Hans P. Barendregt. The Lambda-Calulus. Its Syntax and Semantics, volume 103 of Studies in Logic and The Foundations of Mathematics. North-Holland, 1984. 155  Home/Search   Document Not in Database   Summary   Related Articles   Check

....by P v u ; and alpha equivalence, denoted # , are all defined in the standard way. We follow the convention that, if P 1 , P n occur in a certain mathematical context, then in these processes all bound names are chosen to be di#erent from the free names (cf. the variable convention [2]) except when otherwise mentioned. Processes of the form #x 1 . #x n P are sometimes abbreviated to #x 1 . x n P . As in the # calculus, where the # binds looser than application, we take the view that # binds looser than the parallel composition: #x P Q means #x(P Q) This ....

H. P. Barendregt. The Lambda-Calculus, its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1974. Revised edition.

....The synchroniser ty :iiy for the second argument suspends forever. 7 Uniform Confluence for ffi For proving a uniform confluence result for ffi, we have to consider how uniform confluence behaves with respect to a union of calculi. We first present a variation of the HindleyRosen Lemma [Bar84] for uniform confluence and then apply it to the ffi calculus. But the general results of this Section are also applicable to other unions of calculi such as the callby need calculus [AFMOW95] and the ae calculus [NM95] The union of two calculi (E; j; 1 ) and (E; j; 2 ) is defined by (E; j; ....

Henk P. Barendregt. The Lambda Calculus. Its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. Elsevier, 1984.

....The synchroniser t y :iiy for the second argument suspends forever. 7 Uniform Confluence for ffi For proving a uniform confluence result for ffi, we have to consider how uniform confluence behaves with respect to a union of calculi. We first present a variation of the Hindley Rosen Lemma [Bar84] for uniform confluence and then apply it to the ffi calculus. But the general 13 results of this Section are also applicable to other unions of calculi such as the call by need calculus [AFMOW95] and the ae calculus [NM95] The union of two calculi (E ; j; 1 ) and (E ; j; 2 ) is defined by ....

Henk P. Barendregt. The Lambda Calculus. Its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. Elsevier/North Holland, Amsterdam - New York - Oxford, 1984.

....different notion of labels for the calculus: whereas Hyland Wadsworth labels generalize the finite developments theorem, ours generalize superdevelopments [10] This is shown in Section 5. We conclude in Section 6. 2 2 Preliminaries on SKInT Recall that the syntax of the calculus is [3]: t : x j tt j x t where x ranges over an infinite set of so called variables, and terms s and t that are equivalent are considered equal; we denote terms by s, t, and variables by x, y, z, etc. We shall write = for equivalence; in the first order calculi to come, will ....

....never be applied to any argument list. A distinguishing feature of labeled reduction in the calculus is that if labeled reduction is restricted to labels with at most p nested underlinings, with p fixed, then every such p bounded labeled reduction terminates (local strong normalization [3]) We shall prove this for in Section 3.2, and then for labeled SKInT in Section 3.3. Now, provided we ignore labels, there is a translation J K of SKInT into that preserves reduction; this is basically the J K translation of [7] Figure 8. This translation 5 is as follows, ....

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H. Barendregt. The Lambda Calculus, Its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, Amsterdam, 1984.

....as we can always add s in front in order to bind all universal variables. A ground term has no free existential variable. All terms that are equivalent (i.e. differ only in the name of bound variables) will be dealt with as though they were equal, using Barendregt s naming convention [4]. We consider the following rewrite rules: x s)t s[x : t] x tx t (x not free in t) where s[x : t] denotes the standard capture avoiding substitution. We write , the corresponding one step rewrite relations; if is a rewrite relation, we write its ....

H. Barendregt. The Lambda Calculus, Its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, Amsterdam, 1984.

....always add s in front in order to bind all universal variables. A term is ground iff it has no free (existential) variable. All terms that are equivalent (i.e. differ only in the name of bound variables) will be dealt with as though they were equal, using Barendregt s naming convention [3]. We consider the following rewrite rules: x s)t s[x : t] x tx t (x not free in t) where s[x : t] denotes the standard capture avoiding substitution. We write , the corresponding one step rewrite relations; if is a rewrite relation, we write its ....

H. Barendregt. The Lambda Calculus, Its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, Amsterdam, 1984.

....context. let x = 3 in 1 Hole If its hole is lled with (x 2) we obtain expression let x = 3 in 1 (x 2) In this expression, x in (x 2) is bound by the let construct. That is, the hole lling operation is essentially di erent from capture avoiding substitution in the lambda calculus [2]. Our hole lling operation captures free variables. Moreover, each hole may provide a di erent set of bindings. The following context: let a = 1 in let = A in let b = true in let = B in ( will bind a and b occurring in an expression lled in B to 1 and true, respectively. A piece of code ....

H.P. Barendregt. The Lambda Calculus, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1984. revised edition.

....For example, one can approximate the Halting Problem by computable functions. Scott has observed that every algebra arises from a re exive object R in some cartesian closed category C . We give the construction explicitly as it is needed for analysis later (although the reader is referred to [Bar84] fora comprehensive treatment) Given a cartesian closed C with re exive object R, together with morphisms Fun : R [R ) R] and Gr : R ) R] R such that Gr ; Fun = id [R)R] de ne a algebra hA; i as follows: i) A is the homset Hom C (1; R) 7 (ii) For any object A with f; g : A ....

....of D, DREC and DEAC are explored in [KNO01] one which is immediate from the Exact Correspondence Theorem is that two terms of the calculus have the same denotation precisely when they have the same Nakajima tree. This equality is captured by the maximal consistent sensible theory H (see [Bar84]) Also, almost by construction, every EAC strategy on U is the denotation of some closed term of the calculus. Furthermore, DEAC inherits the order extensionality property which the syntax of the calculus (modulo the theory H ) enjoys. Thus: Theorem 2.2.5 The equational theory of the ....

H. P. Barendregt. The Lambda Calculus, Its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1984.

....a 00 ) Total pca s, where application is a total operation on the carrier set, will be called ca s, and nontotal pca s, where application is not defined everywhere, will be called nca s. Ca s are extensively studied in the context of models of calculus and Combinatory Logic (CL) cf. e.g. [3]) nca s are a little less well off in this respect. They figure in the semantics of programming languages (see the forthcoming book by Mitchell [13] as well as in the formalization of constructive mathematics (see [4] 15] In fact, they are the models of a minimal axiomatic basis for ....

....hA; s; k; Deltai be a pca with unique hnf s. Then both T 1 (A) and T 2 (A) are orthogonal and, a fortiori, confluent. 2 We shall use confluence of its subsystems to prove that T (A) is confluent. For this, we invoke a proposition that is sometimes referred to as the Lemma of Hindley Rosen ( 9] [3]) For i = 1; 2, let us write i for the rewrite relation associated with T i (A) The reflexive closure of i is denoted by j i , its transitive reflexive closure by i . Moreover, we say that 1 and 2 commute, if for all t; t 0 ; t 00 2 T (A[V ) t 1 t 0 t 2 t 00 ) ....

H.P. Barendregt. The Lambda Calculus, its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, revised edition,

....We describe the construction of T (A) for a concrete set A, leaving it to the reader to check that the construction can be captured in a typable operator T (X) To define T (A) we first construct the set, T , of unlabelled finite trees. This is done by a standard construction, e.g. see [2]. The idea is that we represent a finite tree as a set of addresses for the nodes in the tree. The addresses are sequences of positive natural numbers. The root has the empty sequence, for its address; a non empty sequence, n 1 ; n k , represents the node you arrive at by a k stage ....

.... Theorem 5 There is an operator, Hered, with the following property: if A is any set, and V is any subset of the set T (A) then Hered(V ) is the (unique) subset of T (A) such that: Hered(V ) ft : T (A)jt 2 V 8c : ran(Children(t) ffl c 2 Hered(V )g (49) 4 An alternative approach, followed in [2], is to define a partial ordering on sequences of positive natural numbers, corresponding to the order in which nodes are encountered in an appropriate traversal of a tree, and then to define a tree to be any set of sequences that is downwards closed with respect to that partial ordering. 9 ....

H.P. Barendregt. The Lambda Calculus, volume 103 of Studies in Logic and the Foundations of Mathematics. North Holland, 1984.

....) by (X ; T ; S; 0 ) T de nes the type language and X de nes the constraint system. The set S de nes the set of valid constraints used in the type schemes and in type derivations. 0 is the set with the initial type declarations. 3. 1 The term language The term language is just the calculus ([3]) with the additional term let M : x j x:M j MM 0 j let x = M in M 0 3.2 The type language The type language is de ned by T . Let range over an in nite set of type variables, and 0 denote types and denote type schemes. Then the type language is de ned as follows: j ....

H. Barendregt. The Lambda Calculus, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1984.

.... conventions allow us to replace clause 3 by 8a; a 0 ; a 00 2 A saa 0 a 00 aa 00 (a 0 a 00 ) Total pca s (ca s) where application is a total operation on the carrier set, are extensively studied in the context of models of calculus and Combinatory Logic (CL) cf. e.g. [Bar84], HS86] nontotal pca s (nca s) where application is not defined everywhere, are a little less well off in this respect. They figure in the semantics of programming languages (see the forthcoming book by Mitchell [Mit9 ] as well as in the formalization of constructive mathematics (see [Bee85] ....

H.P. Barendregt. The Lambda Calculus, its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, revised edition,

....we show that they can be easily verified for many existing notions of meaninglessness and easily refuted for some notions that are known not to be good characterizations of meaninglessness. 1 Introduction The concept of a meaningless term in a rewrite system originates with the lambda calculus [Bar84, Bar92] There exist terms in the lambda calculus that, in certain precisely definable senses, cannot be distinguished from each other and cannot contribute information to any context in which they are placed. Such terms may intuitively be considered meaningless or undefined, and in a ....

....of their important properties, which in the past have been proved separately. We consider left linear term rewrite systems and lambda calculus, in both finitary and transfinite forms. We assume the reader to be familiar with the basic theory of term rewriting [DJ90, Klo92] and lambda calculus [Bar84, HS86] The basic theory of transfinite rewriting has already been set out [KKSdV95, KKSdV97] We will show the usefulness of our axioms in several ways. They arise naturally from the notion of rewriting as computation of the meaning of terms. The axioms imply two standard lemmas: the ....

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H. P. Barendregt. The Lambda Calculus, Its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. Elsevier Science Publishers B.V., Amsterdam, revised edition, 1984.

....two p.o s and add a (new) common least element below. This equation can be solved, for example, in the ccc of Scott domains with continuous functions, by taking D to be the inverse limit of a suitable projective system of adequate domains, a method due to Scott [32] and well understood now, c.f. [4], p.477. This is the starting point for the model construction. Now we have to interpret in D the constants and , which are subject to axioms which prevent them from being continuous. The reason why cannot be continuous is linked to the fact that 8 represents a very strong form of ....

.... ccc if there are two continuous functions: A : D [D D] 6) D D] D (7) such that A = id (8) In particular is injective and A is surjective. Then there is a standard way to interpret terms of calculus, where parameters in D are allowed (sketched in Section 3, c.f. also [4], Chapter 5, Paragraph 4) and (8) ensures that any two equivalent terms get the same interpretation. If furthermore A = id, then D = D D] the model is called extensional, and any two equivalent terms get the same interpretation. Solving (1) amounts to nding an almost extensional ....

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H.P. Barendregt. The Lambda Calculus, Its Syntax and Semantics, volume 103 of Studies in Logic and The Foundation of Mathematics. NorthHolland, 1984.

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Henk Barendregt. The Lambda Calculus, volume 103 of Studies in Logic and the Foundations of Mathematics. North Holland, 1984.

....so as to contain (whether free or bound) none of the free variables of t 1 ; t n . C[ is called a (strongly) separating context. In the above de nition, we can without loss of generality take s 1 ; s n to be a tuple of distinct new variables. Separability is as de ned in [Bar84], although we nd it technically more convenient to speak of tuples rather than sets of expressions. Note that a tuple containing repeated elements is never separable. Strong separability 5 is our extension of the concept. The terms x and y are separable (by the context ( xy: s 1 s 2 ) but ....

....the rst equality. Translatability is a much stronger property of a rule set than consistency. It is possible to add to the lambda calculus a new function symbol F and the rules for Berry s F , and the resulting system will be consistent (as demonstrated by the fact that it has a model, see [Bar84]) Parallel or is also consistent with lambda calculus. The Mal cev operator is not: adding such an operator allows proving x = y (by a non trivial argument) 6 A uniform translation of strongly separable systems We now demonstrate that every strongly separable orthogonal TRS has a uniform ....

H.P. Barendregt. The Lambda Calculus, volume 103 of Studies in Logic and the Foundations of Mathematics. Elsevier Science Publishers B.V., P.O. Box

....G such that G E Gamma dEe if E is in normal form. Equivalently (due to confluence) we can say that G takes a term and produces the representation of its normal form (if such exist) However, such a term G does provably not exist (see section 6. 6 of Barendregts book on the lambda calculus [1]) Hence, we must relax the condition somewhat. Mayer Goldberg [5] relaxes the condition by restricting the class of terms E that G works for to be the set of proper combinators. Berger and Schwichtenberg [2] relax the condition by requiring E to be in the simply typed lambda calculus and ....

H. P. Barendregt. The lambda Calculus, its syntax and semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, New York, Oxford, 2 edition, 1984.

....that conversion is not validated. Details of the construction, which yield a universal model whose equational theory is precisely that of B ohm tree equality, will be presented in a sequel. 1. 2 Prerequisites We assume familiarity with the untyped calculus for which the standard reference is [3]. Particularly vital topics include the notions of solvability and head normal forms, B ohm trees, standard theories and models. Basic category theory, up to CCCs and adjunctions, is also assumed see for example [14] Some references are made to computability. A knowledge of the standard ....

H. P. Barendregt. The Lambda Calculus, Its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland, 2nd edition, 1984. 53

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Hans P. Barendregt. The Lambda-Calulus. Its Syntax and Semantics, volume 103 of Studies in Logic and The Foundations of Mathematics. North-Holland, 1984. 155

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Henk P. Barendregt. The Lambda Calculus, Its Syntax and Semantics, volume 103 of Studies in Logics and the Foundations of Mathematics. North-Holland, 1981.

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H. P. Barendregt. The Lambda Calculus. Its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1984.

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H.P. Barendregt. The Lambda Calculus, its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. NorthHolland Publishing Company, revised edition, 1984. (Second printing 1985).

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A. I. Mal'cev. The Metamathematics of Algebraic Systems, volume 66 of Studies in Logic and The Foundations of Mathematics. North Holland, 1971.

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H. Barendregt. The Lambda Calculus. Its Syntax and Semantics, volume 103 of Studies in logic and the foundations of mathematics. North-Holland, Amsterdam  New York  Oxford, revised edition, 1985.

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A. I. Mal'cev. The Metamathematics of Algebraic Systems, volume 66 of Studies in Logic and The Foundations of Mathematics. North Holland, 1971.

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