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Realizability Models for Type Theories

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. ...

##
###
A Realizability Model for Impredicative Hoare Type Theory
[chapter]

*
Programming Languages and Systems
*

x Benefits:
s x In this talk: a

doi:10.1007/978-3-540-78739-6_26
dblp:conf/esop/PetersenBNM08
fatcat:vwbf3vu375bh5oxs2aim2h5ndi
*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*. ...##
###
DECIDABLE MODELS OF ω-STABLE THEORIES

2014
*
Journal of Symbolic Logic (JSL)
*

We further characterize the decidable

doi:10.1017/jsl.2013.2
fatcat:w3463o6xjrdulj2ia7drhlyw3y
*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*. ...##
###
Distributions of countable models of theories with continuum many types
[article]

2012
*
arXiv
*
pre-print

,

arXiv:1210.4043v1
fatcat:ghimqf6cz5hwrgmdqnrn2qng64
*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 ∞ ...##
###
DISTRIBUTIONS OF COUNTABLE MODELS OF QUITE O-MINIMAL EHRENFEUCHT THEORIES

2020
*
Eurasian Mathematical Journal
*

We describe Rudin-Keisler preorders and distribution functions of numbers of limit

doi:10.32523/2077-9879-2020-11-3-66-78
fatcat:rqfuu6qfrjdnnbxx2aaeph6kjy
*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: ...##
###
Tame Theories with Hyperarithmetic Homogeneous Models

1989
*
Proceedings of the American Mathematical Society
*

A tame

doi:10.2307/2046925
fatcat:53yxot2vbvc6delfmmewattriq
*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*...##
###
Tame theories with hyperarithmetic homogeneous models

1989
*
Proceedings of the American Mathematical Society
*

A tame

doi:10.1090/s0002-9939-1989-0937851-6
fatcat:u3szo2a4ena6rfdzrwjnwuzqu4
*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*...##
###
Persistently finite, persistently arithmetic theories

1983
*
Proceedings of the American Mathematical Society
*

,xn) E T} is a complete consistent

doi:10.1090/s0002-9939-1983-0715872-0
fatcat:rapb7n3n2faehjcgygk5isu46i
*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*...##
###
Persistently Finite, Persistently Arithmetic Theories

1983
*
Proceedings of the American Mathematical Society
*

,xn) E T} is a complete consistent

doi:10.2307/2045502
fatcat:2ab2lc2uwjgozgc3rb3hfrg37u
*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*...##
###
Distributions of countable models of quite o-minimal Ehrenfeucht theories
[article]

2018
*
arXiv
*
pre-print

We describe Rudin-Keisler preorders and distribution functions of numbers of limit

arXiv:1802.08078v1
fatcat:f2jv3v6kmnd5hfgt2uvwi7da5i
*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. ...##
###
Foundations of recursive model theory

1978
*
Annals of Mathematical Logic
*

However, Fact 3 cannot be similarly extended; we witl also show that there exists a complete decidable

doi:10.1016/0003-4843(78)90030-x
fatcat:2tcklys57nebtfxm4tz5b5vwsq
*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*. ...##
###
Specifications via Realizability

2006
*
Electronical Notes in Theoretical Computer Science
*

We present a system, called RZ,

doi:10.1016/j.entcs.2005.08.007
fatcat:epfkad52fndftjrx7jmyb2ngba
*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. ...##
###
Uncountable categoricity for gross models

2004
*
Proceedings of the American Mathematical Society
*

We prove that if

doi:10.1090/s0002-9939-04-07451-9
fatcat:xcedc2fnkna3zc64mo6m7hbooy
*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)}. ...##
###
The inclusion relations of the countable models of set theory are all isomorphic
[article]

2017
*
arXiv
*
pre-print

Analogous results hold also

arXiv:1704.04480v1
fatcat:qbauh5q3vrd6hk77vpoybldwj4
*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*. ...##
###
Orbits of subsets of the monster model and geometric theories
[article]

2017
*
arXiv
*
pre-print

Let M be the monster

arXiv:1702.01892v2
fatcat:7o5eklujkbhmtaoudbyqle5b3q
*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. ...
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