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B. Plotkin - One of the best experts on this subject based on the ideXlab platform.
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Algebraic Logic and knowledge bases
2021Co-Authors: Elena Aladova, B. Plotkin, Tatjana PlotkinAbstract:Knowledge bases theory provides an important example of the field where applications of universal algebra and Algebraic Logic look very natural, and their interaction with practical problems arising in computer science might be very productive. In this paper we study the equivalence problem for knowledge bases. Our interest is to find out how the informational equivalence is related to the Logical description of knowledge. The main objectives of this paper are Logically-geometrically equivalent and LG-isotypic knowledge bases. We will see that these notions give us a good characterization of knowledge bases.
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Algebraic Logic and Logical geometry two in one
Vestnik St. Petersburg University: Mathematics, 2013Co-Authors: B. PlotkinAbstract:The aim of the paper is to define the notion of isotypic algebras and to formulate a series of new problems related to this notion.
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Algebraic Logic and Logical geometry in arbitrary varieties of algebras
arXiv: Group Theory, 2012Co-Authors: B. PlotkinAbstract:The paper consists of two parts. The first part is devoted to Logic for universal Algebraic geometry. The second one deals with problems and some results. It may be regarded as a brief exposition of some ideas from the book in progress: B.Plotkin, E.Aladova, E.Plotkin, "Algebraic Logic and Logical geometry in arbitrary varieties of algebras"
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Algebraic Logic and Logically geometric types in varieties of algebras
arXiv: Logic, 2011Co-Authors: B. Plotkin, Elena Aladova, Eugene PlotkinAbstract:The main objective of this paper is to show that the notion of type which was developed within the frames of Logic and model theory has deep ties with geometric properties of algebras. These ties go back and forth from universal Algebraic geometry to the model theory through the machinery of Algebraic Logic. We show that types appear naturally as Logical kernels in the Galois correspondence between filters in the Halmos algebra of first order formulas with equalities and elementary sets in the corresponding affine space.
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Algebraic geometry in first-order Logic
Journal of Mathematical Sciences, 2006Co-Authors: B. PlotkinAbstract:In every variety of algebras Θ, we can consider its Logic and its Algebraic geometry. In previous papers, geometry in equational Logic, i.e., equational geometry, has been studied. Here we describe an extension of this theory to first-order Logic (FOL). The Algebraic sets in this geometry are determined by arbitrary sets of FOL formulas. The principal motivation of such a generalization lies in the area of applications to knowledge science. In this paper, the FOL formulas are considered in the context of Algebraic Logic. For this purpose, we define special Halmos categories. These categories in Algebraic geometry related to FOL play the same role as the category of free algebras Θ0 play in equational Algebraic geometry. This paper consists of three parts. Section 1 is of introductory character. The first part (Secs. 2–4) contains background on Algebraic Logic in the given variety of algebras Θ. The second part is devoted to Algebraic geometry related to FOL (Secs. 5–7). In the last part (Secs. 8–9), we consider applications of the previous material to knowledge science.
George Voutsadakis - One of the best experts on this subject based on the ideXlab platform.
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categorical abstract Algebraic Logic tarski congruence systems Logical morphisms and Logical quotients
2015Co-Authors: George VoutsadakisAbstract:A general notion of a congruence system is introduced for π-institutions. Congruence systems in this sense are collections of equivalence relations on the sets of sentences of the π-institution that are preserved both by signature morphisms and by fixed collections of natural transformations from finite tuples of sentences to sentences. Based on this notion of a congruence system, the notion of a Tarski congruence system, generalizing the notion of a Tarski congruence from sentential Logics, is considered. Logical and biLogical morphisms are introduced for π-institutions, also generalizing similar concepts from the theory of sentential Logics, and their relationship with the familiar translations and interpretations of institutions is discussed. Finally, the interplay between these Logical maps and the formation of Logical quotients of π-institutions and the way they transform the Tarski congruence systems is investigated.
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categorical abstract Algebraic Logic closure operators on classes of pofunctors
2012Co-Authors: George VoutsadakisAbstract:Following work of Pa lasi nska and Pigozzi on partially ordered varieties and quasi-varieties of universal algebras, the author recently introduced partially ordered systems (posystems) and partially ordered functors (pofunctors) to cover the case of the Algebraic systems arising in categorical abstract Algebraic Logic. Analogs of the ordered homomorphism theorems of universal algebra were shown to hold in the context of pofunctors. In the present work, operators on classes of pofunctors are introduced and it is shown that classes of pofunctors are closed under the HSP and the SPPU operators, forming analogs of the well-known variety and quasi-variety operators, respectively, of universal algebra.
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categorical abstract Algebraic Logic equivalence of closure systems
2010Co-Authors: George VoutsadakisAbstract:In their famous “Memoirs” monograph, Blok and Pigozzi defined algebraizable deductive systems as those whose consequence relation is equivalent to the Algebraic consequence relation associated with a quasivariety of universal algebras. In characterizing this property, they showed that it is equivalent with the existence of an isomorphism between the lattices of theories of the two consequence relations that commutes with inverse substitutions. Thus emerged the prototypical and paradigmatic result relating an equivalence between two consequence relations established by means of syntactic translations and the isomorphism between corresponding lattices of theories. This result was subsequently generalized in various directions. Blok and Pigozzi themselves extended it to ′
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categorical abstract Algebraic Logic preAlgebraicity and protoAlgebraicity
Studia Logica, 2007Co-Authors: George VoutsadakisAbstract:Two classes of π are studied whose properties are similar to those of the protoAlgebraic deductive systems of Blok and Pigozzi. The first is the class of N-protoAlgebraic π-institutions and the second is the wider class of N-preAlgebraic π-institutions. Several characterizations are provided. For instance, N-preAlgebraic π-institutions are exactly those π-institutions that satisfy monotonicity of the N-Leibniz operator on theory systems and N-protoAlgebraic π-institutions those that satisfy monotonicity of the N-Leibniz operator on theory families. Analogs of the correspondence property of Blok and Pigozzi for π-institutions are also introduced and their connections with preand protoAlgebraicity are explored. Finally, relations of these two classes with the (\({\mathcal{I}}\), N)-Algebraic systems, introduced previously by the author as an analog of the \({\mathcal{S}}\) -algebras of Font and Jansana, and with an analog of the Suszko operator of Czelakowski for π-institutions are also investigated.
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categorical abstract Algebraic Logic more on protoAlgebraicity
Notre Dame Journal of Formal Logic, 2006Co-Authors: George VoutsadakisAbstract:ProtoAlgebraic Logics are characterized by the monotonicity of the Leibniz operator on their theory lattices and are at the lower end of the Leibniz hierarchy of abstract Algebraic Logic. They have been shown to be the most primitive among those Logics with a strong enough Algebraic character to be amenable to Algebraic study techniques. ProtoAlgebraic π -institutions were introduced recently as an analog of protoAlgebraic sentential Logics with the goal of extending the Leibniz hierarchy from the sentential framework to the π -institution framework. Many properties of protoAlgebraic Logics, studied in the sentential Logic framework by Blok and Pigozzi, Czelakowski, and Font and Jansana, among others, have already been adapted in previous work by the author to the categorical level. This work aims at further advancing that study by exploring in this new level some more properties of protoAlgebraic sentential Logics.
Agnes Szendrei - One of the best experts on this subject based on the ideXlab platform.
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the 3rd international conference on boolean algebra lattice theory universal algebra set theory and set theoretical topology blast 2010
Order, 2012Co-Authors: John Harding, Bart Kastermans, Keith A Kearnes, Donald J Monk, Agnes SzendreiAbstract:The 3rd International Conference on Boolean Algebra, Lattice Theory, Universal Algebra, Set Theory and Set-Theoretical Topology—BLAST 2010—was held at the University of Colorado at Boulder on June 2–6, 2010. The meeting brought together 70 researchers from Canada, Columbia, the Czech Republic, England, France, Germany, Hungary, Israel, Luxembourg, Mexico, Morocco, New Zealand, Pakistan, Russia, Serbia, and the United States. The aim of the BLAST conference series is to promote interaction between researchers from diverse areas related to the foundations of mathematics. The acronym BLAST emphasizes that research in Boolean algebra, lattice theory, universal algebra, set theory and set-theoretic topology are within the scope of the conference series, but past BLAST meetings have also highlighted other areas including Algebraic Logic, quantum Logic and point-free topology. Plenary speakers at the 2010 conference included Mohamed Bekkali (Universite Sidi Mohamed Ben Abdellah), Ken Kunen (University of Wisconsin), Ralph McKenzie (Vanderbilt University), David Milovich (Texas A & M International University), Judith Roitman (University of Kansas), Grigor Sargsyan (UCLA), Juris Steprans (York University), and Friedrich Wehrung (Caen University). To encourage cross-disciplinary interaction, four multi-day minicourses were scheduled: Special ultraf ilters on countable sets by Andreas Blass (University of Michigan), TopoLogical games by Gary Gruenhage (Auburn University), Locally moving groups and how they are used by Matatyahu Rubin (Ben-Gurion University), and Universal algebra, Mal’cev conditions, and f inite relational structures by Ross Willard (University of Waterloo).
Josep Font - One of the best experts on this subject based on the ideXlab platform.
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assertional Logics truth equational Logics and the hierarchies of abstract Algebraic Logic
2018Co-Authors: Hugo Albuquerque, Ramon Jansana, Josep Font, Tommaso MoraschiniAbstract:We establish some relations between the class of truth-equational Logics, the class of assertional Logics, other classes in the Leibniz hierarchy, and the classes in the Frege hierarchy. We argue that the class of assertional Logics belongs properly in the Leibniz hierarchy. We give two new characterizations of truth-equational Logics in terms of their full generalized models, and use them to obtain further results on the internal structure of the Frege hierarchy and on the relations between the two hierarchies. Some of these results and several counterexamples contribute to answer a few open problems in abstract Algebraic Logic, and open a new one.
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compatibility operators in abstract Algebraic Logic
Journal of Symbolic Logic, 2016Co-Authors: Hugo Albuquerque, Josep Font, Ramon JansanaAbstract:This paper presents a unified framework that explains and extends the already successful applications of the Leibniz operator, the Suszko operator, and the Tarski operator in recent developments in abstract Algebraic Logic. To this end, we refine Czelakowski’s notion of an S-compatibility operator, and introduce the notion of coherent family of S-compatibility operators, for a sentential Logic S. The notion of coherence is a restricted property of commutativity with inverse images by surjective homomorphisms, which is satisfied by both the Leibniz and the Suszko operators. We generalize several constructions and results already existing for the mentioned operators; in particular, the well-known classes of algebras associated with a Logic through each of them, and the notions of full generalized model of a Logic and a special kind of S-filters (which generalizes the less-known notion of Leibniz filter). We obtain a General Correspondence Theorem, extending the well-known one from the theory of protoAlgebraic Logics to arbitrary Logics and to more general operators, and strengthening its formulation. We apply the general results to the Leibniz and the Suszko operators, and obtain several characterizations of the main classes of Logics in the Leibniz hierarchy by the form of their full generalized models, by old and new properties of the Leibniz operator, and by the behaviour of the Suszko operator. Some of these characterizations complete or extend known ones, for some classes in the hierarchy, thus offering an integrated approach to the Leibniz hierarchy that uncovers some new, nice symmetries.
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update to a survey of abstract Algebraic Logic
Studia Logica, 2009Co-Authors: Josep Font, Ramon Jansana, Don PigozziAbstract:A definition and some inaccurate cross-references in the paper A Survey of Abstract Algebraic Logic, which might confuse some readers, are clarified and corrected; a short discussion of the main one is included. We also update a dozen of bibliographic references.
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On weakening the Deduction Theorem and strengthening Modus Ponens
MLQ, 2004Co-Authors: Félix Bou, Josep Font, José Luis García LaprestaAbstract:This paper studies, with techniques of Abstract Algebraic Logic, the effects of putting a bound on the cardinality of the set of side formulas in the Deduction Theorem, viewed as a Gentzen-style rule, and of adding additional assumptions inside the formulas present in Modus Ponens, viewed as a Hilbert-style rule. As a result, a denumerable collection of new Gentzen systems and two new sentential Logics have been isolated. These Logics are weaker than the positive implicative Logic. We have determined their Algebraic models and the relationships between them, and have classified them according to several standard criteria of Abstract Algebraic Logic. One of the Logics is protoAlgebraic but neither equivalential nor weakly algebraizable, a rare situation where very few natural examples were hitherto known. In passing we have found new, alternative presentations of positive implicative Logic, both in Hilbert style and in Gentzen style, and have characterized it in terms of the restricted Deduction Theorem: it is the weakest Logic satisfying Modus Ponens and the Deduction Theorem restricted to at most 2 side formulas. The Algebraic part of the work has lead to the class of quasi-Hilbert algebras, a
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an abstract Algebraic Logic view of some multiple valued Logics
Beyond two, 2003Co-Authors: Josep FontAbstract:Abstract Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum-Tarski process. The notions of Logical matrix and of Leibniz congruence are among its main building blocks. Its most successful part has been developed mainly by BLOK, PIGOZZI and CZELAKOWSKI, and obtains a deep theory and very nice and powerful results for the so-called protoAlgebraic Logics. I will show how the idea (already explored by WOJCKICI and NOWAK) of defining Logics using a scheme of "preservation of degrees of truth" (as opposed to the more usual one of "preservation of truth") characterizes a wide class of Logics which are not necessarily protoAlgebraic and provide another fairly general framework where recent methods in Abstract Algebraic Logic (developed mainly by JANSANA and myself) can give some interesting results. After the general theory is explained, I apply it to an infinite family of Logics defined in this way from subalgebras of the real unit interval taken as an MV-algebra. The general theory determines the Algebraic counterpart of each of these Logics without having to perform any computations for each particular case, and proves some interesting properties common to all of them. Moreover, in the finite case the Logics so obtained are protoAlgebraic, which implies they have a "strong version" defined from their Leibniz filters; again, the general theory helps in showing that it is the Logic defined from the same subalgebra by the truth-preserving scheme, that is, the corresponding finite-valued Logic in the most usual sense. However, for infinite subalgebras the obtained Logic turns out to be the same for all such subalgebras and is not protoAlgebraic, thus the ordinary methods do not apply. After introducing some (new) more general abstract notions for non-protoAlgebraic Logics I can finally show that this Logic too has a strong version, and that it coincides with the ordinary infinite-valued Logic of Lukasiewicz.
Zhaohua Luo - One of the best experts on this subject based on the ideXlab platform.
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Clone Theory and Algebraic Logic
arXiv: Logic in Computer Science, 2009Co-Authors: Zhaohua LuoAbstract:The concept of a clone is central to many branches of mathematics, such as universal algebra, Algebraic Logic, and lambda calculus. Abstractly a clone is a category with two objects such that one is a countably infinite power of the other. Left and right algebras over a clone are covariant and contravariant functors from the category to that of sets respectively. In this paper we show that first-order Logic can be studied effectively using the notions of right and left algebras over a clone. It is easy to translate the classical treatment of Logic into our setting and prove all the fundamental theorems of first-order theory Algebraically.
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clone theory its syntax and semantics applications to universal algebra lambda calculus and Algebraic Logic
arXiv: Logic in Computer Science, 2008Co-Authors: Zhaohua LuoAbstract:The primary goal of this paper is to present a unified way to transform the syntax of a Logic system into certain initial Algebraic structure so that it can be studied Algebraically. The Algebraic structures which one may choose for this purpose are various clones over a full subcategory of a category. We show that the syntax of equational Logic, lambda calculus and first order Logic can be represented as clones or right algebras of clones over the set of positive integers. The semantics is then represented by structures derived from left algebras of these clones.
