C.-H. L. Ong. A semantic view of classical proofs: type-theoretic, categorical, denotational characterizations. In Proc. Eleventh Annual IEEE Symposium on Logic in Computer Science, pages 230--241. IEEE Computer Society Press, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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. A semantic view of classical proofs: type-theoretic, categorical, and denotational characterizations (preliminary extended abstract). In Proceedings, th Annual IEEE Symposium on Logic in Computer Science, pages 230--241, New Brunswick, New Jersey, 27--30 July 1996. IEEE Computer Society Press.

.... 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. A semantic view of classical proofs: type-theoretic, categorical, and denotational characterizations (preliminary extended abstract). In Proc. of 11th Ann. IEEE Symp. on Logic in Computer Science, LICS'96, pp. 230-241. IEEE CS Press, 1996.

....features not available in the sequential algorithms model, supports this view, and the claim that dropping the bracketing condition gives more than just another fully abstract model for sPCF . Extensions of dialogue games to model the calculus have been considered elsewhere: Ong [13] adds a notion of state, whilst Herbelin [9] suggests the adaptation to the Hyland Ong framework which is used here: i.e. the relaxation of the bracketing condition, although in the context of a more syntax oriented study, in contrast to the present work which aims to place these models in ....

C.-H. L. Ong. A semantic view of classical proofs: type-theoretic, categorical and denotational characterizations. In LICS 96. IEEE Computer Society Press, 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. A semantic view of classical proofs: type-theoretic, categorical, and denotational characterizations (preliminary extended abstract). In Proc. of 11th Ann. IEEE Symp. on Logic in Computer Science, LICS'96, pp. 230-241. IEEE CS Press, 1996.

....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 ....

C.-H. L. Ong. A semantic view of classical proofs: type-theoretic, categorical, and denotational characterizations (preliminary extended abstract). In Proceedings, 11 th Annual IEEE Symposium on Logic in Computer Science, pages 230--241, New Brunswick, New Jersey, 27--30 July 1996. IEEE Computer Society Press.

.... than PCF, including untyped and recursively typed languages [5, 4] languages with control features [21] call by value languages [13, 6] and languages with side e#ects and store [7] In a slightly di#erent vein, games models have also been discovered for the constructive classical type theory [28] and for System F [14] Received July 11, 1997. The author acknowledges the support of Oxford University Computing Laboratory. c # 1997, Association for Symbolic Logic 1079 8986 97 0303 0004 2.60 347 348 GUY McCUSKER In this paper, we give an account of a model of FPC, a type theory with ....

C. H. L. Ong, A semantic view of classical proofs: type-theoretic, categorical and denotational characterizations, In Proceedings, Eleventh Annual IEEE Symposium on Logic in Computer Science [19], pp. 230--241.

.... 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 ....

Ong, C.-H. Luke. (1996). A semantic view of classical proofs: Type-theoretic, categorical, and denotational characterizations (preliminary extended abstract). Pages 230--241 of: Proceedings 11th annual ieee symp. on logic in computer science, lics'96, new brunswick, nj, usa, 27--30 july 1996. Los Alamitos, CA: IEEE Computer Society Press.

.... generalizes some aspects of the present work to arbitrary computational effects in place of continuations (Fuhrmann 1999) A different class of models for the call by name calculus, based on fibrations, was defined by Ong and Ritter and later generalized to the disjunctive case by Pym and Ritter (Ong 1996; Pym and Ritter 1998) The focus of these models is different from ours, as they stress the fibered nature of the calculus with respect to control contexts, and thus they are, in a sense, higherorder. However, these models are rich in algebraic structure, and indeed, the calculus forms an ....

....then the control flow will jump back to the term M in the environment in which it was originally called. At that point, M will evaluate to v. Peter Selinger 26 5. The call by name interpretation of the calculus The calculus was originally introduced as a call by name language (Parigot 1992; Ong 1996), although Ong and Steward have later given it a call by value interpretation (Ong and Stewart 1997) We will first consider the call by name semantics, and leave the call by value semantics for the next section. The operational semantics of the call by name calculus can be given in several ....

C.-H. L. Ong. A semantic view of classical proofs: Type-theoretic, categorical, and denotational characterizations. In Proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science, pages 230--241, 1996.

....adapt the term assignment method (i.e. first part) to the LKQ. It is the dual of the LKT. Then one get q calculus. From its duality, q calculus seems to be the candidate to simulate CBV version of calculus. The CBV version of calculus is provably the one which is known as Ong s v [13, 14]. We hope CBN and CBV will be shown to be dual through further study on q calculus. Our characterization of CBN by LKT is purely syntactical. However, our transform allows n calculus to be interpreted in LKT, hence its coherent semantics. It should be more understood how these syntactical ....

C.-H. L. Ong. A semantics view of classical proofs: type-theoretic, categorical, denotational characterizations. In Proc. 11th IEEE Annual Symposium on Logic in Computer Science, pages 230--241. IEEE Computer Society Press, 1996.

....CBV system deserve further study. One can easily adopt the method of the first part to the LKQ which is the dual of the LKT. Then one can get so called q calculus. Naturally this seems to be the candidate to simulate CBV version of calculus. This is provably the one which is known as Ong s v [14, 15]. Denotational semantics for the CBN CBV functions also deserve further study. Our characterization of CBN CBV is purely syntactical. It should be more understood how these syntactical characterization of CBN CBV relate to their (maybe more intrinsic) characterizations using domains, categories ....

C.-H. L. Ong. A semantics view of classical proofs: type-theoretic, categorical, denotational characterizations. In Proc. 11th IEEE Annual Symposium on Logic in Computer Science, pages 230--241. IEEE Computer Society Press, 1996.

....et al. 10] Griffin [14] and Hofmann [18] Another possibility is to develop a denotational model and reason about programs via their denotations. Cartwright et al. 3] give a (fully abstract) model of Idealised Scheme and some models of the calculus have been considered recently by Ong [23], Hofmann and Streicher [19] and Selinger [30] In this paper I shall rather consider techniques based on the operational behaviour of PCF programs. One advantage of such operationally based techniques is that they require relatively little mathematical overhead. Further arguments in favour of ....

C.-H.L. Ong. A semantic view of classical proofs: type-theoretic, categorical and denotational characterizations. In Proceedings of Symposium on Logic in Computer Science, pages 230--241, 1996.

.... that the reduction rules for C were closely related to classical proof normalization as studied by Prawitz [81] Seldin [86,87] and Stalmarck [90] Griffin s discoveries were followed by a series of papers on classical logic, control operators and the Curry Howard isomorphism, see for example [3,4,18,24,42,62 66,69 72,82]. Most of these works introduce one typed classical calculus, i.e. a typed calculus enriched with control operators, and study its properties with respect to e.g. normalization, confluence and categorical semantics or its applications to e.g. classical theorem proving and witness extraction. ....

....our choice and relate our Delta calculi to some of the alternatives found in the literature. Because of the nature of the paper, we only focus on syntactic issues. Categorical issues and the way they influence the design of a classical calculus have been discussed by other authors elsewhere [42,69]. 8.1 Classical natural deduction and classical calculi Our presentation of classical pure type systems is based on Prawitz s format for classical natural deduction. However there are many other formats which also inspired classical calculi. We review some of these formats here; in order to ....

[Article contains additional citation context not shown here]

C.-H. L. Ong. A semantic view of classical proofs: Type-theoretic, categorical, and denotational characterizations. In Logic in Computer Science, 1996. To appear. Barthe, Hatcliff, Sørensen

....transforming all applications of this rule until they involve only atomic formulae. However the use of the de Morgan dualities is vital here; Prawitz [11, Footnote 1, Page 50] mentions that this technique does not extend to all the connectives (the problematic one being the disjunction) Ong [9] suggests a similar strategy for Parigot s system by rewriting applications of Unfreeze until they are of atomic type, although he advocates it to ensure confluence when considering j reduction. Given that this technique requires the use of the de Morgan dualities when considering all the ....

....system by rewriting applications of Unfreeze until they are of atomic type, although he advocates it to ensure confluence when considering j reduction. Given that this technique requires the use of the de Morgan dualities when considering all the connectives, 3 This property enables Ong [9] to define a categorical model. It is well known that a CCC with an isomorphism A = A collapses to a boolean algebra. bierman I shall not consider it here. 4 From Intuitionistic Linear Logic to Classical Linear Logic I shall extend the natural deduction formulation of ILL from my thesis ....

C.-H.L. Ong. A semantic view of classical proofs: type-theoretic, categorical and denotational characterizations. To appear in LICS'96, December 1995. bierman

....championed the view of contexts as iindices for the terms and types derivable in that context.j We believe this to be relevant not only to type theory but also to the modelling of environments in computer science, and we use it for that purpose in our third approach to continuation semantics. Ong [11] also uses a bration to model environments for his categorical formulation of the calculus [14] As this calculus is an extension of the call byname calculus, Ong can assume every bre to be Cartesian closed. However, for call by value programming languages like ML or Scheme, one cannot ....

C.-H. L. Ong. A semantic view of classical proofs: type-theoretic, categorical, denotational characterizations. In Proc. 11th IEEE Annual Symposium on Logic in Computer Science, pages 230241. IEEE Computer Society Press, 1996.

....(Ax v ) s [ff : z:hfii(xz) A B) x ) B fi (L v ) u [fl : x:s [ff : z:hfii(xz) B fi (cut v ) fi B :u [fl : x:s [ff : z:hfii(xz) B ( 0 abstraction) Table 12: Calculation of (E) rule for CND 4. 2 Relation with Ong s mixed substitution Ong s mixed substitution [17, 18] is defined as follows: Given a context C A of type A with a B typed hole [0] the mixed substitution is in the form of t[fi; C=ff] The meaning of this substitution is recursively replacing every named subterm of s of the shape [ff] u by [fi] C[u] In our calculus it is defined as a ....

....in which reflective transitive closure of cut elimination procedure are the equalities between PNs(as morphisms of category) Actually it is partly revealed. For example the relation between continuation models and calculus models (CBN) 15] between calculus models (CBV) and AJM s games model [17] and between AJM s games model and reduction in MELL [24] are known. It seems these models should be unified to this categorical logic for MELL PN because they are essentially the same as we show in this paper. ....

C.-H. L. Ong. A semantics view of classical proofs: type-theoretic, categorical, denotational characterizations. In Proc. 11th IEEE Annual Symposium on Logic in Computer Science, pages 230--241. IEEE Computer Society Press, 1996.

....passing style(CPS) and classical proofs in calculus style. Proof theoretically speaking, this translation induces a logical embedding of classical logic into intuitionistic logic, which is akin to Kolmogorov s negative translation. Moreover Ong and Stewart presented a variant of calculus [17, 18] with call by value (CBV) reduction strategy. calculus are constructed based on the logical system in classical natural deduction (CND) style. There is also a investigation on the relation with CBN CBV and linear logic. Benton and Wadler discusses the relation between the linear lambda calculus ....

....LKQ LKT inherits computational property of linear logic, it seems reasonable to make it as a subsystem of Proof Net. In this paper, we treat only a term calculus on LKQ which is CBV programming system on classical proofs. We explain how to program in LKQ= in relation with Ong and Stewart(OS) s v [18, 17], since this is also the CBV programming system on classical proofs (in natural deduction style) The derivation rule of LKQ= is given in Table 7. 5.1 Sequents of LKQ= Sequent of LKQ= is: 0 ) 1 ; 5 with 2 Where 0 is a multiset of input terms, 1 is a multiset of named terms, and 5 is at most ....

[Article contains additional citation context not shown here]

C.-H. L. Ong. A semantics view of classical proofs: typetheoretic, categorical, denotational characterizations. In Proc. 11th IEEE Annual Symposium on Logic in Computer Science, pages 230--241. IEEE Computer Society Press, 1996.

....In this interpretation the right premise of (L ) is always axiom, and associated term is always in the form of h: fi] h. Hence we only need the abbreviated form of: hfiit. fi is only needed instead of general rule fi . 4. 4 Relation with Ong s mixed substitution Ong s mixed substitution [17, 18] is defined as follows: Given a context C A of type A with a B typed hole [0] the mixed substitution is in the form of t[fi; C=ff] The meaning of this substitution is recursively replacing every named subterm of s of the shape [ff] u by [fi] C[u] In our calculus mixed substitution ....

....which reflective transitive closure of cut elimination procedure are the equalities between PNs(as morphisms of category) Actually it is partly revealed. For example the relation between continuation models and calculus models (CBN) 14] between calculus models (CBV) and AJM s games model [17] and between AJM s games model and reduction in MELL [24] are known. It seems these models should be unified to this categorical logic for MELL PN because they are isomorphic as we show in this paper. ....

C.-H. L. Ong. A semantics view of classical proofs: type-theoretic, categorical, denotational characterizations. In Proc. 11th IEEE Annual Symposium on Logic in Computer Science, pages 230--241. IEEE Computer Society Press, 1996.

.... [2, 3, 4, 5, 6] Rezus [74, 75] Parigot [66, 67, 68, 69] de Groote [25, 27, 28, 29, 30] Krivine [57] Girard [42] Danos, Joinet, and Schellinx [18] Rehof and S rensen [73] Duba, Harper, and MacQueen[32] Harper and Lillibridge [46, 47] Coquand [15] Berardi, Bezem, and Coquand [11] Ong [65], Underwood [89] and Sato [78] Most of these lines of work study one specific classical natural deduction system or one specific typed calculus enriched with control operators. The authors [10] study a notion of classical pure type system (CPTS) systematizing such calculi similarly to how PTSs ....

C.-H. L. Ong. A semantic view of classical proofs: Type-theoretic, categorical, and denotational characterizations. In Logic in Computer Science, 1996.

....easily adopt the term assignment method (i.e. first part) to the LKQ. It is the dual of the LKT. Then one get q calculus. From its duality, q calculus seems to be the candidate to simulate CBV version of calculus. The CBV version of calculus is provably the one which is known as Ong s v [13, 14]. We hope CBN and CBV will be shown to be dual through further study on q calculus. Our characterization of CBN by LKT is purely syntactical. However, our translation allows n calculus to be interpreted in LKT, hence its coherent semantics. It should be more understood how these syntactical ....

C.-H. L. Ong. A semantics view of classical proofs: type-theoretic, categorical, denotational characterizations. In Proc. 11th IEEE Annual Symposium on Logic in Computer Science, pages 230--241. IEEE Computer Society Press, 1996.

....the various logics. For both IL and CL, formulae are given by the grammar OE : p j OE OE j OE OE j OE oe OE; where p is taken from a countable set of atomic formulae which includes a distinguished member, which denotes falsum. 2 A similar presentation has been given independently by Ong [29]. Identity OE Gamma OE ( L ) Gamma OE Gamma Gamma Weakening Gamma; OE Gamma Gamma; OE; OE Gamma Contraction Gamma; OE Gamma Gamma Gamma OE Delta; OE Gamma Cut Gamma; Delta Gamma Gamma; OE Gamma (L ) Gamma; OE Gamma Gamma; Gamma (L ) ....

....to cases where OE is atomic. This is achieved by both factoring formulae through the de Morgan dualities (thus eliminating certain problematic connectives) and by transformation. For example, an application of the above rule where OE = OE oe is transformed to 6 This property enables Ong [29] to define a categorical model. It is well known that a CCC with an isomorphism A = A collapses to a boolean algebra. OE oe ] OE] oe E ) oe E ) oe I ) OE oe ) Delta Delta Delta RAA (oe I ) OE oe where clearly the size of the formula used in the ....

[Article contains additional citation context not shown here]

C.-H.L. Ong. A semantic view of classical proofs: type-theoretic, categorical and denotational characterizations. To appear in LICS, 1996.

....=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 rules therein are just the equality axioms oriented from left to right) See Remark 3.1 for a ....

C.-H. L. Ong. A semantic view of classical proofs: type-theoretic, categorical, denotational characterizations. In Proc. 11th ieee Symp. Logic in Computer Science, New Jersey, July 1996, pages 230--241. ieee Computer Society Press, 1996.

....logical and computational foundations 2 respectively for functional computation with control, in much the same way as the calculus and pcf respectively do for pure functional computation. Previous work The cbn version of , which we shall henceforth call n , has been presented in [27] which is a purely proof theoretic study. The work reported here sets out the computational relevance of the approach, and should be regarded as a sequel to [27] A reading of the first two sections of [27] is probably 1 The adjective call by value qualifies only the fi redex rule and not ....

....do for pure functional computation. Previous work The cbn version of , which we shall henceforth call n , has been presented in [27] which is a purely proof theoretic study. The work reported here sets out the computational relevance of the approach, and should be regarded as a sequel to [27]. A reading of the first two sections of [27] is probably 1 The adjective call by value qualifies only the fi redex rule and not the reduction strategy. 2 The approach works equally well for both the cbn and cbv regimes. In this paper we choose to focus on the latter (and on pcfv ....

[Article contains additional citation context not shown here]

C.-H. L. Ong. A semantic view of classical proofs: typetheoretic, categorical, denotational characterizations. In Proc. 11th ieee Symp. Logic in Computer Science, New Jersey, July 1996, pages 230--241. ieee Computer Society Press, 1996.

No context found.

C.-H. L. Ong. A semantic view of classical proofs: type-theoretic, categorical, denotational characterizations. In Proc. Eleventh Annual IEEE Symposium on Logic in Computer Science, pages 230--241. IEEE Computer Society Press, 1996.

No context found.

C.-H. L. Ong. A semantics view of classical proofs: type-theoretic, categorical, denotational characterizations. In Proc. 11th IEEE Annual Symposium on Logic in Computer Science, pages 230--241. IEEE Computer Society Press, 1996.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC