Relation algebras of every dimension

1992 ◽  
Vol 57 (4) ◽  
pp. 1213-1229 ◽  
Author(s):  
Roger D. Maddux
Keyword(s):  
Binary Relation ◽  
Sequent Calculus ◽  
Logical Consequence ◽  
Relation Algebra ◽  
Binary Relations ◽  
Relation Algebras

AbstractConjecture (1) of [Ma83] is confirmed here by the following result: if 3 ≤ α < ω, then there is a finite relation algebra of dimension α, which is not a relation algebra of dimension α + 1. A logical consequence of this theorem is that for every finite α ≥ 3 there is a formula of the form S ⊆ T (asserting that one binary relation is included in another), which is provable with α + 1 variables, but not provable with only α variables (using a special sequent calculus designed for deducing properties of binary relations).

Undecidable semiassociative relation algebras

1994 ◽  
Vol 59 (2) ◽  
pp. 398-418 ◽  
Author(s):  
Roger D. Maddux
Keyword(s):  
Word Problem ◽  
Free Algebra ◽  
Relation Algebra ◽  
Binary Relations ◽  
Relation Algebras ◽  
Undecidable Theory ◽  
Algebraic Result

AbstractIf K is a class of semiassociative relation algebras and K contains the relation algebra of all binary relations on a denumerable set, then the word problem for the free algebra over K on one generator is unsolvable. This result implies that the set of sentences which are provable in the formalism ℒw× is an undecidable theory. A stronger algebraic result shows that the set of logically valid sentences in ℒw× forms a hereditarily undecidable theory in ℒw×. These results generalize similar theorems, due to Tarski, concerning relation algebras and the formalism ℒ×.

Groups and Algebras of Binary Relations

2002 ◽  
Vol 8 (1) ◽  
pp. 38-64 ◽  
Author(s):  
Steven Givant ◽  
Hajnal Andréka
Keyword(s):  
Positive Solution ◽  
Relation Algebra ◽  
Binary Relations ◽  
Relation Algebras ◽  
Abstract Theory ◽  
Atomic Relation ◽  
Measurability Condition ◽  
Positive Results ◽  
Representable Relation ◽  
Special Classes

AbstractIn 1941, Tarski published an abstract, finitely axiomatized version of the theory of binary relations, called the theory of relation algebras. He asked whether every model of his abstract theory could be represented as a concrete algebra of binary relations. He and Jónsson obtained some initial, positive results for special classes of abstract relation algebras. But Lyndon showed, in 1950, that in general the answer to Tarski's question is negative. Monk proved later that the answer remains negative even if one adjoins finitely many new axioms to Tarski's system. In this paper we describe a far-reaching generalization of the positive results of Jónsson and Tarski, as well as of some later, related results of Maddux. We construct a class of concrete models of Tarski's axioms—called coset relation algebras—that are very close in spirit to algebras of binary relations, but are built using systems of groups and cosets instead of elements of a base set. The models include all algebras of binary relations, and many non-representable relation algebras as well. We prove that every atomic relation algebra satisfying a certain measurability condition—a condition generalizing the conditions imposed by Jónsson and Tarski—is essentially isomorphic to a coset relation algebra. The theorem raises the possibility of providing a positive solution to Tarski's problem by using coset relation algebras instead of the standard algebras of binary relations.

Symmetry vs. Duality in Logic

2014 ◽  
Vol 8 (4) ◽  
pp. 83-97 ◽  
Author(s):  
Giulia Battilotti
Keyword(s):  
Sequent Calculus ◽  
Logical Consequence ◽  
Quantum States ◽  
The Other ◽  
Human Reasoning ◽  
Symmetric Mode ◽  
The Unconscious ◽  
Structural Rules ◽  
Model Based ◽  
Long Time

The author discusses the problem of symmetry, namely of the orientation of the logical consequence. The author shows that the problem is surprisingly entangled with the problem of “being infinite”. The author presents a model based on quantum states and shows that it features satisfy the requirements of the symmetric mode of Bi-logic, a logic introduced in the '70s by the psychoanalyst I. Matte Blanco to describe the logic of the unconscious. The author discusess symmetry, in the model, to include correlations, in order to obtain a possible approach to displacement. In this setting, the author finds a possible reading of the structural rules of sequent calculus, whose role in computation, on one side, and in the representation of human reasoning, on the other, has been debated for a long time.

scholarly journals THE VARIETY OF COSET RELATION ALGEBRAS

2018 ◽  
Vol 83 (04) ◽  
pp. 1595-1609 ◽  
Author(s):  
STEVEN GIVANT ◽  
HAJNAL ANDRÉKA
Keyword(s):  
Large Class ◽  
Identity Element ◽  
Relation Algebra ◽  
Order Theory ◽  
Relation Algebras ◽  
First Order ◽  
First Order Theory ◽  
Atomic Pair ◽  
Finite Set ◽  
Image Position

AbstractGivant [6] generalized the notion of an atomic pair-dense relation algebra from Maddux [13] by defining the notion of a measurable relation algebra, that is to say, a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the “size” of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). In Andréka--Givant [2], a large class of examples of such algebras is constructed from systems of groups, coordinated systems of isomorphisms between quotients of the groups, and systems of cosets that are used to “shift” the operation of relative multiplication. In Givant--Andréka [8], it is shown that the class of these full coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic and complete measurable relation algebra is isomorphic to a full coset relation algebra.Call an algebra $\mathfrak{A}$ a coset relation algebra if $\mathfrak{A}$ is embeddable into some full coset relation algebra. In the present article, it is shown that the class of coset relation algebras is equationally axiomatizable (that is to say, it is a variety), but that no finite set of sentences suffices to axiomatize the class (that is to say, the class is not finitely axiomatizable).

Undecidability of representability as binary relations

2012 ◽  
Vol 77 (4) ◽  
pp. 1211-1244 ◽  
Author(s):  
Robin Hirsch ◽  
Marcel Jackson
Keyword(s):  
Relation Algebra ◽  
Binary Relations ◽  
Wide Range ◽  
Finite Representability

AbstractIn this article we establish the undecidability of representability and of finite representability as algebras of binary relations in a wide range of signatures. In particular, representability and finite representability are undecidable for Boolean monoids and lattice ordered monoids, while representability is undecidable for Jónsson's relation algebra. We also establish a number of undecidability results for representability as algebras of injective functions.

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.)

Variable Neighborhood Model for Agent Control Introducing Accessibility Relations Between Agents with Linear Temporal Logic

2014 ◽  
Vol 18 (6) ◽  
pp. 937-945
Author(s):  
Seiki Ubukata ◽  
◽  
Tetsuya Murai ◽  
Yasuo Kudo ◽  
Seiki Akama ◽  
...  
Keyword(s):  
Topological Space ◽  
Temporal Logic ◽  
Binary Relation ◽  
Linear Temporal Logic ◽  
Binary Relations ◽  
Cellular Space ◽  
Current Time ◽  
Low Level ◽  
Local Properties ◽  
Agent Control

In general, there are two types of agents, reflex and deliberative. The former does not have the ability for deep planning that produces higher-level actions to attain goals cooperatively, which is the ability of the latter. Can we cause reflex agents to act as though they could plan their actions? In this paper, we propose a variable neighborhood model for reflex agent control, that allows such agents to create plans in order to attain their goals. The model consists of three layers: (1) topological space, (2) agent space, and (3) linear temporal logic. Agents with their neighborhoods move in a topological space, such as a plane, and in a cellular space. Then, a binary relation between agents is generated each time from the agents’ position and neighborhood. We call the pair composed of a set of agents and binary relations the agent space. In order to cause reflex agents to have the ability to attain goals superficially, we consider the local properties of the binary relation between agents. For example, if two agents have a symmetrical relation at the current time, they can struggle to maintain symmetry or they could abandon symmetry at the next time, depending on the context. Then, low-level behavior, that is, the maintenance or abandonment of the local properties of binary relations, grant reflex agents a method for selecting neighborhoods for the next time. As a result, such a sequence of low-level behavior generates seemingly higher-level actions, as though reflex agents could attain a goal with such actions. This low-level behavior is shown through simulation to generate the achievement of a given goal, such as cooperation and target pursuing.

On transitive subrelations of binary relations

2011 ◽  
Vol 76 (4) ◽  
pp. 1429-1440 ◽  
Author(s):  
Christopher S. Hardin
Keyword(s):  
Binary Relation ◽  
Transitive Closure ◽  
Infinite Chain ◽  
Binary Relations ◽  
Transitive Relation

AbstractThe transitive closure of a binary relation R can be thought of as the best possible approximation of R “from above” by a transitive relation. We consider the question of approximating a relation from below by transitive relations. Our main result is that every thick relation (a relation whose complement contains no infinite chain) on a countable set has a transitive thick subrelation. This allows for a solution to a problem arising from previous work by the author and Alan Taylor. We also exhibit a thick relation on an uncountable set with no transitive thick subrelation.

An algebraic approach to binary relations

2015 ◽  
Vol 08 (02) ◽  
pp. 1550017 ◽  
Author(s):  
Ivan Chajda ◽  
Helmut Länger ◽  
Petr Ševčik
Keyword(s):  
Binary Relation ◽  
Binary Operation ◽  
Algebraic Approach ◽  
Binary Relations ◽  
Point Extension

Several attempts were made to assign to a given binary relation a certain binary operation in order to allow an algebraic approach for investigating binary relations. However, the previous attempts by the first two authors were restricted to the case of so-called directed binary relations. In this paper, this restriction is omitted and a general approach is developed. We assign to every binary relation a partial binary operation in such a way that the properties of the relation can be described by properties of the assigned operation. These properties are expressed mostly in terms of existential or strong identities. The partial binary operation can be extended to an everywhere defined binary operation by means of the so-called one-point extension. This enables us to get a genuine algebraic approach similar to that for directed binary relations.

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.

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