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18Review: C. J. Ash, J. Knight, Computable Structures and the Hyperarithmetical Hierarchy (review)Bulletin of Symbolic Logic 7 (3): 383-385. 2001.
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18Uncountable degree spectraAnnals of Pure and Applied Logic 54 (3): 255-263. 1991.We consider a recursive model and an additional recursive relation R on its domain, such that there are uncountably many different images of R under isomorphisms from to some recursive model isomorphic to . We study properties of the set of Turing degrees of all these isomorphic images of R on the domain of
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17Regular relations and the quantifier “there exist uncountably many”Mathematical Logic Quarterly 29 (3): 151-161. 1983.
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16Some effects of Ash–Nerode and other decidability conditions on degree spectraAnnals of Pure and Applied Logic 55 (1): 51-65. 1991.With every new recursive relation R on a recursive model , we consider the images of R under all isomorphisms from to other recursive models. We call the set of Turing degrees of these images the degree spectrum of R on , and say that R is intrinsically r.e. if all the images are r.e. C. Ash and A. Nerode introduce an extra decidability condition on , expressed in terms of R. Assuming this decidability condition, they prove that R is intrinsically r.e. if and only if a natural recursive-syntacti…Read more
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16Dependence relations in computably rigid computable vector spacesAnnals of Pure and Applied Logic 132 (1): 97-108. 2005.We construct a computable vector space with the trivial computable automorphism group, but with the dependence relations as complicated as possible, measured by their Turing degrees. As a corollary, we answer a question asked by A.S. Morozov in [Rigid constructive modules, Algebra and Logic, 28 570–583 ; 379–387 ]
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15Ash C. J. and Knight J.. Computable structures and the hyperarithmetical hierarchy. Studies in logic and the foundations of mathematics, vol. 144. Elsevier, Amsterdam etc. 2000, xv + 346 pp (review)Bulletin of Symbolic Logic 7 (3): 383-385. 2001.
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13On the isomorphism problem for some classes of computable algebraic structuresArchive for Mathematical Logic 61 (5): 813-825. 2022.We establish that the isomorphism problem for the classes of computable nilpotent rings, distributive lattices, nilpotent groups, and nilpotent semigroups is \-complete, which is as complicated as possible. The method we use is based on uniform effective interpretations of computable binary relations into computable structures from the corresponding algebraic classes.
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12Cohesive powers of structuresArchive for Mathematical Logic 1-24. forthcoming.A cohesive power of a structure is an effective analog of the classical ultrapower of a structure. We start with a computable structure, and consider its effective power over a cohesive set of natural numbers. A cohesive set is an infinite set of natural numbers that is indecomposable with respect to computably enumerable sets. It plays the role of an ultrafilter, and the elements of a cohesive power are the equivalence classes of certain partial computable functions determined by the cohesive s…Read more
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12On Cohesive Powers of Linear OrdersJournal of Symbolic Logic 88 (3): 947-1004. 2023.Cohesive powersof computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let$\omega $,$\zeta $, and$\eta $denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of$\omega $. If$\mathcal {L}$is a computable copy of$\omega $that is computably isomorphic to the usual pre…Read more
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11Interpreting a Field in its Heisenberg GroupJournal of Symbolic Logic 87 (3): 1215-1230. 2022.We improve on and generalize a 1960 result of Maltsev. For a field F, we denote by $H(F)$ the Heisenberg group with entries in F. Maltsev showed that there is a copy of F defined in $H(F)$, using existential formulas with an arbitrary non-commuting pair of elements as parameters. We show that F is interpreted in $H(F)$ using computable $\Sigma _1$ formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, Melnikov, R. Miller, and Mo…Read more
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7Computable Structures and the Hyperarithmetical HierarchyBulletin of Symbolic Logic 7 (3): 383-385. 2001.
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4Review: Sergei S. Goncharov, Countable Boolean Algebras and Decidability (review)Journal of Symbolic Logic 63 (3): 1188-1190. 1998.
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2Computability TheoryIn Bharath Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice, Springer. pp. 1933-1961. 2024.Computability theory is the mathematical theory of algorithms, which explores the power and limitations of computation. Classical computability theory formalized the intuitive notion of an algorithm and provided a theoretical basis for digital computers. It also demonstrated the limitations of algorithms and showed that most sets of natural numbers and the problems they encode are not decidable (Turing computable). Important results of modern computability theory include the classification of th…Read more
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2Logic and Algebraic Structures in Quantum Computing (edited book)Cambridge University Press. 2014.Experts in the field explore the connections across physics, quantum logic, and quantum computing.
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Logic in the History and Philosophy of Mathematical PracticeIn Bharath Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice, Springer. pp. 1905-1920. 2024.Mathematical logic is the study of reasoning about mathematical objects and the degree to which mathematical and scientific reasoning can be formalized and mechanized. Logic provides the foundations of mathematics and of theoretical computer science. Classical logic defined truth, developed the theory of infinite numbers, resolved paradoxes of naive set theory, defined what an algorithm is, and established that certain mathematical principles are independent from the rest of mathematics. Modern …Read more
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Countable Nonstandard Models: Following Skolem’s ApproachIn Bharath Sriraman (ed.), Handbook of the History and Philosophy of Mathematical Practice, Springer. pp. 1989-2009. 2024.In 1934, Skolem gave a remarkable construction of a countable nonstandard model of arithmetic. His construction contains ideas of the ultrapower construction which was introduced in model theory 20 years later. However, typical ultrapower constructions produce uncountable models. Skolem’s construction can also be connected with ideas from computability theory, formalized by Turing and others in 1936. The proof of one of Skolem’s key statements can be interpreted using computability-theoretic not…Read more
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George Washington UniversityRegular Faculty
Areas of Interest
Logic and Philosophy of Logic |
General Philosophy of Science |