Cylindric modal logic

1995 ◽  
Vol 60 (2) ◽  
pp. 591-623 ◽  
Author(s):  
Yde Venema
Keyword(s):  
Algebraic Logic ◽  
Equational Theory ◽  
Completeness Theorem ◽  
Cylindric Algebras ◽  
First Order ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Existential Quantification ◽  
Cylindric Set Algebras

AbstractTreating the existential quantification ∃νi as a diamond ♢i and the identity νi = νj as a constant δij, we study restricted versions of first order logic as if they were modal formalisms. This approach is closely related to algebraic logic, as the Kripke frames of our system have the type of the atom structures of cylindric algebras; the full cylindric set algebras are the complex algebras of the intended multidimensional frames called cubes.The main contribution of the paper is a characterization of these cube frames for the finite-dimensional case and, as a consequence of the special form of this characterization, a completeness theorem for this class. These results lead to finite, though unorthodox, derivation systems for several related formalisms, e.g. for the valid n-variable first order formulas, for type-free valid formulas and for the equational theory of representable cylindric algebras. The result for type-free valid formulas indicates a positive solution to Problem 4.16 of Henkin, Monk and Tarski [16].

scholarly journals On weak and strong interpolation in algebraic logics

2006 ◽  
Vol 71 (1) ◽  
pp. 104-118 ◽  
Author(s):  
Gábor Sági ◽  
Saharon Shelah
Keyword(s):  
Algebraic Logic ◽  
Amalgamation Property ◽  
Weak Form ◽  
Interpolation Theorem ◽  
Order Logic ◽  
Cylindric Algebras ◽  
First Order ◽  
Finite Dimensional ◽  
Craig’S Interpolation Theorem ◽  
Superamalgamation Property

AbstractWe show that there is a restriction, or modification of the finite-variable fragments of First Order Logic in which a weak form of Craig's Interpolation Theorem holds but a strong form of this theorem does not hold. Translating these results into Algebraic Logic we obtain a finitely axiomatizable subvariety of finite dimensional Representable Cylindric Algebras that has the Strong Amalgamation Property but does not have the Superamalgamation Property. This settles a conjecture of Pigozzi [12].

scholarly journals Decidability of Definability

2013 ◽  
Vol 78 (4) ◽  
pp. 1036-1054 ◽  
Author(s):  
Manuel Bodirsky ◽  
Michael Pinsker ◽  
Todor Tsankov
Keyword(s):  
Random Graph ◽  
Topological Dynamics ◽  
Homogeneous Structure ◽  
First Order ◽  
Computational Problem ◽  
Random Tournament ◽  
Universal Quantification ◽  
Countably Infinite ◽  
Existential Quantification

AbstractFor a fixed countably infinite structure Γ with finite relational signature τ, we study the following computational problem: input are quantifier-free τ-formulas ϕ0, ϕ1, …, ϕn that define relations R0, R1, …, Rn over Γ. The question is whether the relation R0 is primitive positive definable from R1, …, Rn, i.e., definable by a first-order formula that uses only relation symbols for R1, …, Rn, equality, conjunctions, and existential quantification (disjunction, negation, and universal quantification are forbidden).We show decidability of this problem for all structures Γ that have a first-order definition in an ordered homogeneous structure Δ with a finite relational signature whose age is a Ramsey class and determined by finitely many forbidden substructures. Examples of structures Γ with this property are the order of the rationals, the random graph, the homogeneous universal poset, the random tournament, all homogeneous universal C-relations, and many more. We also obtain decidability of the problem when we replace primitive positive definability by existential positive, or existential definability. Our proof makes use of universal algebraic and model theoretic concepts, Ramsey theory, and a recent characterization of Ramsey classes in topological dynamics.

scholarly journals Undecidable theories of Lyndon algebras

2001 ◽  
Vol 66 (1) ◽  
pp. 207-224 ◽  
Author(s):  
Vera Stebletsova ◽  
Yde Venema
Keyword(s):  
Projective Geometry ◽  
Algebraic Logic ◽  
Equational Theory ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
Projective Geometries ◽  
Interesting Connection

AbstractWith each projective geometry we can associate a Lyndon algebra. Such an algebra always satisfies Tarski's axioms for relation algebras and Lyndon algebras thus form an interesting connection between the fields of projective geometry and algebraic logic. In this paper we prove that if G is a class of projective geometries which contains an infinite projective geometry of dimension at least three, then the class L(G) of Lyndon algebras associated with projective geometries in G has an undecidable equational theory. In our proof we develop and use a connection between projective geometries and diagonal-free cylindric algebras.

On the number of generators of cylindric algebras

1985 ◽  
Vol 50 (4) ◽  
pp. 865-873
Author(s):  
H. Andréka ◽  
I. Németi
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Single Element ◽  
Abstract Model ◽  
Finitely Generated ◽  
Cylindric Algebras ◽  
Abstract Model Theory ◽  
First Order ◽  
Cylindric Set Algebras

The theory of cylindric algebras (CA's) is the algebraic theory of first order logics. Several ideas about logic are easier to formulate in the frame of CA-theory. Such are e.g. some concepts of abstract model theory (cf. [1] and [10]–[12]) as well as ideas about relationships between several axiomatic theories of different similarity types (cf. [4] and [10]). In contrast with the relationship between Boolean algebras and classical propositional logic, CA's correspond not only to classical first order logic but also to several other ones. Hence CA-theoretic results contain more information than their counterparts in first order logic. For more about this see [1], [3], [5], [9], [10] and [12].Here we shall use the notation and concepts of the monographs Henkin-Monk-Tarski [7] and [8]. ω denotes the set of natural numbers. CAα denotes the class of all cylindric algebras of dimension α; by “a CAα” we shall understand an element of the class CAα. The class Dcα ⊆ CAα was defined in [7]. Note that Dcα = 0 for α ∈ ω. The classes Wsα, and Csα were defined in 1.1.1 of [8], p. 4. They are called the classes of all weak cylindric set algebras, regular cylindric set algebras and cylindric set algebras respectively. It is proved in [8] (I.7.13, I.1.9) that ⊆ CAα. (These inclusions are proper by 7.3.7, 1.4.3 and 1.5.3 of [8].)It was proved in 2.3.22 and 2.3.23 of [7] that every simple, finitely generated Dcα is generated by a single element. This is the algebraic counterpart of a property of first order logics (cf. 2.3.23 of [7]). The question arose: for which simple CAα's does “finitely generated” imply “generated by a single element” (see p. 291 and Problem 2.3 in [7]). In terms of abstract model theory this amounts to asking the question: For which logics does the property described in 2.3.23 of [7] hold? This property is roughly the following. In any maximal theory any finite set of concepts is definable in terms of a single concept. The connection with CA-theory is that maximal theories correspond to simple CA's (the elements of which are the concepts of the original logic) and definability corresponds to generation.

Relation algebra reducts of cylindric algebras and an application to proof theory

2002 ◽  
Vol 67 (1) ◽  
pp. 197-213 ◽  
Author(s):  
Robin Hirsch ◽  
Ian Hodkinson ◽  
Roger D. Maddux
Keyword(s):  
Proof Theory ◽  
Predicate Logic ◽  
Algebraic Logic ◽  
Relation Algebra ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
First Order ◽  
Order Language ◽  
Binary Relation Symbol ◽  
Made In

AbstractWe confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal α ≥ 3 there is a logically valid sentence X, in a first-order language ℒ with equality and exactly one nonlogical binary relation symbol E, such that X contains only 3 variables (each of which may occur arbitrarily many times), X has a proof containing exactly α + 1 variables, but X has no proof containing only α variables. This solves a problem posed by Tarski and Givant in 1987.

Copeland algebras

1972 ◽  
Vol 37 (4) ◽  
pp. 646-656 ◽  
Author(s):  
Daniel B. Demaree
Keyword(s):  
Predicate Logic ◽  
Algebraic Logic ◽  
Boolean Algebras ◽  
Primary Concern ◽  
Cylindric Algebras ◽  
First Order ◽  
Polyadic Algebras ◽  
Algebraic Setting ◽  
Finite Set ◽  
Natural Way

It is well known that the laws of logic governing the sentence connectives—“and”, “or”, “not”, etc.—can be expressed by means of equations in the theory of Boolean algebras. The task of providing a similar algebraic setting for the full first-order predicate logic is the primary concern of algebraic logicians. The best-known efforts in this direction are the polyadic algebras of Halmos (cf. [2]) and the cylindric algebras of Tarski (cf. [3]), both of which may be described as Boolean algebras with infinitely many additional operations. In particular, there is a primitive operator, cκ, corresponding to each quantification, ∃υκ. In this paper we explore a version of algebraic logic conceived by A. H. Copeland, Sr., and described in [1], which has this advantage: All operators are generated from a finite set of primitive operations.Following the theory of cylindric algebras, we introduce, in the natural way, the classes of Copeland set algebras (SCpA), representable Copeland algebras (RCpA), and Copeland algebras of formulas. Playing a central role in the discussion is the set, Γ, of all equations holding in every set algebra. The reason for this is that the operations in a set algebra reflect the notion of satisfaction of a formula in a model, and hence an equation expresses the fact that two formulas are satisfied by the same sequences of objects in the model. Thus to say that an equation holds in every set algebra is to assert that a certain pair of formulas are logically equivalent.

The equational theory of CA3 is undecidable

1980 ◽  
Vol 45 (2) ◽  
pp. 311-316 ◽  
Author(s):  
Roger Maddux
Keyword(s):  
Equational Theory ◽  
Cylindric Algebra ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
3 Dimensional ◽  
First Order ◽  
Equational Theories ◽  
Order Language ◽  
Theoretic Proof

There is no algorithm for determining whether or not an equation is true in every 3-dimensional cylindric algebra. This theorem completes the solution to the problem of finding those values of α and β for which the equational theories of CAα and RCAβ are undecidable. (CAα and RCAβ are the classes of α-dimensional cylindric algebras and representable β-dimensional cylindric algebras. See [4] for definitions.) This problem was considered in [3]. It was known that RCA0 = CA0 and RCA1 = CA1 and that the equational theories of these classes are decidable. Tarski had shown that the equational theory of relation algebras is undecidable and, by utilizing connections between relation algebras and cylindric algebras, had also shown that the equational theories of CAα and RCAβ are undecidable whenever 4 ≤ α and 3 ≤ β. (Tarski's argument also applies to some varieties K ⊆ RCAβ with 3 ≤ β and to any variety K such that RCAα ⊆ K ⊆ CAα and 4 ≤ α.)Thus the only cases left open in 1961 were CA2, RCA2 and CA3. Shortly there-after Henkin proved, in one of Tarski's seminars at Berkeley, that the equational theory of CA2 is decidable, and Scott proved that the set of valid sentences in a first-order language with only two variables is recursive [11]. (For a more model-theoretic proof of Scott's theorem see [9].) Scott's result is equivalent to the decidability of the equational theory of RCA2, so the only case left open was CA3.

Relativised quantification: Some canonical varieties of sequence-set algebras

1998 ◽  
Vol 63 (1) ◽  
pp. 163-184 ◽  
Author(s):  
Hajnal Andréka ◽  
Robert Goldblatt ◽  
István Németi
Keyword(s):  
Existential Quantifier ◽  
Cylindric Algebras ◽  
Order Model ◽  
Assigned Value ◽  
First Order ◽  
Cartesian Space ◽  
A Value ◽  
Quantificational Logic ◽  
Image Position ◽  
Existential Quantification

This paper explores algebraic aspects of two modifications of the usual account of first-order quantifiers.Standard first-order quantificational logic is modelled algebraically by cylindric algebras. Prime examples of these are algebras whose members are sets of sequences: given a first-order model U for a language that is based on the set {υκ: κ < α} of variables, each formula φ is represented by the setof all those α-length sequences x = 〈xκ: κ < α〉 that satisfy φ in U. Such a sequence provides a value-assignment to the variables (υκ is assigned value xκ), but it may also be viewed geometrically as a point in the α-dimensional Cartesian spaceαU of all α-length sequences whose terms come from the underlying set U of U. Then existential quantification is represented by the operation of cylindrification. To explain this, define a binary relation Tκ on sequences by putting xTκy if and only if x and y differ at most at their κth coordinate, i.e.,Then for any set X ⊆ αU, the setis the “cylinder” generated by translation of X parallel to the κth coordinate axis in αU. Given the standard semantics for the existential quantifier ∃υκ asit is evident that

Omitting types for finite variable fragments and complete representations of algebras

2008 ◽  
Vol 73 (1) ◽  
pp. 65-89 ◽  
Author(s):  
Hajnal Andréka ◽  
István Németi ◽  
Tarek Sayed Ahmed
Keyword(s):  
Algebraic Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Cylindric Algebras ◽  
First Order ◽  
Complete Representations ◽  
Atomic Relation ◽  
Representations Of Algebras ◽  
Omitting Types ◽  
Binary Relation Symbol

AbstractWe give a novel application of algebraic logic to first order logic. A new, flexible construction is presented for representable but not completely representable atomic relation and cylindric algebras of dimension n (for finite n > 2) with the additional property that they are one-generated and the set of all n by n atomic matrices forms a cylindric basis. We use this construction to show that the classical Henkin-Orey omitting types theorem fails for the finite variable fragments of first order logic as long as the number of variables available is > 2 and we have a binary relation symbol in our language. We also prove a stronger result to the effect that there is no finite upper bound for the extra variables needed in the witness formulas. This result further emphasizes the ongoing interplay between algebraic logic and first order logic.

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