R. Statman. Logical relations and the typed lambda calculus. Information and Control, 65:85--97, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... reduction, the answer is positive, since each step pushes one abstraction to the front. Thus, we can obtain the assertion, if we show termination and con uence for reduction in DLC. Instead of showing that directly using the well known logicalrelations method due to Tait [31] and Statman [30], we will map DLC into the simply typed calculus and use the results there. Let A be a well formed DLC term of type A # , furthermore let 0 be obtained from by dropping all mode information, and A 00 be obtained from A by exchanging all dynamically bound variables U with ....

R. Statman. Logical relations and the typed lambda calculus. Information and Computation, 65, 1985.

.... covers, see Lambek and Scott [LS86] To our knowledge, the first application of this method to type disciplines is given in Appendix C of Lafont [Laf88] In the case of simple types, this method corresponds closely to so called logical relations, described for instance in Plotkin [Plo80] Statman [Sta85], and Mitchell [Mit90] This correspondence is examined in detail. In the case of polymorphic types, a central role is played by relators, i.e. maps that take objects to objects and relations to relations. Sconing may be used to express relators as functors. jcm cs.stanford.edu Department of ....

....one to one correspondence between the morphisms C C 0 and the morphisms 1 C)C 0 , which holds in any cartesian closed category. In the special case of well pointed cartesian closed categories, we may compare sconing to logical relations for Henkin models of simply typed lambda calculus (see [Sta85, Mit90]) We shall do so in the next section. We conclude this section with a binary version of a general sconing framework given by means of comma categories in Proposition 4.2. This binary version (as well as the k ary version) can be obtained from Proposition 4.2 itself. Proposition 4.4 Let A, B, ....

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R. Statman. Logical relations and the typed lambda calculus. Information and Control, 65:85--97, 1985.

....(R ; 1.3. Lambda definability and full completeness. An element f 2 A oe) is called lambda definable if f = T [x:oe . M: for some term x:oe . M: Central to characterizing the lambda definable terms is the following proposition often known as the Fundamental Theorem of Logical Relations [Sta85]. Proposition 1. Let R be a logical relation on A. Then each f = T [x:oe . M: satisfies R. Proof. By definition, T is the unique ccc representation that satisfies J ; T = b T . The logical relation R necessarily satisfies b R; Sigma = b T , so that J ; R; Sigma = b T and therefore R; ....

R. Statman. Logical relations and the typed lambda calculus. Information and Control, 65:85-97, 1985.

....if M j= L ff for every Lauchli model M. ffl Gamma j= L ff if and only if M j= L ff whenever M j= L fl for every fl 2 Gamma. ffl ThL (M = fff j M j= L ffg. ThL (M) is called the theory of M. The hierarchies of permutations f ff g defined by Lauchli models are examples of the logical relations [Sta85] used to study definability in the lambda calculus. Lauchli s results [Lau70] are stated only for full models, but his proofs construct only cyclic models. For generality and smoothness of reasoning we decided to cover both extremes, and everything in between. Clearly, larger groups make for ....

R. Statman. Logical relations and the typed lambda calculus. Information and Control, 65:85--97, 1985.

....semantics of calculi have been based on rewriting theory or proof theory, e.g. normalization or cut elimination, ordinal assignments, Church Rosser, etc. Such techniques, e.g. the familiar Tait Girard computability method (Girard, Lafont, Taylor 1989) or the method of logical relations (Statman 1985; Mitchell 1990) are often based on ingenuity and lack the explanatory power of a model theoretic proof. At the same time, they introduce specialized syntactic notions intrinsic to the technique (e.g. neutral terms, admissible logical relations, etc. but orthogonal to the problem. We use ....

R. Statman, Logical Relations and the Typed Lambda Calculus, Inf. and Control 65 (1985), pp. 85-97.

....Logical relations have proven useful in the study of Henkin lambda models. For example, we may prove the completeness of pure ; conversion (without equational hypotheses) for many speci c classical models, and characterize the lambda de nable elements of certain models using logical relations [Plo80, Sta82, Sta85b, Sta85a]. In [Plo80] Plotkin introduced I relations, which are families of typed relations over a Henkin model, indexed by possible worlds. In this section, we will consider Kripke logical relations, which are the straightforward generalization of I relations to Kripke lambda models. In the classical ....

R. Statman. Logical relations and the typed lambda calculus. Information and Control, 65:85-97, 1985.

....the shape 21 R 0 0 0 0 R 0 0 0 0 9 where the broken arrow indicates the existence of a factorization. Diagrams of this shape arise as Reynolds invariance conditions in his approach to parametricity [27] 2] pp.53 55, as well as in Statman s theory of logical relations [30], and in Wadler s universal Horn conditions which motivated this discussion. We emphasize, however, that the invariance conditions given by diagrams of this shape have deep roots in the left rules of Gentzen sequent calculus for propositional logic prior to any considerations of second order ....

R. Statman. Logical Relations and the Typed Lambda Calculus. Inf. and Control 65(1985), pp. 85-97.

....semantics for the simply typed calculus, and we show that it relates properly to the standard continuous functions semantics: we establish a correspondence for each term M between the intensional meaning of M and the extensional meaning of M . In showing this we make use of logical relations [Sta85]. 2 Computations, Comonads and Algorithms We first present some relevant category theoretic results. Although in the rest of the paper we will be mainly interested in a particular application, we present the background in a rather general way, so that the underlying assumptions can be clearly ....

....types, and at arrow types is defined in the natural logical way, so that an algorithm a relates to a function f iff whenever i relates to e, then the result of applying a to i in Alg relates to the result of applying f to e. In fact, this family of relations constitutes a logical relation [Sta85] between our two models. Proposition 5.16 Algorithm compositions relate to function compositions, in that for all a 2 I( 0 ) a 0 2 I( 0 00 ) and f 2 E( 0 ) f 0 2 E( 0 00 ) a 0 f a 0 0 00 f 0 ) a 0 ffi a) 00 (f 0 ffi f) ....

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R. Statman. Logical relations and the typed lambda calculus. Information and Control, 65:85--97, 1985.

..... Logical relations are used extensively in the study of typed lambda calculus and have applications outside lambda calculus, for example to abstract interpretation [Abr90] and data refinement [Ten94] A good reference for logical relations is [Mit96] An important but more difficult reference is [Sta85]. The Basic Lemma is the key to many of the applications of logical relations. It says that any logical relation over A and B relates the interpretation of each lambda term in A to its interpretation in B. Lemma 1.2 (Basic Lemma) Let R be a logical relation over Henkin models A and B. Then for ....

R. Statman. Logical relations and the typed lambda calculus. Information and Control 65:85--97 (1985).

..... Logical relations are used extensively in the study of typed lambda calculus and have applications outside lambda calculus, for example to abstract interpretation [Abr90] and data refinement [Ten94] A good reference for logical relations is [Mit96] An important but more difficult reference is [Sta85]. The Basic Lemma is the key to many of the applications of logical relations. It says that any logical relation over A and B relates the interpretation of each lambda term in A to its interpretation in B. Lemma 1.2 (Basic Lemma) Let R be a logical relation over Henkin models A and B. Then for ....

R. Statman. Logical relations and the typed lambda calculus. Information and Control 65:85--97 (1985).

..... Logical relations are used extensively in the study of typed lambda calculus and have applications outside lambda calculus, for example to abstract interpretation [Abr90] and data refinement [Ten94] A good reference for logical relations is [Mit96] An important but more difficult reference is [Sta85]. The Basic Lemma is the key to many of the applications of logical relations. It says that any logical relation over A and B relates the interpretation of each lambda term in A to its interpretation in B. An extended version of this paper, which includes proofs, is Report ECS LFCS 99405, ....

R. Statman. Logical relations and the typed lambda calculus. Information and Control 65:85--97 (1985).

.... Gamma Gamma R Gamma Gamma Gamma Gamma 9 where the broken arrow indicates the existence of a factorization. Diagrams of this shape arise as Reynolds invariance conditions in his approach to parametricity [27] 2] pp.53 55, as well as in Statman s theory of logical relations [30], and in Wadler s universal Horn conditions which motivated this discussion. We emphasize, however, that the invariance conditions given by diagrams of this shape have deep roots in the left rules of Gentzen sequent calculus for propositional logic prior to any considerations of second order ....

R. Statman. Logical Relations and the Typed Lambda Calculus. Inf. and Control 65(1985), pp. 85-97.

....the answer is positive, since each step pushes one ffi abstraction to the front. Thus, we can obtain the assertion, if we show termination and confluence for fij reduction in DLC. Instead of showing that directly using the well known logicalrelations method due to Tait [31] and Statman [30], we will map DLC into the simply typed calculus and use the results there. Let A be a well formed DLC term of type A #ff, furthermore let ffl ff 0 be obtained from ff by dropping all mode information, and ffl A 00 be obtained from A by exchanging all dynamically bound variables U with ....

R. Statman. Logical relations and the typed lambda calculus. Information and Computation, 65, 1985.

....interpretations of information hiding requires a theory of semantic equivalence for elements of existential type. In this respect, the use of relations between types has proved very useful. Study of relational properties of types goes back to the logical relations of Plotkin (1973) and Statman (1985) for simply typed lambda calculus and the notion of relational parametricity for polymorphic types due to Reynolds (1983) More relevant is Mitchell s principle for establishing the denotational equivalence of programs involving higher order functions and different implementations of an abstract ....

Statman, R. (1985). Logical relations and the typed lambda calculus. Information and Control 65, 85--97.

....alone is a simple consequence of order independence of ffi abstraction (cf. 3.6) Thus, we can obtain the assertion, if we show termination and confluence for fij reduction in DLC. Instead of showing that directly using the well known logical relations method due to Tait [Tai67] and Statman [Sta85], we will map DLC into the simply typed calculus and use the results there. Let A be a well formed DLC formula of type ff, furthermore let ffl e ff be obtained from ff by dropping all mode information, and ffl A 0 be obtained from A by exchanging all dynamically bound variables U with ....

R. Statman. Logical relations and the typed lambda calculus. Information and Computation, 65, 1985.

....properties of programs. As background references, for a description of F 2 , we recommend [GLT89, PDM89, Sc90, Rey90] for relevant ideas and notation from logic, especially sequent calculus and natural deduction, we recommend [vanD79] A good introduction to logical relations can be found in [Sta85, Mit91]. 2 Constructing proofs of parametricity from type inferences The slogan generally associated with parametricity is that programs evaluated in related environments have related results. We now formalize this notion in a very straightforward manner, using binary relations and second order ....

....on terms. We can safely limit this theory to a set of equations M = N , where M and N are equivalent modulo at most one fi reduction. Observe, then, how computation is related to proof. Proposition 2.4 The logical rules given are all sound. Lemma 2. 5 (Fundamental Theorem of Logical Relations [Sta85, Mit91]) If Gamma E: oe is a derivable type judgement, then Gamma R oe (E 0 ; E 00 ) is a derivable sequent. Proof. By induction on the inference of the type judgement. Definition 2.6 When R oe (E; E) is a provable sequent, we say the term E is parametric. We remark that in this ....

R. Statman. Logical relations and the typed lambda calculus. Information and Control 65 (1985), pp. 85--97.

....It is important to note that the set of simple sequents is obtained by left composition with the three canonical morphisms described above. 3 Logical Relations and Logical Permutations Logical relations play an important role in the recent proof theory and semantics of typed lambda calculi [35, 39, 40, 43]. We begin with logical relations on Henkin models; for further developments see [5, 35, 37, 38] 3.1 Definitions and Examples Consider a simply typed lambda calculus with product types. A Henkin model is a wellpointed cartesian closed category (ccc) Equivalently, a Henkin model is a ....

....M of type oe with free variables (i.e. in context) x : Gamma, sometimes denoted x : Gamma . M : oe. Consider Henkin models A; B, with assignments for variables j A ; j B in the respective models. Let M j A denote the meaning of M in model A w.r.t. the given variable assignment. Theorem 3. 1 ([39, 43, 35]) Let R A Theta B be a logical relation between Henkin models A; B . Let x : Gamma . M : oe. Suppose the interpretations of the free variables are related, i.e. for all i; R(jA (x i ) j B (x i ) Then R( M j A ; M j B ) In particular, if A = B and M is a closed term (i.e. contains no ....

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R. Statman, Logical Relations and the Typed Lambda Calculus, Information and Control 65, (1985), pp. 85-97.

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R. Statman. Logical relations and the typed lambda calculus. Information and Control, 65:85--97, 1985.

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R. Statman. Logical relations and the typed lambda calculus. Information and Control, 65:85--97, 1985.

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R. Statman. Logical relations and the typed lambda-calculus. Inf. and Control, 65:85--97, 1985.

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R. Statman. Logical relations and the typed lambda calculus. Information and Control 65:85-- 97 (1985).

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R. Statman. Logical relations and the typed lambda calculus. Information and Computation, 65, 1985.

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R. Statman. Logical relations and the typed lambda calculus. Information and Computation, 65, 1985.

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R. Statman. Logical relations and the typed lambda calculus. Information and Computation, 65, 1985.

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R. Statman. Logical relations and the typed lambda calculus. Information and Control 65:85-97, 1985.

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