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15
Kleene Algebra with Domain
, 2003
"... We propose Kleene algebra with domain (KAD), an extension of Kleene algebra with two equational axioms for a domain and a codomain operation, respectively. KAD considerably augments the expressibility of Kleene algebra, in particular for the specification and analysis of state transition systems. We ..."
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Cited by 42 (29 self)
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We propose Kleene algebra with domain (KAD), an extension of Kleene algebra with two equational axioms for a domain and a codomain operation, respectively. KAD considerably augments the expressibility of Kleene algebra, in particular for the specification and analysis of state transition systems. We develop the basic calculus, discuss some related theories and present the most important models of KAD. We demonstrate applicability by two examples: First, an algebraic reconstruction of Noethericity and wellfoundedness. Second, an algebraic reconstruction of propositional Hoare logic.
Dynamic Algebras as a wellbehaved fragment of Relation Algebras
 In Algebraic Logic and Universal Algebra in Computer Science, LNCS 425
, 1990
"... The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect ..."
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Cited by 35 (5 self)
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The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect to representable relation algebras, when expressed in their DA form are complete with respect to representable dynamic algebras. Moreover, whereas the theory of RA is undecidable, that of DA is decidable in exponential time. These results follow from representability of the free intensional dynamic algebras. Dept. of Computer Science, Stanford, CA 94305. This paper is based on a talk given at the conference Algebra and Computer Science, Ames, Iowa, June 24, 1988. It will appear in the proceedings of that conference, to be published by SpringerVerlag in the Lecture Notes in Computer Science series. This work was supported by the National Science Foundation under grant number CCR8814921 ...
On induction vs. *continuity
 Proc. Workshop on Logics of Programs 1981, SpringVerlag Lect. Notes in Comput
, 1981
"... Abstract. In this paper we study the relative expressibility of the infinitary *continuity condition (*cant) X ~ V n x and the equational but weaker induction axiom Ond) X ^ [a*](X =[alX) [a*]X in Propositional Dynamic Logic. We show: (1) under ind only, there is a firstorder sentence ..."
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Cited by 23 (12 self)
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Abstract. In this paper we study the relative expressibility of the infinitary *continuity condition (*cant) <a*>X ~ V n <an>x and the equational but weaker induction axiom Ond) X ^ [a*](X =[alX) [a*]X in Propositional Dynamic Logic. We show: (1) under ind only, there is a firstorder sentence distinguishing separable dynamic algebras from standard Kripke models; whereas (2) under the stronger axiom *cant, the class of separable dynamic algebras and the class of standard Kripke models are indistinguishable by any sentence of infinitary firstorder logic. I.
A Proof System for Contact Relation Algebras
"... Contact relations have been studied in the context of qualitative geometry and physics since the early 1920s, and have recently received attention in qualitative spatial reasoning. In this paper, we present a sound and complete proof system in the style of Rasiowa & Sikorski (1963) for relation a ..."
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Cited by 16 (12 self)
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Contact relations have been studied in the context of qualitative geometry and physics since the early 1920s, and have recently received attention in qualitative spatial reasoning. In this paper, we present a sound and complete proof system in the style of Rasiowa & Sikorski (1963) for relation algebras generated by a contact relation. 1 Introduction Contact relations arise in the context of qualitative geometry and spatial reasoning, going back to the work of de Laguna (1922), Nicod (1924), Whitehead (1929), and, more recently, of Clarke (1981), Cohn et al. (1997), Pratt & Schoop (1998, 1999) and others. They are a generalisation of the "overlap relation" , obtained from a "part of" relation, which for the first time was formalised by Lesniewski (1916), (see also Lesniewski, 1983). One of Lesniewski's main concerns was to build a paradoxfree foundation of Mathematics, one pillar of which was mereology 1 or, as it was originally called, the general theory of manifolds or colle...
Macneille completions and canonical extensions
 Transactions of the American Mathematical Society
"... Abstract. Let V be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if V is closed under MacNeille completions, then it is also closed under canonical exten ..."
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Cited by 13 (4 self)
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Abstract. Let V be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if V is closed under MacNeille completions, then it is also closed under canonical extensions. As a corollary we show that in the case of Boolean algebras with operators, any such variety V is generated by an elementary class of relational structures. Our main technical construction reveals that the canonical extension of a monotone bounded lattice expansion can be embedded in the MacNeille completion of any sufficiently saturated elementary extension of the original structure. 1.
Modal Kleene Algebra And Applications  A Survey
, 2004
"... Modal Kleene algebras are Kleene algebras with forward and backward modal operators defined via domain and codomain operations. They provide a concise and convenient algebraic framework that subsumes various other calculi and allows treating quite a variety of areas. We survey ..."
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Cited by 11 (5 self)
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Modal Kleene algebras are Kleene algebras with forward and backward modal operators defined via domain and codomain operations. They provide a concise and convenient algebraic framework that subsumes various other calculi and allows treating quite a variety of areas. We survey
Equational axioms of test algebra
 Computer Science Logic, 11th International Workshop, CSL ’97, volume 1414 of LNCS
, 1997
"... We presentacomplete axiomatization of test algebra ([24, 18, 29]), the twosorted algebraic variant of Propositional Dynamic Logic (PDL, [21, 7]). The axiomatization consists of adding a nite number of equations to any axiomatization of Kleene algebra ([15, 26, 17, 4]) and algebraic translations of ..."
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Cited by 4 (0 self)
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We presentacomplete axiomatization of test algebra ([24, 18, 29]), the twosorted algebraic variant of Propositional Dynamic Logic (PDL, [21, 7]). The axiomatization consists of adding a nite number of equations to any axiomatization of Kleene algebra ([15, 26, 17, 4]) and algebraic translations of the Segerberg ([27]) axioms for PDL. Kleene algebras are not nitely axiomatizable ([25, 6]), so our result does not give us a nite axiomatization of test algebra: in fact, no nite equational axiomatization exists. We alsopresent a singlesorted version of test algebra, using the notion of dynamic negation ([9, 2, 11]), to which the previous results carry over. 1
Algebraization and representation of mereotopological structures
 JoRMiCS
, 2004
"... Abstract. Boolean contact algebras are the abstract counterpart of region–based theories of space, which date back to the early 1920s. In this paper, we survey the development of these algebras and relevant construction and representation theorems. 1 ..."
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Cited by 3 (1 self)
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Abstract. Boolean contact algebras are the abstract counterpart of region–based theories of space, which date back to the early 1920s. In this paper, we survey the development of these algebras and relevant construction and representation theorems. 1
Complete axiomatizations for XPath fragments
 In Proceedings LID (Logic in Databases
, 2008
"... Abstract. We provide complete axiomatizations for several fragments of XPath: sets of equivalences from which every other valid equivalence is derivable. Specifically, we axiomatize downward single axis fragments of Core XPath (that is, Core XPath(↓) and Core XPath( ↓ +)) as well as the full Core XP ..."
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Cited by 3 (1 self)
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Abstract. We provide complete axiomatizations for several fragments of XPath: sets of equivalences from which every other valid equivalence is derivable. Specifically, we axiomatize downward single axis fragments of Core XPath (that is, Core XPath(↓) and Core XPath( ↓ +)) as well as the full Core XPath. We make use of techniques from modal logic. XPath is a language for navigating through XML documents. In this paper, we consider the problem of finding complete axiomatizations for fragments of XPath. By an axiomatization we mean a finite set of valid equivalences between XPath expressions. These equivalences can be thought of as (undirected) rewrite rules. Completeness then means that any two equivalent expressions can be rewritten to each other using the given equivalences. Completeness tells us, in a mathematically precise way, that the given set of equivalences captures everything there is to say about semantic equivalence of XPath expressions. We are aware of two complete axiomatizations for XPath fragments. The first is for the downward, positive and filterfree fragment of XPath [1], a rather limited fragment, and the second [5] concerns Core XPath 2.0, a very expressive language, with nonelementary complexity for query containment (see [4]). In this paper, we study Core XPath 1.0, which was introduced in [7, 8] to capture the navigational core of XPath 1.0. Our main results are: – A complete axiomatization for Core XPath( ↓ +) and for Core XPath(↓), i.e., the fragments with only the descendant and only the child axis, respectively. The axiomatizations are complete both for node expressions and for path expressions. ⋆ This technical report is the full version of a paper accepted for LID 2008 workshop. If you are kindly going to quote it in your work, please check
On the Equational Definition of the Least Prefixed Point
, 2003
"... We propose a method to axiomatize by equations the least pre xed point of an order preserving function. We discuss its domain of application and show that the Boolean Modal Calculus has a complete equational axiomatization. The method relies on the existence of a \closed structure" and its rel ..."
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Cited by 2 (0 self)
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We propose a method to axiomatize by equations the least pre xed point of an order preserving function. We discuss its domain of application and show that the Boolean Modal Calculus has a complete equational axiomatization. The method relies on the existence of a \closed structure" and its relationship to the equational axiomatization of Action Logic is made explicit. The implication operation of a closed strucure is not monotonic in one of its variables; we show that the existence of such a term that does not preserve the order is an essential condition for de ning by equations the least pre xed point. We stress the interplay between closed structures and xed point operators by showing that the theory of Boolean modal algebras is not a conservative extension of the theory of modal algebras. The latter is shown to lack the nite model property.