H. Friedman. Equality between functionals. In R. Parikh, editor, Logic Colloquium '75. NorhHolland, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....as Scheme, a dialect of LISP. Primarily the STC is treated syntactically just as a collection of symbols and rules manipulating the symbols. Unlike usual systems of deduction such as the propositional or rst order calculi, there is no standard semantics for the STC. In some special cases, such as [7], the usual model theoretic de nition of semantics for a theory is complete. However, in general this is not true, an example being the retract theory discussed later. Remedying this de ciency are a cornucopia of potential sound and complete semantics in [9] 13] 12] 4] 14] Limiting their ....

Friedman, H., Equality between functionals, Logic Colloquium 73, ed by R. Parikh, Springer-Verlag, New York, 1975.

.... convertibility of terms (programs) We shall in addition show that two (definable) augmented simplicial maps are equal if and only if their defining terms are convertible, i.e. equal as proofs (bottom right = sign) This will be Theorem 72 and Corollary 73, an S4 variant of Friedman s Theorem [16], which will constitute the main goal of this paper. While Friedman s Theorem in the ordinary, non modal, intuitionistic case can be proved in a relatively straightforward way using logical relations [40] the S4 case is more complex, and seems to require one to establish the existence of a ....

.... many terms, while as soon as some base type A gets interpreted as an infinite set, A A will not be countable, and (A A) A will neither be countable nor even of the cardinality of the powerset of Nonetheless, it can be proved that this interpretation is equationally complete: Theorem 49 ([16]) If the two typed A terms M and N, of the same type F, have the same set theoretic interpretation for every choice of the interpretation of base types, then M and N are equivalent. In fact, there is even a fixed set theoretic interpretation such that, if M and N have the same value in this ....

Harvey Friedman, Equality between functionals., Logic Colloquium 1972-73 (Rohit Parikh, ed.), Lecture Notes in Mathematics, vol. 453, Springer-Verlag, 1975, pp. 22-37.

....and so I chose first to study the linear calculus as a simpler calculus which nevertheless contains initial type constructors. At the time of writing, the research presented here remains the only proof of decidability of the theory of coproducts. One other interesting result is [23] which extends [25] in providing a proof system for deriving a set of equations which is sound and complete for all set theoretic models of a calculus with exponentials and coproducts. This theory is then proved decidable by proving it is equivalent to the one presented here. 1.1 Notational Preliminaries ....

....closed categories because, if the interpretations of two terms are equal in all models, then their interpretations in the free model are equal and thus the terms are fij equal. A different problem is to seek sound and complete equational theories for more restricted classes of models. Friedman [25] proved that fij equality is complete for all models in the category Sets. In [77] simple conditions on an arbitrary cartesian closed category C are given to ensure fij equality is sound and complete for all models in C, while in [23] a proof system is given for deriving a sound and complete ....

H. Friedman. Equality between functionals. Logic Colloquium, 1975.

....can now be used to establish the soundness and completeness of the computational calculus with respect to the conventional CPS denotational model. The result explained below follows from the soundness and completeness of the simply typed calculus with respect to the full type structure [42]. In the conventional semantics for [42, 47] each type denotes a set of elements: the base type o denotes some arbitrary infinite set B, and the type t 1 t 2 denotes the set of functions from the denotations of t 1 to the denotations of t 2 . We define the full type pre structure over B as ....

....and completeness of the computational calculus with respect to the conventional CPS denotational model. The result explained below follows from the soundness and completeness of the simply typed calculus with respect to the full type structure [42] In the conventional semantics for [42, 47], each type denotes a set of elements: the base type o denotes some arbitrary infinite set B, and the type t 1 t 2 denotes the set of functions from the denotations of t 1 to the denotations of t 2 . We define the full type pre structure over B as the collection of nonempty sets D for each ....

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Friedman, H. Equality between functionals. In Parikh, R., editor, Logic Colloquium, Springer Verlag, Berlin (1973) 22--37. 152

....This formula corresponds in turn to Axiom 6 of [6] which makes Church s Theory of Types classical. 3.1 Applicative Structures We will make use of the notion of applicative structures, a well known semantical framework for the simply typed lambda calculus, rst introduced by H. Friedman in [11]. See also [18] De nition 3.4 A typed applicative structure hD; App; Consti consists of a set D for each type , a function App : D D D for each pair ( of types, and an intepretation function Const taking constants of type to elements of D . 9 A typed applicative ....

Harvey Friedman. Equality between functionals. In R. Parikh, editor, Logic Colloquium, volume 453 of Lecture Notes in Mathematics, pages 22-37. Springer, 1975.

.... convertibility of terms (programs) We shall in addition show that two (de nable) augmented simplicial maps are equal if and only if their de ning terms are convertible, i.e. equal as proofs (bottom right = sign) This will be Theorem 69 and Corollary 70, an S4 variant of Friedman s Theorem [16], which will constitute the main goal of this paper. While Friedman s Theorem in the ordinary, non modal, intuitionistic case can be proved in a relatively straightforward way using logical relations [37] the S4 case is more complex, and seems to require one to establish the existence of a ....

....terms, while as soon as some base type A gets interpreted as any in nite set, A A will not be countable, while (A A) A will neither be countable nor even of the cardinality of the powerset of N . Nonetheless, it can be proved that this interpretation is equationally complete: Theorem 48 ([16]) If the two typed terms M and N , of the same type F , have the same set theoretic interpretation for every choice of the interpretation of base types, then M and N are equivalent. In fact, there is even a xed set theoretic interpretation such that, if M and N have the same value in this ....

Harvey Friedman, Equality between functionals., Logic Colloquium

....depend on the whole assignment , but only on the values taken by the latter for the free variables of t A . In particular, when t A is a closed term, t A ] does not depend at all on and will then be written : t A ] Let us recall at last the following soundness theorem (see [Fri75]) if t A 1 = t A 2 , then for any assignment we have : t A ] t 0A ] In the sequel, we restrict ourselves to the case of predicates, that is functions from A 1 : A n to B . First note that a predicate P : A 1 : A n B is representable i its ....

Harvey Friedman. Equality between functionals. LNM, 453:22-37, 1975.

....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 semantic structure of terms representing sequences. These equations define the behavior of the distinguished semantic elements 1 st and 2 nd that take apart the semantic elements representing sequences. A type ....

....normalization of T we research the type inference system and the reconstruction algorithm. To show that the T type system is sensible according to standard mathematical intuitions, we give a simple set theoretical model closely following a model by Hindley [Hin69] and to some extent Friedman [Fri75] for the simply typed lambda calculus. In this model we let types denote sets of equivalence classes of terms by convertibility and let typing statements represent set memberships. We prove a soundness and completeness theorem for the model that verifies typing statements as exactly set ....

Harvey Friedman. Equality between functionals. Lecture Notes in Mathematics, 453:22--37, 1975.

....set theory. 2 above cannot be formalized. However, it has been common to assume that no type is empty. When we make this simplifying assumption, we eliminate one case, and the inference is easily carried out within the appropriate axiom system. This leads to the completeness theorems of [Fri75, Hen50, Sta85a]. The drawback, however, is that in many computer science applications it is not appropriate to assume every type is nonempty (inhabited) This point is discussed in [MMMS87] Related discussions of multi sorted equational logic appear in [GM82, GM86] When we reject the nonemptiness assumption ....

....is well known to researchers in categorical logic. We are grateful to Edmund Robinson and Pino Rosolini for some helpful discussion of this point of view, and refer the reader to [Fou77, Sco80, LS86] for related discussion. In short, the usual de nition of semantics of typed lambda calculus, as in [Bar84, Fri75, Hen50], may be formalized in the language of set theory: a model is a collection of sets satisfying several properties easily described by logical formulas. While we 4 usually interpret this de nition in the standard classical model of set theory, other interpretations are possible. In particular, ....

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H. Friedman. Equality between functionals. In R. Parikh, editor, Logic Colloquium, pages 22-37, Springer-Verlag, 1975.

....theorem, to the effect that that an equation is true in the set theoretic model Set if and only if it is provable in the equational theory presented by the axioms ABC described above. This generalizes the corresponding result for the calculus, obtained by Harvey Friedman in the seminal [Fri75]: it is proved there that equality between simply typed lambda terms in the full function type hierarchy over an infinite set is completely axiomatized by fi and j. There does not appear to be an equational presentation of the theory which supports a confluent rewrite system (cf Section 3) but ....

H. Friedman. Equality between functionals. In Rohit Parikh, ed., Logic Colloquium '73, Lec. Notes in Math. 453, 22--37. Springer-Verlag, 1975.

....It has been an open question for some time whether topological semantics are complete in this sense. Results of the kind given here go back to L. Henkin [9] who in effect showed that non standard, set valued semantics are complete ( 14] for some fine points) An oft cited result of H. Friedman [8] established the strong completeness of standard, set valued semantics for the theory consisting of a single basic type (no constants or equations) In this same vein, G. Plotkin has extended the result to certain categories of posets (see [13] Recently, A. Simpson [19] has shown that the ....

H. Friedman, Equality between functionals, Logic Colloquium 73 (R. Parikh, ed.), Springer-Verlag, New York, 1975.

.... in a number 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 ....

Harvey Friedman. Equality between functionals. In Rohit Parikh, editor, Logic Colloquium '75, Studies in Logic and the Foundations of Mathematics, pages 22--37. North-Holland, 1975.

....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 ....

Harvey Friedman. Equality between functionals. In Rohit Parikh, editor, Logic Colloquium '73, volume 453 of Lect. Notes in Math., pages 22--37. Springer-Verlag, 1975.

....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 ....

H. Friedman. Equality between functionals. In R. Parikh, editor, Logic Colloquium, '73, pages 22--37, Volume 453 of Lect. Notes in Math., Springer-Verlag, 1975.

....(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 ....

....hierarchy over B. Let R be the logical relation over M and M 0 induced by the partial function from jAj to jBj with domain jBj jAj which is the identity on this domain. This logical relation relates the meanings of the symbols in Sigma since B is a subalgebra of A. As Friedman has shown in [7], R is a partial function at all types (in his phrase, a partial homomorphism) Combining this fact with the fundamental theorem of logical relations, it follows that for closed terms M and N , M j= M = N implies M 0 j= M = N . Since B is finitely collapsing, by Theorem 6.7 (fi) j) and the ....

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H. Friedman. Equality between functionals. In R. Parikh, editor, Logic Colloquium '73, volume 453 of Lect. Notes in Math., pages 22--37. SpringerVerlag, 1975.

....was partially supported by ONR Grant NOOO14 93 1 1217. 1 Introduction Logical relations are an important tool used 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, ....

H. Friedman. Equality between functionals. In R. Parikh, editor, Logic Colloquium, volume 453 of Lecture Notes in Math., pages 22--37. Springer-Verlag, 1975.

....1 gives the basic notations used in the paper. The reader not familiar with simply typed calculus should consult Hindley and Seldin [3] Section 2 presents standard models for simply typed calculus, it is based on Henkin [2] Section 3 presents the Completeness theorem, it is based on Friedman [1], Plotkin [5] 6] and Statman [9] Section 4 presents the construction of a model for some equational theories. Section 5 presents Statman s finite completeness theorem. Both section 4 and 5 are based on [8] Section 6 presents the definability conjecture. The notion of definability is taken ....

H. Friedman, Equality Between Functionals, Proceedings of Logical Colloquium 72-73, Lecture Notes in Mathematics, 453, R. Parikh (Ed.), Springer-Verlag, 1975, pp. 22-37.

....syntactically, and (h) states that the meaning of functions can be based solely on their meaning under application. The (b) and (h) axioms turn out to be fundamental: not only are they sound, they also are complete for proving equations that hold in all models of the simply typed l calculus [Friedman, 1975]. In other words, an equation between simply typed l terms is valid in all models if and only if it is provable from (b) and (h) These axioms can also be complete for particular models. Friedman, for example, shows that an equation is valid in the full set theoretic model i.e. one with all ....

....axioms can also be complete for particular models. Friedman, for example, shows that an equation is valid in the full set theoretic model i.e. one with all total functions at function types, defined precisely in Section 2. 2 below over an infinite base type iff it is provable from (b) and (h) [Friedman, 1975]. The completeness theorem holds for other models as well, including ones based on continuous functions [Plotkin, 1982, Riecke, 1995, Statman, 1982, Statman, 1985a] These theorems say nothing, however, about extensions of the l calculus involving constants. For instance, suppose we add the ....

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Friedman, H. (1975). Equality between functionals. In Parikh, R., editor, Logic Colloquium '73, volume 453 of Lect. Notes in Math., pages 22--37. Springer-Verlag.

....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 ) ....

H. Friedman. Equality between functionals. In R. Parikh, editor, Logic Colloquium '75. NorthHolland, 1975.

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H. Friedman. Equality between functionals. In R. Parikh, editor, Logic Colloquium '75. NorhHolland, 1975.

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H. Friedman, Equality Between Functionals, Proceedings of Logical Colloquium 72-73, Lecture Notes in Mathematics, 453, R. Parikh (Ed.), Springer-Verlag, 1975, pp. 22-37.

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Friedman, H., Equality between functionals, Logic Colloquium 73, ed by R. Parikh, Springer-Verlag, New York, 1975.

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H. Friedman. Equality between functionals. In Logic Colloquium '72--73, Proc., pages 22--37. Springer-Verlag.

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H. Friedman. Equality between Functionals. in Lecture Notes in Mathematics 453, SpringerVerlag, R. Parikh ed., pp. 22-37. 18

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Harvey Friedman. Equality between functionals. In R. Parikh, editor, Logic Colloquium, Lecture Notes in Mathematics 453, pages 22--37. Springer, 1975.

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