Results 1 
4 of
4
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 ..."
Abstract

Cited by 13 (6 self)
 Add to MetaCart
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.
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 ..."
Abstract
 Add to MetaCart
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: 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 ..."
Abstract
 Add to MetaCart
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.