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Realizability Models for Type Theories
1999
Electronical Notes in Theoretical Computer Science
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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]
Programming Languages and Systems
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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
2014
Journal of Symbolic Logic (JSL)
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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]
2012
arXiv
pre-print
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, 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
2020
Eurasian Mathematical Journal
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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
1989
Proceedings of the American Mathematical Society
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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
1989
Proceedings of the American Mathematical Society
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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
1983
Proceedings of the American Mathematical Society
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,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
1983
Proceedings of the American Mathematical Society
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,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]
2018
arXiv
pre-print
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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
1978
Annals of Mathematical Logic
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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
2006
Electronical Notes in Theoretical Computer Science
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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
2004
Proceedings of the American Mathematical Society
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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]
2017
arXiv
pre-print
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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]
2017
arXiv
pre-print
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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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