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Journal articles on the topic 'Realizability theory'

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Published: 4 June 2021
Last updated: 18 February 2022

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1

McCarty, Charles. "Realizability and recursive set theory." Annals of Pure and Applied Logic 32 (1986): 153–83. http://dx.doi.org/10.1016/0168-0072(86)90050-3.

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2

Hofstra, Pieter, and Michael A. Warren. "Combinatorial realizability models of type theory." Annals of Pure and Applied Logic 164, no. 10 (October 2013): 957–88. http://dx.doi.org/10.1016/j.apal.2013.05.002.

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3

Soto, Ricardo L., Ana I. Julio, and Jaime H. Alfaro. "Permutative universal realizability." Special Matrices 9, no. 1 (January 1, 2021): 66–77. http://dx.doi.org/10.1515/spma-2020-0123.

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Abstract A list of complex numbers Λ is said to be realizable, if it is the spectrum of a nonnegative matrix. In this paper we provide a new sufficient condition for a given list Λ to be universally realizable (UR), that is, realizable for each possible Jordan canonical form allowed by Λ. Furthermore, the resulting matrix (that is explicity provided) is permutative, meaning that each of its rows is a permutation of the first row. In particular, we show that a real Suleĭmanova spectrum, that is, a list of real numbers having exactly one positive element, is UR by a permutative matrix.
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4

Biryukov, Oleg N. "Parity conditions for realizability of Gauss diagrams." Journal of Knot Theory and Its Ramifications 28, no. 01 (January 2019): 1950015. http://dx.doi.org/10.1142/s0218216519500159.

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We consider a problem of realizability of Gauss diagrams by closed plane curves where the plane curves have only double points of transversal self-intersection. We formulate the necessary and sufficient conditions for realizability. These conditions are based only on the parity of double and triple intersections of the chords in the Gauss diagram.
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5

Harnik, Victor, and Michael Makkai. "Lambek's categorical proof theory and Läuchli's abstract realizability." Journal of Symbolic Logic 57, no. 1 (March 1992): 200–230. http://dx.doi.org/10.2307/2275186.

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In this paper we give an introduction to categorical proof theory, and reinterpret, with improvements, Läuchli's work on abstract realizability restricted to propositional logic (but see [M1] for predicate logic). Partly to make some points of a foundational nature, we have included a substantial amount of background material. As a result, the paper is (we hope) readable with a knowledge of just the rudiments of category theory, the notions of category, functor, natural transformation, and the like. We start with an extended introduction giving the background, and stating what we do with a minimum of technicalities.In three publications [L1, 2, 3] published in the years 1968, 1969 and 1972, J. Lambek gave a categorical formulation of the notion of formal proof in deductive systems in certain propositional calculi. The theory is also described in the recent book [LS]. See also [Sz].The basic motivation behind Lambek's theory was to place proof theory in the framework of modern abstract mathematics. The spirit of the latter, at least for the purposes of the present discussion, is to organize mathematical objects into mathematical structures. The specific kind of structure we will be concerned with is category.In Lambek's theory, one starts with an arbitrary theory in any one of several propositional calculi. One has the (formal) proofs (deductions) in the given theory of entailments A ⇒ B, with A and B arbitrary formulas. One introduces an equivalence relation on proofs under which, in particular, equivalent proofs are proofs of the same entailment; equivalence of proofs is intended to capture the idea of the proofs being only inessentially different. One forms a category whose objects are the formulas of the underlying language of the theory, and whose arrows from A to B, with the latter arbitrary formulas, are the equivalence classes of formal proofs of A ⇒ B.
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6

van den Berg, Benno, and Ieke Moerdijk. "Aspects of predicative algebraic set theory, II: Realizability." Theoretical Computer Science 412, no. 20 (April 2011): 1916–40. http://dx.doi.org/10.1016/j.tcs.2010.12.019.

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7

Chen, Ray-Ming, and Michael Rathjen. "Lifschitz realizability for intuitionistic Zermelo–Fraenkel set theory." Archive for Mathematical Logic 51, no. 7-8 (August 14, 2012): 789–818. http://dx.doi.org/10.1007/s00153-012-0299-2.

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8

Carboni, A. "Some free constructions in realizability and proof theory." Journal of Pure and Applied Algebra 103, no. 2 (September 1995): 117–48. http://dx.doi.org/10.1016/0022-4049(94)00103-p.

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9

RATHJEN, MICHAEL, and ANDREW W. SWAN. "LIFSCHITZ REALIZABILITY AS A TOPOLOGICAL CONSTRUCTION." Journal of Symbolic Logic 85, no. 4 (December 2020): 1342–75. http://dx.doi.org/10.1017/jsl.2021.1.

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AbstractWe develop a number of variants of Lifschitz realizability for $\mathbf {CZF}$ by building topological models internally in certain realizability models. We use this to show some interesting metamathematical results about constructive set theory with variants of the lesser limited principle of omniscience including consistency with unique Church’s thesis, consistency with some Brouwerian principles and variants of the numerical existence property.
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10

Belk, Maria, and Robert Connelly. "Realizability of Graphs." Discrete & Computational Geometry 37, no. 2 (February 2007): 125–37. http://dx.doi.org/10.1007/s00454-006-1284-5.

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11

BERGER, ULRICH, and TIE HOU. "A realizability interpretation of Church's simple theory of types." Mathematical Structures in Computer Science 27, no. 8 (July 22, 2016): 1364–85. http://dx.doi.org/10.1017/s0960129516000104.

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We give a realizability interpretation of an intuitionistic version of Church's Simple Theory of Types (CST) which can be viewed as a formalization of intuitionistic higher-order logic. Although definable in CST we include operators for monotone induction and coinduction and provide simple realizers for them. Realizers are formally represented in an untyped lambda–calculus with pairing and case-construct. The purpose of this interpretation is to provide a foundation for the extraction of verified programs from formal proofs as an alternative to type-theoretic systems. The advantages of our approach are that (a) induction and coinduction are not restricted to the strictly positive case, (b) abstract mathematical structures and results may be imported, (c) the formalization is technically simpler than in other systems, for example, regarding the definition of realizability, which is a simple syntactical substitution, and the treatment of nested and simultaneous (co)inductive definitions.
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12

LONGLEY, JOHN R., and ALEX K. SIMPSON. "A uniform approach to domain theory in realizability models." Mathematical Structures in Computer Science 7, no. 5 (October 1997): 469–505. http://dx.doi.org/10.1017/s0960129597002387.

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We propose a uniform way of isolating a subcategory of predomains within the category of modest sets determined by a partial combinatory algebra (PCA). Given a divergence on a PCA (which determines a notion of partiality), we identify a candidate category of predomains, the well-complete objects. We show that, whenever a single strong completeness axiom holds, the category satisfies appropriate closure properties. We consider a range of examples of PCAs with associated divergences and show that in each case the axiom does hold. These examples encompass models allowing a ‘parallel’ style of computation (for example, by interleaving), as well as models that seemingly allow only ‘sequential’ computation, such as those based on term-models for the lambda-calculus. Thus, our approach provides a uniform approach to domain theory across a wide class of realizability models. We compare our treatment with previous approaches to domain theory in realizability models. It appears that no other approach applies across such a wide range of models.
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13

Dihoum, Eman, and Michael Rathjen. "Preservation of choice principles under realizability." Logic Journal of the IGPL 27, no. 5 (February 8, 2019): 746–65. http://dx.doi.org/10.1093/jigpal/jzz002.

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AbstractEspecially nice models of intuitionistic set theories are realizability models $V({\mathcal A})$, where $\mathcal A$ is an applicative structure or partial combinatory algebra. This paper is concerned with the preservation of various choice principles in $V({\mathcal A})$ if assumed in the underlying universe $V$, adopting Constructive Zermelo–Fraenkel as background theory for all of these investigations. Examples of choice principles are the axiom schemes of countable choice, dependent choice, relativized dependent choice and the presentation axiom. It is shown that any of these axioms holds in $V(\mathcal{A})$ for every applicative structure $\mathcal A$ if it holds in the background universe.1 It is also shown that a weak form of the countable axiom of choice, $\textbf{AC}^{\boldsymbol{\omega , \omega }}$, is rendered true in any $V(\mathcal{A})$ regardless of whether it holds in the background universe. The paper extends work by McCarty (1984, Realizability and Recursive Mathematics, PhD Thesis) and Rathjen (2006, Realizability for constructive Zermelo–Fraenkel set theory. In Logic Colloquium 03, pp. 282–314).
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14

NELSON, SAM. "VIRTUAL CROSSING REALIZATION." Journal of Knot Theory and Its Ramifications 14, no. 07 (November 2005): 931–51. http://dx.doi.org/10.1142/s0218216505004159.

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We study virtual isotopy sequences with classical initial and final diagrams, asking when such a sequence can be changed into a classical isotopy sequence by replacing virtual crossings with classical crossings. An example of a sequence for which no such virtual crossing realization exists is given. A conjecture on conditions for realizability of virtual isotopy sequences is proposed, and a sufficient condition for realizability is found. The conjecture is reformulated in terms of 2-knots and knots in thickened surfaces.
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15

KARAGILA, ASAF. "REALIZING REALIZABILITY RESULTS WITH CLASSICAL CONSTRUCTIONS." Bulletin of Symbolic Logic 25, no. 4 (December 2019): 429–45. http://dx.doi.org/10.1017/bsl.2019.59.

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AbstractJ. L. Krivine developed a new method based on realizability to construct models of set theory where the axiom of choice fails. We attempt to recreate his results in classical settings, i.e., symmetric extensions. We also provide a new condition for preserving well ordered, and other particular type of choice, in the general settings of symmetric extensions.
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16

Frey, Jonas. "Characterizing partitioned assemblies and realizability toposes." Journal of Pure and Applied Algebra 223, no. 5 (May 2019): 2000–2014. http://dx.doi.org/10.1016/j.jpaa.2018.08.012.

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17

Johnson, C. R., C. Marijuán, and M. Pisonero. "Symmetric nonnegative realizability via partitioned majorization." Linear and Multilinear Algebra 65, no. 7 (October 6, 2016): 1417–26. http://dx.doi.org/10.1080/03081087.2016.1242113.

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18

Barrett, David A. "Multiple Realizability, Identity Theory, and the Gradual Reorganization Principle." British Journal for the Philosophy of Science 64, no. 2 (June 1, 2013): 325–46. http://dx.doi.org/10.1093/bjps/axs011.

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19

Bose, N., and Y. Shi. "Network realizability theory approach to stability of complex polynomials." IEEE Transactions on Circuits and Systems 34, no. 2 (February 1987): 216–18. http://dx.doi.org/10.1109/tcs.1987.1086097.

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20

Ziegler, Albert. "Generalizing realizability and Heyting models for constructive set theory." Annals of Pure and Applied Logic 163, no. 2 (February 2012): 175–84. http://dx.doi.org/10.1016/j.apal.2011.06.025.

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21

Grinblat, Andrey, and Viktor Lopatkin. "On realizabilty of Gauss diagrams and constructions of meanders." Journal of Knot Theory and Its Ramifications 29, no. 05 (April 2020): 2050031. http://dx.doi.org/10.1142/s0218216520500315.

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The problem concerning which Gauss diagrams can be realized by knots is an old one and has been solved in several ways. In this paper, we present a direct approach to this problem. We show that the needed conditions for realizability of a Gauss diagram can be interpreted as follows “the number of exits = the number of entrances” and the sufficient condition is based on the Jordan curve theorem. Further, using matrices, we redefine conditions for realizability of Gauss diagrams and then we give an algorithm to construct meanders.
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22

SANTOS, WALTER FERRER, MAURICIO GUILLERMO, and OCTAVIO MALHERBE. "Realizability in ordered combinatory algebras with adjunction." Mathematical Structures in Computer Science 29, no. 3 (April 26, 2018): 430–64. http://dx.doi.org/10.1017/s0960129518000075.

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In this work, we continue our consideration of the constructions presented in the paperKrivine's Classical Realizability from a Categorical Perspectiveby Thomas Streicher. Therein, the author points towards the interpretation of the classical realizability of Krivine as an instance of the categorical approach started by Hyland. The present paper continues with the study of the basic algebraic set-up underlying the categorical aspects of the theory. Motivated by the search of a full adjunction, we introduce a new closure operator on the subsets of the stacks of an abstract Krivine structure that yields an adjunction between the corresponding application and implication operations. We show that all the constructions from ordered combinatory algebras to triposes presented in our previous work can be implemented,mutatis mutandis, in the new situation and that all the associated triposes are equivalent. We finish by proving that the whole theory can be developed using the ordered combinatory algebras with full adjunction or strong abstract Krivine structures as the basic set-up.
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23

Rathjen, Michael. "The disjunction and related properties for constructive Zermelo-Fraenkel set theory." Journal of Symbolic Logic 70, no. 4 (December 2005): 1233–54. http://dx.doi.org/10.2178/jsl/1129642124.

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AbstractThis paper proves that the disjunction property, the numerical existence property. Church's rule, and several other metamathematical properties hold true for Constructive Zermelo-Fraenkel Set Theory, CZF, and also for the theory CZF augmented by the Regular Extension Axiom.As regards the proof technique, it features a self-validating semantics for CZF that combines realizability for extensional set theory and truth.
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24

Crolard, Tristan. "A type theory which is complete for Kreisel's modified realizability." Electronic Notes in Theoretical Computer Science 23, no. 1 (1999): 58–73. http://dx.doi.org/10.1016/s1571-0661(04)00104-5.

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25

Julio, Ana I., Carlos Marijuán, Miriam Pisonero, and Ricardo L. Soto. "On universal realizability of spectra." Linear Algebra and its Applications 563 (February 2019): 353–72. http://dx.doi.org/10.1016/j.laa.2018.11.013.

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26

Johnson, Charles R., Ana I. Julio, and Ricardo L. Soto. "Nonnegative realizability with Jordan structure." Linear Algebra and its Applications 587 (February 2020): 302–13. http://dx.doi.org/10.1016/j.laa.2019.11.016.

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27

Julio, Ana I., and Ricardo L. Soto. "On the universal realizability problem." Linear Algebra and its Applications 597 (July 2020): 170–86. http://dx.doi.org/10.1016/j.laa.2020.03.026.

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28

Julio, Ana I., Carlos Marijuán, Miriam Pisonero, and Ricardo L. Soto. "Universal realizability in low dimension." Linear Algebra and its Applications 619 (June 2021): 107–36. http://dx.doi.org/10.1016/j.laa.2021.02.012.

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29

Kabadi, Santosh N., R. Chandrasekaran, and K. P. K. Nair. "Multiroute flows: Cut-trees and realizability." Discrete Optimization 2, no. 3 (September 2005): 229–40. http://dx.doi.org/10.1016/j.disopt.2005.03.005.

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30

Ferrer Santos, Walter, and Octavio Malherbe. "The category of implicative algebras and realizability." Mathematical Structures in Computer Science 29, no. 10 (September 16, 2019): 1575–606. http://dx.doi.org/10.1017/s0960129519000100.

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AbstractIn this paper, we continue with the algebraic study of Krivine’s realizability, completing and generalizing some of the authors’ previous constructions by introducing two categories with objects the abstract Krivine structures and the implicative algebras, respectively. These categories are related by an adjunction whose existence clarifies many aspects of the theory previously established. We also revisit, reinterpret, and generalize in categorical terms, some of the results of our previous work such as: the bullet construction, the equivalence of Krivine’s, Streicher’s, and bullet triposes and also the fact that these triposes can be obtained – up to equivalence – from implicative algebras or implicative ordered combinatory algebras.
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31

Oosten, Jaap Van. "Two remarks on the Lifschitz realizability topos." Journal of Symbolic Logic 61, no. 1 (March 1996): 70–79. http://dx.doi.org/10.2307/2275598.

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The purpose of this note is to clarify two points about the topos Lif, introduced in [13] as a generalization of Lifschitz' realizability ([9, 12]). Lif is a subtopos of Hyland's Effective topos ([1]). The points I want to make are:Remark 1. Lif is the largest subtopos of satisfying the axiom (O):where denotes partial recursive application, and “∈ Tot” means that e and f range over codes for total recursive functions. One may read (O) as the statement “The union of two -sets is again a -set”. That is, let be a subtopos of . Then (O) is true in for the standard interpretation (the variables range over the natural numbers object of , etc.) if and only if the inclusion ↣ factors through the inclusion Lif ↣ .Remark 2. Like , Lif contains at least two weakly complete internal full subcategories, thus providing us with more models of polymorphism and other impredicative type theories.The principle (O) has some standing in the history of constructive mathematics:- H. Friedman has proved that (O) is equivalent to a formulation of intuitionistic completeness of the intuitionistic predicate calculus for Tarskian semantics; see [8]. This is not to imply that this result is of immediate relevance to Lif: Friedman works in a system of analysis, a theory of lawless sequences with an axiom of “open data” for arithmetical formulas, which at least for the domain of all functions from N to N fails in Lif. However, there might exist a “nonstandard model” of arithmetic and a corresponding system of analysis, in which we may be able to carry out his proof.- Moreover, Remark 2 entails that Lif should provide us with models of synthetic domain theory (for an exposition, see [3]), and one with the nice property that the dual of one of the axioms (axiom 7 in [3]) comes for free, by (O).
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32

Galerne, Bruno, and Raphael Lachièze-Rey. "Random measurable sets and covariogram realizability problems." Advances in Applied Probability 47, no. 03 (September 2015): 611–39. http://dx.doi.org/10.1017/s0001867800048758.

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We provide a characterization of realisable set covariograms, bringing a rigorous yet abstract solution to theS2problem in materials science. Our method is based on the covariogram functional for random measurable sets (RAMS) and on a result about the representation of positive operators on a noncompact space. RAMS are an alternative to the classical random closed sets in stochastic geometry and geostatistics, and they provide a weaker framework that allows the manipulation of more irregular functionals, such as the perimeter. We therefore use the illustration provided by theS2problem to advocate the use of RAMS for solving theoretical problems of a geometric nature. Along the way, we extend the theory of random measurable sets, and in particular the local approximation of the perimeter by local covariograms.
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33

Galerne, Bruno, and Raphael Lachièze-Rey. "Random measurable sets and covariogram realizability problems." Advances in Applied Probability 47, no. 3 (September 2015): 611–39. http://dx.doi.org/10.1239/aap/1444308874.

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We provide a characterization of realisable set covariograms, bringing a rigorous yet abstract solution to the S2 problem in materials science. Our method is based on the covariogram functional for random measurable sets (RAMS) and on a result about the representation of positive operators on a noncompact space. RAMS are an alternative to the classical random closed sets in stochastic geometry and geostatistics, and they provide a weaker framework that allows the manipulation of more irregular functionals, such as the perimeter. We therefore use the illustration provided by the S2 problem to advocate the use of RAMS for solving theoretical problems of a geometric nature. Along the way, we extend the theory of random measurable sets, and in particular the local approximation of the perimeter by local covariograms.
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34

Schwichtenberg, Helmut. "Realizability interpretation of proofs in constructive analysis." Theory of Computing Systems 43, no. 3-4 (July 6, 2007): 583–602. http://dx.doi.org/10.1007/s00224-007-9027-4.

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35

Maschio, Samuele, and Thomas Streicher. "Models of intuitionistic set theory in subtoposes of nested realizability toposes." Annals of Pure and Applied Logic 166, no. 6 (June 2015): 729–39. http://dx.doi.org/10.1016/j.apal.2015.03.002.

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36

De Terán, Fernando, Froilán M. Dopico, D. Steven Mackey, and Vasilije Perović. "Quadratic realizability of palindromic matrix polynomials." Linear Algebra and its Applications 567 (April 2019): 202–62. http://dx.doi.org/10.1016/j.laa.2019.01.003.

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37

Belk, Maria. "Realizability of Graphs in Three Dimensions." Discrete & Computational Geometry 37, no. 2 (February 2007): 139–62. http://dx.doi.org/10.1007/s00454-006-1285-4.

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38

Melikhov, Sergey A. "On isotopic realizability of maps factored through a hyperplane." Sbornik: Mathematics 195, no. 8 (August 31, 2004): 1117–63. http://dx.doi.org/10.1070/sm2004v195n08abeh000839.

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39

Grundman, H. G., and T. L. Smith. "Galois realizability of a central C4-extension of D8." Journal of Algebra 322, no. 10 (November 2009): 3492–98. http://dx.doi.org/10.1016/j.jalgebra.2009.08.015.

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40

Awodey, Steve. "A Brief Introduction to Algebraic Set Theory." Bulletin of Symbolic Logic 14, no. 3 (September 2008): 281–98. http://dx.doi.org/10.2178/bsl/1231081369.

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AbstractThis brief article is intended to introduce the reader to the field of algebraic set theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, applying to various classical, intuitionistic, and constructive set theories. Under this scheme some familiar set theoretic properties are related to algebraic ones, while others result from logical constraints. Conventional elementary set theories are complete with respect to algebraic models, which arise in a variety of ways, such as topologically, type-theoretically, and through variation. Many previous results from topos theory involving realizability, permutation, and sheaf models of set theory are subsumed, and the prospects for further such unification seem bright.
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41

NGUYEN, VIET-HANG. "1-EXTENSIONS AND GLOBAL RIGIDITY OF GENERIC DIRECTION-LENGTH FRAMEWORKS." International Journal of Computational Geometry & Applications 22, no. 06 (December 2012): 577–91. http://dx.doi.org/10.1142/s0218195912500173.

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The problem of deciding the unique realizability of a graph in a Euclidean space with distance and/or direction constraints on the edges of the graph has applications in CAD (Computer-Aided Design) and localization in sensor networks. One approach, which has been proved to be efficient in the similar problem for graphs with distance constraints, is to study operations to construct a uniquely realizable graph from a smaller one. In this paper, we extend a result of Jackson and Jordán on the unique realizability preservingness of the so-called “1-extension” in dimension 2 to all dimensions.
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42

Alfakih, Abdo Y. "On rigidity and realizability of weighted graphs." Linear Algebra and its Applications 325, no. 1-3 (March 2001): 57–70. http://dx.doi.org/10.1016/s0024-3795(00)00281-0.

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43

Melikhov, Sergey A. "Isotopic and continuous realizability of maps in the metastable range." Sbornik: Mathematics 195, no. 7 (August 31, 2004): 983–1016. http://dx.doi.org/10.1070/sm2004v195n07abeh000835.

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44

Jensen, C. U., and A. Prestel. "Unique realizability of finite abelian 2-groups as Galois groups." Journal of Number Theory 40, no. 1 (January 1992): 12–31. http://dx.doi.org/10.1016/0022-314x(92)90025-k.

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45

Batagelj, Vladimir, Tomaž Pisanski, and J. M. S. Sim[otilde]es-Pereira. "An algorithm for tree-realizability of distance matrices∗." International Journal of Computer Mathematics 34, no. 3-4 (January 1990): 171–76. http://dx.doi.org/10.1080/00207169008803874.

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46

Speziale, Charles G., Ridha Abid, and Paul A. Durbin. "On the realizability of reynolds stress turbulence closures." Journal of Scientific Computing 9, no. 4 (December 1994): 369–403. http://dx.doi.org/10.1007/bf01575099.

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47

Konovalov, A. Yu. "The semantics of realizability for the constructive set theory based on hyperarithmetical predicates." Moscow University Mathematics Bulletin 72, no. 3 (May 2017): 129–32. http://dx.doi.org/10.3103/s0027132217030068.

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48

DELANSKY, J. F., and N. K. BOSE. "Real and complex polynomial stability and stability domain construction via network realizability theory." International Journal of Control 48, no. 3 (September 1988): 1343–49. http://dx.doi.org/10.1080/00207178808906250.

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49

Underwood, Robert. "The structure and realizability of R-hopf algebra orders in KCp3." Communications in Algebra 26, no. 11 (January 1998): 3447–62. http://dx.doi.org/10.1080/00927879808826352.

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50

Kiselev, D. D. "Optimal bounds for the Schur index and the realizability of representations." Sbornik: Mathematics 205, no. 4 (April 2014): 522–31. http://dx.doi.org/10.1070/sm2014v205n04abeh004386.

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