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Finite Family Developments
"... Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than o ..."
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Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than or equal to m. A rewrite system R enjoys finite family developments (FFD) if R ω is terminating. We show that the class of higher order pattern rewrite systems enjoys FFD, extending earlier results for the lambda calculus and first order term rewrite systems.
Principal Typings in a Restricted Intersection Type System for Beta Normal Forms with de Bruijn Indices
, 2009
"... The λcalculus with de Bruijn indices assembles each αclass of λterms in a unique term, using indices instead of variable names. Intersection types provide finitary type polymorphism and can characterise normalisable λterms, that is a term is normalisable if and only if it is typeable. To be clos ..."
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The λcalculus with de Bruijn indices assembles each αclass of λterms in a unique term, using indices instead of variable names. Intersection types provide finitary type polymorphism and can characterise normalisable λterms, that is a term is normalisable if and only if it is typeable. To be closer to computations and to simplify the formalisation of the atomic operations involved in βcontractions several calculi of explicit substitution were developed and some of them are written with de Bruijn indices. Versions of explicit substitutions calculi without types and with simple type systems are well investigated in contrast to versions with more elaborated type systems such as intersection types. In previous work, we introduced a de Bruijn version of the λcalculus with an intersection type system and proved it preserves the subject reduction, a basic type system property. In this paper a version with de Bruijn indices of an intersection type system originally introduced to characterise principal typings for βnormal forms (βnf for short) is presented. We present the characterisation in this new system and the corresponding versions for the type inference and the reconstruction of normal forms from principal typings algorithms. We briefly discuss about the failure of the subject reduction property and some possible solutions for it. ∗ Supported by a PhD scholarship at the Universidade de Brasília. † Supported by the Fundação de Apoio à Pesquisa do Distrito Federal [FAPDF 8004/2007] 1 1
The Triumph of Types: Principia Mathematica’s Impact on Computer Science
"... Types now play an essential role in computer science; their ascent originates from Principia Mathematica. Type checking and type inference algorithms are used to prevent semantic errors in programs, and type theories are the native language of several major interactive theorem provers. Some of these ..."
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Types now play an essential role in computer science; their ascent originates from Principia Mathematica. Type checking and type inference algorithms are used to prevent semantic errors in programs, and type theories are the native language of several major interactive theorem provers. Some of these trace key features back to Principia. This lecture examines the influence of Principia Mathematica on modern type theories implemented in software systems known as interactive proof assistants. These proof assistants advance daily the goal for which Principia was designed: to provide a comprehensive formalization of mathematics. For instance, the definitive formal proof of the Four Color Theorem was done in type theory. Type theory is considered seriously now more than ever as an adequate foundation for both classical and constructive mathematics as well as for computer science. Moreover, the seminal work in the history of formalized mathematics is the Automath project of N.G. de Bruijn whose formalism is type theory. In addition we explain how type theories have enabled the use of formalized mathematics as a practical programming language, a connection entirely unanticipated at the time of Principia Mathematica’s creation.
The context calculus λc (Extended Abstract)
, 1999
"... Mirna Bognar Roel de Vrijer Abstract The calculus c serves as a general framework for representing contexts. Essential features are control over variable capturing and the freedom to manipulate contexts before or after hole filling, by a mechanism of delayed substitution. The context calculus ..."
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Mirna Bognar Roel de Vrijer Abstract The calculus c serves as a general framework for representing contexts. Essential features are control over variable capturing and the freedom to manipulate contexts before or after hole filling, by a mechanism of delayed substitution. The context calculus c is given in the form of an extension of the lambda calculus. Many notions of context can be represented within the framework; a particular variation can be obtained by the choice of a socalled pretyping. By way of an example we treat the contexts of Hashimoto & Ohori. 1 Introduction The central notion in this paper is that of context, i.e. an expression with special places, called holes, where other expressions can be placed. For example, in the lambda calculus, (x:2)z, where 2 denotes a hole, is a context. In formal systems with bound variables, such as calculus, a distinctive feature of placing an arbitrary expression into a hole of a context is variable capturing: some free varia...
The Triumph of Types: Creating a Logic of Computational Reality
"... Type theory plays an essential role in computing and information science. It is the native language of several industrial strength interactive theorem provers including Coq, HOL, Isabelle, MetaPRL, Nuprl, PVS, and Twelf. These provers are used for building correct by construction software and for cr ..."
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Type theory plays an essential role in computing and information science. It is the native language of several industrial strength interactive theorem provers including Coq, HOL, Isabelle, MetaPRL, Nuprl, PVS, and Twelf. These provers are used for building correct by construction software and for creating formalized mathematical theories whose logical correctness is assured to the highest standards of certainty ever achieved. Interactive provers have also been used to solve open mathematical problems, e.g. definitively proving the Four Color Theorem, finding constructive proofs of Higman’s Lemma and Kruskal’s Theorem, and explaining the Girard paradox. The accumulation of large libraries of formalized mathematical knowledge using provers has led to the field of mathematical knowledge management. Constructive type theories for constructive and intuitionistic mathematics serve as practical programming languages, a connection imagined forty years ago yet only recently realized. These intellectual contributions are matched by an elegant computing and information technology that integrates programming languages, interactive provers, model check