Exhaustible sets in higher-type computation

We say that a set is exhaustible if it admits algorithmic universal quantification for continuous predicates in finite time, and searchable if there is an algorithm that, given any continuous predicate, either selects an element for which the predicate holds or else tells there is no example. The Cantor space of infinite sequences of binary digits is known to be searchable. Searchable sets are exhaustible, and we show that the converse also holds for sets of hereditarily total elements in the hierarchy of continuous functionals; moreover, a selection functional can be constructed uniformly from a quantification functional. We prove that searchable sets are closed under intersections with decidable sets, and under the formation of computable images and of finite and countably infinite products. This is related to the fact, established here, that exhaustible sets are topologically compact. We obtain a complete description of exhaustible total sets by developing a computational version of a topological Arzela--Ascoli type characterization of compact subsets of function spaces. We also show that, in the non-empty case, they are precisely the computable images of the Cantor space. The emphasis of this paper is on the theory of exhaustible and searchable sets, but we also briefly sketch applications

Effectivity and Density in Domains A Survey

AbstractThis article surveys the main results on effectivity and totality in domain theory and its applications. A more abstract and informative proof of Normann's generalized density theorem for total functionals of finite type over the reals is presented

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

Topological Foundations of Cognitive Science

A collection of papers presented at the First International Summer Institute in Cognitive Science, University at Buffalo, July 1994, including the following papers: ** Topological Foundations of Cognitive Science, Barry Smith ** The Bounds of Axiomatisation, Graham White ** Rethinking Boundaries, Wojciech Zelaniec ** Sheaf Mereology and Space Cognition, Jean Petitot ** A Mereotopological Definition of 'Point', Carola Eschenbach ** Discreteness, Finiteness, and the Structure of Topological Spaces, Christopher Habel ** Mass Reference and the Geometry of Solids, Almerindo E. Ojeda ** Defining a 'Doughnut' Made Difficult, N .M. Gotts ** A Theory of Spatial Regions with Indeterminate Boundaries, A.G. Cohn and N.M. Gotts ** Mereotopological Construction of Time from Events, Fabio Pianesi and Achille C. Varzi ** Computational Mereology: A Study of Part-of Relations for Multi-media Indexing, Wlodek Zadrozny and Michelle Ki

Realizability and recursive mathematics

Section 1: Philosophy, logic and constructivityPhilosophy, formal logic and the theory of computation all bear on problems in the foundations of constructive mathematics. There are few places where these, often competing, disciplines converge more neatly than in the theory of realizability structures. Uealizability applies recursion-theoretic concepts to give interpretations of constructivism along lines suggested originally by Heyting and Kleene. The research reported in the dissertation revives the original insights of Kleene—by which realizability structures are viewed as models rather than proof-theoretic interpretations—to solve a major problem of classification and to draw mathematical consequences from its solution.Section 2: Intuitionism and recursion: the problem of classificationThe internal structure of constructivism presents an interesting problem. Mathematically, it is a problem of classification; for philosophy, it is one of conceptual organization. Within the past seventy years, constructive mathematics has grown into a jungle of fullydeveloped "constructivities," approaches to the mathematics of the calculable which range from strict finitism through hyperarithmetic model theory. The problem we address is taxonomic: to sort through the jungle, set standards for classification and determine those features which run through everything that is properly "constructive."There are two notable approaches to constructivity; these must appear prominently in any proposed classification. The most famous is Brouwer's intuitioniam. Intuitionism relies on a complete constructivization of the basic mathematical objects and logical operations. The other is classical recursive mathematics, as represented by the work of Dekker, Myhill, and Nerode. Classical constructivists use standard logic in a mathematical universe restricted to coded objects and recursive operations.The theorems of the dissertation give a precise answer to the classification problem for intuitionism and classical constructivism. Between these realms arc connected semantically through a model of intuitionistic set theory. The intuitionistic set theory IZF encompasses all of the intuitionistic mathematics that does not involve choice sequences. (This includes all the work of the Bishop school.) IZF has as a model a recursion-theoretic structure, V(A7), based on Kleene realizability. Since realizability takes set variables to range over "effective" objects, large parts of classical constructivism appear over the model as inter¬ preted subsystems of intuitionistic set theory. For example, the entire first-order classical theory of recursive cardinals and ordinals comes out as an intuitionistic theory of cardinals and ordinals under realizability. In brief, we prove that a satisfactory partial solution to the classification problem exists; theories in classical recursive constructivism are identical, under a natural interpretation, to intuitionistic theories. The interpretation is especially satisfactory because it is not a Godel-style translation; the interpretation can be developed so that it leaves the classical logical forms unchanged.Section 3: Mathematical applications of the translation:The solution to the classification problem is a bridge capable of carrying two-way mathematical traffic. In one direction, an identification of classical constructivism with intuitionism yields a certain elimination of recursion theory from the standard mathematical theory of effective structures, leaving pure set theory and a bit of model theory. Not only are the theorems of classical effective mathematics faithfully represented in intuitionistic set theory, but also the arguments that provide proofs of those theorems. Via realizability, one can find set-theoretic proofs of many effective results, and the set-theoretic proofs are often more straightforward than their recursion-theoretic counterparts. The new proofs are also more transparent, because they involve, rather than recursion theory plus set theory, at most the set-theoretic "axioms" of effective mathematics.Working the other way, many of the negative ("cannot be obtained recursively") results of classical constructivism carry over immediately into strong independence results from intuitionism. The theorems of Kalantari and Retzlaff on effective topology, for instance, turn into independence proofs concerning the structure of the usual topology on the intuitionistic reals.The realizability methods that shed so much light over recursive set theory can be applied to "recursive theories" generally. We devote a chapter to verifying that the realizability techniques can be used to good effect in the semantical foundations of computer science. The classical theory of effectively given computational domains a la Scott can be subsumed into the Kleene realizability universe as a species of countable noneffective domains. In this way, the theory of effective domains becomes a chapter (under interpre¬ tation) in an intuitionistic study of denotational semantics. We then show how the "extra information" captured in the logical signs under realizability can be used to give proofs of classical theorems about effective domains.Section 4: Solutions to metamathematical problems:The realizability model for set theory is very tractible; in many ways, it resembles a Boolean-valued universe. The tractibility is apparent in the solutions it offers to a number of open problems in the metamathematics of constructivity. First, there is the perennial problem of finding and delimiting in the wide constructive universe those features that correspond to structures familiar from classical mathematics. In the realizability model, it is easy to locate the collection of classical ordinals and to show that they form, intuitionistically, a set rather than a proper class. Also, one interprets an argument of Dekker and Myhill to prove that the classical powerset of the natural numbers contains at least continuum-many distinct cardinals.Second, a major tenet of Bishop's program for constructivity has been that constructive mathematics is "numerical:" all the properties of constructive objects, including the real numbers, can be represented as properties of the natural numbers. The realizability model shows that Bishop's numericalization of mathematics can, in principle, be accomplished. Every set over the model with decidable equality and every metric space is enumerated by a collection of natural numbers

A geometry of information, I: Nerves, posets and differential forms

The main theme of this workshop (Dagstuhl seminar 04351) is `Spatial Representation: Continuous vs. Discrete'. Spatial representation has two contrasting but interacting aspects (i) representation of spaces' and (ii) representation by spaces. In this paper, we will examine two aspects that are common to both interpretations of the theme, namely nerve constructions and refinement. Representations change, data changes, spaces change. We will examine the possibility of a `differential geometry' of spatial representations of both types, and in the sequel give an algebra of differential forms that has the potential to handle the dynamical aspect of such a geometry. We will discuss briefly a conjectured class of spaces, generalising the Cantor set which would seem ideal as a test-bed for the set of tools we are developing.Comment: 28 pages. A version of this paper appears also on the Dagstuhl seminar portal http://drops.dagstuhl.de/portals/04351

Life, Universe and Everything

The iroha song of human concepts (2021) The iroha is a Japanese poem of a perfect pangram and isogram, containing each character of the Japanese syllabary exactly once. It also mimics an ultimate conceptual engineering, in that there is more and more restricted scope for meaningful expressions, given more and more condensed means of description. This culminates in crystallizations of human values by auto-condensations of meaningful concepts. Instead of distilling Japanese values of 11th century, I try for those of human concepts, given our merging mind, language and culture

Infinity

This essay surveys the different types of infinity that occur in pure and applied mathematics, with emphasis on: 1. the contrast between potential infinity and actual infinity; 2. Cantor's distinction between transfinite sets and absolute infinity; 3. the constructivist view of infinite quantifiers and the meaning of constructive proof; 4. the concept of feasibility and the philosophical problems surrounding feasible arithmetic; 5. Zeno's paradoxes and modern paradoxes of physical infinity involving supertasks