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28
ManySorted Coalgebraic Modal Logic: a Modeltheoretic Study
 Theoretical Informatics and Applications
, 2001
"... This paper gives a semantical underpinning for a manysorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in objectoriented languages. These systems will be described as coalgebras of socalled polynomial functors, built up from constants an ..."
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Cited by 53 (3 self)
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This paper gives a semantical underpinning for a manysorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in objectoriented languages. These systems will be described as coalgebras of socalled polynomial functors, built up from constants and identities, using products, coproducts and powersets. The semantical account involves Boolean algebras with operators indexed by polynomial functors, called MBAOs, for Manysorted Boolean Algebras with Operators, combining standard (categorical) models of modal logic and of manysorted predicate logic.
Reengineering of Configurations Based on Mathematical Concept Analysis
 ACM Transactions on Software Engineering and Methodology
, 1996
"... We apply mathematical concept analysis to the problem of reengineering configurations. Concept analysis will reconstruct a taxonomy of concepts from a relation between objects and attributes. We use concept analysis to infer configuration structures from existing source code. Our tool NORA/RECS will ..."
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Cited by 50 (6 self)
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We apply mathematical concept analysis to the problem of reengineering configurations. Concept analysis will reconstruct a taxonomy of concepts from a relation between objects and attributes. We use concept analysis to infer configuration structures from existing source code. Our tool NORA/RECS will accept source code, where configurationspecific code pieces are controlled by the preprocessor. The algorithm will compute a socalled concept lattice, which —when visually displayed — offers remarkable insight into the structure and properties of possible configurations. The lattice not only displays tinegrained dependencies between configurations, but also visualizes the overall quality of configuration structures according to software engineering principles. In a second step, interferences between configurations can be analyzed in order to restructure or simplify configurations. Interferences showing up in the lattice indicate high coupling and low cohesion between configuration concepts. Source files can then be simplified according to the lattice structure. Finally, we show how governing expressions can be simplified by utilizing an isomorphism theorem of mathematical concept analysis.
The Temporal Logic of Coalgebras via Galois Algebras
, 1999
"... This paper introduces a temporal logic for coalgebras. Nexttime and lasttime operators are dened for a coalgebra, acting on predicates on the state space. They give rise to what is called a Galois algebra. Galois algebras form models of temporal logics like Linear Temporal Logic (LTL) and Computatio ..."
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Cited by 33 (7 self)
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This paper introduces a temporal logic for coalgebras. Nexttime and lasttime operators are dened for a coalgebra, acting on predicates on the state space. They give rise to what is called a Galois algebra. Galois algebras form models of temporal logics like Linear Temporal Logic (LTL) and Computation Tree Logic (CTL). The mapping from coalgebras to Galois algebras turns out to be functorial, yielding indexed categorical structures. This gives many examples, for coalgebras of polynomial functors on sets. Additionally, it will be shown how \fuzzy" predicates on metric spaces, and predicates on presheaves, yield indexed Galois algebras, in basically the same coalgebraic manner. Keywords: Temporal logic, coalgebra, Galois connection, fuzzy predicate, presheaf Classication: 68Q60, 03G05, 03G25, 03G30 (AMS'91); D.2.4, F.3.1, F.4.1 (CR'98). 1 Introduction This paper combines the areas of coalgebra and of temporal logic. Coalgebras are simple mathematical structures (similar, but dual, to...
Towards a Duality Result in the Modal Logic of Coalgebras
 In Coalgebraic Methods in Computer Science, volume 33 of ENTCS
, 2000
"... This paper forms a step in the development of the recently emerged connection between coalgebra and modal logic. It introduces (backandforth) transformations between coalgebras of simple polynomial functors and certain Boolean algebras with operators (BAOs). Categorically, these transformations ta ..."
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Cited by 21 (0 self)
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This paper forms a step in the development of the recently emerged connection between coalgebra and modal logic. It introduces (backandforth) transformations between coalgebras of simple polynomial functors and certain Boolean algebras with operators (BAOs). Categorically, these transformations take the form of an adjunction. The BAO associated with a coalgebra can be used for specification, e.g. of classes in objectoriented languages.
Simulations in Coalgebra
 THEOR. COMP. SCI
, 2003
"... A new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive. Such an order forms a basic building block for a "lax relation lifting", or "relator" as used by other authors. Simulations appear as coalgebras of this lifted functor, an ..."
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Cited by 17 (1 self)
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A new approach to simulations is proposed within the theory of coalgebras by taking a notion of order on a functor as primitive. Such an order forms a basic building block for a "lax relation lifting", or "relator" as used by other authors. Simulations appear as coalgebras of this lifted functor, and similarity as greatest simulation. Twoway similarity is then similarity in both directions. In general, it is different from bisimilarity (in the usual coalgebraic sense), but a su#cient condition is formulated (and illustrated) to ensure that bisimilarity and twoway similarity coincide. Also, suitable conditions are identified which ensures that similarity on a final coalgebra forms an (algebraic) dcpo structure. This involves a close investigation of the iterated applications F (#) and F (1) of a functor F with an order to the initial and final sets.
Normal Linear Regression Models with Recursive Graphical Markov Structure
 J. MULTIVARIATE ANAL
, 1998
"... A multivariate normal statistical model defined by the Markov pr er deter by an acyclic digric admits ar efactorof its likelihood function (LF) into the pr duct of conditional LFs, eachfactor having the for of a classical multivar  linear rear model (# MANOVA model).Her these modelsar extended ..."
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Cited by 13 (5 self)
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A multivariate normal statistical model defined by the Markov pr er deter by an acyclic digric admits ar efactorof its likelihood function (LF) into the pr duct of conditional LFs, eachfactor having the for of a classical multivar  linear rear model (# MANOVA model).Her these modelsar extended in anatur way tonor linear rear models whose LFs continue to admit suchr efactorr frr which maximum likelihoodestimator and likelihoodr (LR) test statistics can beder ed by classical linear methods. The centrdistr  of the LR test statisticfor testing one such multivariv norv linear rear model against another isder ed, and there of theseresesion models to blockr enor linear systems is established. It is shown how a collection of nonnested dependentnor linear rear models (# seemingly unringly ringly can be combined into a single multivariv norvlinear rn grear model by imposing apar set of graphical Markov (# conditional independence) restrictions.
Comparing completeness properties of static analyses and their logics
 Proc. 2006 Asian Programming Languages and Systems Symposium (APLAS’06), volume 4279 of Lecture Notes in Computer Science
, 2006
"... Abstract. Static analyses calculate abstract states, and their logics validate properties of the abstract states. We place into perspective the variety of forwards, backwards, functional, and logical completeness used in abstractinterpretationbased static analysis by giving examples and by proving ..."
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Cited by 11 (4 self)
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Abstract. Static analyses calculate abstract states, and their logics validate properties of the abstract states. We place into perspective the variety of forwards, backwards, functional, and logical completeness used in abstractinterpretationbased static analysis by giving examples and by proving equivalences, implications, and independences. We expose two fundamental Galois connections that underlie the logics for static analyses and reveal a new completeness variant, Ocompleteness. We also show that the key concept underlying logical completeness is covering, which we use to relate the various forms of completeness. When we use a static analysis, like dataflow analysis or model checking, to validate a program for correctness or code improvement, we must carefully define the domain of properties the analysis can calculate so that it includes both the goal properties we seek to validate as well as intermediate properties that lead to the goals. Say we try to validate {?}y: = −y;x: = y +1{isPositive(x)}; our analysis requires properties like isNegative to calculate a sound precondition:
Quantum logic in dagger kernel categories
 Order
"... This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial inject ..."
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Cited by 9 (9 self)
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This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical/ordertheoretic properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, orthomodularity, atomicity and completeness. For instance, the Sasaki hook and andthen connectives are obtained, as adjoints, via the existentialpullback adjunction between fibres. 1
Underapproximating predicate transformers
 In Proc. SAS’06, LNCS
, 2006
"... Abstract. We study the underapproximation of the predicate transformers used to give semantics to the modalities in dynamic and temporal logic. Because predicate transformers operate on state sets, we define appropriate powerdomains for sound approximation. We study four such domains — two are based ..."
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Cited by 9 (4 self)
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Abstract. We study the underapproximation of the predicate transformers used to give semantics to the modalities in dynamic and temporal logic. Because predicate transformers operate on state sets, we define appropriate powerdomains for sound approximation. We study four such domains — two are based on “set inclusion ” approximation, and two are based on “quantification ” approximation — and we apply the domains to synthesize the most precise, underapproximating �pre and pre transformers, in the latter case, introducing a focus operation. We also show why the expected abstractions of post and �post are unsound, and we use the powerdomains to guide us to correct, sound underapproximations. 1
Free modal algebras: a coalgebraic perspective
"... Abstract. In this paper we discuss a uniform method for constructing free modal and distributive modal algebras. This method draws on works by (Abramsky 2005) and (Ghilardi 1995). We revisit the theory of normal forms for modal logic and derive a normal form representation for positive modal logic. ..."
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Cited by 5 (1 self)
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Abstract. In this paper we discuss a uniform method for constructing free modal and distributive modal algebras. This method draws on works by (Abramsky 2005) and (Ghilardi 1995). We revisit the theory of normal forms for modal logic and derive a normal form representation for positive modal logic. We also show that every finitely generated free modal and distributive modal algebra axiomatised by equations of rank 1 is a reduct of a temporal algebra. 1