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Showing papers in "Journal of Symbolic Logic in 2010"


Book ChapterDOI
TL;DR: The work of Dantzig, Fulkerson, Hoffman, Edmonds, Lawler and other pioneers on network flows, matching and matroids acquainted me with the elegant and efficient algorithms that were sometimes possible.
Abstract: Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held. These experiences made me aware that seemingly simple discrete optimization problems could hold the seeds of combinatorial explosions. The work of Dantzig, Fulkerson, Hoffman, Edmonds, Lawler and other pioneers on network flows, matching and matroids acquainted me with the elegant and efficient algorithms that were sometimes possible. Jack Edmonds’ papers and a few key discussions with him drew my attention to the crucial distinction between polynomial-time and superpolynomial-time solvability. I was also influenced by Jack’s emphasis on min-max theorems as a tool for fast verification of optimal solutions, which foreshadowed Steve Cook’s definition of the complexity class NP. Another influence was George Dantzig’s suggestion that integer programming could serve as a universal format for combinatorial optimization problems.

8,644 citations


Journal ArticleDOI
TL;DR: It is shown that class with bounded expansion and (newly defined) classes with bounded local expansion and even (very general) nowhere dense classes are quasi wide and that any homomorphism closed first order definable property restricted to a bounded expansion class is a restricted duality.
Abstract: A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of (distinct) vertices of A intersect. In other words, A is d-scattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Ajtai and Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special classes of structures (such as minor closed classes). This in turn led to the notions of wide, almost wide and quasi-wide classes of graphs. It has been proved previously that minor closed classes and classes of graphs with locally forbidden minors are examples of such classes and thus (relativized) homomorphism preservation theorem holds for them. In this paper we show that (more general) classes with bounded expansion and (newly defined) classes with bounded local expansion and even (very general) nowhere dense classes are quasi wide. This not only strictly generalizes the previous results but it also provides new proofs and algorithms for some of the old results. It appears that bounded expansion and nowhere dense classes are perhaps a proper setting for investigation of wide-type classes as in several instances we obtain a structural characterization. This also puts classes of bounded expansion in the new context. Our motivation stems from finite dualities. As a corollary we obtain that any homomorphism closed first order definable property restricted to a bounded expansion class is a restricted duality.

95 citations


Journal ArticleDOI
TL;DR: It turns out that there exists a continuum of reducts of relational structures up to primitive positive interdefinability, and those locally closed clones over a countable domain which contain all permutations of the domain are classified.
Abstract: We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of equality. It turns out that there exists a continuum of such reducts. Equivalently, expressed in the language of universal algebra, we classify those locally closed clones over a countable domain which contain all permutations of the domain.

61 citations


Journal ArticleDOI
TL;DR: Two new results are led to, leading to two new results: the Lω1ω(Q)-definability assumption may be dropped, and each class is determined by its model of dimension ℵ0.
Abstract: A careful exposition of Zilber's quasiminimal excellent classes and their categoricity is given, leading to two new results: the L?1,?(Q)-definability assumption may be dropped, and each class is determined by its model of dimension ?0.

56 citations


Journal ArticleDOI
TL;DR: It is proved that in an inp-minimal densely ordered group, every definable unary function is a union of finitely many continuous locally monotonic functions.
Abstract: Dp-minimality is a common generalization of weak minimality and weak o-minimality. If T is a weakly o-minimal theory then it is dp-minimal (Fact 2.2), but there are dp-minimal densely ordered groups that are not weakly o-minimal. We introduce the even more general notion of inpminimality and prove that in an inp-minimal densely ordered group, every definable unary function is a union of finitely many continuous locally monotonic functions (Theorem 3.2).

41 citations


Journal ArticleDOI
TL;DR: This article shows that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent, and shows that if Σ⊧φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth of φ.
Abstract: Continuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures (such as fields, group, and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces, and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result? The primary purpose of this article is to show that a certain, interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies an approximated form of strong completeness, whereby Σ⊧ φ (if and) only if Σ⊢ φ ∸2 −n for all n . This approximated form of strong completeness asserts that if Σ⊧ φ , then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth of φ . Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory (in a countable, recursive language) recursive? Say that a complete theory T is decidable if for every sentence φ , the value φ T is a recursive real, and moreover, uniformly computable from φ . If T is incomplete, we say it is decidable if for every sentence φ the real number φ T o is uniformly recursive from φ , where φ T o is the maximal value of φ consistent with T . As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable.

40 citations


Journal ArticleDOI
TL;DR: These characterizations allow us to see not only that some natural classes of sets, including maximal sets, are composed entirely of Schnorr trivials, but also that the Schnorr trivial sets form an ideal in the truth-table degrees but not the weak truth- table degrees.
Abstract: We give several characterizations of Schnorr trivial sets, including a new lowness notion for Schnorr triviality based on truth-table reducibility. These characterizations allow us to see not only that some natural classes of sets, including maximal sets, are composed entirely of Schnorr trivials, but also that the Schnorr trivial sets form an ideal in the truth-table degrees but not the weak truth-table degrees. This answers a question of Downey. Griffiths and LaForte.

38 citations


Journal ArticleDOI
Assaf Rinot1
TL;DR: It is relatively consistent with the existence of a supercompact cardinal that fails, while I[S; λ] is fat for every stationary S ⊆ λ+ that reflects stationarily often.
Abstract: Let λ denote a singular cardinal Zeman, improving a previous result of Shelah, proved that □*λ together with 2λ=λ⁺ implies ♢S for every S⊆λ⁺ that reflects stationarily often In this paper, for a set S⊆λ⁺, a normal subideal of the weak approachability ideal is introduced, and denoted by I[S;λ] We say that the ideal is fat if it contains a stationary set It is proved: 1 if I[S;λ] is fat, then NSλ⁺↾ S is non-saturated; 2 if I[S;λ] is fat and 2λ=λ⁺, then ♢S holds; 3 □*λ implies that I[S;λ] is fat for every S⊆λ⁺ that reflects stationarily often; 4 it is relatively consistent with the existence of a supercompact cardinal that □*λ fails, while I[S;λ] is fat for every stationary S⊆λ⁺ that reflects stationarily often The stronger principle ♢*λ⁺ is studied as well

33 citations


Journal ArticleDOI
TL;DR: In this article, a new deductive system CL12 is presented, which is based on the semantics of computability logic (CL) and proves its soundness and completeness with respect to the semantics.
Abstract: Computability logic (CL) is a recently launched program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth that logic has more traditionally been. Formulas in it represent computational problems, "truth" means existence of an algorithmic solution, and proofs encode such solutions. Within the line of research devoted to finding axiomatizations for ever more expressive fragments of CL, the present paper introduces a new deductive system CL12 and proves its soundness and completeness with respect to the semantics of CL. Conservatively extending classical predicate calculus and offering considerable additional expressive and deductive power, CL12 presents a reasonable, computationally meaningful, constructive alternative to classical logic as a basis for applied theories. To obtain a model example of such theories, this paper rebuilds the traditional, classical-logic based Peano arithmetic into a computability-logic-based counterpart. Among the purposes of the present contribution is to provide a starting point for what, as the author wishes to hope, might become a new line of research with a potential of interesting findings?an exploration of the presumably quite unusual metatheory of CL-based arithmetic and other CL-based applied systems. ?

26 citations


Journal ArticleDOI
TL;DR: In inner model theory there are many theorems of the general form “if there is no inner model of large cardinal hypothesis X then there is an L-like inner model Kx which Y covers V”, which always include GCH and Global Square.
Abstract: §1. Introduction. It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals. Examples include the behaviour of cardinal exponentiation, the extent of the tree property, the extent of stationary reflection, and the existence of non-free almost-free abelian groups. The explanation for this phenomenon lies in inner model theory, in particular core models and covering lemmas. If W is an inner model of V then1. W strongly covers V if every uncountable set of ordinals is covered by a set of the same V -cardinality lying in W.2. W weakly covers V if W computes the successor of every V-singular cardinal correctly.Strong covering implies weak covering.In inner model theory there are many theorems of the general form “if there is no inner model of large cardinal hypothesis X then there is an L-like inner model Kx which Y covers V”. Here the L-like properties of Kx always include GCH and Global Square. Examples include1. X is “0# exists”, Kx is L, Y is “strongly”.2. X is “there is a measurable cardinal”, Kx is the Dodd-Jensen core model, Y is “strongly”.3. X is “there is a Woodin cardinal”, Kx is the core model for a Woodin cardinal, Y is “weakly”.

26 citations


Journal ArticleDOI
TL;DR: The point of view presented here clarifies the relation between Hjorth's theorem and first-order logic.
Abstract: We reformulate, in the context of continuous logic, an oscillation theorem proved by G. Hjorth and give a proof of the theorem in that setting which is similar to, but simpler than, Hjorth's original one. The point of view presented here clarifies the relation between Hjorth's theorem and first-order logic.

Journal ArticleDOI
TL;DR: The satisfaction problem for all prenex formulae in the set-theoretic Bernays-Shönfinkel-Ramsey class is semi-decidable over von Neumann's cumulative hierarchy is strengthened into a decidability result for the same collection offormulae.
Abstract: As is well-known, the Bernays-Schonfinkel-Ramsey class of all prenex ∃*∀*-sentences which are valid in classical first-order logic is decidable. This paper paves the way to an analogous result which the authors deem to hold when the only available predicate symbols are ∈ and =, no constants or function symbols are present, and one moves inside a (rather generic) Set Theory whose axioms yield the well-foundedness of membership and the existence of infinite sets. Here semi-decidability of the satisfiability problem for the BSR class is proved by following a purely semantic approach, the remaining part of the decidability result being postponed to a forthcoming paper.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the ideal of finitely chromatic graphs is not locally Katětov-minimal among ideals not satisfying Fatou's lemma.
Abstract: We investigate the pair-splitting number which is a variation of splitting number, pair-reaping number which is a variation of reaping number and cardinal invariants of ideals on ω. We also study cardinal invariants of Fσ ideals and their upper bounds and lower bounds. As an application, we answer a question of S. Solecki by showing that the ideal of finitely chromatic graphs is not locally Katětov-minimal among ideals not satisfying Fatou's lemma.

Journal ArticleDOI
TL;DR: It is shown that there is a class of partial orderings for which Ramsey's Theorem for pairs is equivalent to ACA0, and Ramsey'sTheorem for singletons for the complete binary tree is stronger than .
Abstract: We show, relative to the base theory RCA₀: A nontrivial tree satisfies Ramsey's Theorem only if it is biembeddable with the complete binary tree. There is a class of partial orderings for which Ramsey's Theorem for pairs is equivalent to ACA₀. Ramsey's Theorem for singletons for the complete binary tree is stronger than BΣ⁰₂, hence stronger than Ramsey's Theorem for singletons for ω. These results lead to extensions of results, or answers to questions, of Chubb, Hirst, and McNicholl [3].

Journal ArticleDOI
TL;DR: It is proved that the degree structures of the d.c.e. and the 3-c. e. Turing degrees are not elementarily equivalent, thus refuting a conjecture of Downey, and any degree u ≤ f is either comparable with both e and d, or incomparable with both.
Abstract: We prove that the degree structures of the d.c.e. and the 3-c.e. Turing degrees are not elementarily equivalent, thus refuting a conjecture of Downey. More specifically, we show that the following statement fails in the former but holds in the latter structure: There are degrees f > e > d > 0 such that any degree u ≤ f is either comparable with both e and d, or incomparable with both.

Journal ArticleDOI
TL;DR: These problems of existence of a Hamiltonian path and existence of an infinite path in a tree are equally complex for automatic and for recursive graphs (-complete), but are much simpler for automatic than for recursion graphs (decidable and -complete, resp.).
Abstract: For automatic and recursive graphs, we investigate the following problems: (A) existence of a Hamiltonian path and existence of an infinite path in a tree (B) existence of an Euler path, bounding the number of ends, and bounding the number of infinite branches in a tree (C) existence of an infinite clique and an infinite version of set cover The complexity of these problems is determined for automatic graphs and, supplementing results from the literature, for recursive graphs. Our results show that these problems (A) are equally complex for automatic and for recursive graphs (Σ11-complete), (B) are moderately less complex for automatic than for recursive graphs (complete for different levels of the arithmetic hierarchy), (C) are much simpler for automatic than for recursive graphs (decidable and Σ11-complete, resp.). §

Journal ArticleDOI
TL;DR: A number of results in effective randomness are proved, including the fact that every PA Turing degree is the join of two random Turing degrees, and the existence of a minimal pair of LR degrees below the LR degree of the halting problem.
Abstract: We prove a number of results in effective randomness, using methods in which π⁰₁ classes play an essential role. The results proved include the fact that every PA Turing degree is the join of two random Turing degrees, and the existence of a minimal pair of LR degrees below the LR degree of the halting problem.

Journal ArticleDOI
TL;DR: The notion of retractability of a definable groupoid is introduced (which is slightly stronger than Hrushovski's notion of eliminability), some criteria for when groupoids are retractable are given, and how retractability relates to both 3-uniqueness and the splitness of finite internal covers are shown.
Abstract: Building on Hrushovski's work in [5], we study definable groupoids in stable theories and their relationship with 3-uniqueness and finite internal covers. We introduce the notion of retractability of a definable groupoid (which is slightly stronger than Hrushovski's notion of eliminability), give some criteria for when groupoids are retractable, and show how retractability relates to both 3-uniqueness and the splitness of finite internal covers. One application we give is a new direct method of constructing non-eliminable groupoids from witnesses to the failure of 3-uniqueness. Another application is a proof that any finite internal cover of a stable theory with a centerless liaison groupoid is almost split.

Journal ArticleDOI
TL;DR: The initial algebra of the meadows of characteristic 0 is determined and a normal form theorem is proved for it and the decidability of the closed term problem for meadows and the computability of their initial object is obtained.
Abstract: A meadow is a commutative ring with an inverse operator satisfying 0 −1 = 0. We determine the initial algebra of the meadows of characteristic 0 and show that its word problem is decidable.

Journal ArticleDOI
TL;DR: In this article, the degree-structures induced by good Borel reducibilities were studied and a dichotomy theorem was proved for the degree structures induced by these degree structures.
Abstract: In [9] we have considered a wide class of “well-behaved” reducibilities for sets of reals. In this paper we continue with the study of Borel reducibilities by proving a dichotomy theorem for the degree-structures induced by good Borel reducibilities. This extends and improves the results of [9] allowing to deal with a larger class of notions of reduction (including, among others, the Baire class ξ functions).

Journal ArticleDOI
TL;DR: A continuous uniform boundedness principle CUB is introduced as a formalization of Tao's notion of a correspondence principle and the strength of this principle and various restrictions thereof are studied in terms of reverse mathematics, i.e., in Terms of the “big five” subsystems of second order arithmetic.
Abstract: In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasi-finitization of the infinite pigeonhole principle IPP, arriving at the “finitary” infinite pigeonhole principle FIPP1. That turned out to not be the proper formulation and so we proposed an alternative version FIPP2. Tao himself formulated yet another version FIPP3 in a revised version of his essay. We give a counterexample to FIPP1 and discuss for both of the versions FIPP2 and FIPP3 the faithfulness of their respective finitization of IPP by studying the equivalences IPP$ FIPP2 and IPP$ FIPP3 in the context of reverse mathematics ([9]). In the process of doing this we also introduce a continuous uniform boundedness principle CUB as a formalization of Tao’s notion of a correspondence principle and study the strength of this principle and various restrictions thereof in terms of reverse mathematics, i.e. in terms of the “big five” subsystems of second order arithmetic.

Journal ArticleDOI
TL;DR: It is proved that if V = L then there is a maximal orthogonal (i.e., mutually singular) set of measures on Cantor space and this provides a natural counterpoint to the well-known theorem of Preiss and Rataj that no analyticSet of measures can be maximal Orthogonal.
Abstract: We prove that if V = L then there is a maximal orthogonal (i.e., mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known theorem of Preiss and Rataj [16] that no analytic set of measures can be maximal orthogonal.

Journal ArticleDOI
TL;DR: In this article, it was shown that (ℝ, +, ·, U) defines sets in every level of the projective hierarchy but does not define all projective sets.
Abstract: An open U ⊆ ℝ is produced such that (ℝ, +, ·, U) defines a Borel isomorph of (ℝ, +, ·, ℕ) but does not define ℕ. It follows that (ℝ, +, ·, U) defines sets in every level of the projective hierarchy but does not define all projective sets. This result is elaborated in various ways that involve geometric measure theory and working over o-minimal expansions of (ℝ, +, ·). In particular, there is a Cantor set E ⊆ ℝ such that (ℝ, +, ·, ℕ) defines a Borel isomorph of (ℝ, +, ·, ℕ) and, for every exponentially bounded o-minimal expansion of (ℝ, +, ·), every subset of ℝ definable in (, E) either has interior or is Hausdorff null.

Journal ArticleDOI
TL;DR: It is shown that the equational theory of L(M) coincides with that of some resp.
Abstract: For a finite von Neumann algebra factor M, the projections form a modular ortholattice L(M). We show that the equational theory of L(M) coincides with that of some resp. all L(ℂn × n) and is decidable. In contrast, the uniform word problem for the variety generated by all L(ℂn × n) is shown to be undecidable.

Journal ArticleDOI
TL;DR: It is shown that some tree properties are detected by the presence of certain combinatorial configurations in the characteristic sequence while other properties manifest themselves in the persistence of complicated configurations under localization.
Abstract: For a first-order formula φ(x;y) we introduce and study the characteristic sequence 〈 Pn: n < ω 〉 of hypergraphs defined by Pn(y1,…,yn) := (∃ x) ⋁i ≤ n φ(x;yi). We show that combinatorial and classification theoretic properties of the characteristic sequence reflect classification theoretic properties of φ and vice versa. The main results are a characterization of NIP and of simplicity in terms of persistence of configurations in the characteristic sequence. Specifically, we show that some tree properties are detected by the presence of certain combinatorial configurations in the characteristic sequence while other properties such as instability and the independence property manifest themselves in the persistence of complicated configurations under localization.

Journal ArticleDOI
TL;DR: Fields in which model theoretic algebraic closure coincides with relative field theoreticgebraic closure are examined, exactly the fields in which algebraic independence is an abstract independence relation in the sense of Kim and Pillay.
Abstract: We examine fields in which model theoretic algebraic closure coincides with relative field theoretic algebraic closure. These are perfect fields with nice model theoretic behaviour. For example they are exactly the fields in which algebraic independence is an abstract independence relation in the sense of Kim and Pillay. Classes of examples are perfect PAC fields, model complete large fields and henselian valued fields of characteristic 0.

Journal ArticleDOI
Keita Yokoyama1
TL;DR: This paper introduces the systems ns-ACA0 and ns-WKL0 of non-standard second-order arithmetic in which they can formalize non- standard arguments in ACA0 and WkL0, respectively, and gives direct transformations from non-Standard proofs in ns- ACA0 or ns- WKL 0 into proofs inACA0 or WKl0.
Abstract: In this paper, we introduce the systems nsAC Ao and ns-WK Lo of non-standard second-order arithmetic in which we can formalize non-standard arguments in ACAo and WKLo, respectively. Then, we give direct transformations from non-standard proofs in ns-ACAo or ns-WK Lo into proofs in ACAo or WKL0. ?

Journal ArticleDOI
TL;DR: In this article, the authors develop several aspects of local and global stability in continuous first order logic, and study type-definable groups and genericity of first-order logic.
Abstract: We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity.

Journal ArticleDOI
TL;DR: As applications, this work provides a characterization of the sets S with computable increasing η-representations using support increasing limitwise monotonic sets on ℚ and note relationships between the class of order-computable sets and theclass of support increasing (support strictly increasing) limitwise Monotonic set on certain domains.
Abstract: We extend the notion of limitwise monotonic functions to include arbitrary computable domains. We then study which sets and degrees are support increasing (support strictly increasing) limitwise monotonic on various computable domains. As applications, we provide a characterization of the sets S with computable increasing η-representations using support increasing limitwise monotonic sets on ℚ and note relationships between the class of order-computable sets and the class of support increasing (support strictly increasing) limitwise monotonic sets on certain domains.

Journal ArticleDOI
TL;DR: It is proved that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an analogous result from [2].
Abstract: The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the weakly o-minimal context. We prove that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an analogous result from [2] for sets and functions definable in models of weakly o-minimal theories. We pay special attention to large subsets of Cartesian products of definable sets, showing that if X,Y and S are non-empty definable sets and S is a large subset of X× Y, then for a large set of tuples 〈\overline{a}₁,…,\overline{a}2k〉 ∈ X{2k}, where k=dim(Y), the union of fibers S\overline{a}₁∪…∪ S\overline{a}2k is large in Y. Finally, given a weakly o-minimal structure ℳ, we find various conditions equivalent to the fact that the topological dimension in ℳ enjoys the addition property.