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Journal articles on the topic 'Computation Theory and Mathematics'

Author: Grafiati

Published: 4 June 2021

Last updated: 29 January 2023

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1

Dean, Walter. "Computational Complexity Theory and the Philosophy of Mathematics†." Philosophia Mathematica 27, no. 3 (October 1, 2019): 381–439. http://dx.doi.org/10.1093/philmat/nkz021.

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Abstract Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof.

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2

Maley, Carlo C. "DNA Computation: Theory, Practice, and Prospects." Evolutionary Computation 6, no. 3 (September 1998): 201–29. http://dx.doi.org/10.1162/evco.1998.6.3.201.

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L. M. Adleman launched the field of DNA computing with a demonstration in 1994 that strands of DNA could be used to solve the Hamiltonian path problem for a simple graph. He also identified three broad categories of open questions for the field. First, is DNA capable of universal computation? Second, what kinds of algorithms can DNA implement? Third, can the error rates in the manipulations of the DNA be controlled enough to allow for useful computation? In the two years that have followed, theoretical work has shown that DNA is in fact capable of universal computation. Furthermore, algorithms for solving interesting questions, like breaking the Data Encryption Standard, have been described using currently available technology and methods. Finally, a few algorithms have been proposed to handle some of the apparently crippling error rates in a few of the common processes used to manipulate DNA. It is thus unlikely that DNA computation is doomed to be only a passing curiosity. However, much work remains to be done on the containment and correction of errors. It is far from clear if the problems in the error rates can be solved sufficiently to ever allow for general-purpose computation that will challenge the more popular substrates for computation. Unfortunately, biological demonstrations of the theoretical results have been sadly lacking. To date, only the simplest of computations have been carried out in DNA. To make significant progress, the field will require both the assessment of the practicality of the different manipulations of DNA and the implementation of algorithms for realistic problems. Theoreticians, in collaboration with experimentalists, can contribute to this research program by settling on a small set of practical and efficient models for DNA computation.

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3

Yamada, Shinichi. "A mathematical theory of randomized computation, I." Proceedings of the Japan Academy, Series A, Mathematical Sciences 64, no. 4 (1988): 115–18. http://dx.doi.org/10.3792/pjaa.64.115.

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4

Yamada, Shinichi. "A mathematical theory of randomized computation, II." Proceedings of the Japan Academy, Series A, Mathematical Sciences 64, no. 5 (1988): 155–58. http://dx.doi.org/10.3792/pjaa.64.155.

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5

Yamada, Shinichi. "A mathematical theory of randomized computation, III." Proceedings of the Japan Academy, Series A, Mathematical Sciences 64, no. 6 (1988): 201–4. http://dx.doi.org/10.3792/pjaa.64.201.

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6

Duhr, Claude. "Function Theory for Multiloop Feynman Integrals." Annual Review of Nuclear and Particle Science 69, no. 1 (October 19, 2019): 15–39. http://dx.doi.org/10.1146/annurev-nucl-101918-023551.

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Precise predictions for collider observables require the computation of higher orders in perturbation theory. This task usually involves the evaluation of complicated multiloop integrals, which typically give rise to complicated special functions. This article discusses recent progress in understanding the mathematics underlying multiloop Feynman integrals and discusses a class of functions that generalizes the logarithm and that often appears in multiloop computations. The same class of functions is an active area of research in modern mathematics, which has led to the development of new powerful tools to compute Feynman integrals. These tools are at the heart of some of the most complicated computations ever performed for a hadron collider.

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Lenstra Jr., H. W., Steven M. Serbin, Stig Larsson, Ohannes Karakashian, J. Thomas King, and Ewald Quak. "Book Review: Mathematics of Computation 1943--1993: A half-century of computational mathematics." Mathematics of Computation 66, no. 219 (July 1, 1997): 1367–75. http://dx.doi.org/10.1090/s0025-5718-97-00877-6.

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8

Conceição, Ana C., and Jéssica C. Pires. "Symbolic Computation Applied to Cauchy Type Singular Integrals." Mathematical and Computational Applications 27, no. 1 (December 31, 2021): 3. http://dx.doi.org/10.3390/mca27010003.

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The development of operator theory is stimulated by the need to solve problems emerging from several fields in mathematics and physics. At the present time, this theory has wide applications in the study of non-linear differential equations, in linear transport theory, in the theory of diffraction of acoustic and electromagnetic waves, in the theory of scattering and of inverse scattering, among others. In our work, we use the computer algebra system Mathematica to implement, for the first time on a computer, analytical algorithms developed by us and others within operator theory. The main goal of this paper is to present new operator theory algorithms related to Cauchy type singular integrals, defined in the unit circle. The design of these algorithms was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. Several nontrivial examples computed with the algorithms are presented. The corresponding source code of the algorithms has been made available as a supplement to the online edition of this article.

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9

Edalat, Abbas. "Domains for Computation in Mathematics, Physics and Exact Real Arithmetic." Bulletin of Symbolic Logic 3, no. 4 (December 1997): 401–52. http://dx.doi.org/10.2307/421098.

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AbstractWe present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chaos; we also show how efficient algorithms have been obtained for computing elementary functions in exact real arithmetic.

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10

Doerr, Benjamin, and Thomas Jansen. "Theory of Evolutionary Computation." Algorithmica 59, no. 3 (November 9, 2010): 299–300. http://dx.doi.org/10.1007/s00453-010-9472-3.

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11

Levesley, Jeremy. "Functions of matrices: Theory and computation." Bulletin of the London Mathematical Society 41, no. 6 (December 2009): 1145–46. http://dx.doi.org/10.1112/blms/bdp112.

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12

TESSON, PASCAL, and DENIS THÉRIEN. "MONOIDS AND COMPUTATIONS." International Journal of Algebra and Computation 14, no. 05n06 (October 2004): 801–16. http://dx.doi.org/10.1142/s0218196704001979.

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This contribution wishes to argue in favor of increased interaction between experts on finite monoids and specialists of theory of computation. Developing the algebraic approach to formal computations as well as the computational point of view on monoids will prove to be beneficial to both communities. We give examples of this two-way relationship coming from temporal logic, communication complexity and Boolean circuits. Although mostly expository in nature, our paper proves some new results along the way.

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13

Green, Frederic. "Review of Mathematics and Computation by Avi Wigderson." ACM SIGACT News 52, no. 3 (October 17, 2021): 6–10. http://dx.doi.org/10.1145/3494656.3494659.

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Mathematics and computation are inextricably entangled3. We couldn't do one without the other. The need to calculate can be traced to early human history, and mathematics developed in large part to enable computation. And computation is necessary to propel mathematics. One often loses sight of the fact that the great mathematicians of the past were also prodigious computers: For example, Gauss, Kummer and the other great pioneers of number theory did vast amounts of computation to arrive at or reinforce many of their insights.

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Vejdemo-Johansson, Mikael. "Blackbox computation of A ∞-algebras." gmj 17, no. 2 (June 2010): 391–404. http://dx.doi.org/10.1515/gmj.2010.005.

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Abstract Kadeishvili's proof of theminimality theorem [T. Kadeishvili, On the homology theory of fiber spaces, Russ. Math. Surv. 35:3 (1980), 231–238] induces an algorithm for the inductive computation of an A ∞-algebra structure on the homology of a dg-algebra. In this paper, we prove that for one class of dg-algebras, the resulting computation will generate a complete A ∞-algebra structure after a finite amount of computational work.

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15

CHANG, EE-CHIEN, SUNG WOO CHOI, DO YONG KWON, HYUNGJU PARK, and CHEE K. YAP. "SHORTEST PATH AMIDST DISC OBSTACLES IS COMPUTABLE." International Journal of Computational Geometry & Applications 16, no. 05n06 (December 2006): 567–90. http://dx.doi.org/10.1142/s0218195906002191.

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An open question in Exact Geometric Computation is whether there are transcendental computations that can be made "geometrically exact". Perhaps the simplest such problem in computational geometry is that of computing the shortest obstacle-avoiding path between two points p,q in the plane, where the obstacles are a collection of n discs. This problem can be solved in O(n2 log n) time in the Real RAM model, but nothing was known about its computability in the standard (Turing) model of computation. We first give a direct proof of the Turing-computability of this problem, provided the radii of the discs are rationally related. We make the usual assumption that the numerical input data are real algebraic numbers. By appealing to effective bounds from transcendental number theory, we further show a single-exponential time upper bound when the input numbers are rational. Our result appears to be the first example of a non-algebraic combinatorial problem which is shown computable. It is also a rare example of transcendental number theory yielding positive computational results.

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Kaddoum, G., Anthony J. Lawrance, P. Chargé, and D. Roviras. "Chaos Communication Performance: Theory and Computation." Circuits, Systems, and Signal Processing 30, no. 1 (October 14, 2010): 185–208. http://dx.doi.org/10.1007/s00034-010-9217-1.

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17

Chen, Naiwu (N N. Chan), and Jiankeng (Li Kimhung) Li. "Theory and computation of restricted linear models." Acta Mathematicae Applicatae Sinica 4, no. 4 (November 1988): 378–86. http://dx.doi.org/10.1007/bf02007242.

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18

AGRAWAL, MANINDRA, BARRY COOPER, and ANGSHENG LI. "Preface to Special Issue: Theory and Applications of Models of Computation (TAMC 2008–2009)." Mathematical Structures in Computer Science 20, no. 5 (October 2010): 705–6. http://dx.doi.org/10.1017/s0960129510000277.

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The Theory and Applications of Models of Computation (TAMC) conference series is both international and interdisciplinary in character, bringing together researchers working in computer science, mathematics (especially logic) and the physical sciences. It is this, together with its predominantly computational and computability theoretic focus, that gives the series its special character.

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19

Burgin, Mark, and Rao Mikkililineni. "Seven Layers of Computation: Methodological Analysis and Mathematical Modeling." Filozofia i Nauka Zeszyt specjalny, no. 10 (May 10, 2022): 11–32. http://dx.doi.org/10.37240/fin.2022.10.zs.1.

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We live in an information society where the usage, creation, distribution, manipulation, and integration of information is a significant activity. Computations allow us to process information from various sources in various forms and use the derived knowledge in improving efficiency and resilience in our interactions with each other and with our environment. The general theory of information tells us that information to knowledge is as energy is to matter. Energy has the potential to create or modify material structures and information has the potential to create or modify knowledge structures. In this paper, we analyze computations as a vital technological phenomenon of contemporary society which allows us to process and use information. This analysis allows building classifications of computations based on their characteristics and explication of new types of computations. As a result, we extend the existing typologies of computations by delineating novel forms of information representations. While the traditional approach deals only with two dimensions of computation—symbolic and sub-symbolic, here we describe additional dimensions, namely, super-symbolic computation, hybrid computation, fused computation, blended computation, and symbiotic computation.

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20

Lee, Ciarán M., and Matty J. Hoban. "Bounds on the power of proofs and advice in general physical theories." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2190 (June 2016): 20160076. http://dx.doi.org/10.1098/rspa.2016.0076.

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Quantum theory presents us with the tools for computational and communication advantages over classical theory. One approach to uncovering the source of these advantages is to determine how computation and communication power vary as quantum theory is replaced by other operationally defined theories from a broad framework of such theories. Such investigations may reveal some of the key physical features required for powerful computation and communication. In this paper, we investigate how simple physical principles bound the power of two different computational paradigms which combine computation and communication in a non-trivial fashion: computation with advice and interactive proof systems. We show that the existence of non-trivial dynamics in a theory implies a bound on the power of computation with advice. Moreover, we provide an explicit example of a theory with no non-trivial dynamics in which the power of computation with advice is unbounded. Finally, we show that the power of simple interactive proof systems in theories where local measurements suffice for tomography is non-trivially bounded. This result provides a proof that Q M A is contained in P P , which does not make use of any uniquely quantum structure—such as the fact that observables correspond to self-adjoint operators—and thus may be of independent interest.

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21

Legatiuk, Dmitrii. "Mathematical Modelling by Help of Category Theory: Models and Relations between Them." Mathematics 9, no. 16 (August 15, 2021): 1946. http://dx.doi.org/10.3390/math9161946.

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The growing complexity of modern practical problems puts high demand on mathematical modelling. Given that various models can be used for modelling one physical phenomenon, the role of model comparison and model choice is becoming particularly important. Methods for model comparison and model choice typically used in practical applications nowadays are computation-based, and thus time consuming and computationally costly. Therefore, it is necessary to develop other approaches to working abstractly, i.e., without computations, with mathematical models. An abstract description of mathematical models can be achieved by the help of abstract mathematics, implying formalisation of models and relations between them. In this paper, a category theory-based approach to mathematical modelling is proposed. In this way, mathematical models are formalised in the language of categories, relations between the models are formally defined and several practically relevant properties are introduced on the level of categories. Finally, an illustrative example is presented, underlying how the category-theory based approach can be used in practice. Further, all constructions presented in this paper are also discussed from a modelling point of view by making explicit the link to concrete modelling scenarios.

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22

Vyas, Saurabh, Matthew D. Golub, David Sussillo, and Krishna V. Shenoy. "Computation Through Neural Population Dynamics." Annual Review of Neuroscience 43, no. 1 (July 8, 2020): 249–75. http://dx.doi.org/10.1146/annurev-neuro-092619-094115.

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Significant experimental, computational, and theoretical work has identified rich structure within the coordinated activity of interconnected neural populations. An emerging challenge now is to uncover the nature of the associated computations, how they are implemented, and what role they play in driving behavior. We term this computation through neural population dynamics. If successful, this framework will reveal general motifs of neural population activity and quantitatively describe how neural population dynamics implement computations necessary for driving goal-directed behavior. Here, we start with a mathematical primer on dynamical systems theory and analytical tools necessary to apply this perspective to experimental data. Next, we highlight some recent discoveries resulting from successful application of dynamical systems. We focus on studies spanning motor control, timing, decision-making, and working memory. Finally, we briefly discuss promising recent lines of investigation and future directions for the computation through neural population dynamics framework.

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23

KONDRAT’EVA, M. V., A. B. LEVIN, A. V. MIKHALEV, and E. V. PANKRAT’EV. "COMPUTATION OF DIMENSION POLYNOMIALS." International Journal of Algebra and Computation 02, no. 02 (June 1992): 117–37. http://dx.doi.org/10.1142/s0218196792000098.

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The consideration of differential versions of Hilbert dimension polynomials is due to A. Einstein [1] and E. Kolchin [2] (one can find the coverage of the theory of differential dimension polynomials in [6]). In this paper we introduce the notion of a dimension polynomial of a subset of ℕm associated with arbitrary partition of the set {1,…, m} into disjoint nonempty subsets (m∈ℕ, ℕ denoting the set of all nonnegative integers). The theory of such polynomials is developed. The importance of our considerations is connected with the fact that the computation of differential and difference dimen sion polynomials may be reduced to the computation of some dimension polynomials of subsets of ℕm where m∈ℕ (see [3, p. 115], [5]). We also give some methods and algorithms for computation of dimension polynomials.

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24

BLUM, LENORE, FELIPE CUCKER, MIKE SHUB, and STEVE SMALE. "COMPLEXITY AND REAL COMPUTATION: A MANIFESTO." International Journal of Bifurcation and Chaos 06, no. 01 (January 1996): 3–26. http://dx.doi.org/10.1142/s0218127496001818.

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Finding a natural meeting ground between the highly developed complexity theory of computer science — with its historical roots in logic and the discrete mathematics of the integers — and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis — with its rich history and longstanding traditions in the continuous mathematics of analysis — presents a compelling challenge. Here we illustrate the issues and pose our perspective toward resolution.

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25

Lauder, Alan G. B. "Deformation theory and the computation of zeta functions." Proceedings of the London Mathematical Society 88, no. 03 (April 14, 2004): 565–602. http://dx.doi.org/10.1112/s0024611503014461.

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26

Al-Khaled, Kamel. "Theory and computation in singular boundary value problems." Chaos, Solitons & Fractals 33, no. 2 (July 2007): 678–84. http://dx.doi.org/10.1016/j.chaos.2006.01.047.

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27

Nord, Gail D., David Jabon, and John Nord. "Activities: The Mathematics of the Global Positioning System." Mathematics Teacher 90, no. 6 (September 1997): 455–60. http://dx.doi.org/10.5951/mt.90.6.0455.

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Teacher's Guide: The Global Positioning System (GPS) is a constellation of twenty-four satellites, orbiting approximately 20 200 km above sea level, that enable receivers to compute their position anywhere on the earth with remarkable accuracy. The mathematical theory and computation involved in the GPS are within the scope of the second-year-algebra curriculum. This activity illustrates an application of mathematics to modern navigation.

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Cooper, S. Barry, Benedikt Lowe, and Peter van Emde Boas. "Theory of Computation at CiE 2005." Theory of Computing Systems 41, no. 1 (July 2007): 1–2. http://dx.doi.org/10.1007/s00224-006-4101-x.

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29

Agler, Jim, John Harland, and Benjamin J. Raphael. "Classical function theory, operator dilation theory, and machine computation on multiply-connected domains." Memoirs of the American Mathematical Society 191, no. 892 (2008): 0. http://dx.doi.org/10.1090/memo/0892.

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30

Geiser, Jürgen. "Embedded Zassenhaus Expansion to Splitting Schemes: Theory and Multiphysics Applications." International Journal of Differential Equations 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/314290.

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We present some operator splitting methods improved by the use of the Zassenhaus product and designed for applications to multiphysics problems. We treat iterative splitting methods that can be improved by means of the Zassenhaus product formula, which is a sequential splitting scheme. The main idea for reducing the computation time needed by the iterative scheme is to embed fast and cheap Zassenhaus product schemes, since the computation of the commutators involved is very cheap, since we are dealing with nilpotent matrices. We discuss the coupling ideas of iterative and sequential splitting techniques and their convergence. While the iterative splitting schemes converge slowly in their first iterative steps, we improve the initial convergence rates by embedding the Zassenhaus product formula. The applications are to multiphysics problems in fluid dynamics. We consider phase models in computational fluid dynamics and analyse how to obtain higher order operator splitting methods based on the Zassenhaus product. The computational benefits derive from the use of sparse matrices, which arise from the spatial discretisation of the underlying partial differential equations. Since the Zassenhaus formula requires nearly constant CPU time due to its sparse commutators, we have accelerated the iterative splitting schemes.

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31

Nien, Chia-Hsing, and Frederick J. Wicklin. "An Algorithm for the Computation of Preimages in Noninvertible Mappings." International Journal of Bifurcation and Chaos 08, no. 02 (February 1998): 415–22. http://dx.doi.org/10.1142/s0218127498000279.

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For discrete dynamical systems generated by iterating a diffeomorphism, every point in the phase space has a unique preimage and it is straightforward to compute geometric structures such as inverse orbits and one-dimensional stable manifolds of periodic points. For noninvertible mappings, however, some points have multiple preimages; others may have no preimages. This makes the computation of inverse orbits difficult, because accurate computations require global knowledge about the way the mapping folds and pleats phase space. In this article we use ideas from singularity theory to examine the geometry of noninvertible mappings. We use the geometry to derive a computational algorithm for efficiently computing preimages in noninvertible mappings.

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32

Okazaki, Hiroyuki, and Yuichi Futa. "Polynomially Bounded Sequences and Polynomial Sequences." Formalized Mathematics 23, no. 3 (September 1, 2015): 205–13. http://dx.doi.org/10.1515/forma-2015-0017.

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Abstract In this article, we formalize polynomially bounded sequences that plays an important role in computational complexity theory. Class P is a fundamental computational complexity class that contains all polynomial-time decision problems [11], [12]. It takes polynomially bounded amount of computation time to solve polynomial-time decision problems by the deterministic Turing machine. Moreover we formalize polynomial sequences [5].

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33

Liu, Xue Ting. "The Judgement for Generalized Positive Definite Matrices in Signal Processing." Advanced Materials Research 121-122 (June 2010): 128–32. http://dx.doi.org/10.4028/www.scientific.net/amr.121-122.128.

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The generalized positive definite matrix is an active research field of special matrix, they have applied in computational mathematics, economics, physics, biology, applied mathematics, numerical computation, signal processing, coding theory, oil investigation in recent years, and so on. In this paper, motivated by [3], we give a simple and convenient judging methodwhich can be used to judge whether an nonnegative real matrix A is an generalized positive definite matrix or not.

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34

Yoosefzadeh, H. R., Hamed R. Tareghian, and M. H. Farahi. "Tri-directional Scheduling Scheme: Theory and Computation." Journal of Mathematical Modelling and Algorithms 9, no. 4 (May 22, 2010): 357–73. http://dx.doi.org/10.1007/s10852-010-9132-2.

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DOUGHERTY, EDWARD R., and ULISSES BRAGA-NETO. "EPISTEMOLOGY OF COMPUTATIONAL BIOLOGY: MATHEMATICAL MODELS AND EXPERIMENTAL PREDICTION AS THE BASIS OF THEIR VALIDITY." Journal of Biological Systems 14, no. 01 (March 2006): 65–90. http://dx.doi.org/10.1142/s0218339006001726.

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Knowing the roles of mathematics and computation in experimental science is important for computational biology because these roles determine to a great extent how research in this field should be pursued and how it should relate to biology in general. The present paper examines the epistemology of computational biology from the perspective of modern science, the underlying principle of which is that a scientific theory must have two parts: (1) a structural model, which is a mathematical construct that aims to represent a selected portion of physical reality and (2) a well-defined procedure for relating consequences of the model to quantifiable observations. We also explore the contingency and creative nature of a scientific theory. Among the questions considered are: Can computational biology form the theoretical core of biology? What is the basis, if any, for choosing one particular model over another? And what is the role of computation in science, and in biology in particular? We examine how this broad epistemological framework applies to important statistical methodologies pertaining to computational biology, such as expression-based phenotype classification, gene regulatory networks, and clustering. We consider classification in detail, as the epistemological issues raised by classification are related to all computational-biology topics in which statistical prediction plays a key role. We pay particular attention to classifier-model validity and its relation to estimation rules.

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Karp, Richard M. "George Dantzig’s impact on the theory of computation." Discrete Optimization 5, no. 2 (May 2008): 174–85. http://dx.doi.org/10.1016/j.disopt.2006.12.004.

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N., S. P., and Richard C. Aiken. "Stiff Computation." Mathematics of Computation 47, no. 176 (October 1986): 755. http://dx.doi.org/10.2307/2008193.

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FU, YUXI. "Non-deterministic structures of computation." Mathematical Structures in Computer Science 25, no. 6 (November 10, 2014): 1295–338. http://dx.doi.org/10.1017/s0960129514000012.

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Divergence and non-determinism play a fundamental role in the theory of computation, and their combined effect on computational equality deserves further study. By looking at the issue from the point of view of both computation and interaction, we are led to a canonical equality for non-deterministic computation, revealing its rich algebraic structure. We study this structure in three ways. First, we construct a complete equational system for finite-state non-deterministic computation. The challenge with such a system is to find an equational alternative to fixpoint inductionà laMilner. We establish a negative result in the form of the non-existence of a finite equational system for the canonical equality of non-deterministic computation to support our approach. We then investigate infinite-state non-deterministic computation in the light of definability and show that every recursively enumerable set is generated by an unobservable process. Finally, we prove that, as far as computation is concerned, the effect produced jointly by divergence and non-determinism is model independent for a large class of process models.We use C-graphs, which are interesting in their own right, as abstract representations of the computational objects throughout the paper.

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Baldoni, V., N. Berline, M. Köppe, and M. Vergne. "INTERMEDIATE SUMS ON POLYHEDRA: COMPUTATION AND REAL EHRHART THEORY." Mathematika 59, no. 1 (September 5, 2012): 1–22. http://dx.doi.org/10.1112/s0025579312000101.

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Dagnino, Francesco. "A Meta-theory for Big-step Semantics." ACM Transactions on Computational Logic 23, no. 3 (July 31, 2022): 1–50. http://dx.doi.org/10.1145/3522729.

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It is well known that big-step semantics is not able to distinguish stuck and non-terminating computations. This is a strong limitation as it makes it very difficult to reason about properties involving infinite computations, such as type soundness, which cannot even be expressed. We show that this issue is only apparent: the distinction between stuck and diverging computations is implicit in any big-step semantics and it just needs to be uncovered. To achieve this goal, we develop a systematic study of big-step semantics: we introduce an abstract definition of what a big-step semantics is, we define a notion of computation by formalizing the evaluation algorithm implicitly associated with any big-step semantics, and we show how to canonically extend a big-step semantics to characterize stuck and diverging computations. Building on these notions, we describe a general proof technique to show that a predicate is sound, that is, it prevents stuck computation, with respect to a big-step semantics. One needs to check three properties relating the predicate and the semantics, and if they hold, the predicate is sound. The extended semantics is essential to establish this meta-logical result but is of no concerns to the user, who only needs to prove the three properties of the initial big-step semantics. Finally, we illustrate the technique by several examples, showing that it is applicable also in cases where subject reduction does not hold, and hence the standard technique for small-step semantics cannot be used.

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Lou, Shan, Xiangqian Jiang, and Paul J. Scott. "Geometric computation theory for morphological filtering on freeform surfaces." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2159 (November 8, 2013): 20130150. http://dx.doi.org/10.1098/rspa.2013.0150.

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Surfaces govern functional behaviours of geometrical products, especially high-precision and high-added-value products. Compared with the mean line-based filters, morphological filters, evolved from the traditional E-system, are relevant to functional performance of surfaces. The conventional implementation of morphological filters based on image-processing does not work for state-of-the-art surfaces, for example, freeform surfaces. A set of novel geometric computation theory is developed by applying the alpha shape to the computation. Divide and conquer optimization is employed to speed up the computational performance of the alpha-shape method and reduce memory usage. To release the dependence of the alpha-shape method on the Delaunay triangulation, a set of definitions and propositions for the search of contact points is presented and mathematically proved based on alpha shape theory, which are applicable to both circular and horizontal flat structuring elements. The developed methods are verified through experimentation.

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Blass, Andreas, and Yuri Gurevich. "Witness algebra and anyon braiding." Mathematical Structures in Computer Science 30, no. 3 (March 2020): 234–70. http://dx.doi.org/10.1017/s0960129520000055.

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AbstractTopological quantum computation employs two-dimensional quasiparticles called anyons. The generally accepted mathematical basis for the theory of anyons is the framework of modular tensor categories. That framework involves a substantial amount of category theory and is, as a result, considered rather difficult to understand. Is the complexity of the present framework necessary? The computations of associativity and braiding matrices can be based on a much simpler framework, which looks less like category theory and more like familiar algebra. We introduce that framework here.

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Farmer, William M., Joshua D. Guttman, and F. Javier Thayer. "Contexts in Mathematical Reasoning and Computation." Journal of Symbolic Computation 19, no. 1-3 (January 1995): 201–16. http://dx.doi.org/10.1006/jsco.1995.1012.

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Demanet, Laurent, and Gabriel Peyré. "Compressive Wave Computation." Foundations of Computational Mathematics 11, no. 3 (February 24, 2011): 257–303. http://dx.doi.org/10.1007/s10208-011-9085-5.

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Junde, Wu. "Special Issue “Quantum computation complexity theory and quantum network theory” (Preface)." International Journal of Theoretical Physics 60, no. 7 (June 2, 2021): 2345. http://dx.doi.org/10.1007/s10773-021-04846-5.

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Larsen, Brett W., and Shaul Druckmann. "Towards a more general understanding of the algorithmic utility of recurrent connections." PLOS Computational Biology 18, no. 6 (June 21, 2022): e1010227. http://dx.doi.org/10.1371/journal.pcbi.1010227.

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Lateral and recurrent connections are ubiquitous in biological neural circuits. Yet while the strong computational abilities of feedforward networks have been extensively studied, our understanding of the role and advantages of recurrent computations that might explain their prevalence remains an important open challenge. Foundational studies by Minsky and Roelfsema argued that computations that require propagation of global information for local computation to take place would particularly benefit from the sequential, parallel nature of processing in recurrent networks. Such “tag propagation” algorithms perform repeated, local propagation of information and were originally introduced in the context of detecting connectedness, a task that is challenging for feedforward networks. Here, we advance the understanding of the utility of lateral and recurrent computation by first performing a large-scale empirical study of neural architectures for the computation of connectedness to explore feedforward solutions more fully and establish robustly the importance of recurrent architectures. In addition, we highlight a tradeoff between computation time and performance and construct hybrid feedforward/recurrent models that perform well even in the presence of varying computational time limitations. We then generalize tag propagation architectures to propagating multiple interacting tags and demonstrate that these are efficient computational substrates for more general computations of connectedness by introducing and solving an abstracted biologically inspired decision-making task. Our work thus clarifies and expands the set of computational tasks that can be solved efficiently by recurrent computation, yielding hypotheses for structure in population activity that may be present in such tasks.

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SPITTERS, BAS, and EELIS VAN DER WEEGEN. "Type classes for mathematics in type theory." Mathematical Structures in Computer Science 21, no. 4 (July 1, 2011): 795–825. http://dx.doi.org/10.1017/s0960129511000119.

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The introduction of first-class type classes in the Coq system calls for a re-examination of the basic interfaces used for mathematical formalisation in type theory. We present a new set of type classes for mathematics and take full advantage of their unique features to make practical a particularly flexible approach that was formerly thought to be unfeasible. Thus, we address traditional proof engineering challenges as well as new ones resulting from our ambition to build upon this development a library of constructive analysis in which any abstraction penalties inhibiting efficient computation are reduced to a minimum.The basis of our development consists of type classes representing a standard algebraic hierarchy, as well as portions of category theory and universal algebra. On this foundation, we build a set of mathematically sound abstract interfaces for different kinds of numbers, succinctly expressed using categorical language and universal algebra constructions. Strategic use of type classes lets us support these high-level theory-friendly definitions, while still enabling efficient implementations unhindered by gratuitous indirection, conversion or projection.Algebra thrives on the interplay between syntax and semantics. The Prolog-like abilities of type class instance resolution allow us to conveniently define a quote function, thus facilitating the use of reflective techniques.

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Blaine, Larry. "Theory vs. Computation in Some Very Simple Dynamical Systems." College Mathematics Journal 22, no. 1 (January 1991): 42. http://dx.doi.org/10.2307/2686737.

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Liebling, Thomas M., Denis Naddef, and Laurence A. Wolsey. "Combinatorial Optimization: Theory and Computation The Aussois Workshop 2004." Mathematical Programming 105, no. 2-3 (November 10, 2005): 157–60. http://dx.doi.org/10.1007/s10107-005-0646-8.

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Bailey, David H., Jonathan M. Borwein, and Richard E. Crandall. "Computation and theory of extended Mordell-Tornheim-Witten sums." Mathematics of Computation 83, no. 288 (January 23, 2014): 1795–821. http://dx.doi.org/10.1090/s0025-5718-2014-02768-3.

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