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Proof and computation

1993 Annals of Pure and Applied Logic  
Germany icians working in the fields of computational aspects of H. Schwichtenberg. Munich. Germany S S. Wainer, Lewis, United Kingdom proofs, logical systems, and algebraic specifications.  ...  Stembraggen, Munich, Gemtany The logical notion of a formal proof in various constructive systems can be viewed as a very explicit way to describe a computation procedure.  ...  t "PROOF AND COMPUTATION" July 20 to August 1, 1993 The "Marktoberdorf Summer School" is a course of two R.  ... 
doi:10.1016/0168-0072(93)90193-h fatcat:2sthqrebordbbavxi45i2ugb7u

Proof, understanding, and computers

Vitaly Tselishchev, Siberian Branch of Russian Academy of Science
2020 Philosophical Problems of Information Technologies and Cyberspace  
The role of understanding in two types of proof -traditional conceptual and computer-based-is considered.  ...  It is shown that the concept of understanding in the framework of the philosophy of mathematics for the case of computer proof is most interesting in the works of the late Wittgenstein.  ...  Воеводский работал над процедурами проверки proof-checkers на основе унивалентных оснований, так чтобы все математические доказательства для публикации сопровождались их машинно-проверяемыми эквивалентами  ... 
doi:10.17726/philit.2020.1.5 fatcat:sschdqixjbdmbf3l2fhk7nkreq

Mixing Computations and Proofs

Michael Beeson
2014 Journal of Formalized Reasoning  
We examine the relationship between proof and computation in mathematics, especially in formalized mathematics.  ...  We compare the various approaches to proofs with a significant computational component, including (i) verifying the algorithms, (ii) verifying the results of the unverified algorithms, and (iii) trusting  ...  Mathematics consists of logic and computation, interwoven in tapestries of proofs.  ... 
doi:10.6092/issn.1972-5787/4552 dblp:journals/jfrea/Beeson16 fatcat:ieften5wunhnnesihejategbku

The Robbins problem: computer proofs and human proofs

Louis H. Kauffman
2001 Kybernetes  
Introduction This paper is devoted to a discussion of the relation between computer proof and human proof. It is a discussion of the relationship of persons and machines.  ...  In this sense, a computer does not , can not produce a proof. The computer is not convinced of anything.  ...  A direct proof of this fact would solve the Robbins problem by a result of Steve Winker [Winker, S. (1992) ].  ... 
doi:10.1108/eum0000000005698 fatcat:2rkbqkdhrbg6xegqvwh2szmvvu

Proof and Computation in Geometry [chapter]

Michael Beeson
2013 Lecture Notes in Computer Science  
We consider the relationships between algebra, geometry, computation, and proof. Computers have been used to verify geometrical facts by reducing them to algebraic computations.  ...  But this does not produce computer-checkable first-order proofs in geometry.  ...  Nevertheless, those proofs, if we could find them, would be proofs and not just computations.  ... 
doi:10.1007/978-3-642-40672-0_1 fatcat:syvmbmvp7je7rpppujtew7jo7a

Proof nets, garbage, and computations [chapter]

S. Guerrini, S. Martini, A. Masini
1997 Lecture Notes in Computer Science  
We study the problem of local and asynchronous computation in the context of multiplicative exponential linear logic (MELL) proof nets.  ...  The proposed reduction system is strongly normalizing and con uent.  ...  We plan to apply the proposed approach to the case of functional languages (pure and typed {calculi) both from a theoretical and practical (implementative) point of view AG97].  ... 
doi:10.1007/3-540-62688-3_36 fatcat:e7d5sctfkfcg5ih24rmieubd5a

Proof nets, garbage, and computations

Stefano Guerrini, Simone Martini, Andrea Masini
2001 Theoretical Computer Science  
We study the problem of local and asynchronous computation in the context of multiplicative exponential linear logic (MELL) proof nets.  ...  The proposed reduction system is strongly normalizing and con uent. The proofs are all based on pure syntactical reasonings.  ...  Introduction Computation as rewriting is a fundamental paradigm in computer science, rooted in the analysis of computation of the early days of computability theory.  ... 
doi:10.1016/s0304-3975(00)00094-3 fatcat:eqte4g7d4zezhcpbaywtrgo2u4

Proof Theory and Computational Analysis

Ulrich Kohlenbach
1997 BRICS Report Series  
In this survey paper we start with a discussion how functionals of finite type can be used for the proof-theoretic extraction of numerical data (e.g. effective<br />uniform bounds and rates of convergence  ...  ) from non-constructive proofs in numerical analysis.  ...  Proof: It is clear from the usual proofs in analysis that 1)-5) are true.  ... 
doi:10.7146/brics.v4i30.18956 fatcat:q2kvu7ah6fhoxky2o6uykd6mvq

Proof theory and computational analysis

Ulrich Kohlenbach
1998 Electronical Notes in Theoretical Computer Science  
In this survey paper we start with a discussion how functionals of nite type can be used for the proof-theoretic extraction of numerical data (e.g. e ective uniform bounds and rates of convergence) from  ...  non-constructive proofs in numerical analysis.  ...  (n) : = 1 36(n+1) 2 and thus the computability of the zero of cos in G 2 R !  ... 
doi:10.1016/s1571-0661(05)80219-1 fatcat:ixrvz2n45jbjffobwhegqgyfxq

New Developments in Proofs and Computations [chapter]

Helmut Schwichtenberg
2008 New Computational Paradigms  
An example is Tait's proof of the existence of normal forms for the simply typed λ-calculus, which uses so-called computability predicates.  ...  One only needs to know that certain basic functionals are computable (including primitive recursion operators in finite types), and that the computable functionals are closed under composition.  ...  Instantiating x with y0 and z with y1 gives |A(y0)| T x 1 ...xn(y0) y1 , which is ..xn s,y . Assume for simplicity that s is closed.  ... 
doi:10.1007/978-0-387-68546-5_14 fatcat:4z2kntxwqfbyhbiowa2zj5coey

DATA INTEGRITY PROOF AND SECURE COMPUTATION IN CLOUD COMPUTING

Ponnuramu
2012 Journal of Computer Science  
This method is used to assure data integrity and secured computations with reduced computational and storage overhead of the client.  ...  One of the important concern is the integrity of data and computations.  ...  Merkle Hash Tree Pre-Computed Tokens Science Publications JCS Proof Of Retrievability (POR) Problem Statement One of the important concern is the integrity of data and computation.  ... 
doi:10.3844/jcssp.2012.1987.1995 fatcat:6ujwuho32jdr7dpukfelzcezle

Computer Runtimes and the Length of Proofs [chapter]

Hector Zenil
2012 Lecture Notes in Computer Science  
It is suggested that theorem provers are subject to the same non-linear tradeoff between time and size as computer programs are, affording the possibility of determining optimal timeouts and waiting times  ...  Just as Calude and Stay suggest that most Turing machines are fully determined qua termination by a small number of computational steps, and that the error margin drops drastically, in [8] we have also  ...  Delahaye with whom I've undertaken related research [7] , studying the output distribution of abstract computing machines.  ... 
doi:10.1007/978-3-642-27654-5_17 fatcat:3sbujvawunaevox465jq66jp2m

Herbrand Proofs and Expansion Proofs as Decomposed Proofs

Benjamin Ralph
2020 Journal of Logic and Computation  
In this paper we construct simple deep inference systems for first-order logic, both with and without cut, such that 'decomposed' proofsproofs where the contractive and non-contractive behaviour of the  ...  Translations between proofs in this system, expansion proofs and Herbrand proofs are given, retaining much of the structure in each direction.  ...  the Engineering and Physical Sciences Research Council Project EP/K018868/1 Efficient and Natural Proof Systems and the Agence Nationale de la Recherche project FISP ANR-15-CE25-0014-01.  ... 
doi:10.1093/logcom/exaa052 fatcat:bpitzcjaivf4lgrbtb5ueld6ye

Program Development by Proof Transformation [chapter]

Ulrich Berger, Helmut Schwichtenberg
1995 Proof and Computation  
.) | M ∀ c x A(x) r (∀ c ) − A(r ) (M ∀ c x A(x) r ) A(r ) Restrictions to → + and ∀ + (non-computational) CV(M) := the set of "computational variables" of a derivation M, relative to a fixed assigment  ...  Seq(M) of a proof M consists of its context and end formula.  ...  Two proofs of the existence of a maximal end segment for n + 1 ∀ c n ∃ l j≤n+1 ∀ j ≤n+1 (seg(j , n + 1) ≤ seg(j, n + 1)).  ... 
doi:10.1007/978-3-642-79361-5_1 fatcat:sueg46547ndrlo3hoywapfytda

Logic and computation—Interactive proof with Cambridge LCF

Tobias Nipkow
1988 Science of Computer Programming  
Sadly, sloppy and inaccurate accounts of supposed new results in logic, with inadequate proofs or no proofs at all, can easily be found in the computer science literature.  ...  Care is taken to present both the computational aspect (i.e. the implementation as ML functions) and the relation to the proof systems of the first part.  ...  Proof rules for some of the operations of the schema calculus are provided and are shown to be sound by reference to the semantics.  ... 
doi:10.1016/0167-6423(88)90007-x fatcat:gc5rp3zc5fbtlnj3ga6quv2dne
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