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

Hyperprojective Hierarchy of QCB_0-spaces

We extend the Luzin hierarchy of qcb$_0$-spaces introduced in [ScS13] to all countable ordinals, obtaining in this way the hyperprojective hierarchy of qcb$_0$-spaces. We generalize all main results of [ScS13] to this larger hierarchy. In particular, we extend the Kleene-Kreisel continuous functionals of finite types to the continuous functionals of countable types and relate them to the new hierarchy. We show that the category of hyperprojective qcb$_0$-spaces has much better closure properties than the category of projective qcb$_0$-space. As a result, there are natural examples of spaces that are hyperprojective but not projective.Comment: Conference version to appear in LNC

Splitting methods in algebraic logic: Proving results on non-atom-canonicity, non-finite axiomatizability and non-first oder definability for cylindric and relation algebras

We deal with various splitting methods in algebraic logic. The word `splitting' refers to splitting some of the atoms in a given relation or cylindric algebra each into one or more subatoms obtaining a bigger algebra, where the number of subatoms obtained after splitting is adjusted for a certain combinatorial purpose. This number (of subatoms) can be an infinite cardinal. The idea originates with Leon Henkin. Splitting methods existing in a scattered form in the literature, possibly under different names, proved useful in obtaining (negative) results on non-atom canonicity, non-finite axiomatizability and non-first order definability for various classes of relation and cylindric algebras. In a unified framework, we give several known and new examples of each. Our framework covers Monk's splitting, Andr\'eka's splitting, and, also, so-called blow up and blur constructions involving splitting (atoms) in finite Monk-like algebras and rainbow algebras.Comment: arXiv admin note: substantial text overlap with arXiv:1502.07701, arXiv:1408.328

Finite relation algebras and omitting types in modal fragments of first order logic

Let 2<n\leq l<m< \omega. Let L_n denote first order logic restricted to the first n variables. We show that the omitting types theorem fails dramatically for the n--variable fragments of first order logic with respect to clique guarded semantics, and for its packed n--variable fragments. Both are modal fragments of L_n. As a sample, we show that if there exists a finite relation algebra with a so--called strong l--blur, and no m--dimensional relational basis, then there exists a countable, atomic and complete L_n theory T and type \Gamma, such that \Gamma is realizable in every so--called m--square model of T, but any witness isolating \Gamma cannot use less than $l$ variables. An $m$--square model M of T gives a form of clique guarded semantics, where the parameter m, measures how locally well behaved M is. Every ordinary model is k--square for any n<k<\omega, but the converse is not true. Any model M is \omega--square, and the two notions are equivalent if M is countable. Such relation algebras are shown to exist for certain values of l and m like for n\leq l<\omega and m=\omega, and for l=n and m\geq n+3. The case l=n and m=\omega gives that the omitting types theorem fails for L_n with respect to (usual) Tarskian semantics: There is an atomic countable L_n theory T for which the single non--principal type consisting of co--atoms cannot be omitted in any model M of T. For n<\omega, positive results on omitting types are obained for L_n by imposing extra conditions on the theories and/or the types omitted. Positive and negative results on omitting types are obtained for infinitary variants and extensions of L_{\omega, \omega}.Comment: arXiv admin note: text overlap with arXiv:1408.3282, arXiv:1502.0770

On finite width questionable representations of orders

In this article, we study "questionable representations" of (partial or total) orders, introduced in our previous article "A class of orders with linear? time sorting algorithm". (Later, we consider arbitrary binary functional/relational structures instead of orders.) A "question" is the first difference between two sequences (with ordinal index) of elements of orders/sets. In finite width "questionable representations" of an order O, comparison can be solved by looking at the "question" that compares elements of a finite order O'. A corollary of a theorem by Cantor (1895)is that all countable total orders have a binary (width 2) questionable representation. We find new classes of orders on which testing isomorphism or counting the number of linear extensions can be done in polynomial time. We also present a generalization of questionable-width, called balanced tree-questionable-width, and show that if a class of binary structures has bounded tree-width or clique-width, then it has bounded balanced tree-questionable-width. But there are classes of graphs of bounded balanced tree-questionable-width and unbounded tree-width or clique-width.Comment: 50 page

Some Hierarchies of QCB_0-Spaces

We define and study hierarchies of topological spaces induced by the classical Borel and Luzin hierarchies of sets. Our hierarchies are divided into two classes: hierarchies of countably based spaces induced by their embeddings into the domain P\omega, and hierarchies of spaces (not necessarily countably based) induced by their admissible representations. We concentrate on the non-collapse property of the hierarchies and on the relationships between hierarchies in the two classes.Comment: 24 page

Problems on neat embeddings solved by rainbow constructions and Monk algebras

This paper is a survey of recent results and methods in (Tarskian) algebraic logic. We focus on cylindric algebras. Fix 2<n<\omega. Rainbow constructions are used to solve problems on classes consisting of algebras having a neat embedding property substantially generalizing seminal results of Hodkinson as well as Hirsch and Hodkinson on atom-canonicity and complete representations, respectively. For proving non-atom-canonicity of infinitely many varieties approximating the variety of representable algebras of dimension n, so-called blow up and blur constructions are used. Rainbow constructions are compared to constructions using Monk-like algebras and cases where both constructions work are given. When splitting methods fail. rainbow constructions are used to show that diagonal free varieties of representable diagonal free algebras of finite dimension n, do no admit universal axiomatizations containing only finitely many variables. Notions of representability, like complete, weak and strong are lifted from atom structures to atomic algebras and investigated in terms of neat embedding properties. The classical results of Monk and Maddux on non-finite axiomatizability of the classes of representable relation and cylindric algebras of finite dimension n are reproved using also a blow up and blur construction. Applications to n-variable fragments of first order logic are given. The main results of the paper are summarized in tabular form at the end of the paper.Comment: arXiv admin note: substantial text overlap with arXiv:1408.328

Well Quasiorders and Hierarchy Theory

We discuss some applications of WQOs to several fields were hierarchies and reducibilities are the principal classification tools, notably to Descriptive Set Theory, Computability theory and Automata Theory. While the classical hierarchies of sets usually degenerate to structures very close to ordinals, the extension of them to functions requires more complicated WQOs, and the same applies to reducibilities. We survey some results obtained so far and discuss open problems and possible research directions.Comment: 37 page

Algebraic analysis of temporal and topological finite variable fragments, using cylindric modal algebras

We study what we call topological cylindric algebras and tense cylindric algebras defined for every ordinal $\alpha$. The former are cylindric algebras of dimension $\alpha$ expanded with $\sf S4$ modalities indexed by $\alpha$. The semantics of representable topological algebras is induced by the interior operation relative to a topology defined on their bases. Tense cylindric algebras are cylindric algebras expanded by the modalities $F$(future) and $P$ (past) algebraising predicate temporal logic. We show for both tense and topological cylindric algebras of finite dimension $n>2$ that infinitely many varieties containing and including the variety of representable algebras of dimension $n$ are not atom canonical. We show that any class containing the class of completely representable algebras having a weak neat embedding property is not elementary. From these two results we draw the same conclusion on omitting types for finite variable fragments of predicate topologic and temporal logic. We show that the usual version of the omitting types theorem restricted to such fragments when the number of variables is $>2$ fails dramatically even if we considerably broaden the class of models permitted to omit a single non principal type in countable atomic theories, namely, the non-principal type consting of co atoms.Comment: arXiv admin note: substantial text overlap with arXiv:1308.6165, arXiv:1307.1016, arXiv:1309.0681, arXiv:1307.4298, arXiv:1401.1103, arXiv:1401.115

Three Lectures on Automatic Structures

This paper grew out of three tutorial lectures on automatic structures given by the first author at the Logic Colloquium 2007. We discuss variants of automatic structures related to several models of computation: word automata, tree automata, Buchi automata, and Rabin automata. Word automata process finite strings, tree automata process finite labeled trees, Buchi automata process infinite strings, and Rabin automata process infinite binary labeled trees. Automatic structures are mathematical objects which can be represented by (word, tree, Buchi, or Rabin) automata. The study of properties of automatic structures is a relatively new and very active area of research.Comment: 43 pages, based on tutorial lectures at Logic Colloquium 200