H. P. Barendregt. Lambda calculi with types. Handbook of logic in computer science (vol. II), pages 117--309, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....rather than logic, and hence depended on a less restrictive, or polymorphic, type systems. For example, Milner s functional language ML in [Milner 84] used a polymorphic type theory (Curry s system) In Type Theory there are attempts at unifying the various formalisms (see [de Bruijn 78] Barendregt 92] and [NK 94] so that results can be carried across theories without duplication of work. It is moreover elegant to have unique formulations of Type Theories. After all, such unification will help to rid of the anarchy present as a result of so many different formulations. In fact, the presence ....

....existing formalisms of natural and programming languages. For further details, the reader is referred to [Kamareddine 94c] 1.1 The System T Omega T Omega was presented in [Kamareddine 94c] and was shown to be an extension of various other systems. We will follow the line of Barendregt in [Barendregt 92] in constructing a tree which will have at its top. All the other systems have been shown to have useful applications related to natural and programming languages. Parsons 79] for example, presents a polymorphic system which can accommodate self referential terms. Reddy 93] presents a ....

Barendregt, H., Lambda calculi with types, Handbook of logic in Computer Science, volume II, ed. Abramsky s., Gabbay D.M., Maibaum T.S.E., Oxford University press, 1992.

....free (because of the equivalences) Hence, one gets: M is SN , M is fi e SN , M is fi e s SN , M is s SN. 2 The formal machinery We assume the reader familiar with the calculus (whose terms are : V j( j( V : take terms modulo ff conversion and use the variable convention VC (as in [Bar 92] which avoids any clash of variables. We use x; y; z; x 1 ; x 2 ; and M;N;P; Q; A; B; A 1 ; to range over V and respectively. We assume the usual definition of substitution and use FV (M) for the set of free variables of M . Because we need to see redexes (ordinary and generalised) ....

Barendregt, H., Lambda calculi with types, Handbook of Logic in Computer Science, volume II, ed. Abramsky S., Gabbay D.M., Maibaum T.S.E., Oxford University Press, 1992.

....we define weight(t) weight(body(t) Definition 2.8 (Statements) A statement is of the form t : ae, t and ae are called the subject and the type of the statement respectively. 7 Convention 2. 9 In a context, we never have two occurrences of v (for the same v) Hence, contexts are what [Barendregt 92] calls bases. We need the following definition over contexts: Definition 2.10 Let Gamma = ae 1 v 1 ) ae 2 v 2 ) Delta Delta Delta (ae k v k ) be a context. Then 1. dom( Gamma) fv 1 ; v 2 ; v k g 2. ae v ) 2 Gamma iff (ae v ) is an item of Gamma. If Gamma is a ....

.... Gamma t : ae Gamma t : ae ae Gamma (tffi)t ( Gammaintroduction) Gamma(ae v ) t : ae Gamma (ae v )t : ae ae 2.2 Properties of Here we list the properties of (that we will establish for extended reduction) without proofs. The reader can refer to [Barendregt 92] for details. Theorem 2.12 (The Church Rosser Theorem) If t fi t 1 and t fi t 2 then there exists t 3 such that t 1 fi t 3 and t 2 fi t 3 2 Lemma 2.13 (Context lemma) 1. 8 Gamma; Gamma 8 t 8 ae [ Gamma Gamma t : ae ) Gamma t : ae] 2. 8 Gamma 8 t 8 ae [ Gamma ....

Barendregt, H., Lambda calculi with types, Handbook of Logic in Computer Science, volume II, ed. Abramsky S., Gabbay D.M., Maibaum T.S.E., Oxford University Press, 1992.

....Some appendices collect the rules of all systems. Few are the prerequisites to have a look to this thesis: some knowledge on first order logic and on typed lambda calculi is useful, as well as some prior experiences with OO languages. Adviced texts about lambda calculi are the ones of Barendregt [Bar91, Bar92], and of Hindley and Seldin [HS86] A good introduction to OO is in Abadi and Cardelli s book [AC96a] An enlighting discussion about type safety for OO languages is in Bruce s paper [Bru97] A pleasant handbook to approach a real OO language is Arnold and Gosling s book on Java [AG96] Part I ....

H.P. Barendregt. Lambda Calculi with Types. S. Abramsky et al. eds., Handbook of Logic in Computer Science, 2:117--309, 1992.

....(and functions) rather than logic, and hence depended on a less restrictive, or polymorphic, type systems. For example, Milner s functional language ML in [35] used a polymorphic type theory (Curry s system) In Type Theory there are attempts at unifying the various formalisms (see [5] [3] and [25] so that results can be carried across theories without duplication of work. It is moreover elegant to have unique formulations of Type Theories. After all, such unification will help to rid of the anarchy present as a result of so many different formulations. In fact, the presence of ....

....unifying fomalisms for various existing formalisms of natural and programming languages. For further details, the reader is referred to [22] 1.1 The system T Omega T Omega was presented in [22] and was shown to be an extension of various other systems. We will follow the line of Barendregt in [3], in constructing a tree which will have at its top. All the other systems have been shown to have useful applications related to natural and programming languages. 36] for example, presents a polymorphic system which can accommodate self referential terms. 37] presents a system based on ....

Barendregt, H., Lambda calculi with types, Handbook of logic in Computer Science, volume II, ed. Abramsky s., Gabbay D.M., Maibaum T.S.E., Oxford University press, 1992.

....theory. An implementation of this item notation with most of the concepts discussed in this paper can be found in [15] Keywords: Item notation, Reduction, Canonical Typing, Term restriction. 1 The formal machinery of the Cube in classical notation In this section we introduce the Cube (see [2]) and the usual necessary notions to manipulate terms and types. The systems of the Cube, are based on a set of pseudo expressions or terms T defined by the following abstract syntax (let range over both Pi and ) T = j 2 j V j T T j V :T :T where V is an infinite collection of variables ....

....pseudo context, then Gamma Delta, for Gamma a type assignment, means Gamma x i : A i for 1 i n. If A B then we also say Gamma 1 : x:A : Gamma 2 Gamma 1 : x:B : Gamma 2 and define on pseudo contexts to be the reflexive transitive closure of . Remark 1. 5 Note that we differ from [2] in that we take a declaration to be x:A rather than x : A. The reason for this is that we want pseudo contexts to be as close as possible to terms. In fact the context Gamma can be mapped to the term Gamma: for example, and definitions of boundness freeness of variables in a term and the ....

H. Barendregt, Lambda calculi with types, Handbook of Logic in Computer Science, volume II, eds. Abramsky S., Gabbay D.M., and Maibaum T.S.E., Oxford University Press, 118-414, 1992.

....This enables one to have more freedom in choosing the reduction path of a term, which can result in smaller terms along the reduction path if a clever reduction strategy is used. Moreover, this gain in reductional breadth is not at the expense of reductional length. We show that the cube of [Barendregt 92] extended with shuffle reduction satisfies all its properties such as Church Rosser, Strong Normalisation and Subject Reduction (the latter depends on allowing definitions in contexts) 1 Introduction 1.1 Term reshuffling and reductional equivalence fi equality of two terms A and B is by the ....

....the Netherlands, email: bloo win.tue.nl Department of Computing Science, 17 Lilybank Gardens, University of Glasgow, Glasgow G12 8QQ, Scotland, email: fairouz dcs.glasgow.ac.uk same address as Bloo. email: wsinrpn win.tue.nl Consider in a typed calculus of the cube as described in [Barendregt 92] the terms A j ( fi: y:fi : f :fi fi :fy)ffx B j ( fi: y:fi : f :fi fi :fy)x)ff Both terms have the term f :ff ff :fx as a reduct, so A = fi B. However, B has two redexes whereas A has only one. Here are the redexes of B and their corresponding results in B: 1. r 1 = fi: y:fi : f ....

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Barendregt, H., Lambda calculi with types, Handbook of Logic in Computer Science, volume II, ed. Abramsky S., Gabbay D.M., Maibaum T.S.E., Oxford University Press, 1992.

....of typable process behaviour, turning possibly diverging computation into a strongly normalising one. As would be imagined by the embeddability of typed l calculi, the proof of SN is non trivial, defying naive structural induction. We adapt methods developed for strongly normalising l calculi [8, 23, 61], combined with process algebraic reasoning techniques [11, 51, 53, 57, 66] As far as we know, this is the first time a compositional principle for ensuring SN has been established for name passing processes with non trivial use of replication. The proof method for SN is applicable to significant ....

....naively represent causality incorporating bound names in (4) there is a circular chain a c b a, although this cycle never arises in actual interaction. How can we then prove termination Simple structural inductions would not be usable for the same reason they do not work in typed l calculi [8, 20]. The idea we use is suggested by SN proofs for typed l calculi, due to, among others, Tait [61] His method employs a semantic interpretation of each type [ s] as a collection of strongly normalising l terms, and shows that all typable terms are indeed in these sets. A key step is to prove that ....

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Barendregt, H., Lambda Calculi with Types, Handbook of Logic in Computer Science, Vol 2, pp.118--310, Clarendon Press, Oxford, 1992.

....of typable process behaviour, turning possibly diverging computation into a strongly normalising one. As would be imagined by the embeddability of typed l calculi, the proof of SN is non trivial, defying naive structural induction. We adapt methods developed for strongly normalising l calculi [7, 15, 42], combined with process algebraic reasoning [9, 35, 37, 41, 47] As far as we know, this is the first time a compositional principle for ensuring SN has been established for name passing processes with non trivial use of replication. The proof method for SN is applicable to extensions of the ....

....naively represent causality incorporating bound names in (1) there is a circular chain a c b a, although this cycle never arises in actual interaction. How can we then prove termination Simple structural inductions would not be usable for the same reason they do not work in typed l calculi [7, 12]. The idea we use is suggested by SN proofs for typed l calculi, due to, among others, Tait [42] His method uses a semantic interpretation of each type [ s] as a collection of strongly normalising l terms, and shows that all typable terms are indeed in these sets. A key step is to prove, for ....

[Article contains additional citation context not shown here]

Barendregt, H., Lambda Calculi with Types, Handbook of Logic in Computer Science, Vol 2, pp.118--310, Clarendon Press, Oxford, 1992.

....because the higher order logics offer increased expressive power, and partly due to a wish to extend the applicative programming paradigm by adding a rewriting relation to fi reduction. The focus of this research is on the combination of terms and its formal theory, the typed lambda calculus [1, 2, 8], with algebraic terms and term rewriting [6] There exist several different formalisms on the integration of typed lambda calculus and rewrite systems, and on the study of the interaction between algebraic rewriting and fi reduction. The earliest one seems to be the work on combinatory ....

H. P. Barendregt. Handbook of Logic in Computer Science, chapter Typed lambda calculi. Oxford Univ. Press, 1993. eds. Abramsky et al.

....E mail: fliquori, ronchig di.unito.it. 3 Instytut Informatyki Uniwersytetu Warszawskiego, ul. Banacha 2, 02 097 Warszawa, Polska. E mail: urzy mimuw.edu.pl. Abstract We study the cube of type assignment systems, as introduced in [10] This cube is obtained from Barendregt s typed cube [1] via a natural type erasing function E, that erases type information from terms. We prove that the systems in the former cube enjoy good computational properties, like subject reduction and strong normalization. We study the relationship between the two cubes, which leads to some unexpected ....

....y Partly supported by HCM project No. ERBCHRXCT920046 Typed Lambda Calculus z Partly supported by grants NSF CCR 9113196, KBN 2 1192 91 01 and by a grant from the Commission of The European Communities ERB CIPA CT92 2266(294) 11, 15] and the calculus of constructions [5, 6] Barendregt [1] gave a compact and appealing presentation of a class of typed systems, arranging them in a cube. In this cube, every vertex represents a di erent typed system. One vertex is the origin and represents the simply typed calculus of Church; the edges represent the introduction of some new rules of ....

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Barendregt, H.P., Lambda Calculi with Types, Handbook of Logic in Computer Science, Abramsky, Gabbai, Maibaum eds., Oxford University Press, 1991.

....identity over booleans. In this paper, we consider typing a la Church. We present what is known as Pure Type Systems or PTSs. Important type systems that are PTSs include Church s simply typed calculus [8] and the calculus of constructions [9] which are also systems of the Barendregt cube [4]. Berardi [5] and Terlouw [47] have independently generalised the method of generating type systems into the pure type systems framework. This generalisation has many advantages. First, it enables one to introduce eight logical systems that are in close correspondence with the systems of the ....

....many advantages. First, it enables one to introduce eight logical systems that are in close correspondence with the systems of the Barendregt cube. Those eight logical systems can each be described as a PTS in such a way that the propositions as types interpretation obtains a canonical system form [4]. Second, the general setting of the PTSs makes it easier to write various proofs about the systems of the cube. In the following of the present paper we will brie y review the classical PTS with variable names and those with de Bruijn indices, essentially to state their isomorphism. This is a ....

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H. Barendregt. Lambda calculi with types, Handbook of Logic in Computer Science, volume II, ed. Abramsky S., Gabbay D.M., Maibaum T.S.E., Oxford University Press, 1992.

....of typable process behaviour, turning possibly diverging computation into a strongly normalising one. As would be imagined by the embeddability of typed l calculi, the proof of SN is non trivial, defying naive structural induction. We adapt methods developed for strongly normalising l calculi [7, 16, 43], combined with process algebraic reasoning [9, 36, 38, 42, 48] As far as we know, this is the first time a compositional principle for ensuring SN has been established for name passing processes with non trivial use of replication. The proof method for SN is applicable to extensions of the ....

....represent causality 2 incorporating bound names in (1) there is a circular chain a c b a, although this cycle never arises in actual interaction. How can we then prove termination Simple structural inductions would not be usable for the same reason they do not work in typed l calculi [7, 13]. The idea we use is suggested by SN proofs for typed l calculi, due to, among others, Tait [43] His method employs a semantic interpretation of each type [ s] as a collection of strongly normalising l terms, and shows that all typable terms are indeed in these sets. A key step is to prove that ....

[Article contains additional citation context not shown here]

Barendregt, H., Lambda Calculi with Types, Handbook of Logic in Computer Science, Vol 2, pp.118--310, Clarendon Press, Oxford, 1992.

....of typable process behaviour, turning possibly diverging computation into a strongly normalising one. As would be imagined by the embeddability of typed l calculi, the proof of SN is non trivial, defying naive structural induction. We adapt methods developed for strongly normalising l calculi [7, 16, 43], combined with process algebraic reasoning [9, 36, 38, 42, 48] As far as we know, this is the first time a compositional principle for ensuring SN has been established for name passing processes with non trivial use of replication. The proof method for SN is applicable to extensions of the ....

....represent causality 2 incorporating bound names in (1) there is a circular chain a c b a, although this cycle never arises in actual interaction. How can we then prove termination Simple structural inductions would not be usable for the same reason they do not work in typed l calculi [7, 13]. The idea we use is suggested by SN proofs for typed l calculi, due to, among others, Tait [43] His method employs a semantic interpretation of each type [ s] as a collection of strongly normalising l terms, and shows that all typable terms are indeed in these sets. A key step is to prove that ....

[Article contains additional citation context not shown here]

Barendregt, H., Lambda Calculi with Types, Handbook of Logic in Computer Science, Vol 2, pp.118--310, Clarendon Press, Oxford, 1992.

....proofs to more complex type disciplines, such as polymorphic dependent types, by adapting the computability preserving operations used to define the set of possible righthand sides of a rule. The generalization to arbitrary positive inductive types is treat Our notations and definitions follow [4] for the lambda calculus, and [10] for the rewriting. 2 Algebraic Terms 2.1 Types, Signatures, Terms, Typing Rules We are first given a set I of inductive types. The set T I of types is generated from the set of inductive types by the constructors for functional types and 2 for product ....

Henk Barendregt. Handbook of Logic in Computer Science, chapter Typed lambda calculi. Oxford Univ. Press, 1993. eds. Abramsky et al.

....a Church style term contains all the relevant type information. However, the Church style polymorphic calculus is often defined as follows F = V j F F j F T j V : T F j V F T = V j T T j 8V T where V is the set of type variables and V is the set of individual variables (see e.g. [Bar92], section 5.1, called there 2 Church) Since individual variables have no type annotations, the type reconstruction problem is no 1 A proof of this fact using syntactic methods is in [Urz96] too. 2 longer trivial. In fact, contrary to the common belief, it occurs that the problem is ....

....of equations unifies. Note also that the resulting equations are simple. This gives us the following theorem Theorem 3.1 (the undecidability of unification) Second order unification problem for simple instances is undecidable. 4 The system F We deal with the Church style system F as defined in [Bar92] (called there 2 Church) The technique presented here is standard and consist in enforcing unification so I decided to present it in a little bit more precise manner. I give insights only in crucial places here. We deal with the following problem concerning the system Definition 3 (the type ....

H. P. Barendregt, Lambda calculi with types, Handbook of Logic in Computer Science (S. Abramsky, D. M. Gabbay, and T. S. E. Mainbaum, eds.), vol. 2, Oxford University Press, 1992, pp. 117--309.

....one obtained using Gandy s method. This connection was rst remarked by de Vrijer [61] Other results on bounds on lengths of reductions are obtained for example by Schwichtenberg [52] for , using a method due to Howard [26] and by Springintveld [56] for two systems of the Barendregt s Cube [7]. Springintveld uses Schwichtenberg s results and Geuvers and Nederhof s [22] translation of calculi in Barendregt s Cube into the simply typed calculus. In [22] a (syntactic) modular proof for the Calculus of Constructions [12] is given (see [23] for more references to normalization proofs for ....

H.P. Barendregt. Lambda Calculi with Types. Handbook of Logic in Computer Science, Vol.2, Oxford University Press, 1992, p. 117-309.

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Barendregt, H., Lambda calculi with types, Handbook of Logic in Computer Science, volume II, ed. Abramsky S., Gabbay D.M., Maibaum T.S.E., Oxford University Press, 1992.

....the inconsistencies in his proposal for logic, Church developed the simple theory of types ( Church 1940] From then till the present day, research on the area has grown and one can find various reformulations of type theories. A taxonomy of type systems has recently been given by Barendregt ( Barendregt 92] A version of Church s simple theory of types can be found in this taxonomy under the name or Church. This has, apart from type variables, so called arrow types of the form oe oe , for each pair of types oe and oe . In higher type theories, arrow types are replaced by dependent ....

....( Pi y:oe :oe in term construction and using an extended version of fi reduction for such a Pi application, however, is not commonly used. Yet, it is desirable. Especially now in the new tradition which attempts to unify and generalise the type systems. See for example the Barendregt cube in [Barendregt 92] and the fine structure of the calculus in [KN 9y] Moreover, one may say that fi reduction has been invented as an expedient in order to forebode a possible substitution. So why does one use a direct substitution as in equation 1 below, which is used almost everywhere) if fi reduction can be ....

[Article contains additional citation context not shown here]

Barendregt, H., Lambda calculi with types, Handbook of Logic in Computer Science, volume II, ed. Abramsky S., Gabbay D.M., Maibaum T.S.E., Oxford University Press, 1992.

..... 35 7.4 Comparison with the type systems of Poll and Severi . 36 1 Introduction In this paper, we introduce and motivate definition mechanisms and generalised reduction, and we study their interrelationship. Most importantly, we show that the Barendregt Cube of [Barendregt 92] extended with these two concepts preserves all its old properties including Church Rosser, Subject Reduction and Strong Normalisation. 1.1 Why generalised reduction In the classical calculus, the notions of redex and of fi reduction are described as follows: Definition 1.1 (Redexes and ....

....x ) x y ) a a f ) yffi)f : a a) a (affi) A x ) x y ) a a f ) yffi)f : x (a a) a = a (a a) a Based on the above discussion, we divide the paper into the following sections: ffl In Section 2, we introduce the item notation. ffl In Section 3, we recall the Cube as in [Barendregt 92] and all its properties. ffl In Section 4, we add to the Cube as in [Barendregt 92] generalised reduction , fi and show that , fi (the reflexive transitive closure of , fi ) generalises fi (Lemma 4.3) such that = fi and fi are the same (Lemma 4.5) This means that almost all the ....

[Article contains additional citation context not shown here]

Barendregt, H., Lambda calculi with types, Handbook of Logic in Computer Science, volume II, ed. Abramsky S., Gabbay D.M., Maibaum T.S.E., Oxford University Press, 1992.

....whole process can be carried out within the proof checker; the proof checker is built upon formal systems whose meta theory is easy to understand. The paper is organized as follows: in section 2, we introduce the relevant mathematical background for the subsequent parts of the paper. In section 3, we specify the nature of equational reasoning and delimit the range of equational problems whose resolution can be automated. In section 4, we discuss the possible approaches to the automation of equational reasoning and present our own solution in terms of oracle types. In section 5, we present ....

....variables. The set T# of # terms is defined as follows: ifx # V ,thenx # T# , iff # F# and t 1 , t Arf # T#,thenf(t 1 , t Arf ) # T# . Amap# : T# # T# is a # substitution if for every f # F# and # terms t 1 , t Arf we have #(f(t 1 , t Arf ) f(#t 1 , #t Arf ) 3 The relation # is defined by t, t # # T# , t # t # if there exists # such that #t = t # . The pre order induced by # is denoted by T # # . Thesetvar(s) of variables of a term s is defined inductively as follows: ifx # V ,thenvar(x) x , var(f(t 1 , t Arf ) # 1#i#n ....

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H.P. Barendregt. Typed #-calculi, Handbook of logic in computer science, Abramsky and al eds, OUP 1992.

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H. P. Barendregt. Lambda calculi with types. Handbook of logic in computer science (vol. II), pages 117--309, 1992.

No context found.

H. P. Barendregt. Lambda calculi with types. Handbook of logic in computer science (vol. II), pages 117--309, 1992.

No context found.

H. Barendregt, Lambda calculi with types, Handbook of logic in computer science (S. Abramsky, D.M. Gabbay, and T.S.E. Maibaum, editors), vol. 2, Oxford University Press, 1992, pp. 118--309.

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H. Barendregt. -calculi with types. Handbook of Logic in Computer Science, II, 1992.

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