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Quantales and temporal logics
 ALGEBRAIC METHODOLOGY AND SOFTWARE TECHNOLOGY (AMAST 2006). LNCS 4019
, 2006
"... We propose an algebraic semantics for the temporal logic CTL∗ and simplify it for its sublogics CTL and LTL. We abstractly represent state and path formulas over transition systems in Boolean left quantales. These are complete lattices with a multiplication that preserves arbitrary joins in its left ..."
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We propose an algebraic semantics for the temporal logic CTL∗ and simplify it for its sublogics CTL and LTL. We abstractly represent state and path formulas over transition systems in Boolean left quantales. These are complete lattices with a multiplication that preserves arbitrary joins in its left argument and is isotone in its right argument. Over these quantales, the semantics of CTL∗ formulas can be encoded via finite and infinite iteration operators; the CTL and LTL operators can be related to domain operators. This yields interesting new connections between representations as known from the modal µcalculus and Kleene/ωalgebra.
Towards a Uniform Relational Semantics for Tabular Expressions
 Proc. of RELMICS 98
, 1998
"... Introduction Parnas et al. [4, 5, 6] have proposed tabular expressions as a means to represent the complex relations that are used to specify or document software systems. The idea is that a tabular expression is much easier to understand and verify than a long linear formula. Tabular expressions a ..."
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Cited by 5 (2 self)
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Introduction Parnas et al. [4, 5, 6] have proposed tabular expressions as a means to represent the complex relations that are used to specify or document software systems. The idea is that a tabular expression is much easier to understand and verify than a long linear formula. Tabular expressions are intended to supplement, not replace, notations used by engineers. They were found to be useful for describing large mathematical relations in practical applications. A semantics of tabular expressions, generalizing previous work and introducing new types of tables, is given in [4]. We show here how a simple and powerful algebra of arrays of relations can be used to further generalize this semantics for some of the types of tabular expressions considered in [4]. This opens up the door to new types of tabular expressions and, more importantly, brings a set of algebraic laws for the manipulation of tables. The algebra we present is based on the known [
Principles for Organizing Semantic Relations in Large Knowledge Bases
 IEEE Transactions on Knowledge and Data Engineering
, 1994
"... This paper defines principles for organizing semantic relations represented by slots in framestructured knowledge bases. We consider not only the ways that slots are used in reasoning about a given domain but also the features of the representation language of the knowledgebased system in which th ..."
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Cited by 5 (1 self)
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This paper defines principles for organizing semantic relations represented by slots in framestructured knowledge bases. We consider not only the ways that slots are used in reasoning about a given domain but also the features of the representation language of the knowledgebased system in which the slots reside. We find that the organization of slots may be based on the knowledgelevel semantics of relations and the symbollevel function of slots that implement the representation language. However, the organization of slots is more understandable if these two fundamental distinctions are explicitly separated. The symbollevel organization of slots depends on the inferencing and expressive capabilities of the knowledge representation system. At the knowledge level, two entirely different classification schemes are identified: one based on linguistic similarities and differences, and another based on the types of concepts being related. As a case study, we show how to use the three o...
Algebraic Separation Logic
, 2010
"... We present an algebraic approach to separation logic. In particular, we give an algebraic characterisation for assertions of separation logic, discuss different classes of assertions and prove abstract laws fully algebraically. After that, we use our algebraic framework to give a relational semantic ..."
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We present an algebraic approach to separation logic. In particular, we give an algebraic characterisation for assertions of separation logic, discuss different classes of assertions and prove abstract laws fully algebraically. After that, we use our algebraic framework to give a relational semantics of the commands of the simple programming language associated with separation logic. On this basis we prove the frame rule in an abstract and concise way. We also propose a more general version of separating conjunction which leads to a frame rule that is easier to prove. In particular, we show how to algebraically formulate the requirement that a command does not change certain variables; this is also expressed more conveniently using the generalised separating conjunction. The algebraic view does not only yield new insights on separation logic but also shortens proofs due to a point free representation. It is largely firstorder and hence enables the use of offtheshelf automated theorem provers for verifying properties at a more abstract level.
An Algebraic Characterization of Cartesian Products of Fuzzy Relations
 Bulletin of Informatics and Cybernetics 29
, 1996
"... This paper provides an algebraic characterization of mathematical structures formed by cartesian products of fuzzy relations with supmin composition. A simple proof of a representation theorem for Boolean relation algebras satisfying Tarski rule and point axiom was given by G. Schmidt and T. Str ..."
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This paper provides an algebraic characterization of mathematical structures formed by cartesian products of fuzzy relations with supmin composition. A simple proof of a representation theorem for Boolean relation algebras satisfying Tarski rule and point axiom was given by G. Schmidt and T. Strohlein, and cartesian products of Boolean relation algebras were investigated by B. J'onsson and A. Tarski. Unlike Boolean relation algebras, fuzzy relation algebras are not Boolean but equipped with semiscalar multiplication. First we present a set of axioms for fuzzy relation algebras and add axioms for cartesian products of fuzzy relation algebras. Second we improve the definition of point relations. Then a representation theorem for such relation algebras is deduced. Keywords : fuzzy relations, cartesian products, relation algebras, representation theorem. 1 Introduction In 1941 Tarski [8] proposed a problem, that is, "Is every relation algebra isomorphic to an algebra of all Bo...
Crispness in Dedekind Categories
"... . This paper studies notions of scalar relations and crispness of relations. 1 Introduction Just after Zadeh's invention of the concept of fuzzy sets [19], Goguen [5] generalized the concepts of fuzzy sets and relations to taking values on arbitrary lattices. On the other hand, the theory of relat ..."
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. This paper studies notions of scalar relations and crispness of relations. 1 Introduction Just after Zadeh's invention of the concept of fuzzy sets [19], Goguen [5] generalized the concepts of fuzzy sets and relations to taking values on arbitrary lattices. On the other hand, the theory of relations, namely relational calculus, has been investigated since the middle of the nineteen century, see [13, 16, 17] for more details. Almost all modern formalisations of relation algebras are affected by the work of Tarski [18]. Mac Lane [12] and Puppe [15] exposed a categorical basis for the calculus of additive relations. Freyd and Scedrov [2] developed and summarized categorical relational calculus, which they called allegories. In relational calculus one calculates with relations in an elementfree style, which makes relational calculus a very useful framework for the study of mathematics [8] and theoretical computer science [1, 7, 11] and also a useful tool for applications. Some element...
A Study on Symmetric Quotients
, 1998
"... Symmetric quotients, introduced in the context of heterogeneous relation algebras, have proven useful for applications comprising for example program semantics and databases. Recently, the increased interest in fuzzy relations has fostered a lot of work concerning relationlike structures with weake ..."
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Symmetric quotients, introduced in the context of heterogeneous relation algebras, have proven useful for applications comprising for example program semantics and databases. Recently, the increased interest in fuzzy relations has fostered a lot of work concerning relationlike structures with weaker axiomatisations. In this paper, we study symmetric quotients in such settings and provide many new proofs for properties previously only shown in the strong theory of heterogeneous relation algebras. Thus we hope to make both the weaker axiomatisations and the many applications of symmetric quotients more accessible to people working on problems in some specific part of the wide spectrum of relation categories.
Stronger Compositions for Retrenchments, and Feature Engineering (2002
"... Abstract. Noting that the usual propositionally based way of composing retrenchments can yield many ‘junk ’ cases, alternative approaches to compositionality are introduced (via notions of tidy, neat, and fastidious retrenchments) that behave better in this regard. These alternatives do however make ..."
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Abstract. Noting that the usual propositionally based way of composing retrenchments can yield many ‘junk ’ cases, alternative approaches to compositionality are introduced (via notions of tidy, neat, and fastidious retrenchments) that behave better in this regard. These alternatives do however make other issues such as associativity harder; the technical details are presented. This technology is used to give a retrenchment account of elementary feature engineering, the full flexibility of which, refinement can struggle to capture. Keywords. Retrenchment, Compositionality, Associativity, Feature Engineering. 1
Grammatica: An Implementation of Algebraic Graph Transformation on Mathematica
"... . Grammatica is a prototype implementation of algebraic graph transformation based on relation algebra. It has been implemented using Mathematica on top of the Combinatorica package, and runs therefore on most platforms. It consists of Mathematica routines for representing, manipulating, display ..."
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. Grammatica is a prototype implementation of algebraic graph transformation based on relation algebra. It has been implemented using Mathematica on top of the Combinatorica package, and runs therefore on most platforms. It consists of Mathematica routines for representing, manipulating, displaying and transforming graphs, as well as routines implementing some relation algebratheoretic operations on graphs. It supports both interactive and automatic application of doublepushout graph productions, being therefore both a teaching aid and a research tool for algebraic graph transformation. 1 Introduction The methods and techniques of graph transformation have been under development for almost three decades now, although little e#orts have been devoted within the graph transformation community to issues of tool support. A few notable exceptions are the systems GraphEd [4], PROGRES [8], Agg [5], and Treebag [3]. Grammatica is a prototype implementation of doublepushout algebra...
An Algebraic Calculus of Database Preferences
"... Abstract. Preference algebra, an extension of the algebra of database relations, is a wellstudied field in the area of personalized databases. It allows modelling user wishes by preference terms; they represent strict partial orders telling which database objects the user prefers over other ones. T ..."
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Abstract. Preference algebra, an extension of the algebra of database relations, is a wellstudied field in the area of personalized databases. It allows modelling user wishes by preference terms; they represent strict partial orders telling which database objects the user prefers over other ones. There are a number of constructors that allow combining simple preferences into quite complex, nested ones. A preference term is then used as a database query, and the results are the maximal objects according to the order it denotes. Depending on the size of the database, this can be computationally expensive. For optimisation, preference queries and the corresponding terms are transformed using a number of algebraic laws. So far, the correctness proofs for such laws have been performed by hand and in a pointwise fashion. We enrich the standard theory of relational databases to an algebraic framework that allows completely pointfree reasoning about complex preferences. This blackbox view is amenable to a treatment in firstorder logic and hence to fully automated proofs using offtheshelf verification tools. We exemplify the use of the calculus with some nontrivial laws, notably concerning socalled preference prefilters which perform a preselection to speed up the computation of the maximal objects proper.