Authors:Guohua Wu; Maxim Zubkov Pages: 467 - 486 Abstract: Publication date: June 2018 Source:Annals of Pure and Applied Logic, Volume 169, Issue 6 Author(s): Guohua Wu, Maxim Zubkov In this paper, we prove Kierstead's conjecture for linear orders whose order types are ∑ q ∈ Q F ( q ) , where F is an extended 0′-limitwise monotonic function, i.e. F can take value ζ. Linear orders in our consideration can have finite and infinite blocks simultaneously, and in this sense our result subsumes a recent result of C. Harris, K. Lee and S.B. Cooper, where only those linear orders with finite blocks are considered. Our result also covers one case of R. Downey and M. Moses' work, i.e. ζ ⋅ η . It covers some instances not being considered in both previous works mentioned above, such as m ⋅ η + ζ ⋅ η + n ⋅ η , for example, where m , n > 0 .

Authors:Saeideh Bahrami; Ali Enayat Pages: 487 - 513 Abstract: Publication date: June 2018 Source:Annals of Pure and Applied Logic, Volume 169, Issue 6 Author(s): Saeideh Bahrami, Ali Enayat We investigate the structure of fixed point sets of self-embeddings of models of arithmetic. Our principal results are Theorems A, B, and C below. In what follows M is a countable nonstandard model of the fragment I Σ 1 of PA (Peano Arithmetic); N is the initial segment of M consisting of standard numbers of M ; I fix ( j ) is the longest initial segment of fixed points of j; Fix ( j ) is the fixed point set of j; K 1 ( M ) consists of Σ 1 -definable elements of M ; and a self-embedding j of M is said to be a proper initial self-embedding if j ( M ) is a proper initial segment of M . Theorem A The following are equivalent for a proper initial segment I of M : (1) I = I fix ( j ) for some self-embedding j of M . (2) I is closed under exponentiation. (3) I = I fix ( j ) for some proper initial self-embedding j of M . Theorem B The following are equivalent for a proper initial segment I of M : (1) I = Fix ( j ) for some self-embedding j of M . (2) I is a strong cut of M and I ≺ Σ 1 M . (3) I = Fix ( j ) for some proper initial self-embedding j of M . Theorem C PubDate: 2018-04-15T07:23:22Z DOI: 10.1016/j.apal.2018.01.004

Authors:Alexei G. Myasnikov; Francis Oger; Mahmood Sohrabi Pages: 514 - 522 Abstract: Publication date: June 2018 Source:Annals of Pure and Applied Logic, Volume 169, Issue 6 Author(s): Alexei G. Myasnikov, Francis Oger, Mahmood Sohrabi We give algebraic characterizations of elementary equivalence between rings with finitely generated additive groups. They are similar to those previously obtained for finitely generated nilpotent groups. Here, the rings are not supposed associative, commutative or unitary.

Authors:Olga Kharlampovich; Alexei Myasnikov Pages: 523 - 547 Abstract: Publication date: June 2018 Source:Annals of Pure and Applied Logic, Volume 169, Issue 6 Author(s): Olga Kharlampovich, Alexei Myasnikov We describe solutions to the problem of elementary classification in the class of group algebras of free groups. We will show that unlike free groups, two group algebras of free groups over infinite fields are elementarily equivalent if and only if the groups are isomorphic and the fields are equivalent in the weak second order logic. We will show that the set of all free bases of a free group F is 0-definable in the group algebra K ( F ) when K is an infinite field, the set of geodesics is definable, and many geometric properties of F are definable in K ( F ) . Therefore K ( F ) “knows” some very important information about F. We will show that similar results hold for group algebras of limit groups.

Authors:Sy-David Friedman; Radek Honzik; Šárka Stejskalová Pages: 548 - 564 Abstract: Publication date: June 2018 Source:Annals of Pure and Applied Logic, Volume 169, Issue 6 Author(s): Sy-David Friedman, Radek Honzik, Šárka Stejskalová Starting from a Laver-indestructible supercompact κ and a weakly compact λ above κ, we show there is a forcing extension where κ is a strong limit singular cardinal with cofinality ω, 2 κ = κ + 3 = λ + , and the tree property holds at κ + + = λ . Next we generalize this result to an arbitrary cardinal μ such that κ < cf ( μ ) and λ + ≤ μ . This result provides more information about possible relationships between the tree property and the continuum function.

Authors:Liming Cai; Jainer Chen; Rod Downey; Mike Fellows Pages: 463 - 465 Abstract: Publication date: May 2018 Source:Annals of Pure and Applied Logic, Volume 169, Issue 5 Author(s): Liming Cai, Jainer Chen, Rod Downey, Mike Fellows

Authors:Tamar Lando Pages: 277 - 311 Abstract: Publication date: Available online 3 January 2018 Source:Annals of Pure and Applied Logic Author(s): Tamar Lando Space, as we typically represent it in mathematics and physics, is composed of dimensionless, indivisible points. On an alternative, region-based approach to space, extended regions together with the relations of ‘parthood’ and ‘contact’ are taken as primitive; points are represented as mathematical abstractions from regions. Region-based theories of space have been traditionally modeled in regular closed (or regular open) algebras, in work that goes back to [5] and [20]. More recently, formal logics for region-based theories of space were developed in, e.g., [3] and [18]. It was shown that these logics have both a nice topological and relational semantics, and that the minimal logic for contact algebras, L m i n c o n t (defined below), is complete for both. The present paper explores the question of completeness of L m i n c o n t and its extensions for individual topological spaces of interest: the real line, Cantor space, the rationals, and the infinite binary tree. A second aim is to study a different, algebraic model of logics for region-based theories of space, based on the Lebesgue measure algebra (or algebra of Borel subsets of the real line modulo sets of Lebesgue measure zero). As a model for point-free space, the algebra was first discussed in [2]. The main results of the paper are that L m i n c o n t is weakly complete for any zero-dimensional, dense-in-itself metric space (including, e.g., Cantor space and the rationals); the extension L m i n c o n t + ( C o n ) is weakly complete for the real line and the Lebesgue measure contact algebra. We also prove that the logic L m i n c o n t + ( U n i v ) is weakly complete for the infinite binary tree.

Authors:Peter Holy; Regula Krapf; Philipp Schlicht Abstract: Publication date: Available online 10 April 2018 Source:Annals of Pure and Applied Logic Author(s): Peter Holy, Regula Krapf, Philipp Schlicht It is well known that pretameness implies the forcing theorem, and that pretameness is characterized by the preservation of the axioms of ZF − , that is ZF without the power set axiom, or equivalently, by the preservation of the axiom scheme of replacement, for class forcing over models of ZF . We show that pretameness in fact has various other characterizations, for instance in terms of the forcing theorem, the preservation of the axiom scheme of separation, the forcing equivalence of partial orders and their dense suborders, and the existence of nice names for sets of ordinals. These results show that pretameness is a strong dividing line between well and badly behaved notions of class forcing, and that it is exactly the right notion to consider in applications of class forcing. Furthermore, for most properties under consideration, we also present a corresponding characterization of the Ord-chain condition.

Authors:Silvio Ghilardi; Alessandro Gianola Abstract: Publication date: Available online 4 April 2018 Source:Annals of Pure and Applied Logic Author(s): Silvio Ghilardi, Alessandro Gianola Wolter in [38] proved that the Craig interpolation property transfers to fusion of normal modal logics. It is well-known [21] that for such logics Craig interpolation corresponds to an algebraic property called superamalgamability. In this paper, we develop model-theoretic techniques at the level of first-order theories in order to obtain general combination results transferring quantifier-free interpolation to unions of theories over non-disjoint signatures. Such results, once applied to equational theories sharing a common Boolean reduct, can be used to prove that superamalgamability is modular also in the non-normal case. We also state that, in this non-normal context, superamalgamability corresponds to a strong form of interpolation that we call “comprehensive interpolation property” (which consequently transfers to fusions).

Authors:Saharon Shelah; Sebastien Vasey Abstract: Publication date: Available online 2 April 2018 Source:Annals of Pure and Applied Logic Author(s): Saharon Shelah, Sebastien Vasey We study abstract elementary classes (AECs) that, in ℵ 0 , have amalgamation, joint embedding, no maximal models and are stable (in terms of the number of orbital types). Assuming a locality property for types, we prove that such classes exhibit superstable-like behavior at ℵ 0 . More precisely, there is a superlimit model of cardinality ℵ 0 and the class generated by this superlimit has a type-full good ℵ 0 -frame (a local notion of nonforking independence) and a superlimit model of cardinality ℵ 1 . We also give a supersimplicity condition under which the locality hypothesis follows from the rest.

Authors:Levon Haykazyan; Rahim Moosa Abstract: Publication date: Available online 30 March 2018 Source:Annals of Pure and Applied Logic Author(s): Levon Haykazyan, Rahim Moosa The correspondence between definable connected groupoids in a theory T and internal generalised imaginary sorts of T, established by Hrushovski in [“Groupoids, imaginaries and internal covers,” Turkish Journal of Mathematics, 2012], is here extended in two ways: First, it is shown that the correspondence is in fact an equivalence of categories, with respect to appropriate notions of morphism. Secondly, the equivalence of categories is shown to vary uniformly in definable families, with respect to an appropriate relativisation of these categories. Some elaborations on Hrushovki's original constructions are also included.

Authors:Beke Abstract: Publication date: Available online 13 March 2018 Source:Annals of Pure and Applied Logic Author(s): T. Beke, J. Rosický We introduce the notion of λ-equivalence and λ-embeddings of objects in suitable categories. This notion specializes to L ∞ λ -equivalence and L ∞ λ -elementary embedding for categories of structures in a language of arity less than λ, and interacts well with functors and λ-directed colimits. We recover and extend results of Feferman and Eklof on “local functors” without fixing a language in advance. This is convenient for formalizing Lefschetz's principle in algebraic geometry, which was one of the main applications of the work of Eklof.

Authors:Alexander C. Block; Benedikt Löwe Abstract: Publication date: Available online 13 March 2018 Source:Annals of Pure and Applied Logic Author(s): Alexander C. Block, Benedikt Löwe Assuming AD + DC , the hierarchy of norms is a wellordered structure of equivalence classes of ordinal-valued maps. We define operations on the hierarchy of norms, in particular an operation that dominates multiplication as an operation on the ranks of norms, and use these operations to establish a considerably improved lower bound for the length of the hierarchy of norms.

Authors:Aaron Stump Abstract: Publication date: Available online 13 March 2018 Source:Annals of Pure and Applied Logic Author(s): Aaron Stump In this paper, it is shown that induction is derivable in a type-assignment formulation of the second-order dependent type theory λ P 2 , extended with the implicit product type of Miquel, dependent intersection type of Kopylov, and a built-in equality type. The crucial idea is to use dependent intersections to internalize a result of Leivant's showing that Church-encoded data may be seen as realizing their own type correctness statements, under the Curry–Howard isomorphism.

Authors:F. Mwesigye; J.K. Truss Abstract: Publication date: Available online 9 March 2018 Source:Annals of Pure and Applied Logic Author(s): F. Mwesigye, J.K. Truss Two structures A and B are n-equivalent if player II has a winning strategy in the n-move Ehrenfeucht–Fraïssé game on A and B. Ordinals and m-coloured ordinals are studied up to n-equivalence for various values of m and n.

Authors:Joseph S. Miller; Mariya I. Soskova Abstract: Publication date: Available online 8 January 2018 Source:Annals of Pure and Applied Logic Author(s): Joseph S. Miller, Mariya I. Soskova We prove that the cototal enumeration degrees are exactly the enumeration degrees of sets with good approximations, as introduced by Lachlan and Shore [17]. Good approximations have been used as a tool to prove density results in the enumerations degrees, and indeed, we prove that the cototal enumerations degrees are dense.

Authors:Terry Abstract: Publication date: Available online 8 January 2018 Source:Annals of Pure and Applied Logic Author(s): C. Terry Given a finite relational language L , a hereditary L -property is a class of finite L -structures which is closed under isomorphism and model theoretic substructure. This notion encompasses many objects of study in extremal combinatorics, including (but not limited to) hereditary properties of graphs, hypergraphs, and oriented graphs. In this paper, we generalize certain definitions, tools, and results form the study of hereditary properties in combinatorics to the setting of hereditary L -properties, where L is any finite relational language with maximum arity at least two. In particular, the goal of this paper is to generalize how extremal results and stability theorems can be combined with well-known techniques and tools to yield approximate enumeration and structure theorems. We accomplish this by generalizing the notions of extremal graphs, asymptotic density, and graph stability theorems using structures in an auxiliary language associated to a hereditary L -property. Given a hereditary L -property H , we prove an approximate asymptotic enumeration theorem for H in terms of its generalized asymptotic density. Further we prove an approximate structure theorem for H , under the assumption of that H has a stability theorem. The tools we use include a new application of the hypergraph containers theorem (Balogh-Morris-Samotij [14], Saxton-Thomason [38]) to the setting of L -structures, a general supersaturation theorem for hereditary L -properties (also new), and a general graph removal lemma for L -structures proved by Aroskar and Cummings in [5]. Similar results in the setting of multicolored graphs and hypergraphs were recently proved independently by Falgas-Ravry, O'Connel, Strömberg, and Uzzell [21].

Authors:Rob Egrot Pages: 235 - 242 Abstract: Publication date: March 2018 Source:Annals of Pure and Applied Logic, Volume 169, Issue 3 Author(s): Rob Egrot Let m and n be cardinals with 3 ≤ m , n ≤ ω . We show that the class of posets that can be embedded into a distributive lattice via a map preserving all existing meets and joins with cardinalities strictly less than m and n respectively cannot be finitely axiomatized.

Authors:Uri Andrews; Andrea Sorbi Pages: 243 - 259 Abstract: Publication date: March 2018 Source:Annals of Pure and Applied Logic, Volume 169, Issue 3 Author(s): Uri Andrews, Andrea Sorbi We study computably enumerable equivalence relations (or, ceers), under computable reducibility ≤, and the halting jump operation on ceers. We show that every jump is uniform join-irreducible, and thus join-irreducible. Therefore, the uniform join of two incomparable ceers is not equivalent to any jump. On the other hand there exist ceers that are not equivalent to jumps, but are uniform join-irreducible: in fact above any non-universal ceer there is a ceer which is not equivalent to a jump, and is uniform join-irreducible. We also study transfinite iterations of the jump operation. If a is an ordinal notation, and E is a ceer, then let E ( a ) denote the ceer obtained by transfinitely iterating the jump on E along the path of ordinal notations up to a. In contrast with what happens for the Turing jump and Turing reducibility, where if a set X is an upper bound for the A-arithmetical sets then X ( 2 ) computes A ( ω ) , we show that there is a ceer R such that R ≥ Id ( n ) , for every finite ordinal n, but, for all k, R ( k ) ≱ Id ( ω ) (here Id is the identity equivalence relation). We show that if a , b are notations of the same ordinal less than ω 2 , then E ( a ) ≡ E ( b ) , but there are notations a , b of ω 2 such that Id ( a ) and Id ( b ) are incomparable. Moreover, there is no non-universal ceer which is an upper bound for all the ceers of the form Id ( a ) where a is a notation for ω 2 .

Authors:Nick Bezhanishvili; Vincenzo Marra; Daniel McNeill; Andrea Pedrini Abstract: Publication date: Available online 17 December 2017 Source:Annals of Pure and Applied Logic Author(s): Nick Bezhanishvili, Vincenzo Marra, Daniel McNeill, Andrea Pedrini In 1938, Tarski proved that a formula is not intuitionistically valid if, and only if, it has a counter-model in the Heyting algebra of open sets of some topological space. In fact, Tarski showed that any Euclidean space R n with n ⩾ 1 suffices, as does e.g. the Cantor space. In particular, intuitionistic logic cannot detect topological dimension in the Heyting algebra of all open sets of a Euclidean space. By contrast, we consider the lattice of open subpolyhedra of a given compact polyhedron P ⊆ R n , prove that it is a locally finite Heyting subalgebra of the (non-locally-finite) algebra of all open sets of P, and show that intuitionistic logic is able to capture the topological dimension of P through the bounded-depth axiom schemata. Further, we show that intuitionistic logic is precisely the logic of formulæ valid in all Heyting algebras arising from polyhedra in this manner. Thus, our main theorem reconciles through polyhedral geometry two classical results: topological completeness in the style of Tarski, and Jaśkowski's theorem that intuitionistic logic enjoys the finite model property. Several questions of interest remain open. E.g., what is the intermediate logic of all closed triangulable manifolds'

Authors:David Fernández-Duque; Joost J. Joosten Abstract: Publication date: Available online 14 December 2017 Source:Annals of Pure and Applied Logic Author(s): David Fernández-Duque, Joost J. Joosten Given a computable ordinal Λ, the transfinite provability logic GLP Λ has for each ξ < Λ a modality [ ξ ] intended to represent a provability predicate within a chain of increasing strength. One possibility is to read [ ξ ] ϕ as ϕ is provable in T using ω-rules of depth at most ξ, where T is a second-order theory extending ACA 0 . In this paper we will formalize such iterations of ω-rules in second-order arithmetic and show how it is a special case of what we call uniform provability predicates. Uniform provability predicates are similar to Ignatiev's strong provability predicates except that they can be iterated transfinitely. Finally, we show that GLP Λ is sound and complete for any uniform provability predicate.

Authors:Merlin Carl; Philipp Schlicht; Philip Welch Abstract: Publication date: Available online 13 December 2017 Source:Annals of Pure and Applied Logic Author(s): Merlin Carl, Philipp Schlicht, Philip Welch We call a subset of an ordinal λ recognizable if it is the unique subset x of λ for which some Turing machine with ordinal time and tape and an ordinal parameter, that halts for all subsets of λ as input, halts with the final state 0. Equivalently, such a set is the unique subset x which satisfies a given Σ 1 formula in L [ x ] . We further define the recognizable closure for subsets of λ by closing under relative recognizability for subsets of λ. We prove several results about recognizable sets and their variants for other types of machines. Notably, we show the following results from large cardinals. • Recognizable sets of ordinals appear in the hierarchy of inner models at least up to the level Woodin cardinals, while computable sets are elements of L. • A subset of a countable ordinal λ is in the recognizable closure for subsets of λ if and only if it is an element of the inner model M ∞ , which is obtained by iterating the least measure of the least fine structural inner model M 1 with a Woodin cardinal through the ordinals.

Authors:Bruno Dinis; Jaime Gaspar Abstract: Publication date: Available online 12 December 2017 Source:Annals of Pure and Applied Logic Author(s): Bruno Dinis, Jaime Gaspar We present a bounded modified realisability and a bounded functional interpretation of intuitionistic nonstandard arithmetic with nonstandard principles. The functional interpretation is the intuitionistic counterpart of Ferreira and Gaspar's functional interpretation and has similarities with Van den Berg, Briseid and Safarik's functional interpretation but replacing finiteness by majorisability. We give a threefold contribution: constructive content and proof-theoretical properties of nonstandard arithmetic; filling a gap in the literature; being in line with nonstandard methods to analyse compactness arguments.

Authors:Logan Axon Abstract: Publication date: Available online 8 December 2017 Source:Annals of Pure and Applied Logic Author(s): Logan Axon Martin-Löf randomness was originally defined and studied in the context of the Cantor space 2 ω . In [1] probability theoretic random closed sets (RACS) are used as the foundation for the study of Martin-Löf randomness in spaces of closed sets. We use that framework to explore Martin-Löf randomness for the space of closed subsets of R and a particular family of measures on this space, the generalized Poisson processes. This gives a novel class of Martin-Löf random closed subsets of R . We describe some of the properties of these Martin-Löf random closed sets; one result establishes that a real number is Martin-Löf random if and only if it is contained in some Martin-Löf random closed set.

Authors:Antonio Di Nola; Serafina Lapenta; Ioana Leuştean Abstract: Publication date: Available online 31 October 2017 Source:Annals of Pure and Applied Logic Author(s): Antonio Di Nola, Serafina Lapenta, Ioana Leuştean We study Łukasiewicz logic enriched by a scalar multiplication with scalars in [ 0 , 1 ] . Its algebraic models, called Riesz MV-algebras, are, up to isomorphism, unit intervals of Riesz spaces with strong unit endowed with an appropriate structure. When only rational scalars are considered, one gets the class of DMV-algebras and a corresponding logical system. Our research follows two objectives. The first one is to deepen the connections between functional analysis and the logic of Riesz MV-algebras. The second one is to study the finitely presented MV-algebras, DMV-algebras and Riesz MV-algebras, connecting them from logical, algebraic and geometric perspective.

Authors:Raine Abstract: Publication date: Available online 20 October 2017 Source:Annals of Pure and Applied Logic Author(s): Raine Rönnholm In this paper we analyze k-ary inclusion–exclusion logic, INEX[k], which is obtained by extending first order logic with k-ary inclusion and exclusion atoms. We show that every formula of INEX[k] can be expressed with a formula of k-ary existential second order logic, ESO[k]. Conversely, every formula of ESO[k] with at most k-ary free relation variables can be expressed with a formula of INEX[k]. From this it follows that, on the level of sentences, INEX[k] captures the expressive power of ESO[k]. We also introduce several useful operators that can be expressed in INEX[k]. In particular, we define inclusion and exclusion quantifiers and so-called term value preserving disjunction which is essential for the proofs of the main results in this paper. Furthermore, we present a novel method of relativization for team semantics and analyze the duality of inclusion and exclusion atoms.

Authors:Chris Le Sueur Abstract: Publication date: Available online 17 October 2017 Source:Annals of Pure and Applied Logic Author(s): Chris Le Sueur In this paper we develop a technique for proving determinacy of classes of the form ω 2 - Π 1 1 + Γ (a refinement of the difference hierarchy on Π 1 1 lying between ω 2 - Π 1 1 and ( ω 2 + 1 ) - Π 1 1 ) from weak principles, establishing upper bounds for the determinacy-strength of the classes ω 2 - Π 1 1 + Σ α 0 for all computable α and of ω 2 - Π 1 1 + Δ 1 1 . This bridges the gap between previously known hypotheses implying determinacy in this region.

Authors:Tapani Hyttinen; Gianluca Paolini Abstract: Publication date: Available online 17 October 2017 Source:Annals of Pure and Applied Logic Author(s): Tapani Hyttinen, Gianluca Paolini Based on Crapo's theory of one point extensions of combinatorial geometries, we find various classes of geometric lattices that behave very well from the point of view of stability theory. One of them, ( K 3 , ≼ ) , is ω-stable, it has a monster model and an independence calculus that satisfies all the usual properties of non-forking. On the other hand, these classes are rather unusual, e.g. in ( K 3 , ≼ ) the Smoothness Axiom fails, and so ( K 3 , ≼ ) is not an AEC.

Authors:Mohammad Golshani; Rahman Mohammadpour Abstract: Publication date: Available online 14 October 2017 Source:Annals of Pure and Applied Logic Author(s): Mohammad Golshani, Rahman Mohammadpour Assuming the existence of a strong cardinal κ and a measurable cardinal above it, we force a generic extension in which κ is a singular strong limit cardinal of any given cofinality, and such that the tree property holds at κ + + .

Authors:Philipp Hieronymi; Travis Nell; Erik Walsberg Abstract: Publication date: Available online 13 October 2017 Source:Annals of Pure and Applied Logic Author(s): Philipp Hieronymi, Travis Nell, Erik Walsberg Let T be a consistent o-minimal theory extending the theory of densely ordered groups and let T ′ be a consistent theory. Then there is a complete theory T ⁎ extending T such that T is an open core of T ⁎ , but every model of T ⁎ interprets a model of T ′ . If T ′ is NIP, T ⁎ can be chosen to be NIP as well. From this we deduce the existence of an NIP expansion of the real field that has no distal expansion.

Authors:Gunnar Wilken Abstract: Publication date: Available online 21 September 2017 Source:Annals of Pure and Applied Logic Author(s): Gunnar Wilken We provide mutual elementary recursive order isomorphisms between classical ordinal notations, based on Skolem hulling, and notations from pure elementary patterns of resemblance of order 2, showing that the latter characterize the proof-theoretic ordinal 1 ∞ of the fragment Π 1 1 – CA 0 of second order number theory, or equivalently the set theory KP ℓ 0 . As a corollary, we prove that Carlson's result on the well-quasi orderedness of respecting forests of order 2 implies transfinite induction up to the ordinal 1 ∞ . We expect that our approach will facilitate analysis of more powerful systems of patterns.

Authors:S. Barry Cooper; Leo Harrington; Alistair H. Lachlan; Steffen Lempp; Robert I. Soare Abstract: Publication date: Available online 31 July 2017 Source:Annals of Pure and Applied Logic Author(s): S. Barry Cooper, Leo Harrington, Alistair H. Lachlan, Steffen Lempp, Robert I. Soare We indicate how to fix an error in the proof of the main theorem of our original paper, pointed out to us by Yong Liu and Keng Meng Ng.

Authors:Ivan Abstract: Publication date: Available online 26 July 2017 Source:Annals of Pure and Applied Logic Author(s): Ivan Tomašić The theory ACFA admits a primitive recursive quantifier elimination procedure. It is therefore primitive recursively decidable.

Authors:Tommaso Moraschini Abstract: Publication date: Available online 26 July 2017 Source:Annals of Pure and Applied Logic Author(s): Tommaso Moraschini In this paper we consider, from a computational point of view, the problem of classifying logics within the Leibniz and Frege hierarchies typical of abstract algebraic logic. The main result states that, for logics presented syntactically, this problem is in general undecidable. More precisely, we show that there is no algorithm that classifies the logic of a finite consistent Hilbert calculus in the Leibniz and in the Frege hierarchies.

Authors:Jan Dobrowolski; Byunghan Kim; Junguk Lee Abstract: Publication date: Available online 4 July 2017 Source:Annals of Pure and Applied Logic Author(s): Jan Dobrowolski, Byunghan Kim, Junguk Lee Let p be a strong type of an algebraically closed tuple over B = acl eq ( B ) in any theory T. Depending on a ternary relation Image 1 satisfying some basic axioms (there is at least one such, namely the trivial independence in T), the first homology group H 1 ⁎ ( p ) can be introduced, similarly to [3]. We show that there is a canonical surjective homomorphism from the Lascar group over B to H 1 ⁎ ( p ) . We also notice that the map factors naturally via a surjection from the ‘relativised’ Lascar group of the type (which we define in analogy with the Lascar group of the theory) onto the homology group, and we give an explicit description of its kernel. Due to this characterization, it follows that the first homology group of p is independent from the choice of Image 1 , and can be written simply as H 1 ( p ) . As consequences, in any T, we show that H 1 ( p ) ≥ 2 ℵ 0 unless H 1 ( p ) is trivial, and we give a criterion for the equality of stp and Lstp of algebraically closed tuples using the notions of the first homology group and a relativised Lascar group. We also argue how any abelian connected compact group can appear as the first homology group of the type of a model.

Authors:Enrique Casanovas; Luis Jaime Corredor Abstract: Publication date: Available online 30 June 2017 Source:Annals of Pure and Applied Logic Author(s): Enrique Casanovas, Luis Jaime Corredor Let M be the monster model of a complete first-order theory T. If D is a subset of M , following D. Zambella we consider e ( D ) = { D ′ ( M , D ) ≡ ( M , D ′ ) } and o ( D ) = { D ′ ( M , D ) ≅ ( M , D ′ ) } . The general question we ask is when e ( D ) = o ( D ) ' The case where D is A-invariant for some small set A is rather straightforward: it just means that D is definable. We investigate the case where D is not invariant over any small subset. If T is geometric and ( M , D ) is an H-structure (in the sense of A. Berenstein and E. Vassiliev) we get some answers. In the case of SU-rank one, e ( D ) is always different from o ( D ) . In the o-minimal case, everything can happen, depending on the complexity of the definable closure. We also study the case of lovely pairs of geometric theories.

Authors:L.B. Ostrovsky; M.E. Zhukovskii Abstract: Publication date: Available online 21 June 2017 Source:Annals of Pure and Applied Logic Author(s): L.B. Ostrovsky, M.E. Zhukovskii We study asymptotical probabilities of first order and monadic second order properties of Bernoulli random graph G ( n , n − a ) . The random graph obeys FO (MSO) zero-one k-law (k is a positive integer) if, for any first order (monadic second order) formulae with quantifier depth at most k, it is true for G ( n , n − a ) with probability tending to 0 or to 1. Zero-one k-laws are well studied only for the first order language and a < 1 . We obtain new zero-one k-laws (both for first order and monadic second order languages) when a > 1 . Proofs of these results are based on the earlier studies of first order equivalence classes and our study of monadic second order equivalence classes. The respective results are of interest by themselves.

Authors:Anton Freund Abstract: Publication date: Available online 20 June 2017 Source:Annals of Pure and Applied Logic Author(s): Anton Freund We describe a “slow” version of the hierarchy of uniform reflection principles over Peano Arithmetic (PA). These principles are unprovable in Peano Arithmetic (even when extended by usual reflection principles of lower complexity) and introduce a new provably total function. At the same time the consistency of PA plus slow reflection is provable in PA + Con ( PA ) . We deduce a conjecture of S.-D. Friedman, Rathjen and Weiermann: Transfinite iterations of slow consistency generate a hierarchy of precisely ε 0 stages between PA and PA + Con ( PA ) (where Con ( PA ) refers to the usual consistency statement).

Authors:Saskia Chambille; Pablo Cubides Kovacsics; Eva Leenknegt Abstract: Publication date: Available online 9 June 2017 Source:Annals of Pure and Applied Logic Author(s): Saskia Chambille, Pablo Cubides Kovacsics, Eva Leenknegt We prove that in a P-minimal structure, every definable set can be partitioned as a finite union of classical cells and regular clustered cells. This is a generalization of previously known cell decomposition results by Denef and Mourgues, which were dependent on the existence of definable Skolem functions. Clustered cells have the same geometric structure as classical, Denef-type cells, but do not have a definable function as center. Instead, the center is given by a definable set whose fibers are finite unions of balls.

Authors:Ari Meir Brodsky; Assaf Rinot Abstract: Publication date: Available online 8 June 2017 Source:Annals of Pure and Applied Logic Author(s): Ari Meir Brodsky, Assaf Rinot We propose a parameterized proxy principle from which κ-Souslin trees with various additional features can be constructed, regardless of the identity of κ. We then introduce the microscopic approach, which is a simple method for deriving trees from instances of the proxy principle. As a demonstration, we give a construction of a coherent κ-Souslin tree that applies also for κ inaccessible. We then carry out a systematic study of the consistency of instances of the proxy principle, distinguished by the vector of parameters serving as its input. Among other things, it will be shown that all known ⋄-based constructions of κ-Souslin trees may be redirected through this new proxy principle.

Abstract: Publication date: Available online 7 June 2017 Source:Annals of Pure and Applied Logic Author(s): M. Hrušák, D. Meza-Alcántara, E. Thümmel, C. Uzcátegui We study several combinatorial properties of (mostly definable) ideals on countable sets. In several cases, we identify critical ideals for such properties in the Katětov order. In particular, the ideal R generated by the homogeneous subsets of the random graph is critical for the Ramsey property. The question as to whether there is a tall definable Ramsey ideal is raised and studied. It is shown that no tall F σ ideal is Ramsey, while there is a tall co-analytic Ramsey ideal.

Authors:Jacob Davis Abstract: Publication date: Available online 3 May 2017 Source:Annals of Pure and Applied Logic Author(s): Jacob Davis Starting from a supercompact cardinal we build a model in which 2 ℵ ω 1 = 2 ℵ ω 1 + 1 = ℵ ω 1 + 3 but there is a jointly universal family of size ℵ ω 1 + 2 of graphs on ℵ ω 1 + 1 . The same technique will work for any uncountable cardinal in place of ω 1 .

Authors:Samaria Montenegro Abstract: Publication date: Available online 21 April 2017 Source:Annals of Pure and Applied Logic Author(s): Samaria Montenegro The main result of this paper is that if M is a bounded PRC field, then T h ( M ) eliminates imaginaries in the language of rings expanded by constant symbols. As corollary of the elimination of imaginaries and the fact that the algebraic closure (in the sense of model theory) defines a pregeometry we obtain that the complete theory of a bounded PRC field is superrosy of U th -rank 1.

Authors:Kaisa Kangas Abstract: Publication date: Available online 6 April 2017 Source:Annals of Pure and Applied Logic Author(s): Kaisa Kangas We show that if M is a Zariski-like structure (see [6]) and the canonical pregeometry obtained from the bounded closure operator (bcl) is non locally modular, then M interprets either an algebraically closed field or a non-classical group.

Authors:David Cerna; Alexander Leitsch; Giselle Reis; Simon Wolfsteiner Abstract: Publication date: Available online 5 April 2017 Source:Annals of Pure and Applied Logic Author(s): David Cerna, Alexander Leitsch, Giselle Reis, Simon Wolfsteiner In this paper we present a procedure allowing the extension of a CERES-based cut-elimination method to intuitionistic logic. Previous results concerning this problem manage to capture fragments of intuitionistic logic, but in many essential cases structural constraints were violated during normal form construction resulting in a classical proof. The heart of the CERES method is the resolution calculus, which ignores the structural constraints of the well known intuitionistic sequent calculi. We propose, as a method of avoiding the structural violations, the generalization of resolution from the resolving of clauses to the resolving of cut-free proofs, in other words, what we call proof resolution. The result of proof resolution is a cut-free proof rather than a clause. Note that resolution on ground clauses is essentially atomic cut, thus using proof resolution to construct cut-free proofs one would need to join the two proofs together and remove the atoms which where resolved. To efficiently perform this joining (and guarantee that the resulting cut-free proof is intuitionistic) we develop the concept of proof subsumption (similar to clause subsumption) and indexed resolution, a refinement indexing atoms by their corresponding positions in the cut formula. Proof subsumption serves as a tool to prove the completeness of the new method CERES-i, and indexed resolution provides an efficient strategy for the joining of two proofs which is in general a nondeterministic search. Such a refinement is essential for any attempt to implement this method. Finally we compare the complexity of CERES-i with that of Gentzen-based methods.