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Measuring the Relative Complexity of Mathematical Constructions and Theorems

Jun Le Goh
2019 Bulletin of Symbolic Logic  
prepared by Nebojša Ikodinović and Marija Boričić.  ...  Abstract We investigate the relative complexity of mathematical constructions and theorems using the frameworks of computable reducibilities and reverse mathematics.  ...  First, we study the computational content of various theorems with reverse mathematical strength around Arithmetical Transfinite Recursion (ATR 0 ) from the point of view of computable reducibilities,  ... 
doi:10.1017/bsl.2019.34 fatcat:xw74gnwgfrfgnlt4cuhm3gnk7m

A theorem on linear point sets

Henry Blumberg
1919 Bulletin of the American Mathematical Society  
Since A and H differ by a set of zero measure, the relative exterior measure of the one is the same at every point as that of the other.  ...  , and of mathematical physics which can be solved by the aid of that theory.  ...  We obtain a particular case of our theorem if we assume A to be a measurable set. Exterior measure will then be replaced by measure, and relative exterior measure by "relative measure."  ... 
doi:10.1090/s0002-9904-1919-03201-7 fatcat:xsyt442lrjct7eqikmedskdpli

Transversal theory, an account of some aspects of combinatorial mathematics

1972 Discrete Mathematics  
Hall's theorem and the notion of duality. The method of 'elementary constructions'. Rado's selection rinciple. Variants, refinemer: ts, and applications of all's theorem. ndependent transversals.  ...  Definitions and examples of complex manifolds. Sheaves and coho- molog;:. Geometry of complex manifolds. Applications for elliptic partial differential equations to deformations. Index. E.D.  ... 
doi:10.1016/0012-365x(72)90119-7 fatcat:7obg7hlcxbetnjkslosv3pnrm4

Page 5 of Mathematical Reviews Vol. 53, Issue 4 [page]

1977 Mathematical Reviews  
A recursive relatedness theorem is proved, showing that any two relative complexity measures are related by a fixed recursive function.  ...  This theorem allows proofs of results for all measures to be obtained from proofs for a particular measure. ““Complexity-determined reducibilities are studied.  ... 

The Digital and the Real Universe. Foundations of Natural Philosophy and Computational Physics

Klaus Mainzer
2019 Philosophies  
, measuring, and computing.  ...  The following article refers to the author's book, "The Digital and the Real World. Computational Foundations of Mathematics, Science, Technology, and Philosophy."  ...  Reverse mathematics enable us to determine the proof-theoretic strength resp. complexity of theorems by classifying them with respect to equivalent theorems and proofs.  ... 
doi:10.3390/philosophies4010003 fatcat:mgv373e4sbhgreenikab6am5qa

Sentence complexity of theorems in Mizar [article]

Jesse Alama
2013 arXiv   pre-print
Friedman, we use Mizar to take on the question of surveying the sentence complexity (measured by quantifier alternation) of mathematical theorems.  ...  We find, as Friedman suggests, that the sentence complexity of most Mizar theorems is universal (Π_1, or ∀), and as one goes higher in the sentence complexity hierarchy the number of Mizar theorems having  ...  We looked at the definitions and theorems across the MML and used prenex normal form as a measure of the complexity of these items.  ... 
arXiv:1311.1915v1 fatcat:saxhyqkr3jbvvdaxcbgjiceyuy

Proof as a Mathematical Object - Proposals for a Research Program

Peteris Daugulis
2020 Baltic Journal of Modern Computing  
The need to develop a theory for measuring value and complexity of mathematical implications and proofs is discussed including motivations, benefits and implementation problems.  ...  The problem of representing logical implications and proofs by mathematical objects is considered.  ...  , measure and quantify mathematical results such as theorems and lemmas.  ... 
doi:10.22364/bjmc.2020.8.2.01 fatcat:qrmomkvvjvbjzpvqnugflwtw4q

Page 350 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 25, Issue [page]

1919 American Mathematical Society. Bulletin of the American Mathematical Society  
We then define therelative exterior measure of A at the point x” as * Cf. Denjoy, Journal de Mathématiques, ser. 7, vol. 1 (1915), p. 130.  ...  , and of mathematical physics which can be solved by the aid of that theory.  ... 

On coordinatization of mathematics [article]

Peteris Daugulis
2018 arXiv   pre-print
The need to develop a theory for measuring value and complexity of mathematical implications and proofs is discussed including motivations, benefits and implementation problems.  ...  The problem of advancing coordinatization of mathematics is considered.  ...  problems, study the network of polynomial reductions, 6) build a map relating theorems and reductions of computational classes for a given domain, 7) introduce measures of cognitive complexity of mathematical  ... 
arXiv:1306.0520v5 fatcat:bkqi7vglmzcezitm6hyg4qc25i

Capacitability for Co-Analytic Sets

Theodore Slaman
2022 New Zealand Journal of Mathematics  
It follows from a theorem of Davies (1952) that if A is an analytic subset of the Cantor middle third set, λ is positive and the Hausdorff s-measure of A is greater than λ, then there is a compact subset  ...  C of A such that the Hausdorff s-measure of C is greater than λ.  ...  For any B, the set of X such that X is Martin-Löf random relative to B has full measure.  ... 
doi:10.53733/170 fatcat:5kuatjaxu5ffhh6lvujxna4t6q

Page 3706 of Mathematical Reviews Vol. , Issue 92g [page]

1992 Mathematical Reviews  
In §3, we solve the Henson and Moore problem by introducing a construction of countably additive measures from internal Banach space-valued measures.  ...  For example, the Hadamard gap theorem, Bloch’s theo- rem, and the two Picard theorems all are proved in the first half of the book, and the prime number theorem is covered near the three-quarter point  ... 

Page 546 of Mathematical Reviews Vol. , Issue 86b [page]

1986 Mathematical Reviews  
This construction has since been used by a number of researchers to obtain new results in many areas including measure and probability theory, potential theory, math- ematical physics and mathematical  ...  {For the entire collection see MR 85d:00025.} MEASURE AND INTEGRATION 30 FUNCTIONS OF A COMPLEX VARIABLE See also 01021. Dettman, John W. (1-OAKL) * Applied complex variables.  ... 

Quantum Model Theory (QMod): Modeling Contextual Emergent Entangled Interfering Entities [chapter]

Diederik Aerts, Sandro Sozzo
2012 Lecture Notes in Computer Science  
In this sense, QMod is a generalization of quantum mechanics, similar to how the general relativity manifold mathematical formalism is a generalization of special relativity.  ...  . for each context corresponding to a measurement, and for its real version it does not need the property of 'linearity of the set of states' to model the quantum effect.  ...  Let us now work out a mathematical construction put forward in the representation theorem proven in Sec. 2.  ... 
doi:10.1007/978-3-642-35659-9_12 fatcat:si22hvr3rzawjplpmheo7rntrq

Page 5416 of Mathematical Reviews Vol. , Issue 93j [page]

1993 Mathematical Reviews  
Summary: “A unified treatment is given of the relative position of L?  ...  /G, and unlike the usual convex hull construction where the convex hulls of two subgroups may be disjoint, the hull for any subgroup of G contains the base point.  ... 

Guest Editorial: Tenth International Conference on Computability, Complexity and Randomness (CCR 2015)

Andrew Lewis-Pye, Wolfgang Merkle
2017 Theory of Computing Systems  
This article is part of the Topical Collection on Special Issue on Computability, Complexity and Randomness (CCR 2015) London School of Economics Department of Mathematics, London School of Economics and  ...  In "Exact constructive and computable dimensions" by Ludwig Staiger, the author provides a systematic development of exact constructive dimension, which quantitatively refines constructive dimension just  ... 
doi:10.1007/s00224-017-9809-2 fatcat:gtfithndffdulndutrqnek3qiq
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