Silvio Valentini

Inductively generated formal topologies

Formal topology aims at developing general topology in intuitionistic and predicative mathematics. Many classical results of general topology have been already brought into the realm of constructive mathematics by using formal topology and also new light on basic topological notions was gained with this approach which allows distinction which are not expressible in classical topology. Here we give a systematic exposition of one of the main tools in formal topology: inductive generation. In fact, many formal topologies can be presented in a predicative way by an inductive generation and thus their properties can be proved inductively. We show however that some natural complete Heyting algebra cannot be inductively defined.

Constructive domain theory as a branch of intuitionistic pointfree topology

Abstract In this paper, the notions of information base and of translation between information bases are introduced; they have a very simple intuitive interpretation and can be taken as an alternative approach to domain theory. Technically, they form a category which is equivalent to the category of Scott domains and approximable mappings. All the definitions and most of the results are inspired by the intuitionistic approach to pointfree topology as developed mainly by Martin-Lof and the first author. As in intuitionistic pointfree topology, constructivity is guaranteed by adopting the framework of Martin-Lofs intuitionistic type theory, equipped with a few abbreviations which allow to use a standard set theoretic notation.

A modal sequent calculus for a fragment of arithmetic

Global properties of canonical derivability predicates (the standard example is Pr() in Peano Arithmetic) are studied here by means of a suitable propositional modal logic GL. A whole book [1] has appeared on GL and we refer to it for more information and a bibliography on GL. Here we propose a sequent calculus for GL and, by exhibiting a good proof procedure, prove that such calculus admits the elimination of cuts. Most of standard results on GL are then easy consequences: completeness, decidability, finite model property, interpolation and the fixed point theorem.

Tychonoff's Theorem in the Framework of Formal Topologies

In this paper we give a constructive proof of the pointfree version of Tychonoff’s theorem within formal topology, using ideas from Coquand’s proof in [7]. To deal with pointfree topology Coquand uses Johnstone’s coverages. Because of the representation theorem in [3], from a mathematical viewpoint these structures are equivalent to formal topologies but there is an essential difference also. Namely, formal topologies have been developed within Martin Lof’s constructive type theory (cf. [15]), which thus gives a direct way of formalizing them (cf. [4]). The most important aspect of our proof is that it is based on an inductive definition of the topological product of formal topologies. This fact allows us to transform Coquand’s proof into a proof by structural induction on the last rule applied in a derivation of a cover. The inductive generation of a cover, together with a modification of the inductive property proposed by Coquand, makes it possible to formulate our proof of Tychonoff’s theorem in constructive type theory. There is thus a clear difference to earlier localic proofs of Tychonoff’s theorem known in the literature (cf. [9], [10], [12], [14]). Indeed we not only avoid to use the axiom of choice, but reach constructiveness in a very strong sense. Namely, our proof of Tychonoff’s theorem supplies an algorithm which, given a cover of the product space, computes a finite subcover, provided that there exists a similar algorithm for each component space. Since type theory has been implemented on a computer (cf. [18]), an eventual strict formalization of our proof will at the same time be a computer program that executes the task of finding a finite subcover. The paper is organized as follows. In the first part we recall the basic definitions and motivations of formal topologies and introduce in this framework

An elementary proof of strong normalization for intersection types

Abstract. We provide a new and elementary proof of strong normalization for the lambda calculus of intersection types. It uses no strong method, like for instance Tait-Girard reducibility predicates, but just simple induction on type complexity and derivation length and thus it is obviously formalizable within first order arithmetic. To obtain this result, we introduce a new system for intersection types whose rules are directly inspired by the reduction relation. Finally, we show that not only the set of strongly normalizing terms of pure lambda calculus can be characterized in this system, but also that a straightforward modification of its rules allows to characterize the set of weakly normalizing terms.

The problem of the formalization of constructive topology

Abstract.Formal topologies are today an established topic in the development of constructive mathematics. One of the main tools in formal topology is inductive generation since it allows to introduce inductive methods in topology. The problem of inductively generating formal topologies with a cover relation and a unary positivity predicate has been solved in [CSSV]. However, to deal both with open and closed subsets, a binary positivity predicate has to be considered. In this paper we will show how to adapt to this framework the method used to generate inductively formal topologies with a unary positivity predicate; the main problem that one has to face in such a new setting is that, as a consequence of the lack of a complete formalization, both the cover relation and the positivity predicate can have proper axioms.

Representation Theorems for Quantales

In this paper we prove that any quantale Q is (isomorphic to) a quantale of suitable relations on Q. As a consequence two isomorphism theorems are also shown with suitable sets of functions of Q into Q. These theorems are the mathematical background one needs in order to give natural and complete semantics for (non-commutative) Linear Logic using relations. Mathematics Subject Classification: 06D05, 06D10, 06D20, 03G25.

THE JUDGEMENT CALCULUS FOR INTUITIONISTIC LINEAR LOGIC: PROOF THEORY AND SEMANTICS

In this paper we propose a new set of rules for a judgement calculus, i.e. a typed lambda calculus, based on Intuitionistic Linear Logic; these rules ease the problem of defining a suitable mathematical semantics. A proof of the canonical form theorem for this new system is given: it assures, beside the consistency of the calculus, the termination of the evaluation process of every well-typed element. The definition of the mathematical semantics and a completeness theorem, that turns out to be a representation theorem, follow. This semantics is the basis to obtain a semantics for the evaluation process of every well-typed program. 1991 MSC: 03B20, 03B40.

Krivine's Intuitionistic Proof of Classical Completeness (for countable languages)

Abstract In 1996, Krivine applied Friedmans A-translation in order to get an intuitionistic version of Godel completeness result for first-order classical logic and (at most) countable languages and models. Such a result is known to be intuitionistically underivable (see J. Philos. Logic 25 (1996) 559), but Krivine was able to derive intuitionistically a weak form of it, namely, he proved that every consistent classical theory has a model. In this paper, we want to analyze the ideas Krivines remarkable result relies on, ideas which where somehow hidden by the heavy formal machinery used in the original proof. We show that the ideas in Krivines proof can be used to intuitionistically derive some (suitable variants of) crucial mathematical results, which were supposed to be purely classical up to now: the Ultrafilter Theorem for countable Boolean algebras, and the maximal ideal theorem for countable rings.

Decidability in Intuitionistic Type Theory is Functionally Decidable

In this paper we show that the usual intuitionistic characterization of the decidability of the propositional function B(x) prop [x : A], i. e. to require that the predicate (∀x ∈ A) (B(x) ∨ ¬ B(x)) is provable, is equivalent, when working within the framework of Martin-Lofs Intuitionistic Type Theory, to require that there exists a decision function ψ: A Boole such that (∀x ∈ A) ((ψ(x) = Booletrue) B(x)). Since we will also show that the proposition x = Booletrue [x: Boole] is decidable, we can alternatively say that the main result of this paper is a proof that the decidability of the predicate B(x)prop [x : A] can be effectively reduced by a function ψ A Boole to the decidability of the predicate ψ(x) = Booletrue [x : A]. All the proofs are carried out within the Intuitionistic Type Theory and hence the decision function ψ, together with a proof of its correctness, is effectively constructed as a function of the proof of (∀x ∈ A)(B(x) ∨ ¬ B(x)). Mathematics Subject Classification: 03B15, 03B20.