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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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3Computability 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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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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18Belegradek, O., Verbovskiy, V. and Wagner, FO, CosetAnnals of Pure and Applied Logic 121 (1): 287. 2003.
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13On 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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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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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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192005–06 Winter Meeting of the Association for Symbolic LogicBulletin of Symbolic Logic 12 (4): 613-624. 2006.
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7Computable Structures and the Hyperarithmetical HierarchyBulletin of Symbolic Logic 7 (3): 383-385. 2001.
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26Sergei S. Goncharov. Countable Boolean algebras and decidability. English translation of Schetnye bulevy algebry i razreshimost′. Siberian school of algebra and logic. Consultants Bureau, New York, London, and Moscow, 1997, xii + 318 pp (review)Journal of Symbolic Logic 63 (3): 1188-1190. 1998.
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20Computability-theoretic categoricity and Scott familiesAnnals of Pure and Applied Logic 170 (6): 699-717. 2019.
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35Σ 1 0 and Π 1 0 equivalence structuresAnnals of Pure and Applied Logic 162 (7): 490-503. 2011.We study computability theoretic properties of and equivalence structures and how they differ from computable equivalence structures or equivalence structures that belong to the Ershov difference hierarchy. Our investigation includes the complexity of isomorphisms between equivalence structures and between equivalence structures
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68Enumerations in computable structure theoryAnnals of Pure and Applied Logic 136 (3): 219-246. 2005.We exploit properties of certain directed graphs, obtained from the families of sets with special effective enumeration properties, to generalize several results in computable model theory to higher levels of the hyperarithmetical hierarchy. Families of sets with such enumeration features were previously built by Selivanov, Goncharov, and Wehner. For a computable successor ordinal α, we transform a countable directed graph into a structure such that has a isomorphic copy if and only if has a com…Read more
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44Degree spectra of the successor relation of computable linear orderingsArchive for Mathematical Logic 48 (1): 7-13. 2009.We establish that for every computably enumerable (c.e.) Turing degree b the upper cone of c.e. Turing degrees determined by b is the degree spectrum of the successor relation of some computable linear ordering. This follows from our main result, that for a large class of linear orderings the degree spectrum of the successor relation is closed upward in the c.e. Turing degrees
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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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4Review: Sergei S. Goncharov, Countable Boolean Algebras and Decidability (review)Journal of Symbolic Logic 63 (3): 1188-1190. 1998.
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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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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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33Spectra of Structures and RelationsJournal of Symbolic Logic 72 (1). 2007.We consider embeddings of structures which preserve spectra: if g: M → S with S computable, then M should have the same Turing degree spectrum (as a structure) that g(M) has (as a relation on S). We show that the computable dense linear order L is universal for all countable linear orders under this notion of embedding, and we establish a similar result for the computable random graph G. Such structures are said to be spectrally universal. We use our results to answer a question of Goncharov, an…Read more
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47Turing degrees of certain isomorphic images of computable relationsAnnals of Pure and Applied Logic 93 (1-3): 103-113. 1998.A model is computable if its domain is a computable set and its relations and functions are uniformly computable. Let be a computable model and let R be an extra relation on the domain of . That is, R is not named in the language of . We define to be the set of Turing degrees of the images f under all isomorphisms f from to computable models. We investigate conditions on and R which are sufficient and necessary for to contain every Turing degree. These conditions imply that if every Turing degre…Read more
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223Frequency computations and the cardinality theoremJournal of Symbolic Logic 57 (2): 682-687. 1992.
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22Partial automorphism semigroupsAnnals of Pure and Applied Logic 156 (2): 245-258. 2008.We study the relationship between algebraic structures and their inverse semigroups of partial automorphisms. We consider a variety of classes of natural structures including equivalence structures, orderings, Boolean algebras, and relatively complemented distributive lattices. For certain subsemigroups of these inverse semigroups, isomorphism of the subsemigroups yields isomorphism of the underlying structures. We also prove that for some classes of computable structures, we can reconstruct a c…Read more
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49Bounding Homogeneous ModelsJournal of Symbolic Logic 72 (1). 2007.A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a d-decidable homogeneous model A, i.e., the elementary diagram De (A) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a single CD theory T such that every homogene…Read more
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58$\Pi _{1}^{0}$ Classes and Strong Degree Spectra of RelationsJournal of Symbolic Logic 72 (3). 2007.We study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable $\Pi _{1}^{0}$ subsets of 2ω and Kolmogorov complexity play a major role in the proof
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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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George Washington UniversityRegular Faculty
Areas of Interest
Logic and Philosophy of Logic |
General Philosophy of Science |