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Realizability Models for Type Theories

Bernhard Reus
1999 Electronical Notes in Theoretical Computer Science  
Realizability semantics does not only provide intuitive models but can also be used for proving independence results of type theories.  ...  This tutorial aims at giving an account on the realizability models for several constructive t ype theories.  ...  Acknowledgements I wish to thank Thomas Streicher for many discussions about the presented topics, especially his own results, and his criticism regarding this tutorial.  ... 
doi:10.1016/s1571-0661(04)00108-2 fatcat:7v24nb5g6nfifec3xmtg7swqum

A Realizability Model for Impredicative Hoare Type Theory [chapter]

Rasmus Lerchedahl Petersen, Lars Birkedal, Aleksandar Nanevski, Greg Morrisett
Programming Languages and Systems  
x Benefits: s x In this talk: a model for Impredicative Hoare Type Theory (HTT) s for reasoning about state and aliasing Impredicative HTT x Full Higher-order Dependent Type Theory: s Γ τ : Type  ...  Type theory for imp. programs x We integrate Hoare Logic into a Type Theory.  ... 
doi:10.1007/978-3-540-78739-6_26 dblp:conf/esop/PetersenBNM08 fatcat:vwbf3vu375bh5oxs2aim2h5ndi

DECIDABLE MODELS OF ω-STABLE THEORIES

URI ANDREWS
2014 Journal of Symbolic Logic (JSL)  
We further characterize the decidable models of ω-stable theories with countably many countable models as those which realize only recursive types.  ...  In particular, we show that for a countable ω-stable T, every countable model of T admits a decidable presentation if and only if all n-types in T are recursive and T has only countably many countable  ...  First, for all n, every n-type (a complete type in n variables) in T must be recursive, as it is realized in a decidable countable model.  ... 
doi:10.1017/jsl.2013.2 fatcat:w3463o6xjrdulj2ia7drhlyw3y

Distributions of countable models of theories with continuum many types [article]

Roman A. Popkov, Sergey V. Sudoplatov
2012 arXiv   pre-print
, for limit models over types and over sequences of types, and for other countable models of theory.  ...  We present distributions of countable models and correspondent structural characteristics of complete theories with continuum many types: for prime models over finite sets relative to Rudin-Keisler preorders  ...  , that all types in X are approximated so that, if the type p ∞ (x) is realized in a model M of expanded theory, then the type q m (y) is realized in M by the principal formula Q m (a, y), where |= p ∞  ... 
arXiv:1210.4043v1 fatcat:ghimqf6cz5hwrgmdqnrn2qng64

DISTRIBUTIONS OF COUNTABLE MODELS OF QUITE O-MINIMAL EHRENFEUCHT THEORIES

Beibut Kulpeshov, Kazakh-British Technical University, Sergey Sudoplatov, Institute of Mathematics and Mathematical Modeling, Sobolev Institute of Mathematics of Siberian Branch of the Russian Academy of Sciences, Novosibirsk State Technical University, Novosibirsk State University
2020 Eurasian Mathematical Journal  
We describe Rudin-Keisler preorders and distribution functions of numbers of limit models for quite o-minimal Ehrenfeucht theories. Decomposition formulas for these distributions are found.  ...  Since for any type p ∈ S(T ) there exists a countable model M of T , realizing p, and the model M realizes exactly countably many types, the availability of a powerful type implies that T is small , that  ...  Furthermore, we choose among s types p j some m types, responsible for the existence of limit models generated by realizations of these types, and obtain For s = 1 we have 6 = 3 + 1 · 3 · 1, for s = 2:  ... 
doi:10.32523/2077-9879-2020-11-3-66-78 fatcat:rqfuu6qfrjdnnbxx2aaeph6kjy

Tame Theories with Hyperarithmetic Homogeneous Models

Terrence Millar
1989 Proceedings of the American Mathematical Society  
A tame theory is a decidable first-order theory with only countably many countable models, and all complete types recursive.  ...  It is shown here that the recursive complexity of countable homogeneous models of tame theories is unbounded in the hyperarithmetic hierarchy. (1) V«e W [ßn is total];  ...  The types must be sufficiently independent to allow for a complex type spectrum for the desired model, and yet dependent enough to prevent uncountably many type spectra for different models of the theory  ... 
doi:10.2307/2046925 fatcat:53yxot2vbvc6delfmmewattriq

Tame theories with hyperarithmetic homogeneous models

Terrence Millar
1989 Proceedings of the American Mathematical Society  
A tame theory is a decidable first-order theory with only countably many countable models, and all complete types recursive.  ...  It is shown here that the recursive complexity of countable homogeneous models of tame theories is unbounded in the hyperarithmetic hierarchy. (1) V«e W [ßn is total];  ...  The types must be sufficiently independent to allow for a complex type spectrum for the desired model, and yet dependent enough to prevent uncountably many type spectra for different models of the theory  ... 
doi:10.1090/s0002-9939-1989-0937851-6 fatcat:u3szo2a4ena6rfdzrwjnwuzqu4

Persistently finite, persistently arithmetic theories

C. J. Ash, T. S. Millar
1983 Proceedings of the American Mathematical Society  
,xn) E T} is a complete consistent theory in L(ci,..., cn). A (complete) type is a (complete) n-ary type for some n.  ...  A persistently finite theory is a complete consistent theory T such that for every complete type r(ii,..., xn) of T, the theory r(ci,..., c") has only finitely many countable models up to isomorphism.  ...  If for a complete decidable theory there is an r.e. list of all its recursive complete types, and the theory has only countably many complete types, then the theory has a decidable homogeneous model realizing  ... 
doi:10.1090/s0002-9939-1983-0715872-0 fatcat:rapb7n3n2faehjcgygk5isu46i

Persistently Finite, Persistently Arithmetic Theories

C. J. Ash, T. S. Millar
1983 Proceedings of the American Mathematical Society  
,xn) E T} is a complete consistent theory in L(ci,..., cn). A (complete) type is a (complete) n-ary type for some n.  ...  A persistently finite theory is a complete consistent theory T such that for every complete type r(ii,..., xn) of T, the theory r(ci,..., c") has only finitely many countable models up to isomorphism.  ...  If for a complete decidable theory there is an r.e. list of all its recursive complete types, and the theory has only countably many complete types, then the theory has a decidable homogeneous model realizing  ... 
doi:10.2307/2045502 fatcat:2ab2lc2uwjgozgc3rb3hfrg37u

Distributions of countable models of quite o-minimal Ehrenfeucht theories [article]

Beibut Kulpeshov, Sergey Sudoplatov
2018 arXiv   pre-print
We describe Rudin-Keisler preorders and distribution functions of numbers of limit models for quite o-minimal Ehrenfeucht theories. Decomposition formulas for these distributions are found.  ...  Furthermore, we choose among s types p j some m types, responsible for the existence of limit models generated by realizations of these types, and obtain 4 m − 1 possibilities for these limit models by  ...  the type p(x), with the least realization of that type; • one limit model over the type p(x), with the set of realizations of p(x) forming an open interval.  ... 
arXiv:1802.08078v1 fatcat:f2jv3v6kmnd5hfgt2uvwi7da5i

Foundations of recursive model theory

Terrence S. Millar
1978 Annals of Mathematical Logic  
However, Fact 3 cannot be similarly extended; we witl also show that there exists a complete decidable theory T and recursive types F~, F2 of T such that no decidable model of T realizes both F~ and r~  ...  Each recu~sive non-principal type of a complete decidable theory is omitl:ed frm':" some decidable model of that theory.  ...  We will say that a set B is tile witness set for a model of a theory T if it is a witness set for the set of types of T realized in that model.  ... 
doi:10.1016/0003-4843(78)90030-x fatcat:2tcklys57nebtfxm4tz5b5vwsq

Specifications via Realizability

Andrej Bauer, Christopher A. Stone
2006 Electronical Notes in Theoretical Computer Science  
We present a system, called RZ, for automatic generation of program specifications from mathematical theories.  ...  We translate mathematical theories to specifications by computing their realizability interpretations in the ML language augmented with assertions (as comments).  ...  Parameterized Theories A theory may be parameterized by one or more models of other theories. For example, a theory Real of the reals may be parameterized in terms of a model N of the naturals.  ... 
doi:10.1016/j.entcs.2005.08.007 fatcat:epfkad52fndftjrx7jmyb2ngba

Uncountable categoricity for gross models

Michael C. Laskowski, Anand Pillay
2004 Proceedings of the American Mathematical Society  
We prove that if for some uncountable κ, T has a unique gross model of cardinality κ, then for any uncountable κ, T has a unique gross model of cardinality κ.  ...  A model M is said to be gross if all infinite definable sets in M have the same cardinality (as M ).  ...  On the other hand, every (partial) finitary type of T realized in M must be realized in N . But for any ϕ(x,ȳ) ∈ L, N omits the type: {P ϕ (ȳ)} ∪ {there are infinitely many realizations of ϕ(x,ȳ)}.  ... 
doi:10.1090/s0002-9939-04-07451-9 fatcat:xcedc2fnkna3zc64mo6m7hbooy

The inclusion relations of the countable models of set theory are all isomorphic [article]

Joel David Hamkins, Makoto Kikuchi
2017 arXiv   pre-print
Analogous results hold also for class theories such as Gödel-Bernays set theory and Kelley-Morse set theory.  ...  A very weak set theory suffices, even finite set theory, provided that one excludes the ω-standard models with no infinite sets and the ω-standard models of set theory with an amorphous set.  ...  Since the lattice is unbounded, this type is finitely realized, and so by ω-saturation, the whole type is realized.  ... 
arXiv:1704.04480v1 fatcat:qbauh5q3vrd6hk77vpoybldwj4

Orbits of subsets of the monster model and geometric theories [article]

Enrique Casanovas, Luis Jaime Corredor
2017 arXiv   pre-print
Let M be the monster model of a complete first-order theory T. If D is a subset of M, following D. Zambella we consider e(D)={D^'| (M,D)≡ (M,D^')} and o(D)={D^'| (M,D) (M,D^')}.  ...  The case where D is A-invariant for some small set A is rather straightforward: it just mean that D is definable. We investigate the case where D is not invariant over any small subset.  ...  By a monster model of T we understand a model M of T whose universe is a proper class and realizes all types over all subsets. Every theory has a monster model and it is unique up to isomorphism.  ... 
arXiv:1702.01892v2 fatcat:7o5eklujkbhmtaoudbyqle5b3q
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