Hendrik Pieter Barendregt. The Lambda-Calculus: Its Syntax and Semantics. North-Holland, Amsterdam, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with the typing of a function application in a simple type system. The typing rule concerns the application of a function, denoted in this case by t to its arguments, denoted t 1 , t n . The study of type systems and their properties is typically based upon the theory of calculus [2]. While calculus does not itself provide the facilities expected of a programming language, it captures the aspects of functional programming essential to reasoning about a type system. One of the most in uential type systems for functional programming languages has been the Hindley Milner [66] ....

.... 1) length(Xs, N) This is equivalent to the de nition: length(V1, V2) V1 = V2 = 0 ; V1 = Xs] length(Xs, N) V2 = N 1 ) As in Prolog, the expression [X Xs] is a syntactic convenience for . X, Xs) where . represents the list constructor symbol. For example, [1,2,3] is shorthand for . 1, 2, 3, Unlike in Prolog, the expression N 1 denotes a function application which is evaluated to an integer. In Prolog, the expression would be left unevaluated as a compound term. Modules. Mercury modules are divided into two sections: an interface ....

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Hendrik Barendregt. The lambda calculus: its syntax and semantics. Elsevier Science Publishers, 1984.

....relation and to define an auxiliary notion of parallel reduction that will simplify some of the inductive arguments in later sections. The main result of this section is the Church Rosser property, by a straightforward adaptation of Tait and Martin Lof s proof for ordinary fi reduction (c.f. Bar84] 3.1 Definition [Parallel reduction] Single step parallel reduction is the least relation closed under the following rules: Top(K1 K2) T fi Top(K2) Fun(A:K) S) T fi [T S T fi S Fun(A:K) T fi Fun(A:K) T S T fi S T All (A S) T fi All (A ....

....S 1 , and S 2 with S fi S 1 and S fi S 2 , there is a type S 3 such that S 1 fi S 3 and S 2 fi S 3 . Psi Psi Gamma Gamma 1 fi R . fi R R Psi Psi . Proof: Straightforward extension of the standard argument (c.f. Bar84] 3.7 Corollary [Church Rosser for fi ] For all types S, S 1 , and S 2 with S fi S 1 and S Gamma fi S 2 , there is a type S 3 such that S 1 fi S 3 and S 2 fi S 3 . The proof of strong normalization for fi has to be deferred until after we have studied the properties of ....

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Hendrik Barendregt. The Lambda Calculus: Its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland, revised edition, 1984.

....p.7 8 white to an arbitrary set of colors (we will make this totally formal in section 2) The references in brackets indicate, where the reader can nd a fully formal development of the respective informal arguments. The colored calculus is a variant of the simply typed calculus [6] see [1, 18] for an introduction) where occurrences of constants and free variable can be annotated with so called colors which are either color constants C = fa; b; g or color variables X = fA; B; g. Whenever colors are irrelevant, we simply omit them. Colors are indicated by subscripts labeling ....

Hendrik P. Barendregt. The Lambda-Calculus: Its Syntax and Semantics. North-Holland, 1980.

....to # calculus. That is, functional nets constitute a programming method which derives much of its simplicity and 1 Lecture Notes, to be given at the APPSEM Summer School,Caminha,Portugal,September 2000. 1 elegance from close connections to a fundamental underlying calculus. # calculus [Chu51, Bar84] is ideally suited as a basis for functional programs, but it can support mutable state only indirectly, and nondeterminism and concurrency not at all. The pair of join calculus and functional nets has much broader applicability functional, imperative and concurrent program constructions are ....

Hendrik P. Barendregt. The Lambda Calculus: its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, revised edition, 1984.

....p.7 8 white to an arbitrary set of colors (we will make this totally formal in section 2) The references in brackets indicate, where the reader can find a fully formal development of the respective informal arguments. The colored calculus is a variant of the simply typed calculus [6] see [1, 18] for an introduction) where occurrences of constants and free variable can be annotated with so called colors which are either color constants C = fa; b; g or color variables X = fA; B; g. Whenever colors are irrelevant, we simply omit them. Colors are indicated by subscripts labeling ....

Hendrik P. Barendregt. The Lambda-Calculus: Its Syntax and Semantics. North-Holland, 1980.

....of r, and use r to denote the reflexive and transitive closure of r . We also use rs to denote the reduction relation r [ s. The notations and are, respectively, shorthands for fiutd and fiutd . The standard definitions of r redex and r normal form are used. See Barendregt [2] for more details. We will use normal form as an abbreviation for fi u td normal form. Also, that a term M 2 is strongly normalizable means that M is fi u td strongly normalizable; i.e. there is no infinite fi u td reduction sequence starting with M . Proposition 2.8 Every term M 2 is ....

Hendrik Pieter Barendregt. The Lambda Calculus: Its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North--Holland, revised edition, 1984.

.... f = N(f,g,i) deferred g deferred h private i = O(h) end) M 2 = module deferred f exported g = P(f,i) deferred h private i = Q end) Notice that although the module components are named, the modules themselves do not bear names, i.e. they are anonymous, like abstractions in the calculus [Bar84]. In the m calculus, we would write the above as: M 1 = ff . w = N(w; x; z) g . x = ffl; h . y = ffl; z = O(y)g M 2 = ff . w = ffl; g . x = P(w; z) h . y = ffl; z = Qg 2 Notice that in the m calculus, each component has separate external and internal names (like in [HL94] The internal ....

Hendrik Pieter Barendregt. The Lambda Calculus: Its Syntax and Semantics. NorthHolland, revised edition, 1984.

....since it helps clarify the meaning of derived constructs and the interactions between them. An analogous approach is well established in the field of functional programming, where elaborate functional programming languages are reduced to # calculus as a kernel language. Classical # calculus [8, 5] is ideally suited as a basis for functional programs, but less well suited for imperative and reactive programs. Extensions of # calculus for modeling these aspects have been devised, but these extension quickly become too complex to seem completely natural. Join calculus, on the other hand, ....

Hendrik P. Barendregt. The Lambda Calculus: its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, revised edition, 1984.

....semantics. This will be based on a reduction relation which relates programs to their values. We will then derive the calculus for reasoning about operational equivalence between terms. 3.1 Preliminaries We begin by de ning some notation. We generally adopt the terminology and notation of [Bar84]. We use the symbol to denote the set of terms, the metavariables M , N , to range over terms, the metavariables x, y, to range over variables and the metavariable X to range over symbolic constants. This set includes the boolean constants TRUE and FALSE. For pure LISP the terms ....

Hendrik Pieter Barendregt. The Lambda Calculus: Its Syntax and Semantics. North-Holland, revised edition, 1984.

....see [4] They are suited to define the shape of graphs that belong to a graph type. e.g. the rules of figure 2 define the shape of structured control flow diagrams D. 2. 4 Graph Reduction Graph transformation can compute functions on graphs, as term rewriting [11] and lambda calculus [2] compute functions on terms. Example 3 (Term Graph Reduction, see [14] In figure 3 we define graph transformation rules for addition on term graphs, which are shared representations of functional expressions. a) Rules for addition (b) A transformation Fig. 3. Term graph reduction In term ....

Hendrik P. Barendregt. The Lambda Calculus---Its Syntax and Semantics. NorthHolland, Amsterdam, 1981.

....between the short term solutions lead to problems in the long run. There are many calculi that could be used as a starting point for a functional language, including combinator calculi and calculi, which have been invented to build foundations of mathematics based on functions [Chu51, CF74, Bar84, HS86] or the more recent and more implementation oriented supercombinatorcalculi [Hug82] The focus is on calculi here since even the basic calculus provides elements that correspond to function definitions and applications, to abstraction, static name binding and nested scopes. Despite the ....

.... assumptions, equivalences between terms can be proven in Church that are not valid in Berkling : 8M 2 term : fv(M) fg ) 8N; P 2 term; x 2 var : x:x:M N)P ) x:M N) M (in Berkling , take M = nx) However, most of the theoretical work on calculi identifies ff equivalent terms (cf. Bar84, Appendix C] so that the most important results carry over to the extended calculi with no or only trivial modifications, which strengthens the point that the conceptual substance of the calculi is not affected by the extensions. If it really becomes necessary to reason about terms that cannot ....

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

....directed subset X D such that e vD F X, we have e vD x for some x 2 X. We list here some technical definitions and basic properties of bound complete algebraic cpos, as well as the properties of Scott open and Scott closed powerdomains. Some of their proofs can be found in Barendregt [3], Gierz, Hofmann, Keimel, Lawson, Mislove, and Scott [10] Schmidt [14] and Stoy [15] Fact A.36 Let D be a domain. Then, 1. if A 2 PC (D) then A 2 PO (D) 2. if A 2 PO (D) then A 2 PC (D) 2 Fact A.37 Let D be a domain. Then, 1. PC (D) is a lattice with X u Y = X Y and X t Y = X [ Y for ....

Hendrik Pieter Barendregt. The Lambda Calculus: Its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North--Holland, revised edition, 1984.

....operates on CPS code, the same ideas apply to the ordinary lambda calculus. In this section we formalize our optimizations as a confluent term rewriting system in a lambda calculus with records. Free and bound variables and substitution are defined as usual; we refer the reader to Barendregt [3] for basic definitions. We write fv(M ) for the free variables of M , and the substitution of M for x in N is written Nfx 7 Mg. We consider terms to be equal modulo renaming of bound variables. An atom is a variable or constant. Function inlining can be implemented by the following two rules. ....

Hendrik P. Barendregt. The Lambda Calculus: Its Syntax and Semantics. North-Holland, Amsterdam, 1981.

....SML NJ. It cleanly exports inlinable portions of one source file through the binary object file into the importing compilation unit. This approach should be applicable to a wide range of languages and compilers with intermediate languages based on # calculi [KKR 86, Pey87] # calculus [Chu41, Bar81] has the advantage over competing intermediate representations of being a well studied logical system. It also proves to be very convenient for expressing the necessary transformations in my prototype. Although the emphasis on # calculus favors functional programming languages as a target for my ....

Hendrik P. Barendregt. The Lambda Calculus: Its Syntax and Semantics. North-Holland, Amsterdam, 1981.

....so called bookkeeping for fan interactions. 1 Introduction A fundamental distinction between the varieties of functional programming languages can be made by considering respective evaluation strategies. In what order are subexpressions evaluated All such languages are founded on the calculus [Chu41, Bar84]. But this calculus, like everyday algebraic calculation, is indeterminate: choosing what subexpression to simplify next is not specified, though the Church Rosser theorem provides assurance that we always get the same answer, if there is one. The designer of a functional programming language must ....

Hendrik Barendregt. The Lambda Calculus: Its Syntax and Semantics. North Holland, 1984.

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Hendrik Pieter Barendregt. The Lambda-Calculus: Its Syntax and Semantics. North-Holland, Amsterdam, 1980.

No context found.

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

No context found.

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

No context found.

Hendrik Pieter Barendregt. The Lambda Calculus: Its Syntax and Semantics. NorthHolland, revised edition, 1984.

No context found.

Hendrik Pieter Barendregt. The Lambda Calculus: Its Syntax and Semantics. North-Holland,

No context found.

Hendrik P. Barendregt. The Lambda-Calculus: Its Syntax and Semantics. North- Holland, Amsterdam, revised edition, 1984.

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Hendrik Barendregt. The Lambda Calculus: Its Syntax and Semantics. North Holland, 1984.

No context found.

Hendrik Pieter Barendregt. The Lambda Calculus: Its Syntax and Semantics. North-Holland, revised edition, 1984.

No context found.

Hendrik Pieter Barendregt. The Lambda Calculus: Its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North--Holland, revised edition, 1984.

No context found.

Hendrik Barendregt. The Lambda Calculus: Its Syntax and Semantics. North Holland, 1984.

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