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Hybrid Logics: Characterization, Interpolation and Complexity
 Journal of Symbolic Logic
, 1999
"... Hybrid languages are expansions of propositional modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work (for example [BS98, BT98a, BT99]) has focussed on a more constrained system called H(#; @). We sho ..."
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Cited by 100 (35 self)
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Hybrid languages are expansions of propositional modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work (for example [BS98, BT98a, BT99]) has focussed on a more constrained system called H(#; @). We show in detail that H(#; @) is modally natural. We begin by studying its expressivity, and provide model theoretic characterizations (via a restricted notion of EhrenfeuchtFrasse game, and an enriched notion of bisimulation) and a syntactic characterization (in terms of bounded formulas). The key result to emerge is that H(#; @) corresponds to the fragment of rstorder logic which is invariant for generated submodels. We then show that H(#; @) enjoys (strong) interpolation, provide counterexamples for its nite variable fragments, and show that weak interpolation holds for the sublanguage H(@). Finally, we provide complexity results for H(@) and other fragments and variants, and sh...
Finitely Representable Databases
, 1995
"... : We study classes of infinite but finitely representable databases based on constraints, motivated by new database applications such as geographical databases. We formally define these notions and introduce the concept of query which generalizes queries over classical relational databases. We prove ..."
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Cited by 55 (8 self)
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: We study classes of infinite but finitely representable databases based on constraints, motivated by new database applications such as geographical databases. We formally define these notions and introduce the concept of query which generalizes queries over classical relational databases. We prove that in this context the basic properties of queries (satisfiability, containment, equivalence, etc.) are nonrecursive. We investigate the theory of finitely representable models and prove that it differs strongly from both classical model theory and finite model theory. In particular, we show that most of the well known theorems of either one fail (compactness, completeness, locality, 0/1 laws, etc.). An immediate consequence is the lack of tools to consider the definability of queries in the relational calculus over finitely representable databases. We illustrate this very challenging problem through some classical examples. We then mainly concentrate on dense order databases, and exhibit...
A Reference Model for Requirements and Specifications
, 2000
"... We define a reference model for applying formal methods to the development of user requirements and their reduction to behavioral specification of a system. The approach is characterized by its focus on the shared phenomena that define the interface between the system and the environment in which it ..."
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Cited by 46 (4 self)
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We define a reference model for applying formal methods to the development of user requirements and their reduction to behavioral specification of a system. The approach is characterized by its focus on the shared phenomena that define the interface between the system and the environment in which it will operate and on how the parts of this interface are controlled. This paper extends our previous work on this model by representing it in higherorder logic and determining some of its key mathematical ramifications. In particular, we introduce a new form of refinement which is pivotal to defining the desired soundness and consistency properties precisely. 1 Introduction There are a collection of artifacts that commonly arise in programming projects. Among these are the program itself, of course, and also the document that describes the requirements of the software. This requirements document may undergo many revisions as the project proceeds. Requirements often fall into two categorie...
Queries with Arithmetical Constraints
 Theoretical Computer Science
, 1997
"... In this paper, we study the expressive power and the complexity of firstorder logic with arithmetic, as a query language over relational and constraint databases. We consider constraints over various domains (N, Z, Q, and R), and with various arithmetical operations (6, +, \Theta, etc.). We first c ..."
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Cited by 30 (3 self)
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In this paper, we study the expressive power and the complexity of firstorder logic with arithmetic, as a query language over relational and constraint databases. We consider constraints over various domains (N, Z, Q, and R), and with various arithmetical operations (6, +, \Theta, etc.). We first consider the data complexity of firstorder queries. We prove in particular that linear queries can be evaluated in AC 0 over finite integer databases, and in NC 1 over linear constraint databases. This improves previously known bounds. We also show that over all domains, enough arithmetic lead to arithmetical queries, therefore, showing the frontiers of constraints for database purposes. We then tackle the problem of the expressive power, with the definability of the parity and the connectivity, which are the most classical examples of queries not expressible in firstorder logic over finite structures. We prove that these two queries are firstorder definable in presence of (enough) ari...
OrderSorted Feature Theory Unification
, 1997
"... Ordersorted feature (OSF) terms provide an adequate representation for objects as flexible records. They are sorted, attributed, possibly nested, structures, ordered thanks to a subsort ordering. Sort definitions offer the functionality of classes imposing structural constraints on objects. These c ..."
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Cited by 18 (2 self)
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Ordersorted feature (OSF) terms provide an adequate representation for objects as flexible records. They are sorted, attributed, possibly nested, structures, ordered thanks to a subsort ordering. Sort definitions offer the functionality of classes imposing structural constraints on objects. These constraints involve variable sorting and equations among feature paths, including selfreference. Formally, sort definitions may be seen as axioms forming an OSF theory. OSF theory unification is the process of normalizing an OSF term, using sortunfolding to enforce structural constraints imposed on sorts by their definitions. It allows objects to inherit, and thus abide by, constraints from their classes. A formal system is thus obtained that logically models record objects with recursive class definitions accommodating multiple inheritance. We show that OSF theory unification is undecidable in general. However, we propose a set of confluent normalization rules which is complete for detecti...
Formal Properties of Modularisation
"... Summary. Modularity of ontologies is currently an active research field, and many different notions of a module have been proposed. In this paper, we review the fundamental principles of modularity and identify formal properties that a robust notion of modularity should satisfy. We explore these pro ..."
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Cited by 11 (4 self)
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Summary. Modularity of ontologies is currently an active research field, and many different notions of a module have been proposed. In this paper, we review the fundamental principles of modularity and identify formal properties that a robust notion of modularity should satisfy. We explore these properties in detail in the contexts of description logic and classical predicate logic and put them into the perspective of wellknown concepts from logic and modular software specification such as interpolation, forgetting and uniform interpolation. We also discuss reasoning problems related to modularity. 1
Borel sets with large squares
 Fundamenta Mathematicae
, 1999
"... Abstract. For a cardinal µ we give a sufficient condition ⊕µ (involving ranks measuring existence of independent sets) for: ⊗µ: if a Borel set B ⊆ R × R contains a µsquare (i.e. a set of the form A × A, A  = µ) then it contains a 2 ℵ0square and even a perfect square. And also for ⊗ ′ µ: if ψ ∈ ..."
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Cited by 9 (2 self)
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Abstract. For a cardinal µ we give a sufficient condition ⊕µ (involving ranks measuring existence of independent sets) for: ⊗µ: if a Borel set B ⊆ R × R contains a µsquare (i.e. a set of the form A × A, A  = µ) then it contains a 2 ℵ0square and even a perfect square. And also for ⊗ ′ µ: if ψ ∈ Lω1,ω has a model of cardinality µ then it has a model of cardinality continuum generated in a “nice”, “absolute”way. Assuming MA+2 ℵ0> µ for transparency, those three conditions (⊕µ, ⊗µ and ⊗ ′ µ) are equivalent, and by this we get e.g. ∧ [2 ℵ0
Completions of µalgebras
 In Proceedings of the Twentieth Annual IEEE Symposium on Logic in Computer Science (LICS 2005
, 2005
"... A µalgebra is a model of a first order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms (f, µx.f) where µx.f is axiomatized as the least prefixed point of f, whose axioms are equations or equational implications. Standard µalgebras are complete meaning ..."
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Cited by 8 (2 self)
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A µalgebra is a model of a first order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms (f, µx.f) where µx.f is axiomatized as the least prefixed point of f, whose axioms are equations or equational implications. Standard µalgebras are complete meaning that their lattice reduct is a complete lattice. We prove that any non trivial quasivariety of µalgebras contains a µalgebra that has no embedding into a complete µalgebra. We focus then on modal µalgebras, i.e. algebraic models of the propositional modal µcalculus. We prove that free modal µalgebras satisfy a condition – reminiscent of Whitman’s condition for free lattices – which allows us to prove that (i) modal operators are adjoints on free modal µalgebras, (ii) least prefixed points of Σ1operations satisfy the constructive relation µx.f = W n≥0 f n (⊥). These properties imply the following statement: the MacNeilleDedekind completion of a free modal µalgebra is a complete modal µalgebra and moreover the canonical embedding preserves all the operations in the class Comp(Σ1, Π1) of the fixed point alternation hierarchy.
Towards Practical Constraint Databases (Extended Abstract)
, 1996
"... ) St ephane Grumbach I.N.R.I.A. Rocquencourt BP 105 78153 Le Chesnay, France stephane.grumbach@inria.fr Jianwen Su Computer Science Department University of California Santa Barbara, California 93106, USA su@cs.ucsb.edu Abstract We develop a framework for (real) constraint databases based on ..."
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Cited by 8 (3 self)
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) St ephane Grumbach I.N.R.I.A. Rocquencourt BP 105 78153 Le Chesnay, France stephane.grumbach@inria.fr Jianwen Su Computer Science Department University of California Santa Barbara, California 93106, USA su@cs.ucsb.edu Abstract We develop a framework for (real) constraint databases based on finite precision arithmetic which fulfills the main requirements of practical constraint databases. First, it allows the manipulation of approximate values, standard in scientific applications. More importantly, it permits the extension of the relational calculus with aggregate functions, while preserving the fundamental property of closed form evaluation with PTIME data complexity. This is an important step since the initial model of [KKR90] cannot be extended to aggregate functions. Moreover, finite precision computation plays a central role in efficient query processing. We introduce the finite precision semantics of queries and prove expressive power results concerning it. We then prese...
Coherence and Consistency in Domains
 IN THIRD ANNUAL SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 1990
"... Almost all of the categories normally used as a mathematical foundation for denotational semantics satisfy a condition known as consistent completeness. The goal of this paper is to explore the possibility of using a different conditionthat of coherencewhich has its origins in topology and log ..."
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Cited by 8 (4 self)
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Almost all of the categories normally used as a mathematical foundation for denotational semantics satisfy a condition known as consistent completeness. The goal of this paper is to explore the possibility of using a different conditionthat of coherencewhich has its origins in topology and logic. In particular, we concentrate on those posets whose principal ideals are algebraic lattices and whose topologies are coherent. These form a cartesian closed category which has fixed points for domain equations. It is shown that a "universal domain" exists. Since the construction of this domain seems to be of general significance, a categorical treatment is provided and applied to other classes of domains. Universal domains constructed in this fashion enjoy an additional property: they are saturated. We show that there is exactly one such domain in each of the classes under consideration.