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....from the usual substitution property for multisorted algebras in two ways: it is a substitution property for relations instead of functions, and it allows relations among objects of different types. Simulation relations resemble the logical relations used in the study of the lambda calculus [Sta85] [Mit86] An important property of logical relations is captured by the so called fundamental theorem of logical relations [Sta85] which states that simulation is preserved by expressions. The substitution property is similar to the defining property of Nipkow s simulation relations [Nip86] The ....

....of functions, and it allows relations among objects of different types. Simulation relations resemble the logical relations used in the study of the lambda calculus [Sta85] Mit86] An important property of logical relations is captured by the so called fundamental theorem of logical relations [Sta85], which states that simulation is preserved by expressions. The substitution property is similar to the defining property of Nipkow s simulation relations [Nip86] The difference is that we provide for message passing (through program operations) and allow objects of one type to be related to ....

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R. Statman. Logical Relations and the Typed -Calculus. Information and Control, 65(2/3):85--97, May/June 1985.

....(C; j C [r=b] j= b = 2 2 s) 33) 5.9 Simulation is Preserved by LOAL Programs The following lemmas are used to show that simulation is preserved by LOAL programs, not just by single invocations of program operations. This property is analogous to the fundamental theorem of logical relations [59]. We show that simulation is preserved by all type safe LOAL expressions in two steps. The first step, Lemma 5.11, assumes that the denotations of LOAL functions are related by a simulation relation (in a way described below) and shows that the possible results of an expression preserve ....

....in the proof of Lemma 5.15, only expressions that do not involve function calls are used. For a given algebra, the denotation of a LOAL function is a mapping from tuples of arguments to sets of possible results. Such mappings are related by analogy to the definition 52 of logical relations [59] [49] That is, if R is family of sorted relations, it is extended to the signatures of LOAL function identifiers as follows: R S T def = n (f 1 ; f 2 ) j q R S r ) f 1 ( q) R T f 2 ( r) o : 34) That is, for all f 1 and f 2 , f 1 is related by R S T to f 2 if and only if whenever ....

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Statman, R. Logical relations and the typed -calculus. Information and Control, 65(2/3):85--97, May/June 1985.

....discussion of how correctness allows us to reason in a useful way about the standard interpretation of terms. The formal proof of correctness of B uses the following theorem, due in this particular form to [Abr90] Proposition 3. 2) but originally due to Plotkin ( Plo80] Proposition 1, see also [Sta85] Fundamental Theorem of Logical Relations) Theorem 2.4.4 (The Binary Logical Relations Theorem) Let I and J be interpretations and let R : I J be a logical relation. Suppose that c I R c J for each and for each c : Then for all oe, for all e : oe, for all ae 2 Env I and ffi 2 Env ....

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

....free variables of s and t; this is equivalent to comparing appropriate closures of s and t. The aim of this article is to show that the relations 6 and j are undecidable. As 6 is a logical relation and a pre order at type B, certain properties of 6 and j follow from the logical relations lemma [13, 17]: Lemma 3 6 is a pre order, and j is an equivalence, at all types. If f; g : A ) B then f 6 g iff f x 6 g x for all closed x : A. The relations 6 and j contain the conversion relation Gamma . If C[ Delta] is a context with a hole of type A, and s 6 t are terms of type A, then C[s] 6 C[t] ....

R. Statman. Logical Relations and the Typed -Calculus. Information and Control 65:85-97, 1985.

....derivations that whenever A : B then 8x2 [ A] S j:8y2 G( A] F ae) x R j;i;ae (A) y ) x R j;i;ae (B) A;B (y) Notice here that [ A] S = B] S whenever A : B. Finally, by induction on typing derivations we prove the following extension of the Fundamental Lemma of Logical Relation [20]. Proposition 7.10 Suppose that Gamma e : A and that j; i; ae are assignments as above. Suppose, further that fl S 2 G( Gamma] S ) and fl F 2 G( Gamma] F ) are such that fl S (x) R j;i;ae ( Gamma(x) fl F (x) for each x 2 dom( Gamma) Then [ Gamma e : A] j S (fl S ) ....

R. Statman. Logical relations and the typed -calculus. Information and Control, 65:85--97, 1985. 44

....done so we know that the set of conversion rules we obtain is complete since, by corollary7 13 we know that if Eq( M ] id; N ] id) then Id(nf(M) nf(N) and then by theorem8 and symmetry and transitivity of = we know that M =N . To prove the theorem we define a Kripke logical relation [14, 18]: M 2 Gamma A u 2 Gamma k Gamma A CV Gamma;A (M;u) 2 Set this relation will correspond to Tait s computability predicate. A derivation of base type is intuitively CV related to a semantic object of base type if they are convertible with each other; or more precisely: M c = Delta;A ....

R. Statman. Logical Relation and the Typed -calculus. Information and Control, 65, 1985.

.... of papers [16, 19, 17, 18] The definition of Standard ML constitutes an extensive experiment in programming language specification based on operational semantics [46, 45] The method of logical relations is fundamental to the study of the typed calculus, as emphasized by Friedman [23] Statman [64], and Plotkin [57] Examples of the use of logical relations in the analysis of programming languages can be found in Plotkin s influential study of PCF [56] and in Mitchell s chapter mentioned above, to name two sources. The metaphor of computations (versus values ) implicit in Plotkin s v ....

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

....called candidates of reducibility. To prove the strong normalization of System F [ Girard, 1972 ] see the lucid account in Girard s book [ Girard et al. 1989 ] Another variation of the same theme called logical relation was independently developed by Plotkin [ Plotkin, 1972 ] see also [ Statman, 1985 ] Logical relations are a very useful tool for proving properties of typed lambda calculi. Denotational semantics prescribes meanings to syntactic entities compositionally. This means that the meaning of a composite term is defined in terms of the meanings of the respective components. If the ....

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

....introduced reducibility (or computability) as a technique for proving strong normalization for the simply typed calculus. Girard [7] introduced the method of the candidates of reducibility a technique for proving strong normalization for the second order typed calculus (and F ) Statman [23] and Mitchell [20] observed that reducibility can be used to prove other properties besides strong normalization, for example, confluence. The above lead to some natural observations: ffl There are some similarities between reducibility and realizability, but they remain somewhat implicit. ffl ....

R. Statman. Logical Relations and the Typed -Calculus. Information and Control, 65(2/3):85--97, 1985.

....in proving some deep results about various typed calculi and their models. A special form of the concept of a logical relation first appeared in Harvey Friedman s seminal paper [4] General logical relations were defined and used extensively in the pioneering work of Plotkin [18] and Statman [19, 21, 20], and later on in a more general setting by Breazu Tannen and Coquand [2] Mitchell [15] Mitchell and Moggi [16] and Abramsky [1] among others. As the name indicates, logical relations are certain kinds of relations, and they are used to prove relational properties of terms. On the other hand, ....

R. Statman. Logical Relations and the Typed -Calculus. Information and Control, 65(2/3):85--97, 1985.

....done so we know that the set of conversion rules we obtain is complete since, by corollary7 we know that if Eq( M ] id; N ] id) then Id(nf(M ) nf(N ) and then by theorem8 and symmetry and transitivity of = we know that M =N . To prove the theorem we define a Kripke logical relation [14, 18]: M 2 Gamma A u 2 Gamma k Gamma A CV Gamma ;A (M; u) 2 Set this relation will correspond to Tait s computability predicate. A derivation of base type is intuitively CV related to a semantic object of base type if they are convertible with each other; or more precisely: M c = Delta;A ....

R. Statman. Logical Relation and the Typed -calculus. Information and Control, 65, 1985.

....is rather well known for proving strong normalization (or normalization) but the fact that it can also be used to prove confluence or other properties does not seem to be as well known. Statman showed that various properties of the simply typed calculus can be obtained using logical relations [18], but John Mitchell seems to be one of the first who realized that reducibility can be used to prove more general properties than strong normalization. The general idea is that if a unary predicate P expressing a property of (typed) terms satisfies the conditions for being a candidate (as ....

R. Statman. Logical Relations and the Typed -calculus. Information and Control, 65(2/3):85-- 97, 1985.

....translations. Complete type information can be used for optimization such as instantiating polymorphic equality to monomorphic equality and choosing efficient representation of data types. It is also useful to prove correctness of compilation through such a method as logical relations [29, 7, 24, 27, 28]. Furthermore, constructing compilers as phases of type preserving translations has a practical advantage for development of compilers as mentioned in [20, 30] When we debug a compiler itself, the code of its intermediate language can be type checked. This has greatly helped to find bugs in our ....

....For this deductive system, we proved that the translation preserves the operational behavior of a program. It can formulated as below. Theorem 2 (Correctness) Let ; e : b and ; e e 0 . Then, e # c if and only if e 0 # c. The proof is based on the method of logical relations [29, 7, 24, 27, 28]. The proof of this theorem appears in Subsection 4.3. 4.1 Tolmach s Lifting Tolmach s lifting is formulated by restricting the rule (let) as follows: Delta; Delta 0 ; Sigma; u:h Delta; Delta 0 i; Gamma e 1 e 0 1 Delta; Sigma; Gamma; x:8 Delta 0 :h Delta; Delta 0 i ) ....

R. Statman. Logical relations and the typed -calculus. Information and Control, 65, 1985.

....G) and applying it to x 2 F 0 . Now we define a family of relations R oe j[ oe] P j Theta [ oe] S by induction on oe as follows. xR o y ( x = y fR oe g ( 8x; y:xR oe y ) app(f; x)R g(y) The following slight adaptation of the Fundamental Theorem of Logical Relations [15] (which was stated for Henkin models rather than cartesian closed categories) is proved by induction on terms. Proposition 4.2 Suppose that Gamma = x 1 : oe 1 ; xn : oe n and Gamma M : Let j be an environment satisfying Gammaand let fl = fl 1 ; fl n ) be such that fl i ....

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

.... attempt to survey the other papers that have explored or used logical relations for either the simply typed or the untyped lambda calculus, in either syntactic or semantic (though usually not category theoretic) versions; important examples include [20] where Kripke relations were introduced) and [28]. Closer to our concerns is [29] where Mitchell and Meyer give a logical relations theorem for the polymorphic lambda calculus, using the Bruce Meyer Mitchell concept of a model [30] It is tricky to compare the Bruce Mitchell Meyer framework with that of PL categories [31, 32] Nevertheless, we ....

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

....translations. Complete type information can be used for optimization such as instantiating polymorphic equality to monomorphic equality and choosing efficient representation of data types. It is also useful to prove correctness of compilation through such a method as logical relations [24, 6, 20, 22, 23]. Furthermore, constructing compilers as phases of type preserving translations has a practical advantage for development of compilers as mentioned in [16, 25] When we debug a compiler itself, a code of an intermediate language can be type checked. This has greatly helped to find bugs in our ....

R. Statman. Logical relations and the typed -calculus. Information and Control, 65, 1985.

....homomphisms are a special instance of a more general notion called a logical relation which serves as the one of the most basic tools for reasoning about types. Many of the properties of logical relations were developed by Tait, Statman, and Howard [ Howard, 1973; Statman, 1982; Statman, 1985a; Statman, 1985b; Statman, 1986; Tait, 1967 ] and they The Semantics of Types in Programming Languages 29 continue to be a topic of interest for applications. A general survey on logical relations is included in [ Mitchell, 1990 ] and [ Burn et al. 1986 ] furnishes an example of how logical relations can ....

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

....in contrast to soundness theorems based on a direct semantics which often establish, for example, that a program of base type yields a value of that base type, if it yields a value at all. To obtain such a result in this setting seems to require the use of an appropriate form of logical relation [26, 33]. An alternative is to consider the typing properties of the call by value cps transform, from which an observational soundness theorem may be extracted; see [5, 18, 24] 3.4 First Class Continuations To account for the continuation passing primitives introduced in Section 2, we extend the ....

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

....study of the typed lambda calculus, one considers only equations between terms of the same type. Logical relations among models of the typed lambda calculus are families of relations indexed by type, so that elements of type T in one model are only related to elements of type T in the other model [Sta85] Similarly, homomorphisms among multi sorted algebras are families of functions indexed by sort, so that the elements of one sort are mapped to elements of the same sort. 3 Leavens s work was supported in part by the ISU Achievement Foundation, the National Science Foundation under Grants ....

R. Statman. Logical Relations and the Typed -Calculus. Information and Control, 65(2/3):85--97, May/June 1985.

....as established in the previous case, Y(r)p q) t v. The rest are entirely similar. The computability argument central to the proof can be traced back to an important paper of Tait [29] A variation of the theme called logical relation was independently developed by Plotkin [20] see also [28]. Logical relations are a very useful tool for proving properties of typed lambda calculi. Plotkin s adequacy result for PCF has been extended to richer computational settings. For example, in [25] it is shown that adequacy holds for a meta language for denotational semantics which has an ....

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

.... Given a standard, call by value operational semantics for Mini ML with the value restriction, and given the stratification between monotypes and polytypes in both Mini ML and ML i , it is possible to modify a standard binary logical relations style argument for the simply typed lambda calculus [48, 15, 40, 45, 46] to show the correctness of the (var) FTV( n=tn ] Delta Delta Delta ( 1 =t1 ] Delta Delta Delta) Delta Delta; Gamma ] fx : 8t1 ; tn : g . x : n =tn ] Delta Delta Delta ( 1 =t1 ] Delta Delta Delta) x[j 1 j] Delta Delta Delta [j n j] int) Delta; ....

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

....the full Type Structure D and any Other Typed Lambda Model E Let E ff be an arbitrary (extensional) lambda model with pure functional types and j b : D b E b be a partial function onto, for each basic type. Let j ff D ff ThetaE ff be the corresponding induced logical relation (cf. Sta 85] and [Mit 90] Proposition 10 All relations j ff are partial functions D ff E ff onto E ff . Proof. We shall simultaneously construct retraction (right inverse) total mappings i ff : E ff D ff with j ff ffi i ff = id ff : Assume i ff and i fi are retractions of j ff and j ....

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

....an algebra, in a sense made precise in the following definitions, are called homomorphic relations. 2 The various notions of one data structure simulating another are defined in terms of relations of this kind. 2 Homomorphic relations are called logical relations when extended to higher types [20]. An independent generalization of logical relations that appears to be closely related to our notion of homomorphic generalized relations is considered in [10] BEHAVIOR REALIZATION ADJUNCTION 11 Definition 2.6 (standard homomorphic relation) Let A and B be (not necessarily comparable) Sigma ....

R. Statman, Logical Relations and the typed -calculus, Information and Control 65 (May/- June, 1985), no. 2/3, 85--97.

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R. Statman. Logical relations and the typed -calculus. Information and Control, 65, 1985.

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

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