Albert R. Meyer. What is a model of the lambda calculus. Information and Control, 52:87--122, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....map f : C C: f(c) jt[x : c]j C ; for all c 2 C. The drawback of the previous definition is that, if C is an arbitrary combinatory algebra, it may happen that map f is not representable. The axioms of a subclass of combinatory algebras, called models or models of lambda calculus (Meyer [30], Scott [44] 3, Def. 5.2.7] were expressly chosen to make coherent the previous definition of interpretation. For every model C, the set Th(C) ft = u : t; u 2 o ; C j= t = ug constitutes a lambda theory. C is a model of the lambda theory T if T = Th(C) We would like to point out here ....

A.R. Meyer, What is a model of the lambda calculus?, Information and Control 52, (1982), 87--122. 12 Salibra

....presented by diagram in Figure 2. The embedding maps untyped terms into terms in the simply typed lambda calculus using constants f and g that can be thought of as a retraction pair used in the interpretation of the simply typed lambda calculus (see Scott [14] Wadsworth [19] and Meyer [10]) From the syntactical point of view, blocks all redexes, replacing ( x:M)N by f(g( x:M) N . The notion of o reduction ( o ) is introduced to play an analogous role to the lambda abstraction marking ( it replaces a blocked redex f(g( x:M) N by the unblocked redex ( x:M)N . In this case ....

Meyer, A.M.: What is a model of lambda calculus. Information and control 122 (1982) 52-87.

....of Victoria University of Wellington and finantial support of Ca Foscari University of Venice. Thanks are due to Robert Goldblatt for discussions and for making the visit to Wellington possible. Typeset by AM S T E X 1 2 ANTONINO SALIBRA notion of an environment model (the name is due to Meyer [29]) originated with Hindley and Longo [24] They are functional domains where terms can be properly interpreted. Meyer describes them as the natural, most general formulation of what might be meant by mathematical models of the untyped lambda calculus . The main result in [29] is a completeness ....

....name is due to Meyer [29] originated with Hindley and Longo [24] They are functional domains where terms can be properly interpreted. Meyer describes them as the natural, most general formulation of what might be meant by mathematical models of the untyped lambda calculus . The main result in [29] is a completeness theorem demonstrating that every lambda theory is the theory associated with some environment model. The drawback of environment models is that they are higher order structures. However, there exists an intrinsic characterization (up to isomorphism) of environment models as a ....

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A.R. Meyer, What is a model of the lambda calculus?, Information and Control 52 (1982), 87--122.

....a b) projection 1 1) b 2.3 A Term Model Semantics for T In this section we give a model for the (non weak) T type system and prove soundness and completeness of type inference. We base the semantics of T on what Albert Meyer 35 has called an environment model of the untyped lambda calculus [Mey82] We define what it means to be a model for T in terms of basic elements and behaviors, that is, the essential properties of all models of T , and then prove the soundness of the T type inference system of Figure 2.4 Section 2.1. We then construct a term model and use it in the proof of ....

....component of a recursive term sequence, the one that performs the recursive unfolding step. Figure 2.6 gives the rules for term interpretations required by all models of T . The rules i, iv, vii, viii and ix are standard rules for any lambda model. See Hindley and Seldin [HS86] or Meyer [Mey82] for discussions about lambda calculus models. The requirements ii and iii are simple extensions to the standard lambda calculus model that characterizes the expected behavior of numerals. See Friedman s model [Fri75] The remaining two requirements v and vi are for characterizing the necessary ....

Albert R. Meyer. What is a model of the lambda calculus? Information and Control, 52:87--122, 1982.

....a classical applicative structure has enough elements to interpret every lambda term. The environment model condition uses the inductive de nition of the meanings of terms, while the combinatory model condition uses equationally de ned elements K and S called combinators (see [Bar84, Chapter 5] or [Mey82]) Since the two are equivalent (for both Kripke and classical applicative structures) we will de ne models using combinators. For Kripke applicative structures, the description of K and S is simpli ed by introducing the notion of global element. A global element a: of A is a mapping w 7 aw ....

....w such that for all a 2 A w 0 and w 0 w; App ; w 0 (i w;w 0 d)a = x: M : a=x] w 0 Combinators and extensionality guarantee that in the . x: M : case, d exists and is unique. This is proved as in the classical setting, using translation into combinators [Bar84, HS86, Mey82] for existence, and extensionality for uniqueness. We can see the importance of partial environments by looking at the lambda abstraction case in a little more detail. The meaning of a lambda abstraction in environment at w is determined by patched environments [a=x] for a 2 A w 0 with ....

A.R. Meyer. What is a model of the lambda calculus ? Information and Control, 52(1), 1982. pages 87-122.

....denotes substitution of N for x in M , where the bound variables of M are renamed to avoid the capture of the free variables of N [2] We write (M = fij N ) if M and N are provably equivalent in this system. Semantics: We use the notion of environment models to give meaning to simply typed terms [5, 6]. Definition 3 A frame is a pair (fM oe g; fap oe; g) where each M oe is a nonempty set, and ap is an application function with ap oe; M oe ThetaM oe M . The elements of function type must satisfy the extensionality property, i.e. for any f; g 2 M oe , f = g iff ....

Albert R. Meyer. What is a model of the lambda calculus? Information and Control, 52:87--122, 1982.

....with v d for all d, and hd; 0i v hd 0 ; 0i iff d v d 0 . The function : D (D) with d = hd; 0i is an injection; the function : D) D, with hd; 0i = d and = is the corresponding projection. Given these elements, we assign meanings to terms using an environment model [2, 5, 7] in the usual way. Constants of base type mean the obvious elements in the domains, and constants of higher type mean lifted functions. The equations A Y [ MN ] ae = A Y [ M ] ae) A Y [ N ] ae) A Y [ x Pi M ] ae = f; where f(d) A Y [ M ] ae[x 7 d] specify the meanings of ....

A. R. Meyer. What is a model of the lambda calculus ? Information and Control, 52:87--122, 1982.

....(y: M y) with y not free in M , we say that P is in fij normal form. It is a theorem that any term P is equivalent to a unique term Q in fij normal form (see [1] we use fij nf(P ) to denote the fij normal form of P . We use the notion of environment models to give meaning to simply typed terms [7, 11]. Definition 2.1 A frame is a pair (fM oe g; fap oe; g) where each M oe is a nonempty set, and ap is an application function with ap oe; M oe Theta M oe M . The elements of function type must satisfy the extensionality property, i.e. for any f; g 2 M oe , f = g ....

A. R. Meyer. What is a model of the lambda calculus? Information and Control, 52:87--122, 1982.

.... system (q ats) is a structure h A; eval i such that eval is a proper partial function from A to 6 Although the first ( non syntactic ) model for the Lambda Calculus was obtained in the late 60 s, the question of what constitute a model was only properly answered in the early 80 s, see e.g. [Mey82, Koy84]. A A i.e. dom(eval) ae A. A q aswd (q ats) is pointed if (or (A Gamma dom(eval) respectively) is a singleton set. Given a q aswd A = h A; Delta; i, we define an associated partial function eval : A A A by eval(a) def = 8 : x 7 a Delta x if a ; undefined else. The structure ....

....f , selects a unique representative Gr(f) from rep(f ) the set of representatives of f . ffl The standard expansion is specified by the graph function: Gr st (f) def = f (fi; b) fi fin B b 2 f(fi) g. h DA ; Fun; Gr st ; Gamma ] st Gamma i is a functional model (see [Mey82]) which we shall call the standard pse model. Note that Fun : DA [DA r DA ] is the induced map from the applicative structure h DA ; Delta i. ffl The second expansion is applicable only to free pse algebras generated from a base set A, say. The defining graph function is Gr lazy (f) ....

A. Meyer. What is a model of the lambda calculus? Information and Control, 52:87--122, 1982.

....Any term is bh equivalent to a unique long bh normal form [Jensen and Pietrzykowski, 1976] These terms are useful because the body has type i, making certain inductions simpler to perform. 2. 2 Models We use the notion of environment models to give meaning to simply typed terms [Friedman, 1975, Meyer, 1982] Definition 2.1 A frame is a pair (fM s g;fap s;t g) where each M s is a nonempty set, and each ap s;t is an application function with ap s;t : M s t Theta M s M t . We omit the types from ap when they are clear from context. The functions ap s;t must satisfy the ....

Meyer, A. R. (1982). What is a model of the lambda calculus? Information and Control, 52:87--122.

....part of every element. In the presence of extensionality we do not need and that is why in the case of extensionality, combinatory algebras are models of the calculus. Both environment models and combinatory models are equivalent to each other and for a proof of this, the reader is referred to [Meyer 1981]. These are not the only kinds of models provided for the calculus. The two kinds of models cited above together with the term models are algebraic, there are others which have a built in structure. It is easy to work with such models as one does not get involved with the cumbersome syntax) ....

A. Meyer, What is a model of the Lambda Calculus? Unpublished ms., M.I.T. Lab., Computer Science, 1981.

.... s least upper bounds, and that the resulting CPO is algebraic. Furthermore, application is continuous. In section 5 we study the general notion of model of a functional call by value programming language with numbers and pairing. Our approach builds on the work of Milner [Milner, 1977] and Meyer [Meyer, 1982]. We begin by defining the notion of an FLD domain (functional programming language domain) These are reflexive domains with an extensional partial ordering v reflecting degrees of definedness. Next we define a notion of FLEM (functional language environment model) for interpreting expressions in ....

....; G f Pi(d) Pi(d) Dg = f Pi(d) Pi(d) Dg = D: 2 5 Constructing and Characterizing Models In this section we study the general notion of model for a functional call by value programming language with numbers and pairing. Our approach builds on the work of Milner [Milner, 1977] and Meyer [Meyer, 1982]. We begin by defining the notion of an FLD domain (functional programming language domain) These are reflexive domains with an extensional partial ordering v reflecting degrees of definedness. Next we define a notion of FLEM (functional language environment model) for interpreting expressions ....

Meyer, A. R. (1982). What is a model of the lambda calculus? Information and Computation, 52:87--122.

....a higher order syntax. In a type free logic, the same term may appear in different contexts as predicates or functions of different arities and even as atomic formulas. Thus the same intension can be associated with different extensions in different contexts. For instance, in the Lambda Calculus [40], a term is considered a function or an object depending on its syntactic position. In HiLog, the same symbol may denote a predicate, a function, or an atomic formula. Semantics of a type free logic has to maintain the distinctions between various extensions associated with the same intension. ....

Meyer, A.R. [1982] What Is a Model of the Lambda Calculus, Information and Control, 52 (1982), pp. 87--122.

....the two redexes will be shared. 6 Conclusion The two combinatory logics, CL v and CL q , presented in this paper are interesting for several reasons. The simple logic, CL v , may be useful for developing an algebraic model for the v calculus, as CL was used to create a model of the calculus [11]. An open problem is the construction of a set of axioms, corresponding to A fi , for CL v that are equivalent to the inference rule i 0 . The logic CL q is useful for creating implementations of by value languages. Burge [2] rediscovered combinators for computer science. Turner [14] popularized ....

A. Meyer. What is a model of the lambda calculus? Information and Control, 52:87--122, 1982.

....are generalizations of algebraic lattices within the realm of partially ordered sets and a veritable plethora of models for the calculus has been found in categories of domains. An obvious question then is whether models for the calculus can be found in other categories. Following Meyer (cf. [6]) by a functional domain, we mean an object X within a cartesian closed category C with internal hom functor ) which satisfies the property that X ) X is a retract of X in C . 1 The model D1 is an example of such an object; indeed, D1 (D1 ) D1 ) So, one would like to know whether there ....

....The ones we introduced above we shall call special. In the third and fourth sections we also take the time to discuss in some detail the precise effect of our results on models of the untyped lambda calculus in cartesian closed topological categories. This discourse is motivated by Meyer s ideas [6]. In particular, we show that an environment model based on a compact special functional domain in the category of k spaces must be trivial in the sense that its semantic map is constant (3.11) In the fourth section we discuss briefly the concept of combinatory models in concrete cartesian ....

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Meyer, Albert, What is a model of the lambda calculus?, Information and Control 57 (1982), 87 -- 122.

.... 1 =x 1 ; u n =x n ] v 0 [v 1 =x 1 ; v n =x n ] in particular, if T u i = v i for every i 2 f0; ng, then T u 0 [u 1 =x 1 ; u n =x n ] v 0 [v 1 =x 1 ; v n =x n ] 3 COR models In this section we define the class of COR models, like the one defined in [Mey82] for the calculus. We will prove the completeness and correctness of this class of models in relation to COR theories. Definition 3.1 value model. A value model of COR is a poset (D; whose elements are named values, and a function [ from COR terms and valuation functions ae to D such that ....

....(extensional) COR theory. Moreover, if u converts to v then u v is valid. Proof: The complete lattice structure of D corresponds to axioms and rules ( Ref lex) Antisim) and (Trans) Axioms (fi) and (ff) can be proved from the substitution lemma (for a complete proof see [Mey82]) The Psi monotonicity and pointwise order in D D can be used to prove ( monot) and the Phi monotonicity and pointwise order to prove (Apl monot) For a quasi extensional model, using the V definition it is easy to see V[ u [ v) w) ae = Phi(V [u] ae [V [v] ae ) V [w] ae ) and V[ u(w) ....

A. R. Meyer. What is a model of the lambda calculus? Information and Control, 52:87--122, 1982.

....connection can be found in Chapter 9 of [7] Mislove Theorem 5.4 There is a one to one correspondence between models of the untyped lambda calculus and reflexive objects in ccc s. 2 We remark that there are other notions of what a lambda model should be. Of particular note are the results of [34] which elaborate the relationships between a number of approaches to defining this concept, as well as the detailed discussion in [7] These involve concepts such as combinatory algebras and combinatory models. For us, the question is how to find an example of a non degenerate reflexive object in ....

Meyer, A., What is a model of the lambda calculus?, Information and Control 57 (1982), pp. 87--122.

....remarks about what we feel has been achieved and what new challenges still need to be confronted. 2 Inheritance as implicit coercion. A simple analogy will help explain our translation based technique. Consider how the ordinary untyped calculus is interpreted semantically in such sources as [Sco80, Mey82, Koy82, Bar84]. One begins by postulating the existence of a semantic domain D and a pair of arrows Phi: D (D D) and Psi: D D) D such that Phi ffi Psi is the identity on D D. Certain conditions are required of D D to insure that enough functions are present. To interpret an untyped term, ....

A. R. Meyer. What is a model of the lambda calculus? Information and Control, 52:87--122, 1982.

.... in the early 80 s [19, 18, 20, 5] we proposed a paradigm for semantics based compiler correctness, and over the last several years we have begun putting this paradigm into practice [22, 14] In this paradigm, the source and target languages are given denotational semantics in the same theory [11], so that most of the steps of the correctness proof can be done Work supported by the National Science Foundation under grant numbers CCR9014603 and CCR 9304144. within the theory. Furthermore, the compiler is given by a compositional (syntax directed) translation, so properties of the ....

Albert R. Meyer. What Is a Model of the Lambda Calculus? Information and Control, 52:87--122, 1982.

....P is Elf s concrete syntax for abstraction in the framework and binds a variable x of type A in the object P. x = x M N = app M N x: M = lam ( x:term] M ) For example, x: y: x = lam [x:term] lam [y:term] x: As far as we know, this representation is due to Wadsworth [Wad76] and used by Meyer [Mey82] in the construction of an environment model of the untyped calculus. The notation used there is Psi for lam and Phi for app. From the representation above we can read off the type of the constructors, leading to the following signature T . term : type. name term M lam : term term) ....

Albert R. Meyer. What is a model of the lambda calculus. Information and Control, 52:87--122, 1982.

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Albert R. Meyer. What is a model of the lambda calculus. Information and Control, 52:87--122, 1982.

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A. R. Meyer. What is a model of the lambda-calculus? Information and Control, 52:87--122, 1982.

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A. R. Meyer. What is a model of the lambda-calculus? Information and Control, 52:87--122, 1982.

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Albert R. Meyer. What is a model of the lambda calculus? Inform. and Control, 52(1):87-- 122, 1982.

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Albert R. Meyer. What is a model of the lambda calculus. Information and Control, 52:87--122, 1982.

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Meyer, A.R., "What Is a Model of the Lambda Calculus ?" Information and Control 52, pp. 87-122. 1982.

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A. Meyer, "What is a Model of the Lambda Calculus?", Info. and Comp. 52, 1982, 87-122.

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A. R. Meyer. What is a model of the lambda calculus? Information and Control, 52:87--122, 1982.

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A. Meyer, What is a model of the Lambda Calculus? Unpublished ms., M.I.T. Lab., Computer Science, 1981.

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A. R. Meyer. What is a model of the lambda calculus? Information and Control, 52:87--122, 1982.

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Meyer, A.: What is a Model of the Lambda Calculus? Information and Control 52 (1982) 87--122

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