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Reviewing the classical and the de Bruijn notation for λcalculus and pure type systems
 Logic and Computation
, 2001
"... This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentat ..."
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This article is a brief review of the type free λcalculus and its basic rewriting notions, and of the pure type system framework which generalises many type systems. Both the type free λcalculus and the pure type systems are presented using variable names and de Bruijn indices. Using the presentation of the λcalculus with de Bruijn indices, we illustrate how a calculus of explicit substitutions can be obtained. In addition, de Bruijn's notation for the λcalculus is introduced and some of its advantages are outlined.
AUTOMATH and Pure Type Systems
, 1996
"... We study the position of Automath systems within the framework of the Pure Type Systems as discussed in [3]. ..."
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We study the position of Automath systems within the framework of the Pure Type Systems as discussed in [3].
Pure Type Systems with de Bruijn indices
"... Nowadays, type theory has many applications and is used in many different disciplines. Within computer science, logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. A general framework called Pure Type Systems (PTSs for short) ..."
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Nowadays, type theory has many applications and is used in many different disciplines. Within computer science, logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. A general framework called Pure Type Systems (PTSs for short) has been introduced independently by Terlouw and Berardi in 1988 and 1989, in order to provide a unified formalism in which many type systems can be represented. In particular, PTSs allow the representation of the simple theory of types, the polymophic theory of types, the dependent theory of types and various other wellknown type systems such as the Edinburgh Logical Frameworks LF and the Automath system. Pure Type Systems are usually presented using variable names. In this article, we present a formulation of PTSs with de Bruijn indices. De Bruijn indices [6] avoid the problems caused by variable names during the implementation of type systems. We show that PTSs with variable names and PTSs with de Bruijn indices are isomorphic. This isomorphism enables us to answer questions about PTSs with de Bruijn indices including confluence, termination (strong normalisation) and safety (subject reduction).
Belief Revision In Type Theory
"... This paper explores belief revision for belief states in which an agent's beliefs as well as his justifications for these beliefs are explicitly represented in the context of type theory. This allows for a deductive perspective on belief revision which can be implemented using existing machinery for ..."
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This paper explores belief revision for belief states in which an agent's beliefs as well as his justifications for these beliefs are explicitly represented in the context of type theory. This allows for a deductive perspective on belief revision which can be implemented using existing machinery for deductive reasoning.
Variants of the Basic Calculus of Constructions
, 2004
"... In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version i ..."
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In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version in early papers on the subject by Seldin. None of these results is very deep, but it seems useful to collect them in one place.
The ChurchRosser Property for . . .
 Seventh Annual IEEE Symposium on Logic in Computer Science
, 1992
"... In this paper we investigate the ChurchRosser property (CR) for Pure Type Systems with fij reduction. For Pure Type Systems with only fi reduction, CR on well typed terms follows immediately from CR on the so called 'pseudoterms' and subject reduction. For fijreduction, CR on the set of pseudo ..."
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In this paper we investigate the ChurchRosser property (CR) for Pure Type Systems with fij reduction. For Pure Type Systems with only fi reduction, CR on well typed terms follows immediately from CR on the so called 'pseudoterms' and subject reduction. For fijreduction, CR on the set of pseudoterms is just false, as was shown by [Nederpelt 1973]. Here we prove that CR (for fij) on the welltyped terms of a fixed type holds, which is the maximum we can expect in view of Nederpelts counterexample. The proof is given for a large class of Pure Type systems that contains e.g. LF (for which CR for fij was proved by [Salvesen 1989] and [Coquand 1991]), F, F! and the Calculus of Constructions. In the proof, one key lemma (a very weak form of CR for fij on pseudoterms) takes a central position. It is remarkable that in the proof of this key lemma the counterexample to CR for fij is essentially used. 1 Introduction Simply typed lambda calculus (from now on denoted by !) and polymorphic ...
Type Systems for Dummies
"... We extend Pure Type Systems with a function turning each term M of type A into a dummy ∣M ∣ of the same type ( ∣ ⋅ ∣ is not an identity, in that M ≠ ∣M∣). Intuitively, a dummy represents an unknown, canonical object of the given type: dummies are opaque (cannot be internally inspected), and irrele ..."
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We extend Pure Type Systems with a function turning each term M of type A into a dummy ∣M ∣ of the same type ( ∣ ⋅ ∣ is not an identity, in that M ≠ ∣M∣). Intuitively, a dummy represents an unknown, canonical object of the given type: dummies are opaque (cannot be internally inspected), and irrelevant in the sense that dummies of a same type are convertible to each other. This latter condition makes convertibility in PTS with dummies (DPTS) stronger than usual, hence raising not trivial consistency issues. DPTS offer an alternative approach to (proof) irrelevance, tagging irrelevant information at the level of terms and not of types, and avoiding the annoying syntactical duplication of products, abstractions and applications into an explicit and an implicit version, typical of systems like ICC ∗. Categories and Subject Descriptors F.4.1 [Mathematical Logic
Parameters in Pure Type Systems
"... Abstract. In this paper we study the addition of parameters to typed�calculus with definitions. We show that the resulting systems have nice properties and illustrate that parameters allow for a better finetuning of the strength of type systems as well as staying closer to type systems used in pra ..."
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Abstract. In this paper we study the addition of parameters to typed�calculus with definitions. We show that the resulting systems have nice properties and illustrate that parameters allow for a better finetuning of the strength of type systems as well as staying closer to type systems used in practice in theorem provers and programming languages. 1 What are parameters? Parameters occur when functions are only allowed to occur when provided with arguments. As we will show below, both in mathematics and in programming languages the use of parameters is abundant and closely connected to the use of constants and definitions. If we want to be able to use type systems in accordance with practice and yet described in a precise manner, we therefore need parameters, constants, and definitions in type theory as well. Parameters, constants and and���� � ��������� definitions in theorem proving It is interesting to note that