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Modelling general recursion in type theory

2005
*
Mathematical Structures in Computer Science
*

*In*this work, we present a method to formalise

*general*

*recursive*algorithms

*in*

*type*

*theory*. ... Hence,

*general*

*recursive*algorithms have no direct formalisation

*in*

*type*

*theory*since they contain

*recursive*calls that satisfy no syntactic condition guaranteeing termination. ... Formally, it is

*Modelling*

*general*

*recursion*

*in*

*type*

*theory*675 defined by

*recursion*on the length of Γ: ()β ≡ β and (x ∈ α; Γ )β ≡ (x ∈ α)((Γ )β). ...

##
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Towards a theory of abstract data types: A discussion on problems and tools
[chapter]

1980
*
Lecture Notes in Computer Science
*

This paper aims to show that,

doi:10.1007/3-540-09981-6_4
fatcat:vqke4bgb7bbkhltzioe4utian4
*in*order to capture a quite relevant feature such as the*recursiveness*of abstract data*types*,*Model**Theory*works better than Category*Theory*. ... Finally, we consider our own definition of abstract data*type*, based on*model*-theoretic notions; we analyze this definition*in*the frame of the proposed formalization of*recursiveness*, and illustrate the ... level of*generality*~ i.e.*in**model**theory*rather than*in*category*theory*. ...##
###
Page 4414 of Mathematical Reviews Vol. , Issue 86j
[page]

1986
*
Mathematical Reviews
*

Then the

*theory*of the semigroups*in*T has no*model*companion. For (c) this result is new. A sufficient condition is found for a*theory*of semigroups to have no*model*companion. ... First more structural information is obtained for*theories*with fewer than 2° countable*models*; second this information is used to build*theories**in*L, characterizing these countable*models*up to isomorphism ...##
###
Page 5413 of Mathematical Reviews Vol. , Issue 87j
[page]

1987
*
Mathematical Reviews
*

His proof shows that every

*type**in*L, that is realizable*in*every*model*of some L;-*theory*is*generated*by an L,-formula (*generally*speaking, containing additional variables). ... Summary: “We define finite forcing conditions, forcing relations,*generic*sets and*generic**models*for a consistent*theory**in*lattice- valued*model**theory*, and prove the existence theorem for*generic**models*...##
###
A Theory of Class
[chapter]

1997
*
OOIS'96
*

The

doi:10.1007/978-1-4471-0973-0_4
dblp:conf/oois/Simons96
fatcat:4qbgtblthjbo7awknq6z6eepoa
*theory*is*general*,*in*that it encompasses many different approaches to*type*abstraction, such as*type*constructors,*generic*parameters, classes, inheritance and polymorphism. ... The*theory*is elegant,*in*that it is based on a simple generalisation of F-bounds. The*theory*has timely implications for emerging OMG standards and future language designs. ...*In*this*theory*, a*recursive**type*is the least fixed point of the*generator*for a class interface. The two concepts are therefore properly linked. ...##
###
Page 2857 of Mathematical Reviews Vol. , Issue 87f
[page]

1987
*
Mathematical Reviews
*

A 6-

*model*is an w-*model**in*which the notion of well-foundedness is absolute. The authors discuss the connection between 3-*models*and*recursion**in*normal*type*-2 functionals. ... Set*recursion*was introduced by Normann*in*an ear- lier paper [*Generalized**recursion**theory*, II (Oslo, 1977), 303-320, North-Holland, Amsterdam, 1978; MR 80j:03063]. ...##
###
Page 3676 of Mathematical Reviews Vol. , Issue 82i
[page]

1982
*
Mathematical Reviews
*

Properties of nonsplit

*types**in*stable*theories*. (Russian) Sibirsk. Mat. Z. 22 (1981), no. 1, 27-34, 228. ... The authors consider the*models*of a first-order*theory*§*in*which some given formulas define a*model*of another*theory*7. They show that these*models*are the*models*of a*theory*S’. ...##
###
Page 5250 of Mathematical Reviews Vol. , Issue 91J
[page]

1991
*
Mathematical Reviews
*

They have also proved the existence of a simple set with a finitely

*generated**model**in*/,,,.). ... Summary (translated from the Russian): “We prove the following theorem: Let*IN*be a countable homogeneous*model*of a complete decidable*theory*with a computable family of*types*; then IM is 0’-strongly ...##
###
Page 1554 of Mathematical Reviews Vol. 58, Issue 3
[page]

1979
*
Mathematical Reviews
*

The value of

*recursively*saturated and resplendent*models*lies*in*their abundance (any consistent*theory*having infinite*models*has resplendent*models**in*all cardinalities) and the fact that they possess ... They study a notion of reducibility and complexity of*generalized*spectra. J. A. Makowsky (Jerusalem) 58 + 10398 Schlipf, John Stewart Toward*model**theory*through*recursive*saturation. J. ...##
###
An inductive-recursive universe generic for small families
[article]

2022
*
arXiv
*
pre-print

As a trivial consequence, we show that their observational

arXiv:2202.05529v1
fatcat:pdzxmm7hzzffxetf6sgfp5cuae
*type**theory*admits interpretations*in*Grothendieck topoi suitable for use as internal languages. ... We show that it is possible to construct a universe*in*all Grothendieck topoi with injective codes a la Pujet and Tabareau which is nonetheless*generic*for small families. ... Importantly, while induction-*recursion**generally*has remarkable proof-theoretic strength, small induction-*recursion*is a fairly innocuous reasoning principle and can be encoded*in*extensional*type**theory*...##
###
Page 4964 of Mathematical Reviews Vol. , Issue 86k
[page]

1986
*
Mathematical Reviews
*

At the same time, logicians will recognize the present

*theory*as a simultaneous refinement and*generalization*of first-order definability,*in*the spirit of abstract*model**theory*. ... The*models*of this*theory*include (of course) ordi- nary*recursion*on the natural numbers,*recursion**in*higher*types*, positive elementary induction [the author, Elementary induction on abstract structures ...##
###
Higher-Order Model Checking: From Theory to Practice

2011
*
2011 IEEE 26th Annual Symposium on Logic in Computer Science
*

The

doi:10.1109/lics.2011.15
dblp:conf/lics/Kobayashi11
fatcat:uc23ml5ekbc23ojg3herppprdq
*model*checking of higher-order*recursion*schemes (higher-order*model*checking for short) has been actively studied*in*the last decade, and has seen significant progress*in*both*theory*and practice. ... This short article aims to provide an overview of the recent progress*in*higher-order*model*checking and discuss future directions. ... The order of a*recursion*scheme is the highest order of the simple*types*of function symbols, where the order of a simple*type*κ is given by: The tree*generated*by G 0 is shown*in*Figure 1 , which consists ...##
###
Higher-Order Model Checking: An Overview

2015
*
2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science
*

Higher-order

doi:10.1109/lics.2015.9
dblp:conf/lics/Ong15
fatcat:55bfxqlkuzhbtgmzc32m46ooku
*model*checking is about the*model*checking of trees*generated*by*recursion*schemes. The past fifteen years or so have seen considerable progress*in*both*theory*and practice. ... Outline:*In*Section II, we introduce two families of*generators*of infinite structures:*recursion*schemes and higher-order ... to the*model*checking of untyped*recursion*scheme (which is undecidable*in**general*). ...##
###
Interpreting Polymorphic FPC into Domain Theoretic Models of Parametric Polymorphism
[chapter]

2006
*
Lecture Notes in Computer Science
*

This is the first

doi:10.1007/11787006_32
fatcat:2spwgqhc4na6po4ehii2asihwa
*model*of a language with parametric polymorphism,*recursive*terms and*recursive**types**in*a non-linear setting. ... Using recent results about solutions to*recursive*domain equations*in*parametric*models*of PILLY , we show how to interpret FPC*in*these. ...*recursive**types*may be constructed as*in*domain*theory*. ...##
###
Intensional Type Theory with Guarded Recursive Types qua Fixed Points on Universes

2013
*
2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
*

Moreover, we present a

doi:10.1109/lics.2013.27
dblp:conf/lics/BirkedalM13
fatcat:jbot625tcbem3h2ffar5lfcdqa
*general**model*construction for constructing*models*of the intensional*type**theory*with guarded*recursive*functions and*types*. ...*In*particular, we find that the functor category Grpd ω op from the preordered set of natural numbers to the category of groupoids is a*model*of intensional*type**theory*with guarded*recursive**types*. ...*In*this paper we consider more*general**models*of*type**theory*, encompassing intensional*type**theory*. ...
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