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52
Observational logic
 In Algebraic Methodology and Software Technology (AMAST'98
, 1999
"... Abstract. We present an institution of observational logic suited for statebased systems specifications. The institution is based on the notion of an observational signature (which incorporates the declaration of a distinguished set of observers) and on observational algebras whose operations are r ..."
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Cited by 53 (10 self)
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Abstract. We present an institution of observational logic suited for statebased systems specifications. The institution is based on the notion of an observational signature (which incorporates the declaration of a distinguished set of observers) and on observational algebras whose operations are required to be compatible with the indistinguishability relation determined by the given observers. In particular, we introduce a homomorphism concept for observational algebras which adequately expresses observational relationships between algebras. Then we consider a flexible notion of observational signature morphism which guarantees the satisfaction condition of institutions w.r.t. observational satisfaction of arbitrary firstorder sentences. From the proof theoretical point of view we construct a sound and complete proof system for the observational consequence relation. Then we consider structured observational specifications and we provide a sound and complete proof system for such specifications by using a general, institutionindependent result of [6]. 1
Shelah’s stability spectrum and homogeneity spectrum in finite diagrams
 Arch. Math. Logic
"... Spectrum theorems, as well as the equivalence between the order property and instability in the framework of Finite Diagrams. Finite Diagrams is a context which generalizes the first order case. Localized versions of these theorems are presented. Our presentation is based on several papers; the poin ..."
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Cited by 20 (16 self)
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Spectrum theorems, as well as the equivalence between the order property and instability in the framework of Finite Diagrams. Finite Diagrams is a context which generalizes the first order case. Localized versions of these theorems are presented. Our presentation is based on several papers; the point of view is contemporary and some of the proofs are new. The treatment of local stability in Finite Diagrams is new.
Categoricity over P for first order T or categoricity for φ ∈ Lω1ω can stop at ℵk while holding for
 0, · · · , ℵk−1. Israel Journal of Mathematics, 70:219–235
, 1990
"... In the 1950’s, Los conjectured that if T was countable first order theory in a language L then if it was categorical in some uncountable power then it was categorical in all uncountable powers. In [7], Morley proved this. Buoyed by this success, more general forms of the Los conjecture were consider ..."
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Cited by 20 (3 self)
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In the 1950’s, Los conjectured that if T was countable first order theory in a language L then if it was categorical in some uncountable power then it was categorical in all uncountable powers. In [7], Morley proved this. Buoyed by this success, more general forms of the Los conjecture were considered. In [10], Shelah showed that if T was any first order theory categorical in some power greater than T  then T was categorical in all powers greater than T . Keisler took up the investigation of the Lω1ω case (see [5]) and gave a sufficient condition for the Morley analysis to work in this situation. Unfortunately, this condition was not necessary. (See the counterexample due to Marcus, [6]) To make Leo happy Partially supported by the BSF, grant 323 Supported by a grant from the NSERC
Classification Theory for Abstract Elementary Classes
 In Logic and Algebra, Yi Zhang editor, Contemporary Mathematics 302, AMS,(2002), 165–203
, 2002
"... In this paper some of the basics of classification theory for abstract elementary classes are discussed. Instead of working with types which are sets of formulas (in the firstorder case) we deal instead with Galois types which are essentially orbits of automorphism groups acting on the structure. S ..."
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Cited by 18 (4 self)
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In this paper some of the basics of classification theory for abstract elementary classes are discussed. Instead of working with types which are sets of formulas (in the firstorder case) we deal instead with Galois types which are essentially orbits of automorphism groups acting on the structure. Some of the most basic results in classification theory for non elementary classes are presented. The motivating point of view is Shelah's categoricity conjecture for L# 1 ,# . While only very basic theorems are proved, an effort is made to present number of different technologies: Flavors of weak diamond, models of weak set theories, and commutative diagrams. We focus in issues involving existence of Galois types, extensions of types and Galoisstability.
SIMPLE HOMOGENEOUS MODELS
, 2002
"... Geometrical stability theory is a powerful set of modeltheoretic tools that can lead to structural results on models of a simple firstorder theory. Typical results offer a characterization of the groups definable in a model of the theory. The work is carried out in a universal domain of the theor ..."
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Cited by 16 (2 self)
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Geometrical stability theory is a powerful set of modeltheoretic tools that can lead to structural results on models of a simple firstorder theory. Typical results offer a characterization of the groups definable in a model of the theory. The work is carried out in a universal domain of the theory (a saturated model) in which the Stone space topology on ultrafilters of definable relations is compact. Here we operate in the more general setting of homogeneous models, which typically have noncompact Stone topologies. A structure M equipped with a class of finitary relations R is strongly λ−homogeneous if orbits under automorphisms of (M, R) have finite character in the following sense: Given α an ordinal < λ ≤ M  and sequences ā = { ai: i < α}, ¯ b = { bi: i < α} from M, if (ai1,..., ain) and (bi1,..., bin) have the same orbit, for all n and i1 < · · · < in < α, then f(ā) = ¯ b for some automorphism f of (M, R). In this paper strongly λ−homogeneous models (M, R) in which the elements of R induce a symmetric and transitive notion of independence with bounded character are studied. This notion of independence, defined using a combinatorial condition called “dividing”, agrees with forking independence when (M, R) is saturated. The concept central to the development of geometrical stability theory for saturated structures, namely the canonical base, is also shown to exist in this setting. These results broaden the scope of the
Interpolation, Preservation, and Pebble Games
 Journal of Symbolic Logic
, 1996
"... Preservation and interpolation results are obtained for L1! and sublogics L ` L1! such that equivalence in L can be characterized by suitable backandforth conditions on sets of partial isomorphisms. 1 Introduction In the heyday of infinitary logic in the 1960's and 70's, most attention was focuse ..."
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Cited by 12 (5 self)
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Preservation and interpolation results are obtained for L1! and sublogics L ` L1! such that equivalence in L can be characterized by suitable backandforth conditions on sets of partial isomorphisms. 1 Introduction In the heyday of infinitary logic in the 1960's and 70's, most attention was focused on L!1! and its fragments (see e.g. Keisler [19]), since countable formulas seemed best behaved. The past decade has seen a renewed interest in L1! and its finite variable fragments L (k) 1! (for 2 k ! !) and the modal fragment L \Pi 1! (see e.g. Ebbinghaus and Flum [17] on the former and Barwise and Moss [9] on the latter), due to various connections with topics in computer science. These logics form a hierarchy of increasingly powerful logics L \Pi 1! ae L (2) 1! ae L (3) 1! ae : : : ae L (k) 1! ae : : : ae L1! ; with each of these inclusions being proper. Moreover, there is a useful and elegant algebraic characterization of equivalence in L in each of these logics L, from b...
Polish group actions: dichotomies and generalized elementary embeddings
 J. Amer. Math. Soc
, 1997
"... The results in this paper involve two different topics in the descriptive theory of Polish group actions. The book BeckerKechris [6] is an introduction to that theory. Our two topics—and two collections of theorems—are rather unrelated, but the proofs for both topics are essentially the same. ..."
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Cited by 9 (0 self)
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The results in this paper involve two different topics in the descriptive theory of Polish group actions. The book BeckerKechris [6] is an introduction to that theory. Our two topics—and two collections of theorems—are rather unrelated, but the proofs for both topics are essentially the same.
Shelah’s categoricity conjecture from a successor for tame abstract elementary classes
 The Journal of Symbolic Logic
, 2006
"... elementary classes. Theorem 0.1. Suppose that K is a χtame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ ≥ Max{χ, LS(K) +}. If K is categorical in λ and λ +, then K is categorical in λ ++. Combining this theorem with ..."
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Cited by 9 (3 self)
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elementary classes. Theorem 0.1. Suppose that K is a χtame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ ≥ Max{χ, LS(K) +}. If K is categorical in λ and λ +, then K is categorical in λ ++. Combining this theorem with some results from [Sh 394], we derive a form of Shelah’s Categoricity Conjecture for tame abstract elementary classes: Corollary 0.2. Suppose K is a χtame abstract elementary class satisfying the amalgamation and joint embedding properties. Let µ0:= Hanf(K). If χ ≤ ℶ (2 µ 0) + and K is categorical in some λ +> ℶ (2 µ 0) +, then K is categorical in µ for all µ> ℶ (2 µ 0) +.
Ranks and pregeometries in finite diagrams
 Ann. Pure Appl. Logic
, 2000
"... ABSTRACT. The study of classes of models of a finite diagram was initiated by S. Shelah in 1969. A diagram D is a set of types over the empty set, and the class of models of the diagram D consists of the models of T which omit all the types not in D. In this work, we introduce a natural dependence r ..."
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Cited by 8 (1 self)
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ABSTRACT. The study of classes of models of a finite diagram was initiated by S. Shelah in 1969. A diagram D is a set of types over the empty set, and the class of models of the diagram D consists of the models of T which omit all the types not in D. In this work, we introduce a natural dependence relation on the subsets of the models for the ℵ0stable case which share many of the formal properties of forking. This is achieved by considering a rank for this framework which is bounded when the diagram D is ℵ0stable. We can also obtain pregeometries with respect to this dependence relation. The dependence relation is the natural one induced by the rank, and the pregeometries exist on the set of realizations of types of minimal rank. Finally, these concepts are used to generalize many of the classical results for models of a totally transcendental firstorder theory. In fact, strong analogies arise: models are determined by their pregeometries or their relationship with their pregeometries; however the proofs are different, as we do not have compactness. This is illustrated with positive results (categoricity) as well as negative results (construction of nonisomorphic models).