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.

Non-finite-axiomatizability results in algebraic logic

1992 ◽  
Vol 57 (3) ◽  
pp. 832-843 ◽  
Author(s):  
Balázs Biró
Keyword(s):  
Algebraic Logic ◽  
Relation Algebra ◽  
Equational Theory ◽  
Negative Answer ◽  
Relation Algebras ◽  
Representable Relation Algebra ◽  
Finite Set ◽  
Variable Symbol ◽  
Axiom Systems ◽  
Representable Relation

This paper deals with relation, cylindric and polyadic equality algebras. First of all it addresses a problem of B. Jónsson. He asked whether relation set algebras can be expanded by finitely many new operations in a “reasonable” way so that the class of these expansions would possess a finite equational base. The present paper gives a negative answer to this problem: Our main theorem states that whenever Rs+ is a class that consists of expansions of relation set algebras such that each operation of Rs+ is logical in Jónsson's sense, i.e., is the algebraic counterpart of some (derived) connective of first-order logic, then the equational theory of Rs+ has no finite axiom systems. Similar results are stated for the other classes mentioned above. As a corollary to this theorem we can solve a problem of Tarski and Givant [87], Namely, we claim that the valid formulas of certain languages cannot be axiomatized by a finite set of logical axiom schemes. At the same time we give a negative solution for a version of a problem of Henkin and Monk [74] (cf. also Monk [70] and Németi [89]).Throughout we use the terminology, notation and results of Henkin, Monk, Tarski [71] and [85]. We also use results of Maddux [89a].Notation. RA denotes the class of relation algebras, Rs denotes the class of relation set algebras and RRA is the class of representable relation algebras, i.e. the class of subdirect products of relation set algebras. The symbols RA, Rs and RRA abbreviate also the expressions relation algebra, relation set algebra and representable relation algebra, respectively.For any class C of similar algebras EqC is the set of identities that hold in C, while Eq1C is the set of those identities in EqC that contain at most one variable symbol. (We note that Henkin et al. [85] uses the symbol EqC in another sense.)

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.

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.

Step by step – Building representations in algebraic logic

1997 ◽  
Vol 62 (1) ◽  
pp. 225-279 ◽  
Author(s):  
Robin Hirsch ◽  
Ian Hodkinson
Keyword(s):  
Algebraic Logic ◽  
Relation Algebra ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
Step Method ◽  
Finite Case ◽  
Representable Relation Algebra ◽  
Important Open Problem ◽  
Player Game ◽  
Representable Relation

AbstractWe consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Finte relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is ω-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable.An important open problem from algebraic logic is addressed by devising another two-player game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras.Other instances of this approach are looked at, and include the step by step method.

scholarly journals On complete representations and minimal completions in algebraic logic, both positive and negative results

2021 ◽  
Author(s):  
Tarek Sayed Ahmed
Keyword(s):  
Algebraic Logic ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
Polyadic Algebras ◽  
Negative Results ◽  
Complete Representations

Fix a finite ordinal \(n\geq 3\) and let \(\alpha\) be an arbitrary ordinal. Let \(\mathsf{CA}_n\) denote the class of cylindric algebras of dimension \(n\) and \(\sf RA\) denote the class of relation algebras. Let \(\mathbf{PA}_{\alpha}(\mathsf{PEA}_{\alpha})\) stand for the class of polyadic (equality) algebras of dimension \(\alpha\). We reprove that the class \(\mathsf{CRCA}_n\) of completely representable \(\mathsf{CA}_n$s, and the class \(\sf CRRA\) of completely representable \(\mathsf{RA}\)s are not elementary, a result of Hirsch and Hodkinson. We extend this result to any variety \(\sf V\) between polyadic algebras of dimension \(n\) and diagonal free \(\mathsf{CA}_n\)s. We show that that the class of completely and strongly representable algebras in \(\sf V\) is not elementary either, reproving a result of Bulian and Hodkinson. For relation algebras, we can and will, go further. We show the class \(\sf CRRA\) is not closed under \(\equiv_{\infty,\omega}\). In contrast, we show that given \(\alpha\geq \omega\), and an atomic \(\mathfrak{A}\in \mathsf{PEA}_{\alpha}\), then for any \(n<\omega\), \(\mathfrak{Nr}_n\A\) is a completely representable \(\mathsf{PEA}_n\). We show that for any \(\alpha\geq \omega\), the class of completely representable algebras in certain reducts of \(\mathsf{PA}_{\alpha}\)s, that happen to be varieties, is elementary. We show that for \(\alpha\geq \omega\), the the class of polyadic-cylindric algebras dimension \(\alpha\), introduced by Ferenczi, the completely representable algebras (slightly altering representing algebras) coincide with the atomic ones. In the last algebras cylindrifications commute only one way, in a sense weaker than full fledged commutativity of cylindrifications enjoyed by classical cylindric and polyadic algebras. Finally, we address closure under Dedekind-MacNeille completions for cylindric-like algebras of dimension \(n\) and \(\mathsf{PA}_{\alpha}\)s for \(\alpha\) an infinite ordinal, proving negative results for the first and positive ones for the second.

An undecidability result for relation algebras

1979 ◽  
Vol 44 (1) ◽  
pp. 111-115 ◽  
Author(s):  
Wolfgang Schönfeld
Keyword(s):  
Algebraic Logic ◽  
Equational Theory ◽  
Main Tool ◽  
Binary Relations ◽  
Relation Algebras ◽  
Finite Structures ◽  
Elementary Calculus ◽  
Primitive Recursive ◽  
Undecidability Result ◽  
Order Predicate Calculus

The elementary calculus of binary relations as developed by Tarski in [5] may be thought of as a certain part of the first-order predicate calculus. Though less expressive, its theory (i.e. the set of its valid sentences) was shown to be undecidable by Tarski in [6]. Translated into algebraic logic this means that the equational theory of the class of relation algebras is undecidable. Similarly it can be proved that the same holds for the (sub-) class of proper relation algebras.The idea in Tarski's proof is to describe a pairing function by which any quantifier prefix may be contracted. In this note we apply a different method to treat the case of finite structures. We prove theTheorem. The equational theory of the class of finite proper relation algebras is undecidable.This result was announced in [4]. The main tool is representing the graph of primitive recursive functions via the cardinalities in finite simple models of equations.

Complete representations in algebraic logic

1997 ◽  
Vol 62 (3) ◽  
pp. 816-847 ◽  
Author(s):  
Robin Hirsch ◽  
Ian Hodkinson
Keyword(s):  
Boolean Algebra ◽  
Algebraic Logic ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
Fixed Dimension ◽  
Complete Representations ◽  
Representable Relation

AbstractA boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary.

Algebraic Logic, Where Does it Stand Today?

2005 ◽  
Vol 11 (4) ◽  
pp. 465-516 ◽  
Author(s):  
Tarek Sayed Ahmed
Keyword(s):  
Proof Theory ◽  
Algebraic Logic ◽  
Theory Model ◽  
Historical Background ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
Open Problems ◽  
The Subject ◽  
Omitting Types ◽  
Modern Perspective

AbstractThis is a survey article on algebraic logic. It gives a historical background leading up to a modern perspective. Central problems in algebraic logic (like the representation problem) are discussed in connection to other branches of logic, like modal logic, proof theory, model-theoretic forcing, finite combinatorics, and Gödel's incompleteness results. We focus on cylindric algebras. Relation algebras and polyadic algebras are mostly covered only insofar as they relate to cylindric algebras, and even there we have not told the whole story. We relate the algebraic notion of neat embeddings (a notion special to cylindric algebras) to the metalogical ones of provability, interpolation and omitting types in variants of first logic. Another novelty that occurs here is relating the algebraic notion of atom-canonicity for a class of boolean algebras with operators to the metalogical one of omitting types for the corresponding logic. A hitherto unpublished application of algebraic logic to omitting types of first order logic is given. Proofs are included when they serve to illustrate certain concepts. Several open problems are posed. We have tried as much as possible to avoid exploring territory already explored in the survey articles of Monk [93] and Németi [97] in the subject.

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].

The contributions of Alfred Tarski to algebraic logic

1986 ◽  
Vol 51 (4) ◽  
pp. 899-906 ◽  
Author(s):  
J. Donald Monk
Keyword(s):  
Algebraic Logic ◽  
Boolean Algebras ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
Boolean Algebras With Operators ◽  
Alfred Tarski ◽  
The Subject ◽  
Closure Algebras

One of the most extensive parts of Tarski's contributions to logic is his work on the algebraization of the subject. His work here involves Boolean algebras, relation algebras, cylindric algebras, Boolean algebras with operators, Brouwerian algebras, and closure algebras. The last two are less developed in his work, although his contributions are basic to other work in those subjects. At any rate, not being conversant with the latest developments in those fields, we shall concentrate on an exposition of Tarski's work in the first four areas, trying to put them in the perspective of present-day developments.For useful comments, criticisms, and suggestions, the author is indebted to Steven Givant, Leon Henkin, Wilfrid Hodges, Bjarni Jónsson, Roger Lyndon, and Robert Vaught.

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