scholarly journals General Recursion and Formal Topology

10.29007/hl75 ◽  
2018 ◽  
Author(s):  
Claudio Sacerdoti Coen ◽  
Silvio Valentini
Keyword(s):  
Type Theory ◽  
New Approach ◽  
Formal Topology

It is well known that general recursion cannot be expressed within Martin-Löf's type theory and that various approaches have been proposed to overcome this problem still maintaining the termination of the computation of the typable terms. In this work we propose a new approach to this problem based on the use of inductively generated formal topologies.

scholarly journals COMPUTER PROOFS PRACTICE AND HUMAN UNDERSTANDING: EPISTEMOLOGICAL ISSUES

2021 ◽  
pp. 5-19
Author(s):  
Lev D. Lamberov ◽  
Keyword(s):  
Type Theory ◽  
Point Of View ◽  
Foundations Of Mathematics ◽  
Traditional Concept ◽  
New Approach ◽  
Informal Mathematics ◽  
Computer Proofs ◽  
Number Of Publications ◽  
Univalent Foundations ◽  
Epistemological Point

In recent decades, some epistemological issues have become especially acute in mathematics. These issues are associated with long proofs of various important mathematical results, as well as with a large and constantly increasing number of publications in mathematics. It is assumed that (at least partially) these difficulties can be resolved by referring to computer proofs. However, computer proofs also turn out to be problematic from an epistemological point of view. With regard to both proofs in ordinary (informal) mathematics and computer proofs, the problem of their surveyability appears to be fundamental. Based on the traditional concept of proof, it must be surveyable, otherwise it will not achieve its main goal — the formation of conviction in the correctness of the mathematical result being proved. About 15 years ago, a new approach to the foundations of mathematics began to develop, combining constructivist, structuralist features and a number of advantages of the classical approach to mathematics. This approach is built on the basis of homotopy type theory and is called the univalent foundations of mathematics. Due to itspowerful notion of equality, this approach can significantly reduce the length of formalized proofs, which outlines a way to resolve the epistemological difficulties that have arisen

Tychonoff's theorem in the framework of formal topologies

1997 ◽  
Vol 62 (4) ◽  
pp. 1315-1332 ◽  
Author(s):  
Sara Negri ◽  
Silvio Valentini
Keyword(s):  
Type Theory ◽  
Constructive Proof ◽  
Topological Product ◽  
Essential Difference ◽  
Constructive Type Theory ◽  
Constructive Type ◽  
Definition Of ◽  
The Axiom Of Choice ◽  
Inductive Property ◽  
Formal Topology

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 Löf's constructive type theory (cf. [16]), 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, 27]). 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.

FORMAL PROOFS OF FUNCTIONAL BSP PROGRAMS

2003 ◽  
Vol 13 (03) ◽  
pp. 365-376 ◽  
Author(s):  
FRÉDÉRIC GAVA
Keyword(s):  
Type Theory ◽  
Great Difficulty ◽  
Parallel Applications ◽  
Formal Proofs ◽  
Higher Order Logic ◽  
New Approach ◽  
Bulk Synchronous Parallel ◽  
Parallel Data ◽  
Wide Range ◽  
The Given

The Bulk Synchronous Parallel ML (BSML) is a functional language for BSP programming, a model of computing which allows parallel programs to be ported to a wide range of architectures. It is based on an extension of the ML language by parallel operations on a parallel data structure called parallel vector, which is given by intention. We present a new approach to certifying BSML programs in the context of type theory. Given a specification and a program, an incomplete proof of the specification (of which algorithmic contents corresponds to the given program) is built in the type theory, in which gaps would correspond to the proof obligation. This development demonstrates the usefulness of higher-order logic in the process of software certification of parallel applications. It also shows that the proof of rather complex parallel algorithms may be done with inductive types without great difficulty by using existing certified programs. This work has been implemented in the Coq Proof Assistant, applied on non-trivial examples and is the basis of a certified library of BSML programs.

On the Henstock Strong Variational Integral

1971 ◽  
Vol 14 (1) ◽  
pp. 87-99 ◽  
Author(s):  
B. S. Thomson
Keyword(s):  
Type Theory ◽  
Space Theory ◽  
Variational Integral ◽  
Locally Compact ◽  
New Approach ◽  
Simplified Approach ◽  
Compact Spaces ◽  
Locally Compact Spaces ◽  
Lebesgue Integration

The theory of integration in division spaces introduced by Henstock ([3], [4]) serves to unite and simplify much of the classical material on nonabsolute integration as well as to provide a new approach to Lebesgue integration. In this paper we sketch a simplified approach to the division space theory and show how it can lead rapidly to the standard Lebesgue-type theory without a substantial departure from the usual methods; some applications to integration in locally compact spaces are briefly developed.

scholarly journals The axiom of multiple choice and models for constructive set theory

2014 ◽  
Vol 14 (01) ◽  
pp. 1450005 ◽  
Author(s):  
Benno van den Berg ◽  
Ieke Moerdijk
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Multiple Choice ◽  
Compactness Theorem ◽  
Cantor Space ◽  
Algebraic Set ◽  
Constructive Set Theory ◽  
Inductive Types ◽  
Formal Topology ◽  
Derived Rules

We propose an extension of Aczel's constructive set theory CZF by an axiom for inductive types and a choice principle, and show that this extension has the following properties: it is interpretable in Martin-Löf's type theory (hence acceptable from a constructive and generalized-predicative standpoint). In addition, it is strong enough to prove the Set Compactness theorem and the results in formal topology which make use of this theorem. Moreover, it is stable under the standard constructions from algebraic set theory, namely exact completion, realizability models, forcing as well as more general sheaf extensions. As a result, methods from our earlier work can be applied to show that this extension satisfies various derived rules, such as a derived compactness rule for Cantor space and a derived continuity rule for Baire space. Finally, we show that this extension is robust in the sense that it is also reflected by the model constructions from algebraic set theory just mentioned.

scholarly journals Structuralism, Invariance, and Univalence

2018 ◽  
Author(s):  
Steve Awodey
Keyword(s):  
Type Theory ◽  
Homotopy Theory ◽  
Mathematical Practice ◽  
New Approach ◽  
Constructive Type Theory ◽  
Homotopy Type Theory ◽  
Computational Implementation ◽  
Geometric Content ◽  
Abstract Homotopy Theory ◽  
New System

The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the univalence axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. This powerful addition to homotopy type theory gives the new system of foundations a distinctly structural character.

The O–C diagrams of minor planets − a new approach to modelling the rotation

1999 ◽  
Vol 173 ◽  
pp. 185-188
Author(s):  
Gy. Szabó ◽  
K. Sárneczky ◽  
L.L. Kiss
Keyword(s):  
Error Analysis ◽  
Time Dependence ◽  
Theoretical Work ◽  
Minor Planet ◽  
Minor Planets ◽  
New Approach ◽  
Preliminary Results ◽  
Ccd Observations

AbstractA widely used tool in studying quasi-monoperiodic processes is the O–C diagram. This paper deals with the application of this diagram in minor planet studies. The main difference between our approach and the classical O–C diagram is that we transform the epoch (=time) dependence into the geocentric longitude domain. We outline a rotation modelling using this modified O–C and illustrate the abilities with detailed error analysis. The primary assumption, that the monotonity and the shape of this diagram is (almost) independent of the geometry of the asteroids is discussed and tested. The monotonity enables an unambiguous distinction between the prograde and retrograde rotation, thus the four-fold (or in some cases the two-fold) ambiguities can be avoided. This turned out to be the main advantage of the O–C examination. As an extension to the theoretical work, we present some preliminary results on 1727 Mette based on new CCD observations.

A New Approach to the Demonstration of Oxidase-Localization Using Tannic Acid Or Catechin

1973 ◽  
Vol 31 ◽  
pp. 698-699
Author(s):  
V. Mizuhira ◽  
Y. Futaesaku
Keyword(s):  
Cross Section ◽  
Tannic Acid ◽  
Elastic Fiber ◽  
The Other ◽  
New Approach ◽  
Tissue Block ◽  
Active Reaction ◽  
The Cross

Previously we reported that tannic acid is a very effective fixative for proteins including polypeptides. Especially, in the cross section of microtubules, thirteen submits in A-tubule and eleven in B-tubule could be observed very clearly. An elastic fiber could be demonstrated very clearly, as an electron opaque, homogeneous fiber. However, tannic acid did not penetrate into the deep portion of the tissue-block. So we tried Catechin. This shows almost the same chemical natures as that of proteins, as tannic acid. Moreover, we thought that catechin should have two active-reaction sites, one is phenol,and the other is catechole. Catechole site should react with osmium, to make Os- black. Phenol-site should react with peroxidase existing perhydroxide.

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