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16
The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations
, 1991
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Step by Step  Building Representations in Algebraic Logic
 Journal of Symbolic Logic
, 1995
"... We 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 characterised according to the outcome of certain games. The Lyndon conditions defini ..."
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Cited by 28 (15 self)
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We 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 characterised 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. Countable relation algebras with homogeneous representations are characterised by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is !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 twoplayer game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this ap...
Expressive Power and Complexity in Algebraic Logic
 Journal of Logic and Computation
, 1997
"... Two complexity problems in algebraic logic are surveyed: the satisfaction problem and the network satisfaction problem. Various complexity results are collected here and some new ones are derived. Many examples are given. The network satisfaction problem for most cylindric algebras of dimension four ..."
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Cited by 20 (2 self)
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Two complexity problems in algebraic logic are surveyed: the satisfaction problem and the network satisfaction problem. Various complexity results are collected here and some new ones are derived. Many examples are given. The network satisfaction problem for most cylindric algebras of dimension four or more is shown to be intractable. Complexity is tiedin with the expressivity of a relation algebra. Expressivity and complexity are analysed in the context of homogeneous representations. The modeltheoretic notion of interpretation is used to generalise known complexity results to a range of other algebraic logics. In particular a number of relation algebras are shown to have intractable network satisfaction problems. 1 Introduction A basic problem in theoretical computing and applied logic is to select and evaluate the ideal formalism to represent and reason about a given application. Many different formalisms are adopted: classical firstorder logic, modal and temporal logics (either...
Canonical Varieties with No Canonical Axiomatisation
 Trans. Amer. Math. Soc
, 2003
"... We give a simple example of a variety V of modal algebras that is canonical but cannot be axiomatised by canonical equations or firstorder sentences. We then show that the variety RRA of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every ..."
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Cited by 12 (7 self)
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We give a simple example of a variety V of modal algebras that is canonical but cannot be axiomatised by canonical equations or firstorder sentences. We then show that the variety RRA of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every axiomatisation of these varieties involves infinitely many noncanonical sentences. Using probabilistic methods...
Relation Algebras with nDimensional Relational Bases
 Annals of Pure and Applied Logic
, 1999
"... We study relation algebras with ndimensional relational bases in the sense of Maddux. Fix n with 3 n !. Write Bn for the class of nonassociative algebras with an n dimensional relational basis, and RAn for the variety generated by Bn . We de ne a notion of representation for algebras in RAn , ..."
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Cited by 9 (2 self)
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We study relation algebras with ndimensional relational bases in the sense of Maddux. Fix n with 3 n !. Write Bn for the class of nonassociative algebras with an n dimensional relational basis, and RAn for the variety generated by Bn . We de ne a notion of representation for algebras in RAn , and use it to give an explicit (hence recursive) equational axiomatisation of RAn , and to reprove Maddux's result that RAn is canonical. We show that the algebras in Bn are precisely those that have a complete representation. Then we prove that whenever 4 n < l !, RA l is not nitely axiomatisable over RAn . This con rms a conjecture of Maddux. We also prove that Bn is elementary for n = 3; 4 only.
A Relational Approach To Optimization Problems
, 1996
"... The main contribution of this thesis is a study of the dynamic programming and greedy strategies for solving combinatorial optimization problems. The study is carried out in the context of a calculus of relations, and generalises previous work by using a loop operator in the imperative programming s ..."
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Cited by 6 (0 self)
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The main contribution of this thesis is a study of the dynamic programming and greedy strategies for solving combinatorial optimization problems. The study is carried out in the context of a calculus of relations, and generalises previous work by using a loop operator in the imperative programming style for generating feasible solutions, rather than the fold and unfold operators of the functional programming style. The relationship between fold operators and loop operators is explored, and it is shown how to convert from the former to the latter. This fresh approach provides additional insights into the relationship between dynamic programming and greedy algorithms, and helps to unify previously distinct approaches to solving combinatorial optimization problems. Some of the solutions discovered are new and solve problems which had previously proved difficult. The material is illustrated with a selection of problems and solutions that is a mixture of old and new. Another contribution is the invention of a new calculus, called the graph calculus, which is a useful tool for reasoning in the relational calculus and other nonrelational calculi. The graph
Finite integral relation algebras
 Universal Algebra and Lattice Theory, Lecture Notes in Mathematics 1149
, 1985
"... Please note that this paper does not exist. It consists entirely of excerpts from the ..."
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Cited by 3 (0 self)
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Please note that this paper does not exist. It consists entirely of excerpts from the
Bare canonicity of representable cylindric and polyadic algebras
"... We show that for finite n ≥ 3, every firstorder axiomatisation of the varieties of representable ndimensional cylindric algebras, diagonalfree cylindric algebras, polyadic algebras, and polyadic equality algebras contains an infinite number of noncanonical formulas. We also show that the class o ..."
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We show that for finite n ≥ 3, every firstorder axiomatisation of the varieties of representable ndimensional cylindric algebras, diagonalfree cylindric algebras, polyadic algebras, and polyadic equality algebras contains an infinite number of noncanonical formulas. We also show that the class of structures for each of these varieties is nonelementary. The proofs employ algebras derived from random graphs.
EPIMORPHISMS IN CYLINDRIC ALGEBRAS AND DEFINABILITY IN FINITE VARIABLE LOGIC
, 2008
"... Abstract. The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CAn of ndimensional cylindric algebras and the class of ..."
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Abstract. The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CAn of ndimensional cylindric algebras and the class of representable algebras in CAn for finite n> 1, solving Problem 10 of [28] for finite n. By a result of Németi, this shows that the Bethdefinability property fails for the finitevariable fragments of first order logic as long as the number n of variables available is> 1 and we allow models of size ≥ n + 2, but holds if we allow only models of size ≤ n + 1. We also use our results in the present paper to prove that several results in the literature concerning injective algebras and definability of polyadic operations in CAn are best possible. We raise several open problems. §0. INTRODUCTION AND THE MAIN RESULTS In algebra, the properties of epimorphisms (in the categorial sense) being surjective and the amalgamation property in a class of algebras are well investigated, see e.g. [1] and [37]. In algebraic logic these properties turn out