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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 ...##
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Neutrally Expandable Models of Arithmetic
[article]

2017
*
arXiv
*
pre-print

We show that cofinal extensions

arXiv:1712.06503v1
fatcat:vjlkes252bfnhdjj6t6itulrjy
*of*prime*models*are neutrally expandable, and ω_1-like neutrally expandable*models*exist, while no*recursively*saturated*model*is neutrally expandable. ... We study the existence and non-existence*of*neutral sets in various*models**of*PA. ... Acknowledgements Jim Schmerl's comments on a preliminary version*of*this paper allowed us to improve some*of*the results and the overall presentation. Thank you Jim. ...##
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Page 3176 of Mathematical Reviews Vol. , Issue 2003e
[page]

2003
*
Mathematical Reviews
*

*generic*functions by

*recursion*on this

*type*.” ... “Our axiomatizations form a powerful foundation for

*generic*programming with dependent

*types*by introducing a

*type*

*of*codes for indexed

*inductive*-

*recursive*definitions and making it possible to define ...

##
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A Finite Axiomatization of Inductive-Recursive Definitions
[chapter]

1999
*
Lecture Notes in Computer Science
*

In this article we give a finite axiomatization

doi:10.1007/3-540-48959-2_11
fatcat:zbpezf6ayvazhhobinsmwgioju
*of**inductive*-*recursive*definitions. We prove consistency by constructing a set-theoretic*model*which makes use*of*one Mahlo cardinal. ...*Induction*-*recursion*is a schema which formalizes the principles for introducing new sets in Martin-Löf's*type*theory. ... In*general*,*induction*-*recursion*allows that a simultaneously defined function T : U → D for an arbitrary fixed*type*D may participate in the*inductive**generation**of*the set U. • The modified non-*inductive*...##
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Constructing a universe for the setoid model
[chapter]

2021
*
Lecture Notes in Computer Science
*

To this aim, we present the construction

doi:10.1007/978-3-030-71995-1_1
fatcat:chq62pby25dabp5rleyvveztdi
*of*a (non-univalent) universe*of*setoids within the setoid*model*, first as an*inductive*-*recursive*definition, which is then translated to an*inductive*-*inductive*... AbstractThe setoid*model*is a*model**of*intensional*type*theory that validates certain extensionality principles, like function extensionality and propositional extensionality, the latter being a limited ...*Inductive*-*recursive*setoid universe In this section we give a first definition*of*the setoid universe, as a direct*generalization**of*the simple*inductive*-*recursive*definition just shown. ...##
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Inductive and Coinductive Components of Corecursive Functions in Coq

2008
*
Electronical Notes in Theoretical Computer Science
*

Bove proposed in her thesis an elegant reformulation

doi:10.1016/j.entcs.2008.05.018
fatcat:qrgv6n7f2re23cacun2fcs3r5i
*of*the method*of*accessibility predicates that widens the range*of*terminative*recursive*functions formalisable in Constructive*Type*Theory. ... In Constructive*Type*Theory,*recursive*and corecursive definitions are subject to syntactic restrictions which guarantee termination for*recursive*functions and productivity for corecursive functions. ... When defining an*inductive*data*type*, we need to introduce constructors to*generate*the elements*of*the new*type*. ...##
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Inductive and Coinductive Components of Corecursive Functions in Coq
[article]

2008
*
arXiv
*
pre-print

Bove proposed in her thesis an elegant reformulation

arXiv:0807.1524v1
fatcat:37dqfspppja4vleyskh7qfngau
*of*the method*of*accessibility predicates that widens the range*of*terminative*recursive*functions formalisable in Constructive*Type*Theory. ... In Constructive*Type*Theory,*recursive*and corecursive definitions are subject to syntactic restrictions which guarantee termination for*recursive*functions and productivity for corecursive functions. ... When defining an*inductive*data*type*, we need to introduce constructors to*generate*the elements*of*the new*type*. ...##
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From the Closed Classical Algorithmic Universe to an Open World of Algorithmic Constellations
[chapter]

2013
*
Computing Nature
*

Then we explain how new

doi:10.1007/978-3-642-37225-4_16
fatcat:yfy26okdojggxiruih6cpya34q
*models**of*algorithms turned the classical closed algorithmic universe into the open world*of*algorithmic constellations, allowing higher flexibility and expressive power, supporting ... constructivism and creativity in mathematical*modeling*. ... Acknowledgements The authors would like to thank Andree Ehresmann, Hector Zenil and Marcin Schroeder for useful and constructive comments on the previous version*of*this work. ...##
###
Page 6056 of Mathematical Reviews Vol. , Issue 2001I
[page]

2001
*
Mathematical Reviews
*

In this article a

*generalization**of**inductive*definitions,*inductive*-*recursive*definitions, is introduced. ...*Inductive*-*recursive*definitions can be parametrized with parameters ranging over any*type**of*the logical framework. A*general*elimination rule is introduced. ...##
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A general formulation of simultaneous inductive-recursive definitions in type theory

2000
*
Journal of Symbolic Logic (JSL)
*

We extend previously given schematic formulations

doi:10.2307/2586554
fatcat:ycevxjircjbn7k5lgpzdfqisp4
*of**inductive*definitions in*type*theory to encompass a*general*notion*of*simultaneous*induction*-*recursion*. ... In this paper we argue that there is an underlying*general*notion*of*simultaneous*inductive*-*recursive*definition which is implicit in Martin-Löf's intuitionistic*type*theory. ... In section 3 we give the*general*schema for simultaneous*inductive*-*recursive*de nitions in*type*theory. ...##
###
Synthesizing Inductive Lemmas for Reasoning with First-Order Logic with Least Fixpoints
[article]

2021
*
arXiv
*
pre-print

In this paper, we undertake a foundational study

arXiv:2009.10207v2
fatcat:sbg75szo5rbxxkkhxvytrukjoe
*of*automatically finding proofs that use*induction*to reason in these logics. ... We implement our procedures and evaluate them over a class*of*theorems involving heap datastructures that require*inductive*proofs. ... in the given theory; (c) is*inductive*on every relevant*model*from the*Type*−3*models*(line 8c) -since*Type*−3*models*witness the failure*of*an*induction*proof*of*earlier proposal, this constraint enforces ...##
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Page 551 of Mathematical Reviews Vol. 47, Issue 3
[page]

1974
*
Mathematical Reviews
*

Theorem 3.10: Let 7’, and 7’, be

*inductive*; every*generic**model**of*7’, is a*generic**model**of*T’, if and only if every component*of*7',, is a component*of*Tay. ... Theorem 3.8: Let 7’ be an*inductive*theory, and let J be an irreducible ideal*of*Z,; if J is a component*of*T'y then the*generic**models**of*Y are precisely the*generic**models**of*7 that are*models**of*Y; ...##
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Experience, generations, and limits in machine learning

2004
*
Theoretical Computer Science
*

The

doi:10.1016/j.tcs.2003.12.005
fatcat:6dzytvnyynbc3kj5zl2rzqurx4
*models*use*recursive*, subrecursive, and super-*recursive*algorithms. ... That yields three basic*models*for learning systems: polynomially bounded turing machines, Turing machines, and*inductive*Turing machines*of*the ÿrst order. ... In addition, di erent*types**of*cells facilitate*modeling*the brain neuron structure by*inductive*Turing machines. ...##
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Experience, generations, and limits in machine learning

2004
*
Theoretical Computer Science
*

The

doi:10.1016/s0304-3975(03)00632-7
fatcat:zzcjrqzsabgmflrbdl6knlx3vu
*models*use*recursive*, subrecursive, and super-*recursive*algorithms. ... That yields three basic*models*for learning systems: polynomially bounded turing machines, Turing machines, and*inductive*Turing machines*of*the ÿrst order. ... In addition, di erent*types**of*cells facilitate*modeling*the brain neuron structure by*inductive*Turing machines. ...##
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Partiality and recursion in interactive theorem provers – an overview

2014
*
Mathematical Structures in Computer Science
*

However, most mature theorem provers lack a direct treatment

doi:10.1017/s0960129514000115
fatcat:y3gdgilgd5bvlev3haw4l3fyxi
*of*partial and*general**recursive*functions; overcoming this weakness has been the objective*of*intensive research during the last decades. ... In this article, we review several techniques that have been proposed in the literature to simplify the formalization*of*partial and*general**recursive*functions in interactive theorem provers. ... constructs a total*model*for any tail-*recursive*equation. ...
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