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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...
Finite Schematizable Algebraic Logic
, 1997
"... In this work, we attempt to alleviate three (more or less) equivalent negative results. These are (i) nonaxiomatizability (by any nite schema) of the valid formula schemas of rst order logic, (ii) nonaxiomatizability (by nite schema) of any propositional logic equivalent with classical rst ..."
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Cited by 9 (1 self)
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In this work, we attempt to alleviate three (more or less) equivalent negative results. These are (i) nonaxiomatizability (by any nite schema) of the valid formula schemas of rst order logic, (ii) nonaxiomatizability (by nite schema) of any propositional logic equivalent with classical rst order logic (i.e., modal logic of quanti cation and substitution), and (iii) nonaxiomatizability (by nite schema) of the class of representable cylindric algebras (i.e., of the algebraic counterpart of rst order logic). Here we present two nite schema axiomatizable classes of algebras that contain, as a reduct, the class of representable quasipolyadic algebras and the class of representable cylindric algebras, respectively. We establish positive results in the direction of nitary algebraization of rst order logic without equality as well as that with equality. Finally, we will indicate how these constructions can be applied to turn negative results (i), (ii) above to positive ones.
A Short Proof of Representability of Fork Algebras
 Bulletin of IGPL
, 1997
"... In this paper a strong relationship is demonstrated between fork algebras and quasiprojective relation algebras. With the help of Tarski's classical representation theorem for quasiprojective relation algebras, a short proof is given for the representation theorem of fork algebras. As a byproduct ..."
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Cited by 3 (0 self)
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In this paper a strong relationship is demonstrated between fork algebras and quasiprojective relation algebras. With the help of Tarski's classical representation theorem for quasiprojective relation algebras, a short proof is given for the representation theorem of fork algebras. As a byproduct, we will discuss the difference between relative and absolute representation theorems. Fork algebras, due to their expressive power and applicability in computing science, have been intensively studied in the last four years. Their literature is alive and productive. See e.g. Baum, Frias, Haeberer, Veloso [33], [34], [3], [7], [8], [6], [9], [10] and SainSimon [27]. As described in the textbook [15] 2.7.46, in algebra, there are two kinds of representation theorems: absolute and relative representation. For fork algebras absolute representation was proved almost impossible in [18], [27], [26] 1 . However, relative representation is still possible and is quite useful (see e.g. [2]). The...
Applications of Alfred Tarski's Ideas in Database Theory
 Proceedings of the 15th International Workshop on Computer Science Logic. LNCS 2142
, 2001
"... Many ideas of Alfred Tarski  one of the founders of modern logic  find application in database theory. We survey some of them with no attempt at comprehensiveness. Topics discussed include the genericity of database queries; the relational algebra, the Tarskian definition of truth for the relation ..."
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Many ideas of Alfred Tarski  one of the founders of modern logic  find application in database theory. We survey some of them with no attempt at comprehensiveness. Topics discussed include the genericity of database queries; the relational algebra, the Tarskian definition of truth for the relational calculus, and cylindric algebras, relation algebras and computationally complete query languages; real polynomial constraint databases; and geometrical query languages.