A Finite First-Order Presentation of Set Theory

We present a first-order formalization of set theory which has a finite number of axioms. Its syntax is similar to that often used in textbooks: it provides an encoding of the comprehension symbol. We prove that this formalization is a "conservative extension" of Zermelo's set theory. In fact the proof is more general and applies to other variants of Zermelo's set theory like ZF. This formalization rests upon an encoding of the comprehension binder in a language of explicit substitution. This presentation of set theory is also described as a deduction modulo system and the proof of equivalence is done within this formalism

Enumeration Reducibility in Closure Spaces with Applications to Logic and Algebra

In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of theorems in many finitely axiomatisable theories is nonrecursive, but the set of theorems for any finitely axiomatisable complete theory is recursive. Finitely presented groups might have an nonrecursive word problem, but finitely presented simple groups have a recursive word problem. In this article we introduce a topological framework based on closure spaces to show that many of these proofs can be obtained in a similar setting. We will show in particular that these statements can be generalized to cover arbitrary structures, with no finite or recursive presentation/axiomatization. This generalizes in particular work by Kuznetsov and others. Examples from first order logic and symbolic dynamics will be discussed at length

The Structure of First-Order Causality

Game semantics describe the interactive behavior of proofs by interpreting formulas as games on which proofs induce strategies. Such a semantics is introduced here for capturing dependencies induced by quantifications in first-order propositional logic. One of the main difficulties that has to be faced during the elaboration of this kind of semantics is to characterize definable strategies, that is strategies which actually behave like a proof. This is usually done by restricting the model to strategies satisfying subtle combinatorial conditions, whose preservation under composition is often difficult to show. Here, we present an original methodology to achieve this task, which requires to combine advanced tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of the monoidal category of definable strategies of our model, by the means of generators and relations: those strategies can be generated from a finite set of atomic strategies and the equality between strategies admits a finite axiomatization, this equational structure corresponding to a polarized variation of the notion of bialgebra. This work thus bridges algebra and denotational semantics in order to reveal the structure of dependencies induced by first-order quantifiers, and lays the foundations for a mechanized analysis of causality in programming languages

On bipartite Rokhsar-Kivelson points and Cantor deconfinement

Quantum dimer models on bipartite lattices exhibit Rokhsar-Kivelson (RK) points with exactly known critical ground states and deconfined spinons. We examine generic, weak, perturbations around these points. In d=2+1 we find a first order transition between a ``plaquette'' valence bond crystal and a region with a devil's staircase of commensurate and incommensurate valence bond crystals. In the part of the phase diagram where the staircase is incomplete, the incommensurate states exhibit a gapless photon and deconfined spinons on a set of finite measure, almost but not quite a deconfined phase in a compact U(1) gauge theory in d=2+1! In d=3+1 we find a continuous transition between the U(1) resonating valence bond (RVB) phase and a deconfined staggered valence bond crystal. In an appendix we comment on analogous phenomena in quantum vertex models, most notably the existence of a continuous transition on the triangular lattice in d=2+1.Comment: 9 pages; expanded version to appear in Phys. Rev. B; presentation improve

The Complexity of Combinations of Qualitative Constraint Satisfaction Problems

The CSP of a first-order theory $T$ is the problem of deciding for a given finite set $S$ of atomic formulas whether $T \cup S$ is satisfiable. Let $T_1$ and $T_2$ be two theories with countably infinite models and disjoint signatures. Nelson and Oppen presented conditions that imply decidability (or polynomial-time decidability) of $\mathrm{CSP}(T_1 \cup T_2)$ under the assumption that $\mathrm{CSP}(T_1)$ and $\mathrm{CSP}(T_2)$ are decidable (or polynomial-time decidable). We show that for a large class of $\omega$-categorical theories $T_1, T_2$ the Nelson-Oppen conditions are not only sufficient, but also necessary for polynomial-time tractability of $\mathrm{CSP}(T_1 \cup T_2)$ (unless P=NP).Comment: Version 2: stronger main result with better presentation of the proof; multiple improvements in other proofs; new section structure; new example

Presentation of a Game Semantics for First-Order Propositional Logic

Game semantics aim at describing the interactive behaviour of proofs by interpreting formulas as games on which proofs induce strategies. In this article, we introduce a game semantics for a fragment of first order propositional logic. One of the main difficulties that has to be faced when constructing such semantics is to make them precise by characterizing definable strategies - that is strategies which actually behave like a proof. This characterization is usually done by restricting to the model to strategies satisfying subtle combinatory conditions such as innocence, whose preservation under composition is often difficult to show. Here, we present an original methodology to achieve this task which requires to combine tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of definable strategies by the means of generators and relations: those strategies can be generated from a finite set of ``atomic'' strategies and that the equality between strategies generated in such a way admits a finite axiomatization. These generators satisfy laws which are a variation of bialgebras laws, thus bridging algebra and denotational semantics in a clean and unexpected way

Enumeration Reducibility in Closure Spaces with Applications to Logic and Algebra

In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of theorems in many finitely axiomatisable theories is nonrecursive, but the set of theorems for any finitely axiomatisable complete theory is recursive. Finitely presented groups might have an nonrecursive word problem, but finitely presented simple groups have a recursive word problem. In this article we introduce a topological framework based on closure spaces to show that many of these proofs can be obtained in a similar setting. We will show in particular that these statements can be generalized to cover arbitrary structures, with no finite or recursive presentation/axiomatization. This generalizes in particular work by Kuznetsov and others. Examples from first order logic and symbolic dynamics will be discussed at length

Geometric methods in the study of Pride groups and relative presentations

Combinatorial group theory is the study of groups given by presentations. Algebraic and geometric methods pervade this area of mathematics and it is the latter which forms the main theme of this thesis. In particular, we use diagrams and pictures over presentations to study problems in the domain of finitely presented groups. Our thesis is split into two distinct halves, though the techniques used in each are very similar. In Chapters 2 - 4 we study Pride groups with the aim to solve their word and conjugacy problems. We also study the second homotopy module of a natural presentation of a Pride group. Chapters 6 and 7 are devoted to the study of relative presentations, with particular attention being paid to those of the form . Determining when such presentations are aspherical is our main objective. Chapter 1 covers the basic material that is used throughout this thesis. The main topics of interest are free groups; presentations of groups; the word, conjugacy, and isomorphism problems for finitely presented groups; first and second order Dehn functions of finitely presented groups; diagrams and pictures over finite presentations; and the second homotopy module of a finite presentation. The reader may skip Chapter 1 if they are familiar with this material. A Pride group is a finitely presented group which can be defined by means of a finite simplicial graph; this is done in Chapter 2. Examples of Pride groups are given in Section 2.1. This section also contains the statements of Conditions (I), (II), (H-I), (H-II), and the asphericity condition. We will always assume that a Pride group satisfies one of these conditions. In Section 2.2 we survey the known results that appear in the literature, while in Section 2.3 we present our original results. We obtain isoperimetric functions for a vertex-finite Pride group G which satisfies (I), (II), (H-I) or (H-II). Sufficient conditions are then obtained for G to have a soluble word problem. Solutions of the conjugacy problem for G are also obtained. However, we require that G satisfies some extra conditions. We calculate a generating set for the second homotopy module of the natural presentation of a non-spherical Pride group, i.e. one which satisfies the asphericity condition. Using this generating set, we obtain an upper bound for the second order Dehn function of a non-spherical vertex-free Pride group. We also obtain information about the second order Dehn function of an arbitrary non-spherical Pride group. Chapter 3 contains various technical results that are needed in Chapter 4. The main focus is that of diagrams over the standard presentation of a vertex-finite Pride group. We study simply-connected r-diagrams in Section 3.1 and in Section 3.2 we study annular r-diagrams. Propositions 3.1.1, 3.2.1, 3.2.2, and Theorems 3.2.1 and 3.2.2 are the main results of this chapter. Chapters 4 and 5 are devoted to the proofs of our main results. Proofs of our results for the word and conjugacy problems of a vertex-finite Pride group are contained in Chapter 4, while Chapter 5 contains proofs of our results about the second homotopy module of a non-spherical Pride group. Chapter 5 also contains a study of pictures over the natural presentation of such a group. In Chapter 6, we turn our attention to relative presentations. Our interest lies in determining when such presentations are aspherical. Relevant background material and definitions are given in this chapter and pictures over relative presentations are also studied. Five tests which are used to determine whether or not a relative presentation is aspherical are given in Section 6.4. Chapter 6 also contains a brief survey of known results in this area. In Chapter 7, the final chapter of this thesis, we present our original contribution to the area of aspherical relative presentations. In particular, we determine when the relative presentation is aspherical where n is greater than or equal to 4 and a, b are elements of H each of order at least 3. There are four exceptional cases for which asphericity cannot be determined

Nuclear chiral dynamics and thermodynamics

This presentation reviews an approach to nuclear many-body systems based on the spontaneously broken chiral symmetry of low-energy QCD. In the low-energy limit, for energies and momenta small compared to a characteristic symmetry breaking scale of order 1 GeV, QCD is realized as an effective field theory of Goldstone bosons (pions) coupled to heavy fermionic sources (nucleons). Nuclear forces at long and intermediate distance scales result from a systematic hierarchy of one- and two-pion exchange processes in combination with Pauli blocking effects in the nuclear medium. Short distance dynamics, not resolved at the wavelengths corresponding to typical nuclear Fermi momenta, are introduced as contact interactions between nucleons. Apart from a set of low-energy constants associated with these contact terms, the parameters of this theory are entirely determined by pion properties and low-energy pion-nucleon scattering observables. This framework (in-medium chiral perturbation theory) can provide a realistic description of both isospin-symmetric nuclear matter and neutron matter. The importance of three-body forces is emphasized, and the role of explicit Delta(1232)-isobar degrees of freedom is investigated in detail. Nuclear chiral thermodynamics is developed and a calculation of the nuclear phase diagram is performed. This includes a successful description of the first-order phase transition from a nuclear Fermi liquid to an interacting Fermi gas and the coexistence of these phases below a critical temperature T_c. Density functional methods for finite nuclei based on this approach are also discussed. Effective interactions, their density dependence and connections to Landau Fermi liquid theory are outlined. Finally, the density and temperature dependence of the chiral (quark) condensate is investigated.Comment: 63 pages, 45 figure

Independence property and hyperbolic groups

We prove that existentially closed $CSA$-groups have the independence property. This is done by showing that there exist words having the independence property relatively to the class of torsion-free hyperbolic groups.Comment: v3: 10 pages (11pt), a few typos corrected, minor rearrangements (e.g. Fact 2.3 and Lemma 2.5); v2: 8 pages (10pt), a false statement in the proof of Fact 2.4 is replaced with a true one; v1: 8 page