility is preserved by coreflexions. We then design a notion of unfolding, which has good properties in the accessible case: its is a right adjoint and is a universal covering, i.e., initial among the morphisms that have the unique lifting property with respect to path shapes. As an application, we prove that the universal covering of a groupoid, a standard construction in algebraic topology, coincides with an unfolding, when the category of path shapes is well chosen

A Radon-Nikod\'ym Theorem for Valuations

We enquire under which conditions, given two $\sigma$-finite, $\omega$-continuous valuations $\nu$ and $\mu$, $\nu$ has density with respect to $\mu$. The answer is that $\nu$ has to be absolutely continuous with respect to $\mu$, plus a certain Hahn decomposition property, which happens to be always true for measures.Comment: 22 pages, 2 figure