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12
On Stepwise Explicit Substitution
, 1993
"... This paper starts by setting the ground for a lambda calculus notation that strongly mirrors the two fundamental operations of term construction, namely abstraction and application. In particular, we single out those parts of a term, called items in the paper, that are added during abstraction and a ..."
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Cited by 41 (30 self)
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This paper starts by setting the ground for a lambda calculus notation that strongly mirrors the two fundamental operations of term construction, namely abstraction and application. In particular, we single out those parts of a term, called items in the paper, that are added during abstraction and application. This item notation proves to be a powerful device for the representation of basic substitution steps, giving rise to different versions of fireduction including local and global fi reduction. In other words substitution, thanks to the new notation, can be easily formalised as an object language notion rather than remaining a meta language one. Such formalisation will have advantages with respect to various areas including functional application and the partial unfolding of definitions. Moreover our substitution is, we believe, the most general to date. This is shown by the fact that our framework can accommodate most of the known reduction strategies, which range from local to...
The Barendregt Cube with Definitions and Generalised Reduction
, 1997
"... In this paper, we propose to extend the Barendregt Cube by generalising reduction and by adding definition mechanisms. We show that this extension satisfies all the original properties of the Cube including Church Rosser, Subject Reduction and Strong Normalisation. Keywords: Generalised Reduction, ..."
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Cited by 37 (17 self)
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In this paper, we propose to extend the Barendregt Cube by generalising reduction and by adding definition mechanisms. We show that this extension satisfies all the original properties of the Cube including Church Rosser, Subject Reduction and Strong Normalisation. Keywords: Generalised Reduction, Definitions, Barendregt Cube, Church Rosser, Subject Reduction, Strong Normalisation. Contents 1 Introduction 3 1.1 Why generalised reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Why definition mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 The item notation for definitions and generalised reduction . . . . . . . . . . 4 2 The item notation 7 3 The ordinary typing relation and its properties 10 3.1 The typing relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Properties of the ordinary typing relation . . . . . . . . . . . . . . . . . . . . 13 4 Generalising reduction in the Cube 15 4.1 The generalised...
A unified approach to Type Theory through a refined λcalculus
, 1994
"... In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. ..."
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Cited by 14 (13 self)
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In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic.
Canonical typing and Πconversion
, 1997
"... In usual type theory, if a function f is of type oe ! oe and an argument a is of type oe, then the type of fa is immediately given to be oe and no mention is made of the fact that what has happened is a form of ficonversion. A similar observation holds for the generalized Cartesian product typ ..."
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Cited by 3 (3 self)
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In usual type theory, if a function f is of type oe ! oe and an argument a is of type oe, then the type of fa is immediately given to be oe and no mention is made of the fact that what has happened is a form of ficonversion. A similar observation holds for the generalized Cartesian product types, \Pi x:oe : . In fact, many versions of type theory assume that fi holds of both types and terms, yet only a few attempt to study the theory where terms and types are really treated equally and where ficonversion is used for both. A unified treatment however, of types and terms is becoming indispensible especially in the approaches which try to generalise many systems under a unique one. For example, [Barendregt 91] provides the Barendregt cube and the Pure Type Systems (PTSs) which are a generalisation of many type theories. Yet even such a generalisation does not use ficonversion for both types and terms. This is unattractive, in a calculus where types have the same syntax as terms (such as the calculi of the cube or the PTSs). For example, in those systems, even though compatibility holds for the typing of abstraction, it does not hold for the typing of application. That is, even though M : N ) y:P :M : \Pi y:P :N holds, the following does not hold: Based on this observation, we present a calculus in which the conversion rules apply to types as well as terms. Abstraction and application, moreover, range over both types and terms. We extend the calculus with a canonical type operator in order to associate types to terms. The type of fa will then be Fa, where F is the type of f and the statement \Gamma ` t : oe from usual type theory is split in two statements in our system: \Gamma ` t and (\Gamma; t) = oe. Such a splitting enables us to discuss the two questio...
A Behavioural Model for Klopâ€™s Calculus
 Logic, Model and Computer Science, ENTCS
, 2006
"... Replace this file withprentcsmacro.sty for your meeting, or withentcsmacro.sty for your meeting. Both can be ..."
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Cited by 2 (1 self)
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Replace this file withprentcsmacro.sty for your meeting, or withentcsmacro.sty for your meeting. Both can be
Refining Reduction in the lambda calculus
, 1996
"... We introduce a calculus notation which enables us to detect in a term, more fi redexes than in the usual notation. On this basis, we define an extended fireduction which is yet a subrelation of conversion. The Church Rosser property holds for this extended reduction. Moreover, we show that we c ..."
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We introduce a calculus notation which enables us to detect in a term, more fi redexes than in the usual notation. On this basis, we define an extended fireduction which is yet a subrelation of conversion. The Church Rosser property holds for this extended reduction. Moreover, we show that we can transform generalised redexes into usual ones by a process called "term reshuffling". Keywords: Item notation, Redexes, Church Rosser. Contents 1 Introduction 3 1.1 The item notation and visible redexes . . . . . . . . . . . . . . . . . . . . . . 4 1.2 The system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Generalising redexes and fireduction 7 2.1 Extending redexes from segments to couples . . . . . . . . . . . . . . . . . . . 7 2.2 Extending fireduction and the Church Rosser theorem . . . . . . . . . . . . . 8 3 Term reshuffling 10 3.1 Partitioning terms into bachelor and wellbalanced segments . . . . . . . . . . 11 3.2 The reshuffling procedure...
Beyond betaReduction in Church's ...
, 1996
"... In this paper, we shall write ! using a notation, item notation, which enables one to make more redexes visible, and shall extend fireduction to all visible redexes. We will prove that ! written in item notation and accommodated with extended reduction, satisfies all its original properties (such a ..."
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In this paper, we shall write ! using a notation, item notation, which enables one to make more redexes visible, and shall extend fireduction to all visible redexes. We will prove that ! written in item notation and accommodated with extended reduction, satisfies all its original properties (such as Church Rosser, Subject Reduction and Strong Normalisation). The notation itself is very simple: if I translates classical terms to our notation, then I(t 1 t 2 ) j (I(t 2 )ffi)I(t 1 ) and I( v:ae :t) j (ae v )I(t). For example, t j (( x7 :X4 :( x6 :X3 : x5 :X1!X2 :x 5 x 4 )x 3 )x 2 )x 1 , can be written in our item notation as I(t) j (x 1 ffi)(x 2 ffi)(X 4 x7 )(x 3 ffi)(X 3 x6 )((X 1 ! X 2 ) x5 )(x 4 ffi)x 5 where the visible redexes are based on all the matching fficouples. So here, the redexes are based on (x 2 ffi)(X 4 x7 ), (x 3 ffi)(X 3 x6 ) and (x 1 ffi)((X 1 ! X 2 ) x5 ). In classical notation however, only the redexes based on ( x7 :X4 : \Gamma \Gamma)x 2 and ( x6 :X3 : \Gamm...
Chapter 1 Syllabus
"... Rewriting System where the concepts of normalisation and confluence are exposed. Section 2.3 introduces two historic rewriting systems: the calculus which is not normalising and the calculus of Petri nets which is not confluent. 2.1 Monoids A monoid is a set M with a law of composition M \Theta M ..."
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Rewriting System where the concepts of normalisation and confluence are exposed. Section 2.3 introduces two historic rewriting systems: the calculus which is not normalising and the calculus of Petri nets which is not confluent. 2.1 Monoids A monoid is a set M with a law of composition M \Theta M ! M which is associative, and having a unit element. The free monoid on a set \Sigma, noted \Sigma , is the set of words or finite sequences of elements of \Sigma. Composition in \Sigma is defined by concatenation and unit is the empty word e. A presentation of a monoid M consists of 1. a set \Sigma of generators, 2. a set R of relations r s where r and s are words in \Sigma . such that M is (isomorphic to) the quotient of \Sigma by the congruence associated to R. A rewriting system is a presentation (\Sigma; R) where every relation is explictly oriented and considered as a rewrite rule. Example. The monoid M 4 of symmetries of the square 1 is presented by \Sigma = fae; oeg and...
On Weak and Strong Normalisations
, 1996
"... With the help of continuations, we first construct a transformation T which transforms every  term t into a Iterm T (t). Then we apply the conservation theorem in calculus to show that t is strongly normalisable if T (t) has a finormal form. In this way, we succeed in establishing the equivalenc ..."
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With the help of continuations, we first construct a transformation T which transforms every  term t into a Iterm T (t). Then we apply the conservation theorem in calculus to show that t is strongly normalisable if T (t) has a finormal form. In this way, we succeed in establishing the equivalence between weak and strong normalisation theorems in various typed calculi. This not only enhances the understanding between weak and strong normalisations, but also presents an elegant approach to proving strong normalisation theorems via the notion of weak normalisations. 1. Introduction In calculus or some other rewriting systems, a term is said to be weakly normalisable if it can be reduced to a normal form in some way while a term is strongly normalisable if every reduction sequence from it terminates with a normal form. Clearly, there exist weakly normalisable terms in calculus which are not strongly normalisable. For instance, (x:y:y)\Omega for\Omega = (x:xx)(x:xx) has a normal ...
Postponement, Conservation and Preservation of Strong Normalisation for Generalised Reduction
"... Postponement of K contractions and the conservation theorem do not hold for ordinary but have been established by de Groote for a mixture of with another reduction relation. In this paper, de Groote's results are generalised for a single reduction relation e which generalises . We show morever, t ..."
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Postponement of K contractions and the conservation theorem do not hold for ordinary but have been established by de Groote for a mixture of with another reduction relation. In this paper, de Groote's results are generalised for a single reduction relation e which generalises . We show morever, that e has the Preservation of Strong Normalisation property. Keywords: Generalised reduction, Postponement of Kcontractions, Generalised Conservation, Preservation of Strong Normalisation. 1 The calculus with generalized reduction In the term (( x : y :N)P )Q, the abstraction starting with x and the argument P form the redex ( x : y :N)P . When this redex is contracted, the abstraction starting with y and Q will in turn form a redex. It is important to note that Q (or some residual of Q) is the only argument that the abstraction (or some residual of the abstraction) starting with y can ever have. This fact has been exploited by many researchers. Reduction has been ex...