## First Year Report: (2007)

### BibTeX

@MISC{07firstyear,

author = {},

title = {First Year Report:},

year = {2007}

}

### OpenURL

### Abstract

3.1 Presentation of the λ-Calculus..................... 5

### Citations

1358 | 2004): A Structural Approach to Operational Semantics
- Plotkin
(Show Context)
Citation Context ...lt for the system E which is a type system integrating expansion. Since expansion involves operations and computational steps transforming types, we may also think about an operational semantics (see =-=[57]-=-). In the framework of realisability semantics, I being a function from A to 2 Λ and �−�I an interpretation function from Type E to 2 Λ , if σ, τ ∈ Type E and σ = τ is an equality of figure 3, we cert... |

1161 |
Lambda Calculus: its Syntax and Semantics
- Barendregt
- 1984
(Show Context)
Citation Context ...ed and generalized. For example, two well known methods to prove the confluence of the λ-calculus are the method of parallel reductions and the method of finiteness of developments, both presented in =-=[4]-=-. Another remarkable improvement has been the introduction of type systems in which proofs are introduced as part of the defined theory. This led to the discovery that types in type system can be asso... |

993 |
Term Rewriting and All That
- Baader, Nipkow
- 1998
(Show Context)
Citation Context ...ing period of the thesis. Finally, we conclude in Section 9. 2 Definitions and notations We assume familiarity with the notations of term rewriting, lambda calculus and type theory as can be found in =-=[4, 5, 54, 3]-=-. Usually, we let n, m, i, j, k range over the set of natural numbers Nat. When we define a metavariable, say a, to range over a set, we implicitly define, the matavariable ai, ∀i ≥ 0, to range over t... |

787 | On understanding types, data abstraction and polymorphism
- Cardelli, Wegner
- 1985
(Show Context)
Citation Context ...ither a program calculating a term of type σ from a term of type σ or a program calculating a term of type τ from a term of type τ. The polymorphism of an intersection type is said to be “parametric” =-=[9]-=-, since a program which possesses an intersection type works “uniformly” on the range of the types given by the intersection. This is in contrast to “ad-hoc” polymorphism which is the polymorphism of ... |

544 | Lambda Calculi with Types
- Barendregt
- 1992
(Show Context)
Citation Context ...ame time, the theory of functionality of Curry and Feys [18] has been a notable development in the nowadays called the simply typed λ-calculus (which can be expressed in the Curry or the Church style =-=[5]-=-). The simply typed λ-calculus is a type system based on functional types (simple types are built from a set of type variables, a set of type constants and a functional type). If we use → to denote th... |

491 |
Denotational semantics: The Scott-Strachey approach to programming language theory
- Stoy
- 1977
(Show Context)
Citation Context ...though realisability semantics is a pleasant and natural semantics to work with, there exist other semantics which may be investigated. Carlier suggests in [10] that a the denotational semantics (see =-=[61]-=-) should be built for the system E which is a type system integrating expansion. Since expansion involves operations and computational steps transforming types, we may also think about an operational ... |

289 |
An unsolvable problem of elementary number theory
- Church
(Show Context)
Citation Context ... for logic and functions. As noticed in section 3.1, the system defined by Church in [12] turns out to be inconsistent. To solve the problem, Church used type free λ-calculus to investigate functions =-=[13]-=-, and in order to deal with logic and functions he added simple types to λ-calculus giving us the “simple theory of types”. The simple theory of types is based on a decorated (terms decorated with typ... |

256 |
Combinatory Reduction Systems
- Klop
- 1980
(Show Context)
Citation Context ...ll known method uses the result of finiteness of “reduction of residuals” or finiteness of “developments”. This result has been proved for various calculi and various reductions, by different methods =-=[18, 6, 51, 35, 36, 54]-=-. This result turned to be useful in proofs of the Church-Rosser property [18, 6]. Given an initial set of redexes of a λ-term M, a development of M is a reduction whose each step reduces only a resid... |

248 |
Interprétation Fonctionnelle et Élimination des Coupures de l’Arithmétique d’Ordre Supérieur
- Girard
- 1972
(Show Context)
Citation Context ... Based on realisability, Tait [62] developed a method called reducibility to prove properties of the λ-calculus (such as normalization properties). Since then, this method has been improved by Girard =-=[29, 30]-=-, Koletsos [52] and Gallier [22, 23, 24, 25] amongst others. So far we have discussed the λ-calculus, type systems and the proof methods known as realisability/reducibility. A system of λ-calculus wit... |

231 | Une extension de l’interpretation de Gödel, à l’analyse, et son application l’élimination des coupures dans l’analyse et la théorie des types - Girard - 1971 |

221 |
Intensional interpretations of functionals of finite type I
- Tait
- 1967
(Show Context)
Citation Context ...nitially, realisability has been designed as a method to interpret formulae, i.e. in the semantics domain. This semantics enables to stress the constructivity of systems. Based on realisability, Tait =-=[62]-=- developed a method called reducibility to prove properties of the λ-calculus (such as normalization properties). Since then, this method has been improved by Girard [29, 30], Koletsos [52] and Gallie... |

219 |
A filter lambda model and the completeness of type assignment
- Barendregt, Coppo, et al.
- 1983
(Show Context)
Citation Context ... functions. Since, his model has been simplified and others have been built. We may cite for example the syntactical model (called λ-model) developed by Hindley and Longo in [39]. The filter model of =-=[7]-=- and the “simple semantics” of [38] turn out to be λ-models. We have to notice that the results obtained in [14] are stated for a subsystem of the λ-calculus called λI-calculus. This calculus is defin... |

128 |
Types et Modèles
- Lambda-Calcul
- 1990
(Show Context)
Citation Context ...references to definitions or lemmas in [47]. The principal result given in [53] is the equality between the sets Λ and CR. The method used to prove this result is a combination of the methods used in =-=[54, 6]-=- (reducibility and developments). As far as we know, the only new result is the proof of the confluence of developments (proof of theorem 3.12 in [53]). In [53], the results and proofs were rather inf... |

116 | Explicit provability and constructive semantics
- Artemov
(Show Context)
Citation Context ...ed to typable terms. This association is known as the Curry-Howard or the Curry-De Bruijn-Howard isomorphism. Brouwer, Heyting and Kolmogorov already suggested this isomorphism in their BHK-semantics =-=[2, 63, 64]-=- which interprets formulae of intuitionistic logic by proofs, which are constructive methods based on functions. Kleene [49] also proposed an interpretation, called realisability, which stresses the c... |

113 |
The principal type scheme of an object in combinatory logic
- Hindley
- 1969
(Show Context)
Citation Context ...gned many types. Existence of a principal type scheme come from the fact that the terms represent some “abstract notions”, it “shows an internal coherence between all functional characters” of a term =-=[34, 17]-=-. If we consider as interpretation of types, the set of terms which can be typed by this type (a realisability semantics) w.r.t. a type system, we can ask ourselves what it means that a term belongs t... |

109 | A Set of Postulates for the Foundation of Logic”, The - Church - 1932 |

87 | The essence of principal typings
- Wells
- 2002
(Show Context)
Citation Context ... principal type scheme. Since, the definition of principal type scheme has been improved. The definition has been extended to principal typing, taking care of the context in which a term is typed. In =-=[68]-=-, Wells gives a new definition, independent from the type system in which the term is typed, of the principal typing of a term. His definition of a principal typing is as follows: • Let S ⊲ M : t if t... |

85 |
On the interpretation of intuitionistic number theory
- Kleene
- 1945
(Show Context)
Citation Context ...Kolmogorov already suggested this isomorphism in their BHK-semantics [2, 63, 64] which interprets formulae of intuitionistic logic by proofs, which are constructive methods based on functions. Kleene =-=[49]-=- also proposed an interpretation, called realisability, which stresses the connection between recursive functions and intuitionism. Many applications have been found to realisability. Initially, reali... |

82 |
An introduction
- Intuitionism
- 1956
(Show Context)
Citation Context ...om Brouwer was the instigator, is to reject all the nonconstructive principles of mathematics. In order to illustrate the viewpoint of a constructivist, we give a common example which can be found in =-=[64, 33]-=-: given A a mathematical statement which has not been proved or refuted, let p be a natural number equal to 1 if A is verified and 0 otherwise. But, while A has not been decided, there does not exist ... |

79 |
Some properties of conversion
- Church, Rosser
- 1936
(Show Context)
Citation Context ...important questions need to be answered about the λ-calculus taken as part as a logical system: • Some forms of consistency has been proved since the conception of λ-calculus such as the Theorem 1 of =-=[14]-=- of which a stronger form is now well known as the Church-Rosser theorem. A binary relation R on Λ satisfies the Church-Rosser property if: ∀M, M1, M2, MR ∗ M1 ∧ MR ∗ M2 ⇒ ∃M3, M1R ∗ M3 ∧ M2R ∗ M3. Le... |

61 | Typability and Type Checking in System F are Equivalent and Undecidable
- Wells
- 1999
(Show Context)
Citation Context ....1) found a term which is not typable in the system Fω: M = (λx.z(x(λf.λu.fu))(x(λv.λg.gv)))(λy.yyy) but which turns to be typable in the rank-3 3 restriction of intersection types. • Wells proved in =-=[67]-=- that type inference in system F is undecidable. However, in [48], Kfoury and Wells defined an intersection type system for which every finite-rank restriction has a decidable type inference. • Wells ... |

51 | Principality and decidable type inference for finiterank intersection types - Kfoury, Wells - 1999 |

49 |
Principal type schemes and λ-calculus semantics
- Coppo, Dezani-Ciancaglini, et al.
- 1980
(Show Context)
Citation Context ...way to express 3polymorphism in [15] using intersection types. These intersection types are lists of usages. Because of the ramified structure of these types, Coppo, Dezani and Venneri introduced in =-=[17]-=- an operation called expansion in order to restore the principal typing property in such systems (in fact, to be able to calculate any typing from a principal one). Since then, this operation has been... |

37 |
Lambda calculus models and extensionality
- Hindley, Longo
- 1980
(Show Context)
Citation Context ...e lattices with continuous functions. Since, his model has been simplified and others have been built. We may cite for example the syntactical model (called λ-model) developed by Hindley and Longo in =-=[39]-=-. The filter model of [7] and the “simple semantics” of [38] turn out to be λ-models. We have to notice that the results obtained in [14] are stated for a subsystem of the λ-calculus called λI-calculu... |

33 | Typed lambda-calculus in classical Zermelo-Fraenkel set theory
- Krivine
(Show Context)
Citation Context ...y has been applied to many fields connected to the λ-calculus. Realisability in proof semantics Realisability is used, among other things, in model theory. We can cite for example the work of Krivine =-=[54, 55]-=-, the work of Hindley [38, 37] or the work of Kamareddine and Nour [21, 44]. In [44] (or in [38]), a type is interpreted by a set of λ-terms depending on the form of the type. These interpretations ar... |

31 |
Highlight of the history of the lambda calculus
- Rosser
- 1982
(Show Context)
Citation Context ...e. Since then, important properties of this calculus have been widely studied in the literature. These properties include the confluence property (also associated with the consistency of the calculus =-=[59]-=-) or the normalization property (also associated with processes of evaluation of the calculus). In order to prove these properties, different methods have been developed and then improved and generali... |

21 |
Coppo and Mariangiola Dezani-Ciancaglini. Structured Communications with Concurrent Constraints
- Mario
- 2009
(Show Context)
Citation Context ...gs, polymorphism. The most popular way to express polymorphism is using the quantifier ∀ as is the case in the system F of Girard. Coppo and Dezani introduced another way to express 3polymorphism in =-=[15]-=- using intersection types. These intersection types are lists of usages. Because of the ramified structure of these types, Coppo, Dezani and Venneri introduced in [17] an operation called expansion in... |

15 |
Degrees, reductions and representability in the lambda calculus
- Barendregt, Bergstra, et al.
- 1976
(Show Context)
Citation Context ...references to definitions or lemmas in [47]. The principal result given in [53] is the equality between the sets Λ and CR. The method used to prove this result is a combination of the methods used in =-=[54, 6]-=- (reducibility and developments). As far as we know, the only new result is the proof of the confluence of developments (proof of theorem 3.12 in [53]). In [53], the results and proofs were rather inf... |

15 |
Reductions of residuals are finite
- Hindley
- 1978
(Show Context)
Citation Context ... rule (η). Let r ∈ {β, η}. In rule (r), the λ-term on the left of the →r is called an r-redex or, just a redex when no ambiguity arises, and the λ-term on the right of the →r is called its contractum =-=[36]-=-. We define R r to be the set of λ-terms which are rredexes. We define →βη to be →β ∪ →η and R βη = R β ∪ R η . Let r ∈ {β, η, βη}. We prefer the notation =r instead of =→r for the symmetric, reflexiv... |

14 |
Church-Rosser theorem for typed functional systems
- Koletsos
(Show Context)
Citation Context ...lity, Tait [62] developed a method called reducibility to prove properties of the λ-calculus (such as normalization properties). Since then, this method has been improved by Girard [29, 30], Koletsos =-=[52]-=- and Gallier [22, 23, 24, 25] amongst others. So far we have discussed the λ-calculus, type systems and the proof methods known as realisability/reducibility. A system of λ-calculus with types allows ... |

12 | A complete characterization of complete intersection-type preorders
- Dezani-Ciancaglini, Honsell, et al.
(Show Context)
Citation Context ...n [37], but using, among others, a semantics called “simple semantics”, which is a realisability semantics as defined by Krivine. Most of the recent intersection type systems involve a subtyping rule =-=[1, 19, 20]-=-. This rule plays a significant part in the power of these type systems. This rule uses a defined preorder relation on types. As it is explained in, for example, [1], 3 the notion of rank is, for exam... |

12 |
A Modern Perspective on Type Theory: From its Origins until Today, volume 29 of Applied Logic Series, chapter 4: Propositions as Types and Pure Type Systems
- Kamareddine, Laan, et al.
- 2005
(Show Context)
Citation Context ...rties . . . . . . . . . . . . . . . . 21 8 A reflection on what to come 23 9 Conclusion 24 21 Introduction After the discovery of some controversial results in analysis during the nineteenth century =-=[42]-=-, a number of mathematicians and logicians have taken the formalization of Mathematics seriously. A new area of research was then developed whose aim is to give solid foundations for mathematics and t... |

12 |
A completeness result for a realisability semantics for an intersection type system
- Kamareddine, Nour
(Show Context)
Citation Context ...y in proof semantics Realisability is used, among other things, in model theory. We can cite for example the work of Krivine [54, 55], the work of Hindley [38, 37] or the work of Kamareddine and Nour =-=[21, 44]-=-. In [44] (or in [38]), a type is interpreted by a set of λ-terms depending on the form of the type. These interpretations are defined in order to interpret types by saturated sets. A set is said to b... |

11 | Intersection types and lambda models
- Alessi, Barbanera, et al.
- 2006
(Show Context)
Citation Context ...n [37], but using, among others, a semantics called “simple semantics”, which is a realisability semantics as defined by Krivine. Most of the recent intersection type systems involve a subtyping rule =-=[1, 19, 20]-=-. This rule plays a significant part in the power of these type systems. This rule uses a defined preorder relation on types. As it is explained in, for example, [1], 3 the notion of rank is, for exam... |

8 |
Church-Rosser property and intersection types. Australian Journal of Logic, 6:37–54
- Koletsos, Stavrinos
- 2008
(Show Context)
Citation Context ...there are different names to refer to the Church-Rosser property. For example, we can find the name Church-Rosser in [59, 26, 27, 4] to refer to the property which we presented in the section 3.1. In =-=[52, 53]-=- a λ-term is said to be possessing the Church-Rosser property if it belongs to the set CR. We can also find the name confluence in, for example, [26, 27]. Sometimes the following property on R (a bina... |

7 | Résultats de complétude pour des classes de types du système AF2”, Informatique Théorique et Application
- FARKH, NOUR
- 1998
(Show Context)
Citation Context ...y in proof semantics Realisability is used, among other things, in model theory. We can cite for example the work of Krivine [54, 55], the work of Hindley [38, 37] or the work of Kamareddine and Nour =-=[21, 44]-=-. In [44] (or in [38]), a type is interpreted by a set of λ-terms depending on the form of the type. These interpretations are defined in order to interpret types by saturated sets. A set is said to b... |

7 |
The completeness theorem for typing lambda-terms
- Hindley
- 1983
(Show Context)
Citation Context ...een simplified and others have been built. We may cite for example the syntactical model (called λ-model) developed by Hindley and Longo in [39]. The filter model of [7] and the “simple semantics” of =-=[38]-=- turn out to be λ-models. We have to notice that the results obtained in [14] are stated for a subsystem of the λ-calculus called λI-calculus. This calculus is defined as the λ-calculus but requires t... |

6 | On the correspondence between proofs and λ-terms
- Gallier
- 1995
(Show Context)
Citation Context ...eveloped a method called reducibility to prove properties of the λ-calculus (such as normalization properties). Since then, this method has been improved by Girard [29, 30], Koletsos [52] and Gallier =-=[22, 23, 24, 25]-=- amongst others. So far we have discussed the λ-calculus, type systems and the proof methods known as realisability/reducibility. A system of λ-calculus with types allows different expressive power de... |

4 |
The simple semantics for Coppe-Dezani-Sallé types
- Hindley
(Show Context)
Citation Context ...ation of the terms typable by this type) of an intersection type system. This result is stated for all the lambda models [39], including the so-called filter model. Hindley proved a similar result in =-=[37]-=-, but using, among others, a semantics called “simple semantics”, which is a realisability semantics as defined by Krivine. Most of the recent intersection type systems involve a subtyping rule [1, 19... |

4 |
Typed lambda-calculi with one binder
- Kamareddine
- 2005
(Show Context)
Citation Context ... built in. This is a goal that we will work on (September 2007 – July 2009). • There are other goals that may have to be studied along the way, these include extensions of PTS with: – Unified binders =-=[40, 32]-=-, – Π-application and abbreviations [43, 41] and – type inclusion. 9 Conclusion As is always the case in any Ph.D., the first year is a big learning experience where the student delves into the numero... |

4 | On pi-conversion in the lambda-cube and the combination with abbreviations
- Kamareddine, Bloo, et al.
- 1999
(Show Context)
Citation Context ...on (September 2007 – July 2009). • There are other goals that may have to be studied along the way, these include extensions of PTS with: – Unified binders [40, 32], – Π-application and abbreviations =-=[43, 41]-=- and – type inclusion. 9 Conclusion As is always the case in any Ph.D., the first year is a big learning experience where the student delves into the numerous domains, understands definitions, unpacks... |

4 | Canonical Typing and \Pi--conversion in the Barendregt Cube
- Kamareddine, Nederpelt
- 1996
(Show Context)
Citation Context ...on (September 2007 – July 2009). • There are other goals that may have to be studied along the way, these include extensions of PTS with: – Unified binders [40, 32], – Π-application and abbreviations =-=[43, 41]-=- and – type inclusion. 9 Conclusion As is always the case in any Ph.D., the first year is a big learning experience where the student delves into the numerous domains, understands definitions, unpacks... |

4 |
Type reconstruction in f[omega
- Urzyczyn
- 1997
(Show Context)
Citation Context ...express polymorphism as in system F designed by Girard [29, 30]. Advantages of intersection type systems over type systems with the ∀ quantifier are well explained in [11] and include: • Urzyczyn, in =-=[65]-=-, (theorem 3.1) found a term which is not typable in the system Fω: M = (λx.z(x(λf.λu.fu))(x(λv.λg.gv)))(λy.yyy) but which turns to be typable in the rank-3 3 restriction of intersection types. • Well... |

3 |
H.: Proving Properties of Typed λ-Terms Using Realisability
- Gallier
- 1995
(Show Context)
Citation Context ...eveloped a method called reducibility to prove properties of the λ-calculus (such as normalization properties). Since then, this method has been improved by Girard [29, 30], Koletsos [52] and Gallier =-=[22, 23, 24, 25]-=- amongst others. So far we have discussed the λ-calculus, type systems and the proof methods known as realisability/reducibility. A system of λ-calculus with types allows different expressive power de... |

3 |
H.: Typing Untyped λ-Terms, or Realisability strikes again
- Gallier
- 1998
(Show Context)
Citation Context ...eveloped a method called reducibility to prove properties of the λ-calculus (such as normalization properties). Since then, this method has been improved by Girard [29, 30], Koletsos [52] and Gallier =-=[22, 23, 24, 25]-=- amongst others. So far we have discussed the λ-calculus, type systems and the proof methods known as realisability/reducibility. A system of λ-calculus with types allows different expressive power de... |

3 | Reducibility: A ubiquitous method in lambda calculus with intersection types
- Ghilezan, Likavec
(Show Context)
Citation Context ...n which associates to a λ-term its β-normal form. Church-Rosser In the literature, there are different names to refer to the Church-Rosser property. For example, we can find the name Church-Rosser in =-=[59, 26, 27, 4]-=- to refer to the property which we presented in the section 3.1. In [52, 53] a λ-term is said to be possessing the Church-Rosser property if it belongs to the set CR. We can also find the name conflue... |

2 |
Expansion Algebra: a Foundational Theory with Applications to Type Systems and Type-Based Program Analysis
- Carlier
- 2008
(Show Context)
Citation Context ...xpansion in order to restore the principal typing property in such systems (in fact, to be able to calculate any typing from a principal one). Since then, this operation has been extensively improved =-=[11, 10]-=-. In Section 2, we give some common notations used in this report. In Section 3, we introduce the syntax of the λ-calculus formalized by Church in 1932 [12] and some principal properties of the λ-calc... |

2 |
On girard’s ”candidats de reductibilité”. 2002. Available athttp: //www.cis.upenn.edu/~jean/gbooks/logic.html (last visited 2007–05–15
- Gallier
(Show Context)
Citation Context |

2 |
Ghilezan and Viktor Kunčak. Confluence of untyped lambda calculus via simple types. Lecture
- Silvia
(Show Context)
Citation Context ...alization of the reducibility method in this framework to more practical type systems and term rewriting systems would be both interesting and useful. Moreover, an investigation of the method used in =-=[26]-=- might lead to a simplification of the method used in [53] and in [47]. • It may be asked why is it that every time a calculus is introduced, proving its properties (especially Confluence, Subject Red... |

2 |
2006, ‘Lambda Types on the Lambda Calculus with Abbreviations
- Guidi
(Show Context)
Citation Context ... built in. This is a goal that we will work on (September 2007 – July 2009). • There are other goals that may have to be studied along the way, these include extensions of PTS with: – Unified binders =-=[40, 32]-=-, – Π-application and abbreviations [43, 41] and – type inclusion. 9 Conclusion As is always the case in any Ph.D., the first year is a big learning experience where the student delves into the numero... |