Journal of Logic and Analysis 

About: Journal of Logic and Analysis is an academic journal published by Journal of Logic and Analysis. The journal publishes majorly in the area(s): Mathematics Subject Classification & Constructive. It has an ISSN identifier of 1759-9008. It is also open access. Over the lifetime, 113 publications have been published receiving 744 citations.

A Lambda Calculus for Real Analysis

Abstract: Stone Duality is a new paradigm for general topology in which computable continuous functions are described directly, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, and the reasoning looks remarkably like a sanitised form of that in classical topology. This is an introduction to ASD for the general mathematician, with application to elementary real analysis. This language is applied to the Intermediate Value Theorem: the solution of equations for continuous functions on the real line. As is well known from both numerical and constructive considerations, the equation cannot be solved if the function "hovers" near 0, whilst tangential solutions will never be found. In ASD, both of these failures, and the general method of finding solutions of the equation when they exist, are explained by the new concept of overtness. The zeroes are captured, not as a set, but by higher-type modal operators. Unlike the Brouwer degree of a mapping, these are naturally defined and (Scott) continuous across singularities of a parametric equation. Expressing topology in terms of continuous functions rather than using sets of points leads to treatments of open and closed concepts that are very closely lattice- (or de Morgan-) dual, without the double negations that are found in intuitionistic approaches. In this, the dual of compactness is overtness. Whereas meets and joins in locale theory are asymmetrically finite and infinite, they have overt and compact indices in ASD. Overtness replaces metrical properties such as total boundedness, and cardinality conditions such as having a countable dense subset. It is also related to locatedness in constructive analysis and recursive enumerability in recursion theory.

Integrals and valuations

Abstract: We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of measures(valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are elementary and tightly connected to the Riesz space structure. 2000 Mathematics Subject Classification 06D22, 28C05

Relative computability and uniform continuity of relations

Abstract: A type-2 computable real function is necessarily continuous; and this remains true for relative, i.e. oracle-based, computations. Conversely, by the Weierstrass Approximation Theorem, every continuous f :[0,1]→ℝ is computable relative to some oracle. In their search for a similar topological characterization of relatively computable multi- valued functions f :[0,1]⇒ℝ (aka relations), Brattka and Hertling (1994) have considered two notions: weak continuity (which is weaker than relative computability) and strong continuity (which is stronger than relative computability). Observing that uniform continuity plays a crucial role in the Weierstrass Theorem, we propose and compare several notions of uniform continuity for relations. Here, due to the additional quantification over values y ∈ f ( x ), new ways arise of (linearly) ordering quantifiers — yet none turns out as satisfactory. We are thus led to a concept of uniform continuity based on the Henkin quantifier ; and prove it necessary for relative computability of compact real relations. In fact iterating this condition yields a strict hierarchy of notions each necessary — and the ω-th level also sufficient — for relative computability.

The probability distribution as a computational resource for randomness testing

Abstract: When testing a set of data for randomness according to a probability distribution that depends on a parameter, access to this parameter can be considered as a computational resource. We call a randomness test Hippocratic if it is not permitted to access this resource. In these terms, we show that for Bernoulli measures p , 0 6 p 6 1 and the Martin-L¨ of randomness model, Hippocratic randomness of a set of data is the same as ordinary randomness. The main idea of the proof is to first show that from Hippocrates-random data one can Turing compute the parameter p. However, we show that there is no single Hippocratic randomness test such that passing the test implies computing p, and in particular there is no universal Hippocratic randomness test. 2000 Mathematics Subject Classification 03D28, 68Q30 (primary); 60F15 (sec- ondary)

Convergence in formal topology: a unifying notion

Abstract: Several variations on the definition of a Formal Topology exist in the literature. They differ on how they express convergence, the formal property corresponding to the fact that open subsets are closed under finite intersections. We introduce a general notion of convergence of which any previous definition is a special case. This leads to a predicative presentation and inductive genera- tion of locales (formal covers), commutative quantales (convergent covers) and suplattices (basic covers) in a uniform way. Thanks to our abstract treatment of convergence, we are able to specify categorically the precise sense according to which our inductively generated structures are free, thus refining Johnstone's coverage theorem. We also obtain a natural and predicative version of a fundamental result by Joyal and Tierney: convergent covers (commutative quantales) correspond to commutative co-semigroups over the category of basic covers (suplattices).