Additive G\"{o}del Logic

We study an extension of \g propositional logic whose corresponding algebra is an ordered Abelian group. Then we expand the ideas to first-order case of this logic

An Integrated Programming and Development Environment for Adiabatic Quantum Optimization

Adiabatic quantum computing is a promising route to the computational power afforded by quantum information processing. The recent availability of adiabatic hardware has raised challenging questions about how to evaluate adiabatic quantum optimization programs. Processor behavior depends on multiple steps to synthesize an adiabatic quantum program, which are each highly tunable. We present an integrated programming and development environment for adiabatic quantum optimization called JADE that provides control over all the steps taken during program synthesis. JADE captures the workflow needed to rigorously specify the adiabatic quantum optimization algorithm while allowing a variety of problem types, programming techniques, and processor configurations. We have also integrated JADE with a quantum simulation engine that enables program profiling using numerical calculation. The computational engine supports plug-ins for simulation methodologies tailored to various metrics and computing resources. We present the design, integration, and deployment of JADE and discuss its potential use for benchmarking adiabatic quantum optimization programs by the quantum computer science community.Comment: 28 pages, 17 figures, feedback welcomed, even if it's criticism; v2 manuscript updated based on reviewer feedback; v3 manuscript updated based on reviewer feedback, title modifie

A brief history of algebraic logic from neat embeddings to rainbow constructions

We take a long magical tour in algebraic logic, starting from classical results on neat embeddings due to Henkin, Monk and Tarski, all the way to recent results in algebraic logic using so--called rainbow constructions invented by Hirsch and Hodkinson. Highlighting the connections with graph theory, model theory, and finite combinatorics, this article aspires to present topics of broad interest in a way that is hopefully accessible to a large audience. The paper has a survey character but it contains new approaches to old ones. We aspire to make our survey fairly comprehensive, at least in so far as Tarskian algebraic logic, specifically, the theory of cylindric algebras, is concerned. Other topics, such as abstract algebraic logic, modal logic and the so--called (central) finitizability problem in algebraic logic will be dealt with; the last in some detail. Rainbow constructions are used to solve problems adressing classes of cylindric--like algebras consisting of algebras having a neat embedding property. The hitherto obtained results generalize seminal results of Hirsch and Hodkinson on non--atom canonicity, non--first order definabiity and non--finite axiomatizability, proved for classes of representable cylindric algebras of finite dimension$>2$. We show that such results remain valid for cylindric algebras possesing relativized {\it clique guarded} representations that are {\it only locally} well behaved. The paper is written in a way that makes it accessible to non--specialists curious about the state of the art in Tarskian algebraic logic. Reaching the boundaries of current research, the paper also aspires to be informative to the practitioner, and even more, stimulates her/him to carry on further research in main stream algebraic logic

Characterizing large cardinals through Neeman's pure side condition forcing

We show that some of the most prominent large cardinal notions can be characterized through the validity of certain combinatorial principles at $\omega_2$ in forcing extensions by the pure side condition forcing introduced by Neeman. The combinatorial properties that we make use of are natural principles, and in particular for inaccessible cardinals, these principles are equivalent to their corresponding large cardinal properties. Our characterizations make use of the concepts of internal large cardinals introduced in this paper, and of the classical concept of generic elementary embeddings.Comment: 28 page

Categoricity of theories in L_{kappa^* omega}, when kappa^* is a measurable cardinal. Part II

We continue the work of [KlSh:362] and prove that for lambda successor, a lambda-categorical theory T in L_{kappa^*, omega} is mu-categorical for every mu, mu <= lambda which is above the (2^{LS(T)})^+-beth cardinal

A Lindstr\"om theorem for intuitionistic propositional logic

It is shown that propositional intuitionistic logic is the maximal (with respect to expressive power) abstract logic satisfying a certain topological property reminiscent of compactness, the Tarski union property and preservation under asimulations

A model theoretic solution to a problem of L\'{a}szl\'{o} Fuchs

Problem 5.1 in page 181 of [Fuc15] asks to find the cardinals $\lambda$ such that there is a universal abelian $p$-group for purity of cardinality $\lambda$, i.e., an abelian $p$-group $U_\lambda$ of cardinality $\lambda$ such that every abelian $p$-group of cardinality $\leq \lambda$ purely embeds in $U_\lambda$. In this paper we use ideas from the theory of abstract elementary classes to show: $\textbf{Theorem.}$ Let $p$ be a prime number. If $\lambda^{\aleph_0}=\lambda$ or $\forall \mu < \lambda( \mu^{\aleph_0} < \lambda)$, then there is a universal abelian $p$-group for purity of cardinality $\lambda$. Moreover for $n\geq 2$, there is a universal abelian $p$-group for purity of cardinality $\aleph_n$ if and only if $2^{\aleph_0} \leq \aleph_n$. As the theory of abstract elementary classes has barely been used to tackle algebraic questions, an effort was made to introduce this theory from an algebraic perspective.Comment: 10 page

Model Checking Existential Logic on Partially Ordered Sets

We study the problem of checking whether an existential sentence (that is, a first-order sentence in prefix form built using existential quantifiers and all Boolean connectives) is true in a finite partially ordered set (in short, a poset). A poset is a reflexive, antisymmetric, and transitive digraph. The problem encompasses the fundamental embedding problem of finding an isomorphic copy of a poset as an induced substructure of another poset. Model checking existential logic is already NP-hard on a fixed poset; thus we investigate structural properties of posets yielding conditions for fixed-parameter tractability when the problem is parameterized by the sentence. We identify width as a central structural property (the width of a poset is the maximum size of a subset of pairwise incomparable elements); our main algorithmic result is that model checking existential logic on classes of finite posets of bounded width is fixed-parameter tractable. We observe a similar phenomenon in classical complexity, where we prove that the isomorphism problem is polynomial-time tractable on classes of posets of bounded width; this settles an open problem in order theory. We surround our main algorithmic result with complexity results on less restricted, natural neighboring classes of finite posets, establishing its tightness in this sense. We also relate our work with (and demonstrate its independence of) fundamental fixed-parameter tractability results for model checking on digraphs of bounded degree and bounded clique-width.Comment: accepted at CSL-LICS 201

Neat embeddings as adjoint situations

We view the neat reduct operator as a functor that lessens dimensions from CA_{\alpha+\omega} to CA_{\alpha} for infinite ordinals \alpha. We show that this functor has no right adjoint. Conversely for polyadic algebras, and several reducts thereof, like Sain's algebras, we show that the analagous functor is an equivalence.Comment: arXiv admin note: substantial text overlap with arXiv:1303.738

Polish G-spaces and continuous logic

We analyse logic actions of Polish groups which arise in continuous logic. We extend the generalised model theory of H.Becker to the case of Polish G-spaces when G is an arbitrary Polish group.Comment: 54 pages, Section 3 is correcte