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An inductive-recursive universe generic for small families [article]

Daniel Gratzer
2022 arXiv   pre-print
As a trivial consequence, we show that their observational 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  ... 
arXiv:2202.05529v1 fatcat:pdzxmm7hzzffxetf6sgfp5cuae

Neutrally Expandable Models of Arithmetic [article]

Athar Abdul-Quader, Roman Kossak
2017 arXiv   pre-print
We show that cofinal extensions 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.  ... 
arXiv:1712.06503v1 fatcat:vjlkes252bfnhdjj6t6itulrjy

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  ... 

A Finite Axiomatization of Inductive-Recursive Definitions [chapter]

Peter Dybjer, Anton Setzer
1999 Lecture Notes in Computer Science  
In this article we give a finite axiomatization 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  ... 
doi:10.1007/3-540-48959-2_11 fatcat:zbpezf6ayvazhhobinsmwgioju

Constructing a universe for the setoid model [chapter]

Thorsten Altenkirch, Simon Boulier, Ambrus Kaposi, Christian Sattler, Filippo Sestini
2021 Lecture Notes in Computer Science  
To this aim, we present the construction 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.  ... 
doi:10.1007/978-3-030-71995-1_1 fatcat:chq62pby25dabp5rleyvveztdi

Inductive and Coinductive Components of Corecursive Functions in Coq

Yves Bertot, Ekaterina Komendantskaya
2008 Electronical Notes in Theoretical Computer Science  
Bove proposed in her thesis an elegant reformulation 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.  ... 
doi:10.1016/j.entcs.2008.05.018 fatcat:qrgv6n7f2re23cacun2fcs3r5i

Inductive and Coinductive Components of Corecursive Functions in Coq [article]

Yves Bertot, Ekaterina Komendantskaya (INRIA Sophia Antipolis)
2008 arXiv   pre-print
Bove proposed in her thesis an elegant reformulation 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.  ... 
arXiv:0807.1524v1 fatcat:37dqfspppja4vleyskh7qfngau

From the Closed Classical Algorithmic Universe to an Open World of Algorithmic Constellations [chapter]

Mark Burgin, Gordana Dodig-Crnkovic
2013 Computing Nature  
Then we explain how new 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.  ... 
doi:10.1007/978-3-642-37225-4_16 fatcat:yfy26okdojggxiruih6cpya34q

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.  ... 

A general formulation of simultaneous inductive-recursive definitions in type theory

Peter Dybjer
2000 Journal of Symbolic Logic (JSL)  
We extend previously given schematic formulations 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.  ... 
doi:10.2307/2586554 fatcat:ycevxjircjbn7k5lgpzdfqisp4

Synthesizing Inductive Lemmas for Reasoning with First-Order Logic with Least Fixpoints [article]

Adithya Murali, Lucas Peña, Eion Blanchard, Christof Löding, P. Madhusudan
2021 arXiv   pre-print
In this paper, we undertake a foundational study 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  ... 
arXiv:2009.10207v2 fatcat:sbg75szo5rbxxkkhxvytrukjoe

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;  ... 

Experience, generations, and limits in machine learning

Mark Burgin, Allen Klinger
2004 Theoretical Computer Science  
The 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.  ... 
doi:10.1016/j.tcs.2003.12.005 fatcat:6dzytvnyynbc3kj5zl2rzqurx4

Experience, generations, and limits in machine learning

M BURGIN
2004 Theoretical Computer Science  
The 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.  ... 
doi:10.1016/s0304-3975(03)00632-7 fatcat:zzcjrqzsabgmflrbdl6knlx3vu

Partiality and recursion in interactive theorem provers – an overview

ANA BOVE, ALEXANDER KRAUSS, MATTHIEU SOZEAU
2014 Mathematical Structures in Computer Science  
However, most mature theorem provers lack a direct treatment 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.  ... 
doi:10.1017/s0960129514000115 fatcat:y3gdgilgd5bvlev3haw4l3fyxi
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