Authors:Paulo Oliva; Thomas Powell Pages: 887 - 921 Abstract: Publication date: May 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 5 Author(s): Paulo Oliva, Thomas Powell We introduce a new, demand-driven variant of Spector's bar recursion in the spirit of the Berardi–Bezem–Coquand functional of [4]. The recursion takes place over finite partial functions u, where the control parameter ω, used in Spector's bar recursion to terminate the computation at sequences s satisfying ω ( s ˆ ) < s , now acts as a guide for deciding exactly where to make bar recursive updates, terminating the computation whenever ω ( u ˆ ) ∈ dom ( u ) . We begin by exploring theoretical aspects of this new form of recursion, then in the main part of the paper we show that demand-driven bar recursion can be directly used to give an alternative functional interpretation of classical countable choice. We provide a short case study as an illustration, in which we extract a new bar recursive program from the proof that there is no injection from N → N to N , and compare this with the program that would be obtained using Spector's original variant. We conclude by formally establishing that our new bar recursor is primitive recursively equivalent to the original Spector bar recursion, and thus defines the same class of functionals when added to Gödel's system T .

Authors:Thomas Gilton; John Krueger Pages: 922 - 1016 Abstract: Publication date: May 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 5 Author(s): Thomas Gilton, John Krueger Mitchell's theorem on the approachability ideal states that it is consistent relative to a greatly Mahlo cardinal that there is no stationary subset of ω 2 ∩ cof ( ω 1 ) in the approachability ideal I [ ω 2 ] . In this paper we give a new proof of Mitchell's theorem, deriving it from an abstract framework of side condition methods.

Authors:Paul Shafer Pages: 1017 - 1031 Abstract: Publication date: May 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 5 Author(s): Paul Shafer An element a of a lattice cups to an element b > a if there is a c < b such that a ∪ c = b . An element of a lattice has the cupping property if it cups to every element above it. We prove that there are non-zero honest elementary degrees that do not have the cupping property, which answers a question of Kristiansen, Schlage-Puchta, and Weiermann. In fact, we show that if b is a sufficiently large honest elementary degree, then b has the anti-cupping property, which means that there is an a with 0 < E a < E b that does not cup to b. For comparison, we also modify a result of Cai to show, in several versions of the degrees of relative provability that are closely related to the honest elementary degrees, that in fact all non-zero degrees have the anti-cupping property, not just sufficiently large degrees.

Authors:Robert Goldblatt; Ian Hodkinson Pages: 1032 - 1090 Abstract: Publication date: May 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 5 Author(s): Robert Goldblatt, Ian Hodkinson There has been renewed interest in recent years in McKinsey and Tarski's interpretation of modal logic in topological spaces and their proof that S4 is the logic of any separable dense-in-itself metric space. Here we extend this work to the modal mu-calculus and to a logic of tangled closure operators that was developed by Fernández-Duque after these two languages had been shown by Dawar and Otto to have the same expressive power over finite transitive Kripke models. We prove that this equivalence remains true over topological spaces. We extend the McKinsey–Tarski topological ‘dissection lemma’. We also take advantage of the fact (proved by us elsewhere) that various tangled closure logics with and without the universal modality ∀ have the finite model property in Kripke semantics. These results are used to construct a representation map (also called a d-p-morphism) from any dense-in-itself metric space X onto any finite connected locally connected serial transitive Kripke frame. This yields completeness theorems over X for a number of languages: (i) the modal mu-calculus with the closure operator ◇; (ii) ◇ and the tangled closure operators 〈 t 〉 (in fact 〈 t 〉 can express ◇); (iii) ◇ , ∀ ; (iv) ◇ , ∀ , 〈 t 〉 ; (v) the derivative operator 〈 d 〉 ; (vi) 〈 d 〉 and the associated tangled closure operators 〈 d t 〉 ; (vii) 〈 d 〉 , ∀ ; (viii) 〈 d 〉 , ∀ , 〈 d t 〉 . Soundness also holds, if: (a) for languages with ∀, X is connected; (b) for languages with 〈 d 〉 , X validates the well-known axiom G 1 . For countable languages without ∀, we prove strong completeness. We also show that in the presence of ∀, strong completeness fails if X is compact and locally connected.

Authors:Vincent Guingona; Cameron Donnay Hill; Lynn Scow Pages: 1091 - 1111 Abstract: Publication date: May 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 5 Author(s): Vincent Guingona, Cameron Donnay Hill, Lynn Scow We use the notion of collapse of generalized indiscernible sequences to classify various model theoretic dividing lines. In particular, we use collapse of n-multi-order indiscernibles to characterize op-dimension n; collapse of function-space indiscernibles (i.e. parameterized equivalence relations) to characterize rosy theories; and finally, convex equivalence relation indiscernibles to characterize NTP2 theories.

Authors:Sean Cox; Philipp Lücke Pages: 1112 - 1131 Abstract: Publication date: May 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 5 Author(s): Sean Cox, Philipp Lücke Given an uncountable regular cardinal κ, a partial order is κ-stationarily layered if the collection of regular suborders of P of cardinality less than κ is stationary in P κ ( P ) . We show that weak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal κ is weakly compact if and only if every partial order satisfying the κ-chain condition is κ-stationarily layered. We prove a similar result for strongly inaccessible cardinals. Moreover, we show that the statement that all κ-Knaster partial orders are κ-stationarily layered implies that κ is a Mahlo cardinal and every stationary subset of κ reflects. This shows that this statement characterizes weak compactness in canonical inner models. In contrast, we show that it is also consistent that this statement holds at a non-weakly compact cardinal.

Authors:Leonardo Manuel Cabrer; Daniele Mundici Pages: 1132 - 1151 Abstract: Publication date: May 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 5 Author(s): Leonardo Manuel Cabrer, Daniele Mundici Differently from boolean logic, in Łukasiewicz infinite-valued propositional logic Ł∞ the theory Θ max , v consisting of all formulas satisfied by a model v ∈ [ 0 , 1 ] n is not the only one having v as its unique model: indeed, there is a smallest such theory Θ min , v , the germinal theory at v, which in general is strictly contained in Θ max , v . The Lindenbaum algebra of Θ max , v is promptly seen to coincide with the subalgebra of the standard MV-algebra [ 0 , 1 ] generated by the coordinates of v. The description of the Lindenbaum algebras of germinal theories in two variables is our main aim in this paper. As a basic prerequisite of independent interest, we prove that for any models v and w the germinal theories Θ min , v and Θ min , w have isomorphic Lindenbaum algebras iff v and w have the same orbit under the action of the affine group over the integers.

Authors:A. Ivanov; B. Majcher-Iwanow Pages: 749 - 775 Abstract: Publication date: April 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 4 Author(s): A. Ivanov, B. Majcher-Iwanow We extend the generalised model theory of H. Becker from [2] to the case of Polish G-spaces when G is an arbitrary Polish group. Our approach is inspired by logic actions of Polish groups which arise in continuous logic.

Authors:Bjørn Kjos-Hanssen; Frank Stephan; Sebastiaan A. Terwijn Pages: 804 - 823 Abstract: Publication date: April 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 4 Author(s): Bjørn Kjos-Hanssen, Frank Stephan, Sebastiaan A. Terwijn We give solutions to two of the questions in a paper by Brendle, Brooke-Taylor, Ng and Nies. Our examples derive from a 2014 construction by Khan and Miller as well as new direct constructions using martingales. At the same time, we introduce the concept of i.o. subuniformity and relate this concept to recursive measure theory. We prove that there are classes closed downwards under Turing reducibility that have recursive measure zero and that are not i.o. subuniform. This shows that there are examples of classes that cannot be covered with methods other than probabilistic ones. It is easily seen that every set of hyperimmune degree can cover the recursive sets. We prove that there are both examples of hyperimmune-free degree that can and that cannot compute such a cover.

Authors:Stephen Flood Pages: 824 - 839 Abstract: Publication date: April 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 4 Author(s): Stephen Flood The theory of simplicial graph decompositions studies the infinite graphs that are built from a sequence of irreducible graphs which are attached together at complete subgraphs. In this paper, we study the logical complexity of deciding if a graph is prime decomposable. A large part of this analysis involves determining which ordinals must appear in these types of decompositions. A result of Diestel says that every countable simplicial tree decomposition can be rearranged to have length at most ω. We show that no such ordinal bound can be found for the lengths of non-tree decompositions. More generally, we show that for each ordinal σ, there is a decomposable graph whose shortest simplicial decomposition has length exactly σ. Adapting this argument, we show that the index set of decomposable computable graphs DECOMP is Π 1 1 hard by showing that WO is 1-reducible to DECOMP .

Authors:Zvonko Iljazović; Lucija Validžić Pages: 840 - 859 Abstract: Publication date: April 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 4 Author(s): Zvonko Iljazović, Lucija Validžić We examine conditions under which a semicomputable set in a computable metric space contains computable points. We prove that computable points in a semicomputable set S are dense if S is a manifold (possibly with boundary) or S has the topological type of a polyhedron. Moreover, we find conditions under which a point in some set has a computable compact neighbourhood in that set. In particular, we show that a point x in a semicomputable set has a computable compact neighbourhood if x has a neighbourhood homeomorphic to Euclidean space.

Authors:Rupert Hölzl; Christopher P. Porter Pages: 860 - 886 Abstract: Publication date: April 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 4 Author(s): Rupert Hölzl, Christopher P. Porter We study the possible growth rates of the Kolmogorov complexity of initial segments of sequences that are random with respect to some computable measure on 2 ω , the so-called proper sequences. Our main results are as follows: (1) We show that the initial segment complexity of a proper sequence X is bounded from below by a computable function (that is, X is complex) if and only if X is random with respect to some computable, continuous measure. (2) We prove that a uniform version of the previous result fails to hold: there is a family of complex sequences that are random with respect to a single computable measure such that for every computable, continuous measure μ, some sequence in this family fails to be random with respect to μ. (3) We show that there are proper sequences with extremely slow-growing initial segment complexity, that is, there is a proper sequence the initial segment complexity of which is infinitely often below every computable function, and even a proper sequence the initial segment complexity of which is dominated by all computable functions. (4) We prove various facts about the Turing degrees of such sequences and show that they are useful in the study of certain classes of pathological measures on 2 ω , namely diminutive measures and trivial measures.

Authors:Murdoch J. Gabbay; Michael Gabbay Pages: 501 - 621 Abstract: Publication date: March 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 3 Author(s): Murdoch J. Gabbay, Michael Gabbay We give a semantics for the λ-calculus based on a topological duality theorem in nominal sets. A novel interpretation of λ is given in terms of adjoints, and λ-terms are interpreted absolutely as sets (no valuation is necessary).

Authors:Gabriel Conant Pages: 622 - 650 Abstract: Publication date: March 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 3 Author(s): Gabriel Conant Let R = ( R , ⊕ , ≤ , 0 ) be an algebraic structure, where ⊕ is a commutative binary operation with identity 0, and ≤ is a translation-invariant total order with least element 0. Given a distinguished subset S ⊆ R , we define the natural notion of a “generalized” R -metric space, with distances in S. We study such metric spaces as first-order structures in a relational language consisting of a distance inequality for each element of S. We first construct an ordered additive structure S ⁎ on the space of quantifier-free 2-types consistent with the axioms of R -metric spaces with distances in S, and show that, if A is an R -metric space with distances in S, then any model of Th ( A ) logically inherits a canonical S ⁎ -metric. Our primary application of this framework concerns countable, universal, and homogeneous metric spaces, obtained as generalizations of the rational Urysohn space. We adapt previous work of Delhommé, Laflamme, Pouzet, and Sauer to fully characterize the existence of such spaces. We then fix a countable totally ordered commutative monoid R , with least element 0, and consider U R , the countable Urysohn space over R . We show that quantifier elimination for Th ( U R ) is characterized by continuity of addition in R ⁎ , which can be expressed as a first-order sentence of R in the language of ordered monoids. Finally, we analyze an example of Casanovas and Wagner in this context.

Authors:Sebastien Vasey Pages: 651 - 692 Abstract: Publication date: March 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 3 Author(s): Sebastien Vasey In the setting of abstract elementary classes (AECs) with amalgamation, Shelah has proven a downward categoricity transfer from categoricity in a successor and Grossberg and VanDieren have established an upward transfer assuming in addition a locality property for Galois types that they called tameness. We further investigate categoricity transfers in tame AECs. We use orthogonality calculus to prove a downward transfer from categoricity in a successor in AECs that have a good frame (a forking-like notion for types of singletons) on an interval of cardinals: Theorem 0.1 Let K be an AEC and let LS ( K ) ≤ λ < θ be cardinals. If K has a type-full good [ λ , θ ] -frame and K is categorical in both λ and θ + , then K is categorical in all μ ∈ [ λ , θ ] . We deduce improvements on the threshold of several categoricity transfers that do not mention frames. For example, the threshold in Shelah's transfer can be improved from ℶ ℶ ( 2 LS ( K ) ) + to ℶ ( 2 LS ( K ) ) + assuming that the AEC is LS ( K ) -tame. The successor hypothesis can also be removed from Shelah's result by assuming in addition either that the AEC has primes over sets of the form M ∪ { a } or (using an unpublished claim of Shelah) that the weak generalized continuum hypothesis holds.

Authors:Agata Ciabattoni; Nikolaos Galatos; Kazushige Terui Pages: 693 - 737 Abstract: Publication date: March 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 3 Author(s): Agata Ciabattoni, Nikolaos Galatos, Kazushige Terui We continue our program of establishing connections between proof-theoretic and order-algebraic properties in the setting of substructural logics and residuated lattices. Extending our previous work that connects a strong form of cut-admissibility in sequent calculi with closure under MacNeille completions of corresponding varieties, we now consider hypersequent calculi and more general completions; these capture logics/varieties that were not covered by the previous approach and that are characterized by Hilbert axioms (algebraic equations) residing in the level P 3 of the substructural hierarchy. We provide algebraic foundations for substructural hypersequent calculi and an algorithm to transform P 3 axioms/equations into equivalent structural hypersequent rules. Using residuated hyperframes we link strong analyticity in the resulting calculi with a new algebraic completion, which we call hyper-MacNeille.

Authors:Martin Hyland; Guy McCusker; Nikos Tzevelekos First page: 233 Abstract: Publication date: February 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 2 Author(s): Martin Hyland, Guy McCusker, Nikos Tzevelekos

Authors:Chrysida Galanaki; Christos Nomikos; Panos Rondogiannis Pages: 234 - 253 Abstract: Publication date: February 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 2 Author(s): Chrysida Galanaki, Christos Nomikos, Panos Rondogiannis Intensional logic programming is an extension of logic programming based on intensional logic, which includes as special cases both temporal and modal logic programming. In [13], M. Orgun and W.W. Wadge provided a general framework for capturing the semantics of intensional logic programming languages. They demonstrated that if the intensional operators of a language obey some simple semantic properties, then the programs of the language are guaranteed to have a minimum model semantics. One key property involved in the construction of [13] is the monotonicity of intensional operators. In this paper we consider intensional logic programming from a game-theoretic perspective. In particular we define a two-person game and demonstrate that it can be used in order to define a model for any given intensional program of the class introduced in [13]. Moreover, this model is shown to be identical to the minimum model constructed in [13]. More importantly, we demonstrate that the game is even applicable to intensional languages with non-monotonic operators. In this way we provide the first (to our knowledge) general framework for capturing the semantics of non-monotonic intensional logic programming.

Authors:Valentin Blot Pages: 254 - 277 Abstract: Publication date: February 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 2 Author(s): Valentin Blot We build a realizability model for Peano arithmetic based on winning conditions for HON games. Our winning conditions are sets of desequentialized interactions which we call positions. We define a notion of winning strategies on arenas equipped with winning conditions. We prove that the interpretation of a classical proof of a formula is a winning strategy on the arena with winning condition corresponding to the formula. Finally we apply this to Peano arithmetic with relativized quantifications and give the example of witness extraction for Π 2 0 -formulas.

Authors:Thomas Seiller Pages: 278 - 320 Abstract: Publication date: February 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 2 Author(s): Thomas Seiller In two previous papers [33,37], we exposed a combinatorial approach to the program of Geometry of Interaction, a program initiated by Jean-Yves Girard [16]. The strength of our approach lies in the fact that we interpret proofs by simpler structures – graphs – than Girard's constructions, while generalising the latter since they can be recovered as special cases of our setting. This third paper extends this approach by considering a generalisation of graphs named graphings, which is in some way a geometric realisation of a graph on a measured space. This very general framework leads to a number of new models of multiplicative-additive linear logic which generalise Girard's geometry of interaction models and opens several new lines of research. As an example, we exhibit a family of such models which account for second-order quantification without suffering the same limitations as Girard's models.

Authors:Julian Gutierrez; Paul Harrenstein; Michael Wooldridge Pages: 373 - 403 Abstract: Publication date: February 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 2 Author(s): Julian Gutierrez, Paul Harrenstein, Michael Wooldridge In this paper we study techniques for reasoning about game-like concurrent systems, where the components of the system act rationally and strategically in pursuit of logically-specified goals. Specifically, we start by presenting a computational model for such concurrent systems, and investigate its computational, mathematical, and game-theoretic properties. We then define and investigate a branching-time temporal logic for reasoning about the equilibrium properties of game-like concurrent systems. The key operator in this temporal logic is a novel path quantifier [ N E ] φ , which asserts that φ holds on all Nash equilibrium computations of the system.

Authors:Ichiro Hasuo; Naohiko Hoshino Pages: 404 - 469 Abstract: Publication date: February 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 2 Author(s): Ichiro Hasuo, Naohiko Hoshino While much of the current study on quantum computation employs low-level formalisms such as quantum circuits, several high-level languages/calculi have been recently proposed aiming at structured quantum programming. The current work contributes to the semantical study of such languages by providing interaction-based semantics of a functional quantum programming language; the latter is, much like Selinger and Valiron's, based on linear lambda calculus and equipped with features like the ! modality and recursion. The proposed denotational model is the first one that supports the full features of a quantum functional programming language; we prove adequacy of our semantics. The construction of our model is by a series of existing techniques taken from the semantics of classical computation as well as from process theory. The most notable among them is Girard's Geometry of Interaction (GoI), categorically formulated by Abramsky, Haghverdi and Scott. The mathematical genericity of these techniques—largely due to their categorical formulation—is exploited for our move from classical to quantum.

Authors:Rutger Kuyper; Joseph S. Miller Abstract: Publication date: Available online 10 March 2017 Source:Annals of Pure and Applied Logic Author(s): Rutger Kuyper, Joseph S. Miller For a class C of sets, let us say that a set A is C stabilising if A △ X ∈ C for every X ∈ C . We prove that the Martin-Löf stabilising sets are exactly the K-trivial sets, as are the weakly 2-random stabilising sets. We also show that the 1-generic stabilising sets are exactly the computable sets.

Authors:Grigory K. Olkhovikov Abstract: Publication date: Available online 9 March 2017 Source:Annals of Pure and Applied Logic Author(s): Grigory K. Olkhovikov The paper continues the line of [6], [7], and [8]. This results in a model-theoretic characterization of expressive powers of arbitrary finite sets of guarded connectives of degree not exceeding 1 and regular connectives of degree 2 over the language of bounded lattices.

Authors:Daniel Perrucci; Marie-Françoise Roy Abstract: Publication date: Available online 8 March 2017 Source:Annals of Pure and Applied Logic Author(s): Daniel Perrucci, Marie-Françoise Roy We describe a new quantifier elimination algorithm for real closed fields based on Thom encoding and sign determination. The complexity of this algorithm is elementary recursive and its proof of correctness is completely algebraic. In particular, the notion of connected components of semialgebraic sets is not used.

Authors:Will Boney; Rami Grossberg Abstract: Publication date: Available online 2 March 2017 Source:Annals of Pure and Applied Logic Author(s): Will Boney, Rami Grossberg We develop a notion of forking for Galois-types in the context of Elementary Classes (AECs). Under the hypotheses that an AEC K is tame, type-short, and failure of an order-property, we consider Definition 1 Let M 0 ≺ N be models from K and A be a set. We say that the Galois-type of A over N does not fork over M 0 , written A ⫝ M 0 N , iff for all small a ∈ A and all small N − ≺ N , we have that Galois-type of a over N − is realized in M 0 . Assuming property (E) (Existence and Extension, see Definition 3.3) we show that this non-forking is a well behaved notion of independence, in particular satisfies symmetry and uniqueness and has a corresponding U-rank. We find conditions for a universal local character, in particular derive superstability-like property from little more than categoricity in a “big cardinal”. Finally, we show that under large cardinal axioms the proofs are simpler and the non-forking is more powerful. In [10], it is established that, if this notion is an independence notion, then it is the only one.

Authors:Dan Hathaway Abstract: Publication date: Available online 1 March 2017 Source:Annals of Pure and Applied Logic Author(s): Dan Hathaway For each a ∈ ω ω , we define a Baire class one function f a : ω ω → ω ω which encodes a in a certain sense. We show that for each Borel g : ω ω → ω ω , f a ∩ g = ∅ implies a ∈ Δ 1 1 ( c ) where c is any code for g. We generalize this theorem for g in a larger pointclass Γ. Specifically, when Γ = Δ 2 1 , a ∈ L [ c ] . Also for all n ∈ ω , when Γ = Δ 3 + n 1 , a ∈ M 1 + n ( c ) .

Authors:Carmi Merimovich Abstract: Publication date: Available online 28 February 2017 Source:Annals of Pure and Applied Logic Author(s): Carmi Merimovich The extender based Magidor-Radin forcing is being generalized to supercompact type extenders.

Authors:Somayyeh Tari Abstract: Publication date: Available online 28 February 2017 Source:Annals of Pure and Applied Logic Author(s): Somayyeh Tari Continuous extension cells, or CE-cells, are cells whose defining functions have continuous extensions on closure of their domains. An o-minimal structure has the CE-cell decomposition property if any cell decomposition has a refinement by CE-cells. If the o-minimal structure M has the CE-cell decomposition property, then it has the open cell property. In other words, every definable open set in M is a finite union of definable open cells. Here, we show that the open cell property does not imply the CE-cell decomposition property. Also, after introducing an existence of limit property, we show that the CE-cell decomposition property is equivalent to the open cell property and the existence of limit property.

Authors:Sonia L'Innocente; Carlo Toffalori; Gena Puninski Abstract: Publication date: Available online 16 February 2017 Source:Annals of Pure and Applied Logic Author(s): Sonia L'Innocente, Carlo Toffalori, Gena Puninski We will prove that the theory of all modules over the ring of algebraic integers is decidable.

Authors:Yurii Khomskii; Giorgio Laguzzi Abstract: Publication date: Available online 16 February 2017 Source:Annals of Pure and Applied Logic Author(s): Yurii Khomskii, Giorgio Laguzzi We investigate two closely related partial orders of trees on ω ω : the full-splitting Miller trees and the infinitely often equal trees, as well as their corresponding σ-ideals. The former notion was considered by Newelski and Rosłanowski while the latter involves a correction of a result of Spinas. We consider some Marczewski-style regularity properties based on these trees, which turn out to be closely related to the property of Baire, and look at the dichotomies of Newelski-Rosłanowski and Spinas for higher projective pointclasses. We also provide some insight concerning a question of Fremlin whether one can add an infinitely often equal real without adding a Cohen real, which was recently solved by Zapletal.

Authors:Junhua Abstract: Publication date: April 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 4 Author(s): Junhua Yu Justification logics serve as “explicit” modal logics in a way that, formula ϕ is a modal theorem if and only if there is a justification theorem, called a realization of ϕ, gained by replacing modality occurrences in ϕ by (justification) terms with structures explicitly explaining their evidential contents. In justification logics, terms stand for justifications of (propositions expressed by) formulas, and as a kind of atomic terms, constants stand for that of (justification) axioms. Kuznets has shown that in order to realize (i.e., offer a realization of) some modal theorems, it is necessary to employ a self-referential constant, that is, a constant that stands for a justification of an axiom containing an occurrence of the constant itself. Based on existing works, including some of the author's, this paper treats the collection of modal theorems that are non-self-referentially realizable as a fragment (called non-self-referential fragment) of the modal logic, and verifies: (1) that fragment is not closed in general under modus ponens; and (2) that fragment is not “conservative” in general when going from a smaller modal logic to a larger one.

Authors:Yun Fan Abstract: Publication date: March 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 3 Author(s): Yun Fan We introduce a property of Turing degrees: being uniformly non- low 2 . We prove that, in the c.e. Turing degrees, there is an incomplete uniformly non- low 2 degree, and not every non- low 2 degree is uniformly non- low 2 . We also build some connection between (uniform) non- low 2 -ness and computable Lipschitz reducibility ( ≤ c l ), as a strengthening of weak truth table reducibility: (1) If a c.e. Turing degree d is uniformly non- low 2 , then for any non-computable Δ 2 0 real there is a c.e. real in d such that both of them have no common upper bound in c.e. reals under cl-reducibility. (2) A c.e. Turing degree d is non- low 2 if and only if for any Δ 2 0 real there is a real in d which is not cl-reducible to it.

Abstract: Publication date: February 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 2 Author(s): Paul-André Melliès Tensorial logic is a primitive logic of tensor and negation which refines linear logic by relaxing the hypothesis that linear negation is involutive. Thanks to this mild modification, tensorial logic provides a type-theoretic account of game semantics where innocent strategies are portrayed as temporal refinements of traditional proof-nets in linear logic. In this paper, we study the algebraic and combinatorial structure of negation in a non-commutative variant of tensorial logic. The analysis is based on a 2-categorical account of dialogue categories, which unifies tensorial logic with linear logic, and discloses a primitive symmetry between proofs and anti-proofs. The micrological analysis of tensorial negation reveals that it can be decomposed into a series of more elementary components: an adjunction L ⊣ R between the left and right negation functors L and R; a pair of linear distributivity laws κ and κ which refines the linear distributivity law between ⊗ and ⅋ in linear logic, and generates the Opponent and Proponent views of innocent strategies between dialogue games; a pair of axiom and cut combinators adapted from linear logic; an involutive change of frame ( − ) ⁎ reversing the point of view of Prover and of Denier on the logical dispute, and reversing the polarity of moves in the dialogue game associated to the tensorial formula.

Authors:Laird Abstract: Publication date: February 2017 Source:Annals of Pure and Applied Logic, Volume 168, Issue 2 Author(s): J. Laird Computational effects which provide access to the flow of control (such as first-class continuations, exceptions and delimited continuations) are important features of higher-order programming languages. There are fundamental differences between them in terms of operational behaviour, expressiveness and implementation, so that understanding how they combine and relate to each other is a challenging objective, with a key role for semantics in making this precise. This paper develops operational and denotational semantics for a hierarchy of programming languages which include combinations of locally declared control prompts to which a program can escape, with first-class continuations which may either capture their enclosing prompts, or be delimited by them. We describe two different hierarchies of models, both based on categories of games and strategies with a computational monad, but obtained using different methodologies. By relaxing combinations of behavioural constraints on strategies with control flow represented by annotation with control pointers we are able to give direct and explicit characterizations of control operators and their effects, including examples characterizing their macro-expressiveness. By constructing a parallel hierarchy of models by applying sequences of monad transformers, and relating these to the direct interpretation of control effects, we obtain games interpretations of higher-level abstractions such as continuations and exceptions, which can be used as the basis for equational reasoning about programs.

Abstract: Publication date: Available online 31 January 2017 Source:Annals of Pure and Applied Logic Author(s): András Pongrácz Let ( H n , E ) denote the Henson graph, the unique countable homogeneous graph whose age consists of all finite K n -free graphs. In this note the reducts of the Henson graphs with a constant are determined up to first-order interdefinability. It is shown that up to first-order interdefinability ( H 3 , E , 0 ) has 13 reducts and ( H n , E , 0 ) has 16 reducts for n ≥ 4 .

Authors:Matthew Harrison-Trainor Abstract: Publication date: Available online 20 January 2017 Source:Annals of Pure and Applied Logic Author(s): Matthew Harrison-Trainor A set A is coarsely computable with density r ∈ [ 0 , 1 ] if there is an algorithm for deciding membership in A which always gives a (possibly incorrect) answer, and which gives a correct answer with density at least r. To any Turing degree a we can assign a value Γ T ( a ) : the minimum, over all sets A in a, of the highest density at which A is coarsely computable. The closer Γ T ( a ) is to 1, the closer a is to being computable. Andrews, Cai, Diamondstone, Jockusch, and Lempp noted that Γ T can take on the values 0, 1/2, and 1, but not any values in strictly between 1/2 and 1. They asked whether the value of Γ T can be strictly between 0 and 1/2. This is the Gamma question. Replacing Turing degrees by many-one degrees, we get an analogous question, and the same arguments show that Γ m can take on the values 0, 1/2, and 1, but not any values strictly between 1/2 and 1. We will show that for any r ∈ [ 0 , 1 / 2 ] , there is an m-degree a with Γ m ( a ) = r . Thus the range of Γ m is [ 0 , 1 / 2 ] ∪ { 1 } . Benoit Monin has recently announced a solution to the Gamma question for Turing degrees. Interestingly, his solution gives the opposite answer: the only possible values of Γ T are 0, 1/2, and 1.

Authors:Fan Yang; Jouko Väänänen Abstract: Publication date: Available online 20 January 2017 Source:Annals of Pure and Applied Logic Author(s): Fan Yang, Jouko Väänänen We consider team semantics for propositional logic, continuing [34]. In team semantics the truth of a propositional formula is considered in a set of valuations, called a team, rather than in an individual valuation. This offers the possibility to give meaning to concepts such as dependence, independence and inclusion. We associate with every formula ϕ based on finitely many propositional variables the set 〚 ϕ 〛 of teams that satisfy ϕ. We define a maximal propositional team logic in which every set of teams is definable as 〚 ϕ 〛 for suitable ϕ. This requires going beyond the logical operations of classical propositional logic. We exhibit a hierarchy of logics between the smallest, viz. classical propositional logic, and the maximal propositional team logic. We characterize these different logics in several ways: first syntactically by their logical operations, and then semantically by the kind of sets of teams they are capable of defining. In several important cases we are able to find complete axiomatizations for these logics.

Authors:Anton Freund Abstract: Publication date: Available online 6 January 2017 Source:Annals of Pure and Applied Logic Author(s): Anton Freund As Paris and Harrington have famously shown, Peano Arithmetic does not prove that for all numbers k , m , n there is an N which satisfies the statement PH ( k , m , n , N ) : For any k-coloring of its n-element subsets the set { 0 , … , N − 1 } has a large homogeneous subset of size ≥m. At the same time very weak theories can establish the Σ 1 -statement ∃ N PH ( k ‾ , m ‾ , n ‾ , N ) for any fixed parameters k , m , n . Which theory, then, does it take to formalize natural proofs of these instances? It is known that ∀ m ∃ N PH ( k ‾ , m , n ‾ , N ) has a natural and short proof (relative to n and k) by Σ n − 1 -induction. In contrast, we show that there is an elementary function e such that any proof of ∃ N PH ( e ( n ) ‾ , n + 1 ‾ , n ‾ , N ) by Σ n − 2 -induction is ridiculously long. In order to establish this result on proof lengths we give a computational analysis of slow provability, a notion introduced by Sy-David Friedman, Rathjen and Weiermann. We will see that slow uniform Σ 1 -reflection is related to a function that has a considerably lower growth rate than F ε 0 but dominates all functions F α with α < ε 0 in the fast-growing hierarchy.

Authors:Luck Abstract: Publication date: Available online 6 January 2017 Source:Annals of Pure and Applied Logic Author(s): Luck Darnière We introduce topological notions of polytopes and simplexes, the latter being expected to fulfil in p-adically closed fields the function of real simplexes in the classical results of triangulation of semi-algebraic sets over real closed fields. We prove that the faces of every p-adic polytope are polytopes and that they form a rooted tree with respect to specialisation. Simplexes are then defined as polytopes whose faces tree is a chain. Our main result is a construction allowing to divide every p-adic polytope in a complex of p-adic simplexes with prescribed faces and shapes.

Authors:Yasuo Yoshinobu Abstract: Publication date: Available online 6 January 2017 Source:Annals of Pure and Applied Logic Author(s): Yasuo Yoshinobu We introduce a property of posets which strengthens ( ω 1 + 1 ) -strategic closedness. This property is defined using a variation of the Banach–Mazur game on posets, where the first player chooses a countable set of conditions instead of a single condition at each turn. We prove PFA is preserved under any forcing over a poset with this property. As an application we reproduce a proof of Magidor's theorem about the consistency of PFA with some weak variations of the square principles. We also argue how different this property is from ( ω 1 + 1 ) -operational closedness, which we introduced in our previous work, by observing which portions of MA + ( ω 1 -closed ) are preserved or destroyed under forcing over posets with either property.

Authors:Erin Caulfield Abstract: Publication date: Available online 6 January 2017 Source:Annals of Pure and Applied Logic Author(s): Erin Caulfield We construct a class of finite rank multiplicative subgroups of the complex numbers such that the expansion of the real field by such a group is model-theoretically well-behaved. As an application we show that a classification of expansions of the real field by cyclic multiplicative subgroups of the complex numbers due to Hieronymi does not even extend to expansions by subgroups with two generators.

Authors:Ali Enayat; Tin Lok Wong Abstract: Publication date: Available online 2 January 2017 Source:Annals of Pure and Applied Logic Author(s): Ali Enayat, Tin Lok Wong We develop machinery to make the Arithmetized Completeness Theorem more effective in the study of many models of I Δ 0 + B Σ 1 + exp , including all countable ones, by passing on to the conservative extension WKL 0 ⁎ of I Δ 0 + B Σ 1 + exp . Our detailed study of the model theory of WKL 0 ⁎ leads to the simplification and improvement of many results in the model theory of Peano arithmetic and its fragments pertaining to the construction of various types of end extensions and initial segments.

Authors:Piotr Szewczak; Boaz Tsaban Abstract: Publication date: Available online 24 August 2016 Source:Annals of Pure and Applied Logic Author(s): Piotr Szewczak, Boaz Tsaban We construct Menger subsets of the real line whose product is not Menger in the plane. In contrast to earlier constructions, our approach is purely combinatorial. The set theoretic hypothesis used in our construction is far milder than earlier ones, and holds in almost all canonical models of set theory of the real line. On the other hand, we establish productive properties for versions of Menger's property parameterized by filters and semifilters. In particular, the Continuum Hypothesis implies that every productively Menger set of real numbers is productively Hurewicz, and each ultrafilter version of Menger's property is strictly between Menger's and Hurewicz's classic properties. We include a number of open problems emerging from this study.

Authors:A.D. Brooke-Taylor; Fischer S.D. Friedman D.C. Montoya Abstract: Publication date: Available online 24 August 2016 Source:Annals of Pure and Applied Logic Author(s): A.D. Brooke-Taylor, V. Fischer, S.D. Friedman, D.C. Montoya We provide a model where u ( κ ) < 2 κ for a supercompact cardinal κ. [10] provides a sketch of how to obtain such a model by modifying the construction in [6]. We provide here a complete proof using a different modification of [6] and further study the values of other natural generalizations of classical cardinal characteristics in our model. For this purpose we generalize some standard facts that hold in the countable case as well as some classical forcing notions and their properties.

Authors:Nathanael Ackerman; Cameron Freer; Aleksandra Kwiatkowska; Rehana Patel Abstract: Publication date: Available online 24 August 2016 Source:Annals of Pure and Applied Logic Author(s): Nathanael Ackerman, Cameron Freer, Aleksandra Kwiatkowska, Rehana Patel We consider the space of countable structures with fixed underlying set in a given countable language. We show that the number of ergodic probability measures on this space that are S ∞ -invariant and concentrated on a single isomorphism class must be zero, or one, or continuum. Further, such an isomorphism class admits a unique S ∞ -invariant probability measure precisely when the structure is highly homogeneous; by a result of Peter J. Cameron, these are the structures that are interdefinable with one of the five reducts of the rational linear order ( Q , < ) .