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The Lascar Group and the Strong Types of Hyperimaginaries

Byunghan Kim
2013 Notre Dame Journal of Formal Logic  
This is an expository note on the Lascar group. We also study the Lascar group over hyperimaginaries, and make some new observations on the strong types over those.  ...  definable bounded equivalence relation (over A). Choose a model M containing c. Note that F type-defines a bounded equivalence relation on M |M | too.  ...  Note that for a type Ψ(x, M ) type-defining ν −1 ({id}), Ψ(x, y) must be the finest bounded type-definable equivalence relation on tp(M/A).  ... 
doi:10.1215/00294527-2143916 fatcat:yo6veora5befhjeppvdjpfvd5y

Normal hyperimaginaries [article]

Enrique Casanovas, Joris Potier
2013 arXiv   pre-print
In the second part, we show that all closed sets in Kim-Pillay spaces are equivalent to hyperimaginaries and we use this to introduce an approximation of φ-types for bounded hyperimaginaries.  ...  We present a new proof of Lascar-Pillay's theorem on bounded hyperimaginaries based on properties of normal hyperimaginaries.  ...  Given normal e and d ∈ bdd(∅), there are a κ-tuple m and a 0-type-definable equivalence relation F on κ-tuples such that, if E is the 0-type-definable equivalence relation ≡ e on κ-tuples, then m E ∼ e  ... 
arXiv:1112.2049v3 fatcat:uryyxpa2ljfhhfyjzgwtvpk2ua

A type directed translation of MLF to system F

Daan Leijen
2007 Proceedings of the 2007 ACM SIGPLAN international conference on Functional programming - ICFP '07  
Based on insight gained from this translation, we also define "Rigid MLF" (MLF = ), a restriction of MLF where all bound values have a System F type.  ...  We believe that this partly because it is unclear how the rich language of MLF types relate to standard System F types.  ...  The equivalence relation in Figure 4 defines an equivalence on types that abstracts from syntactical artifacts, like the order of the quantifiers.  ... 
doi:10.1145/1291151.1291169 dblp:conf/icfp/Leijen07 fatcat:c5cs4kzavreytntgyg76zkdhzi

A type directed translation of MLF to system F

Daan Leijen
2007 SIGPLAN notices  
Based on insight gained from this translation, we also define "Rigid MLF" (MLF = ), a restriction of MLF where all bound values have a System F type.  ...  We believe that this partly because it is unclear how the rich language of MLF types relate to standard System F types.  ...  The equivalence relation in Figure 4 defines an equivalence on types that abstracts from syntactical artifacts, like the order of the quantifiers.  ... 
doi:10.1145/1291220.1291169 fatcat:uiq4ho5lezavpnxylkf4ntdcvi

Topological dynamics and the complexity of strong types [article]

Krzysztof Krupiński, Anand Pillay, Tomasz Rzepecki
2015 arXiv   pre-print
All the results which we obtain for bounded, invariant equivalence relations carry over to the case of bounded index, invariant subgroups of definable groups.  ...  Let C be a monster model of a countable theory, p ∈ S(∅), and E be a bounded, (invariant) Borel (or, more generally, analytic) equivalence relation on p(C).  ...  A bounded, type-definable equivalence relation in a countable theory is smooth.  ... 
arXiv:1510.00340v1 fatcat:ldzie6nwd5fqrjt3yt5ekt3m2q

Subtyping constrained types [chapter]

Valery Trifonov, Scott Smith
1996 Lecture Notes in Computer Science  
This paper develops several subtyping relations on polymorphic constrained types of a general form that allows recursive constraints and multiple bounds on type variables.  ...  A constrained type is a type that comes with a set of subtyping constraints on variables occurring in the type.  ...  Amadio and Cardelli [3] present a type system µ with recursive types (modeled by regular trees) and a subtyping relation on them equivalent to ≤ tree .  ... 
doi:10.1007/3-540-61739-6_52 fatcat:fvmhiolphrhwzkovlqkc3yawt4

EQUIVALENCE RELATIONS INVARIANT UNDER GROUP ACTIONS

TOMASZ RZEPECKI
2018 Journal of Symbolic Logic (JSL)  
AbstractWe extend some recent results about bounded invariant equivalence relations and invariant subgroups of definable groups: we show that type-definability and smoothness are equivalent conditions  ...  in a wider class of relations than heretofore considered, which includes all the cases for which the equivalence was proved before.As a by-product, we show some analogous results in purely topological  ...  • a ∅-type-definable, countably supported set X, • a bounded, invariant equivalence relation E on X, • a type-definable and E-saturated set Y ⊆ X.  ... 
doi:10.1017/jsl.2017.88 fatcat:4sbonpjtove7ri2kxtguuoe66e

A note on Lascar strong types in simple theories [article]

Byunghan Kim
1996 arXiv   pre-print
In this paper, we prove for such T, the notion of Lascar Strong type coincides with the notion of a strong type,over an arbitrary set.  ...  Recall that an equivalence relation E (on C n , not necessarily definable,) is said to be bounded if it has strictly less thanκ many equivalence classes.  ...  (Pillay and Poizat [PPo] ) Even in stable T , an ∞-definable bounded equivalence relation need not be equivalent to a conjunction of finite definable equivalence relations: The model M consists of the  ... 
arXiv:math/9608216v1 fatcat:x5bts3xzszgyfj4qras735br3q

General Bindings and Alpha-Equivalence in Nominal Isabelle [chapter]

Christian Urban, Cezary Kaliszyk
2011 Lecture Notes in Computer Science  
Our extension includes new definitions of α-equivalence and establishes automatically the reasoning infrastructure for α-equated terms.  ...  It provides a proving infrastructure for reasoning about programming language calculi involving named bound variables (as opposed to de-Bruijn indices).  ...  terms are definable as datatype in Isabelle/HOL and our relation for α-equivalence is an equivalence relation).  ... 
doi:10.1007/978-3-642-19718-5_25 fatcat:nthyd76evjg5bntpwzleohnrka

SIMPLE ALMOST HYPERDEFINABLE GROUPS

ITAÏ BEN-YAACOV
2006 Journal of Mathematical Logic  
In particular, we prove the existence of stabilisers of Lascar strong types and of the connected and locally components of subgroups, and that in a one-based theory a group is bounded-by-abelian-by-bounded  ...  The next level comes in [BTW01] , where we have to replace the type-definable equivalence relation by which we divide by an almost type-definable one, namely a "nice" union of type-definable relations  ...  Then there is a type-definable equivalence relation E(x, y) on p such that E(a, b) if and only if H(a) and H(b) are commensurate. Proof.  ... 
doi:10.1142/s0219061306000529 fatcat:epmsjczi3jdrpk7fqsd7hkhxeq

General Bindings and Alpha-Equivalence in Nominal Isabelle

Christian Urban, Cezary Kaliszyk, Gilles Barthe
2012 Logical Methods in Computer Science  
Our extension includes new definitions of alpha-equivalence and establishes automatically the reasoning infrastructure for alpha-equated terms.  ...  It provides a proving infrastructure for reasoning about programming language calculi involving named bound variables (as opposed to de-Bruijn indices).  ...  Stephanie Weirich suggested to separate the subgrammars of kinds and types in our Core-Haskell example. Ramana Kumar and Andrei Popescu helped us with comments for an earlier version of this paper.  ... 
doi:10.2168/lmcs-8(2:14)2012 fatcat:elyvg7fhzjcnrbh6cba6n5j73u

Cyclic Evaluation of Transitivity of Reciprocal Relations

B. De. Baets, H. De. Meyer, B. De. Schuymer
2006 Social Choice and Welfare  
A general framework for studying the transitivity of reciprocal relations is presented.  ...  An upper bound function acting upon the ordered weights encountered provides an upper bound for the 'sum minus 1' of these weights.  ...  A reciprocal relation Q on A is g-stochastic transitive if and only if it is cycle-transitive w.r.t. the upper bound function U g defined by  ... 
doi:10.1007/s00355-006-0093-3 fatcat:nf2as4ypgzghjmmioqod3ftzuq

A descending chain condition for groups definable in o-minimal structures

Alessandro Berarducci, Margarita Otero, Yaa'cov Peterzil, Anand Pillay
2005 Annals of Pure and Applied Logic  
Equivalently, G has a smallest (necessarily normal) type-definable subgroup G 00 of bounded index and G/G 00 equipped with the "logic topology" is a compact Lie group.  ...  We prove that if G is a group definable in a saturated o-minimal structure, then G has no infinite descending chain of type-definable subgroups of bounded index.  ...  The first two authors thank Dikran Dikranjan for useful information on compact groups.  ... 
doi:10.1016/j.apal.2005.01.002 fatcat:pmfn2atkljazpfzkl2ly3sg6fq

Provable conditions in computational complexity theory

Daryel Sachse-Åkerlind
1985 Bulletin of the Australian Mathematical Society  
The requirement of being provably equivalent defines an equivalence relation on the set of all algorithms. We call the equivalence classes provable equivalence classes.  ...  We show that for natural complexity questions about / , what can be proved about f depends on the provable equivalence class to which the defining algorithm for / belongs.  ...  Various results are developed relating the two types of provable complexity classes to each other and to ordinary complexity classes.  ... 
doi:10.1017/s0004972700002367 fatcat:a6ibdnwsyvd4lavm4go6cmwcyq

Towards Taming Java Wildcards and Extending Java with Interval Types [article]

Moez A. AbdelGawad
2018 arXiv   pre-print
In this work we build on our concurrent work, in which we model Java subtyping using a partial graph product, to suggest how wildcard types in Java can be generalized, and simplified, to interval types  ...  Of the complex features of generic nominally-typed OO type systems, wildcard types and variance annotations are probably the hardest to fully grasp.  ...  Also, so as to have the lower bound always on the left of a type argument expression and the upper bound always on the right of a type argument expression (as is the customary mathematical notation for  ... 
arXiv:1805.10931v3 fatcat:mkyg44uxp5evrif5bhgjlmcyqa
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