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Modelling general recursion in type theory

ANA BOVE, VENANZIO CAPRETTA
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).  ... 
doi:10.1017/s0960129505004822 fatcat:po4vp4jk7zbznirmwyohmdkhty

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

A. Bertoni, G. Mauri, P. Miglioli
1980 Lecture Notes in Computer Science  
This paper aims to show that, 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.  ... 
doi:10.1007/3-540-09981-6_4 fatcat:vqke4bgb7bbkhltzioe4utian4

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]

Anthony J. H. Simons
1997 OOIS'96  
The 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.  ... 
doi:10.1007/978-1-4471-0973-0_4 dblp:conf/oois/Simons96 fatcat:4qbgtblthjbo7awknq6z6eepoa

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]

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

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

Naoki Kobayashi
2011 2011 IEEE 26th Annual Symposium on Logic in Computer Science  
The 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  ... 
doi:10.1109/lics.2011.15 dblp:conf/lics/Kobayashi11 fatcat:uc23ml5ekbc23ojg3herppprdq

Higher-Order Model Checking: An Overview

Luke Ong
2015 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science  
Higher-order 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).  ... 
doi:10.1109/lics.2015.9 dblp:conf/lics/Ong15 fatcat:55bfxqlkuzhbtgmzc32m46ooku

Interpreting Polymorphic FPC into Domain Theoretic Models of Parametric Polymorphism [chapter]

Rasmus Ejlers Møgelberg
2006 Lecture Notes in Computer Science  
This is the first 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.  ... 
doi:10.1007/11787006_32 fatcat:2spwgqhc4na6po4ehii2asihwa

Intensional Type Theory with Guarded Recursive Types qua Fixed Points on Universes

Lars Birkedal, Rasmus Ejlers Mogelberg
2013 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science  
Moreover, we present a 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.  ... 
doi:10.1109/lics.2013.27 dblp:conf/lics/BirkedalM13 fatcat:jbot625tcbem3h2ffar5lfcdqa
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