C.-H. L. Ong and C. A. Stewart. A curry-howard foundation for functional computation with control. In Proceedings of the 24th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages. ACM press, January 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....a CBV variant of Parigot s # calculus, namely #v . Our #v is general CBV language in the sense that one can simulate CBV # calculus with continuations(catch throw) and exception handling(handle raise) by our #v . However these investigations are not new, since Ong and Stewart describe these in [16]. What we do here is to improve Ong Stewart s CBV # calculus to be compatible to the q protocol. Specifically, we introduce only two symmetric reduction rules, namely #v and # v , and corresponding substitution, namely # and substitution. This shows the duality of values and evaluation contexts ....

....Clearly the Church Rosser property is only meaningful in this setting. Our point here is that the concept of CBV is not build on the reduction system as an evaluation order. 3. 2 Relation to Ong Stewart s CBV # calculus Now we demonstrate how our #v is di#erent from Ong Stewart s CBV # calculus[15, 16]. In a word, we pack n 1 length of reduction sequence into single reduction. Consider our general # v redex: #. #] E[#. #] M ] We assume E consists of n fold singular context, i.e. E = K 0 . Kn 1 . In the style of Ong Stewart s # reduction rule, the reduction proceeds as ....

C.-H. L. Ong and C. A. Stewart. A Curry-Howard foundation for functional computation with control. In Conference Record of POPL '97: The 24th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 215--227, Paris, France, 15--17 January 1997.

....respectively. The reduction rules are given in Figure 9 (substitutions [ ws) w] and [ sw) w] are de ned as in [Par92] Note that the rules are the same for the and top calculi. n is Parigot s original calculus, while our presentation of v is similar to Ong and Stewart [OS97]. Both sets of reduction rules are well typed and enjoy subject reduction. Instead of showing a correspondence between the top calculi and the C calculi as in [dG94] we have searched for an isomorphic calculus. This turns out to be interesting in its own right since it extends the ....

....For example, C top is CM C( k:M( f:kf) which is in fact a reduction rule for F [Fel88] This work fails in relating to C in an untyped framework, since it does not express continuations as abortive functions. It says in fact that F behaves as C in the simply typed case. Ong and Stewart [OS97] also do not consider the abort step in Felleisen s rules. This could be justi ed because in a simply typed setting these steps are of type . Therefore, it seems we have a mismatch. While the aborts are essential in the reduction semantics, they are irrelevant in the corresponding proof. We ....

C.-H. Luke Ong and C. A. Stewart. A Curry-Howard foundation for functional computation with control. In POPL'97, pages 215-227. 1997.

....functional programming [11, 13, 14] As we have seen in the present paper, the category of knowing (but well bracketed) strategies captures Idealized Algol. If we conversely retain innocence but weaken the bracketing condition then we get a model of PCF extended with non local control operators [8, 23]. Thus we begin to develop a semantic taxonomy of constraints on strategies mirroring the presence or absence of various kinds of computational features. ....

C.-H. L. Ong and C. A. Stewart. A Curry-Howard foundation for functional computation with control (summary). Preprint, July 1996.

.... typed calculi, see, e.g. 27, 28, 8] and applied to the compilation and optimization of typed languages, see, e.g. 19, 44] Grin s discovery initiated a series of studies on the computational content of classical proofs where CPS translations are a frequently employed tool, see, e.g. [13, 33, 38, 39, 34, 12, 4, 42, 24, 25, 35, 36, 6]. Inductive and coinductive types, see, e.g. 31, 29, 20, 15, 40] are syntactic representations for initial algebras (such as natural numbers and lists) resp. nal coalgebras (such as conatural numbers and streams) in typed calculi. Despite being pervasive in the type theoretical literature ....

C.-H. L. Ong and C. A. Stewart. A Curry-Howard foundation for functional computation with control. In Conf. Record of 24th ACM SIGPLAN-SIGACT Symp. on Principles of Programming Languages, POPL'97, pp. 215-227, ACM Press, 1997.

....complex, continuation style behaviour, such as prompts [3] The diculty of predicting on an ad hoc basis how control e ects will interact suggests that more formal ways of reasoning about them would be useful. One possibility is equational reasoning using control calculi such as C [2] or [8]. The counterexample in Section 3 shows the limitations of these calculi, however, in that their equational theories are not consistent with the presence of exceptions. There are many other ways to model or reason about control, but one which deserves mention here is game semantics. The results ....

....which holds in RC . More signi cantly, this is an equivalence which is at the basis of several equational theories of control , such as Felleisen s C calculus [2] and Parigot s calculus (which has been proposed as a a metalanguage for functional computation with control by Ong and Stewart [8]) Proposition 3. For any E[ S, M : T in RC : E[callcc M ] RC callcc k S)T :E[M y:k E[y] This equivalence is a typed version of the rule C lift which is a key axiom of Sabry and Felleisen s equational theory of the calculus with callcc [13] where it is shown to be sound using a ....

C.-H. L. Ong and C. Stewart. A Curry-Howard foundation for functional computation with control. In Proceedings of ACM SIGPLAN-SIGACT syposium on Principles of Programming Languages, Paris, January

.... calculus, and the call by value calculus is an extension of Moggi s c calculus [19] which is a standard calculus for reasoning about call by value programming. Additionally, the following fact is well known: Curry Howard correspondences exist between calculi and classical logics [21, 20]. The fact that control operators enable calculi to correspond with classical logics was rst discovered by Grin [10] Proof theoretically, the duality is a kind of De Morgan duality [2] iv Yoshihiko Kakutani Recursion and iteration Recursion is indispensable for programming languages and ....

....simply M = n N when A, and are obvious from context. So is the call by value case too: M = v N means M : A j is equal to N : A j in the v calculus. Remark 1. The n calculus and the v calculus are variants of Parigot s calculi [21] and Ong Stewart s v calculus [20]. Parigot has shown that his calculus has Church Rosser property and the strong normalization property, and Ong and Stewart as well. Additionally, although it is natural from the coincidence of the CPS translations, we stress that the v calculus is an extension of Moggi s c calculus. ....

C. H. L. Ong and C. A. Stewart. A Curry-Howard foundation for functional computation with control. In Proceedings of ACM SIGPLAN-SIGACT Symposium on Principle of Programming Languages, Paris, January

....represented using a third order constant. Since cases of third and higher order encodings are very rare in comparison with those of second order, a second order representation is given as well and equivalence to the third order representation is proven formally. 1 Introduction The calculus [Par92,OS97,Bie98], a proof theory for the implicational fragment of classical logic, has been established as a general tool to reason about functional programming languages with control, e.g. continuations and exceptions. It is basically an extension of the calculus by a second binder. Some of its properties like ....

....As it will be seen, the calculus is one of the rare examples that can best be represented in a way that involves a third order constructor. This representation will enable a natural implementation of the structural or mixed substitution that has been discussed controversially in the literature [OS97,SR98,dG98,Bie98]. This work was supported by the Graduiertenkolleg Logik in der Informatik (GKLI) Munich, and the Office of Technology in Education, Carnegie Mellon University. The remainder of this article is organized as follows: In Sect. 2 we will introduce the calculus in its original formulation by ....

[Article contains additional citation context not shown here]

C.-H. L. Ong and C. A. Stewart. A Curry-Howard foundation for functional computation with control. In Proceedings of ACM SIGPLAN-SIGACT Symposium on Principle of Programming Languages, pages 215--227, Paris, Januar 1997. ACM Press.

....The extent to which control operators alter the sorts of programs which can be written is underlined by their types, which are derivable in classical but not intuitionistic logic. This extension to the Curry Howard correspondence between proofs and programs has been formalised elsewhere [5] [14]; here it suggests that the same basic games model will give a semantics to PCF with a simple escape mechanism, which can be extended to the higher order control operators) and to proofs of classical logic. The fact that features of classical reasoning can be identified in games, and are the same ....

C.-H. L. Ong and C. A. Stewart. A Curry-Howard foundation for functional computation with control. to appear in proceedings POPL, 1997.

.... typed calculi, see, e.g. 27, 28, 8] and applied to the compilation and optimization of typed languages, see, e.g. 19, 44] Grin s discovery initiated a series of studies on the computational content of classical proofs where CPS translations are a frequently employed tool, see, e.g. [13, 33, 38, 39, 34, 12, 4, 42, 24, 25, 35, 36, 6]. Inductive and coinductive types, see, e.g. 31, 29, 20, 15, 40] are syntactic representations for initial algebras (such as natural numbers and lists) resp. nal coalgebras (such as conatural numbers and streams) in typed calculi. Despite being pervasive in the type theoretical literature ....

C.-H. L. Ong and C. A. Stewart. A Curry-Howard foundation for functional computation with control. In Conf. Record of 24th ACM SIGPLAN-SIGACT Symp. on Principles of Programming Languages, POPL'97, pp. 215-227, ACM Press, 1997.

....study its correspondence with t protocol. Note that LKT is a proof theoretical dual to LKQ. Hence it also means that the duality of CBN and CBV can be explained through the proof theoretical duality of LKT and LKQ. As for a term calculus for CBV CND, there is a noteworthy work by Ong and Stewart[11, 12]. The main difference is that we restrict ourselves to the # redex in CBV evaluation context which can be determined uniquely in a named term. This new reduction rule also removes the sequential reduction nature of Ong Stewart s # reduction. It recovers the diamond property of parallel ....

....E[ means in M , replace all subterms of the form [#] L by the term [#] E[L] The third type substitution of the form: M [X : B] means in M , replace all type variable X by the type B . 3. 2 Relation to Ong Stewart s CBV # calculus Now we demonstrate how our #v is di#erent from CBV # calculus[11, 12]. In a word, we pack n 1 length of sequential reduction sequence into single reduction. Consider our general # v redex: #. #] E[#. #] M ] We assume E consists of n fold singular context, i.e. E = K 0 # K 1 # . # Kn 1 . In the style of Ong Stewart s # reduction rule, the reduction ....

[Article contains additional citation context not shown here]

C.-H. L. Ong and C. A. Stewart. A Curry-Howard foundation for functional computation with control. In Conference Record of POPL '97: The 24th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pages 215--227, Paris, France, 15--17 January 1997.

....3.2 theorem prover [FF90] and the ideas of Fourman and Phoa [PF92] 2 EPSRC grants GR L89532 Notions of computability for general datatypes and GR J84205 Frameworks for programming language semantics and logic. 3 Namely, the category of PERs on the van Oosten Longley combinatory algebra [Oos97, Lon98] 2 As a somewhat separate issue, we will also add some support for data abstraction (abstypes and opaque signature constraints) Thus, we believe that we will cover most of the features of ML occurring commonly in programming practice. Of course, with future advances in semantic ....

.... from our point of view which embody a notion of functional programming with control (e.g. PCF catch; PCF; PCF with first order callcc) These languages su#ce for modelling exceptions in Java, for instance, and from a semantic point of view they are now very well understood (see e.g. CCF94, OS97, Lai98, Lon98] We will define a language L 2 by isolating a (syntactically defined) class of safe uses of exceptions in ML corresponding to the above languages. Thus, the known semantic models will guide us in choosing a sublanguage with pleasant logical properties, which will integrate ....

C.-H.L. Ong and C.A. Stewart. A Curry-Howard foundation for functional computation with control. In Proc. Symposium on Principles of Programming Languages, pages 215--227. ACM Press, 1997.

.... calculus. Besides, we have only investigated here M. Parigot call by name calculus. C. H. Ong and C. A. Stewart (1996; 1997) have proposed a call by value calculus. It is likely that a call by value ct calculus can be derived from their work. Notice that P. De Groote (1994b) and C. H. Ong (1996; 1997) separate the and the [ in their calculus. Nevertheless, this separation does not define a catch throw mechanism (since in ff:t the type of t is ) We did not consider tag abstraction as in the work of H. Nakano, Y. Kamayema and M. Sato, since there is no need for tag abstraction in the ....

Ong, C.-H. L., & Stewart, C. A. 1997 (15--17 Jan.). A Curry-Howard foundation for functional computation with control. Pages 215--227 of: Conference record of POPL '97: The 24th ACM SIGPLAN-SIGACT symposium on principles of programming languages.

....rules of calculus captures the mechanism of functional programming languages with control[3, 4, 6] However we can not apply an arbitrary reduction for implementation of programming language. Usually we fix a reduction strategy. A call by value calculus v was first considered by Ong and Stewart [11]. The v calculus contains another reduction rule so called symmetric structural reduction such that: N(ff:M) ff:M [ ff]w : ff] N w) Note that a subsystem is not always confluent even if the whole system is confluent. Therefore, the confluence of does not yield the confluence of v , ....

....QED Since the transitive and reflexive closure of is identical to the transitive closure of ) we have the confluence of calculus. Theorem 3. calculus is confluent. 4 Parallel Computation in Call by Value Calculus A call by value version of calculus was first provided by Ong and Stewart [11]. As compared with the call by name system, one can adopt some reduction rules more in the call by value system; so called symmetric structural reduction [12] such that N (ff:M) ff:M [ ff]w : ff] Nw) It is known that adding such reduction rules breaks down the confluence unless the above ....

[Article contains additional citation context not shown here]

C. -H. L. Ong and C. A. Stewart: "A Curry-Howard Foundation for Functional Computation with Control", Proc. 24th Annual ACM SIGPLAN-SIGACT Symposium of Principles of Programming Languages, 1997.

....that it corresponds to the double negationelimination rule (the : elim rule) hence the system gives computational meaning to classical proofs. After him, many researchers formulated typed calculi which corresponds to classical logic, such as Parigot s calculus[8] its callby value variant[7], de Groote s exception calculus, and classical catch throw calculus NK c=t [10] These are extension of the isomorphism to classical logic from a computational viewpoint, but we extend it from a logical one. The purpose of our research is to find the reduction rule (or the normalization of proof, ....

Ong, C.-H. L. and C. A. Stewart: "A Curry-Howard Foundation for Functional Computation with Control", Proc. 24th ACM Symposium on Principles of Programming Languages, 1997.

....reduce to and to (hx = i )f =xg. iii) The axiom ( cong) may look complicated, but its e ect is simply to commute the contexts E and D. The mediating abstractions on both sides of ensure that typability is preserved by the commuting operation. The axiom was rst introduced in [17] in the context of a call by value version of calculus for representing classical propositional proofs. Observe that the equality axiom obtained from ( cong) by replacing by = is admissible in the autonomous theory: E[ A :D[h iu] B :h i(E[ A :D[h iu] B ....

C.-H. L. Ong and C. A. Stewart. A Curry-Howard foundation for functional computation with control. In Proceedings of ACM SIGPLAN-SIGACT Symposium on Principle of Programming Languages, Paris, January 1997, pages 215-227. ACM Press, 1997.

....A ] B may look bewildering and a clutter, but its effect is simply to commute the contexts E B [ and D [ note that the mediating abstractions on both sides of the congruence ensure that typability is preserved by the operation. To our knowledge, the axiom was first introduced in [22] in the context of a call by value version of calculus for representing classical propositional proofs. The four congruence axioms may seem ad hoc, but they capture the commutation of contexts that correspond to the naturality conditions of autonomous categories; the details are set out in ....

....axioms above are subject to the strong typability side condition. ii) Hence Gamma C[z]ft =z c g = C[t] B, subject to the strong typability side condition, is provable in any pre autonomous theory augmented by ( eq) 2. 4 Autonomous Theories 21 The axiom (i eq) was first introduced in [22] as an axiom of a (call by value) proof system for classical propositional logic (see also [21] Proof A remark on notation. We find it useful to qualify an instance of an equality axiom by a variable in square brackets as in [ff] say) Its meaning should be clear by reference to Figure 5 (the ....

C.-H. L. Ong and C. A. Stewart. A Curry-Howard foundation for functional computation with control. In Proceedings of ACM SIGPLAN-SIGACT Symposium on Principle of Programming Languages, Paris, January 1997. ACM Press, 1997.

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C.-H. L. Ong and C. A. Stewart. A curry-howard foundation for functional computation with control. In Proceedings of the 24th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages. ACM press, January 1997.

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C.-H. L. Ong and C. A. Stewart. A Curry-Howard foundation for functional computation with control. In Proceeding of POPL'97, 1997.

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C. H. L. Ong and C. A. Stewart. A Curry-Howard foundation for functional computation with control. In Proceedings of ACM SIGPLAN-SIGACT Symposium on Principle of Programming Languages, Paris, January 1997. ACM Press, 1997.

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C.-H. L. Ong and C. A. Stewart. A Curry-Howard foundation for functional computation with control. In Proceedings of ACM SIGPLAN-SIGACT Symposium on Principle of Programming Languages, pages 215-227, Paris, Januar 1997. ACM Press.

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C.-H. L. Ong and C. A. Stewart. A Curry-Howard foundation for functional computation with control. In Proceedings of ACM SIGPLAN-SIGACT Symposium on Principle of Programming Languages, pages 215-227, Paris, Januar 1997. ACM Press.

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L. Ong, C. Stewart. A Curry-Howard Foundation for Functional Computation with Control (1997). In Proceedings of POPL'97.

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C. -H. L. Ong and C. A. Stewart: "A Curry-Howard Foundation for Functional Computation with Control", 24th ACM Symposium on Principles of Programming Languages, 1997. 16

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C. H. L. Ong, C. A. Stewart, A Curry-Howard Foundation for functional computation with control, Proceedings of ACM SIGPLAN-SIGACT Symposium on Principle of Programming Languages, Paris, ACM Press, January (1997).

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L. Ong, C. Stewart. A Curry-Howard Foundation for Functional Computation with Control (1997). In Proceedings of POPL'97.

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