Chetan Murthy. A computational analysis of Girard's translation and LC. In Proceedings of Seventh Symposium on Logic in Comp. Sci., pages 90--101, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....sequent style intuitionistic logic (i.e. LJ) is investigated by Zucker[26] Pottinger[20] and recently Mints[10] We extend this to classical case, by using LKQ and the q protocol. As for the relation between classical logic and CPS translation, Murthy s pioneering work is also noteworthy[11]. He shows that one can interpret Girard s LC[7] of which the negative fragment is LKT) by means of CPS calculi with intuitionistic extract method. We conclude how our approach confronts to the Selinger s work on co control category[24] in the last section 6. 2 Background In this section, we ....

....choose the # v redex in the application from left to right(LR) order. Of course, the opposite order should be studied in its own right. This phenomena is known in the previous study of CPS; the CBV right to left(RL) evaluation method. This kind of CPS translation was shown, for example, by Murthy[11]. One can adopt the RL evaluation method in our #v . For this, we first modify the evaluation context as follows: E : ME Ev This modification leads us a RL version of our #v . Then, we modify the translation (in order to keep the simulation theorem) as follows: MN : let y = in S = N ....

Chetan R. Murthy. A computational analysis of Girard's translation and LC. In Proceedings, Seventh Annual IEEE Symposium on Logic in Computer Science, pages 90--101, Santa Cruz, California, 22--25 June 1992. IEEE Computer Society Press.

....the # calculus with let construct. Instead we develop direct isomorphism between LKQ and a familiar CBV # calculus with a let construct equipped with a classical extension. As for the relation between constructive classical logic and CPS translation, Murthy s pioneering work is also noteworthy[8]. He shows that one can interpret Girard s LC[5] of which the negative fragment is LKT) by means of CPS calculi with intuitionistic extract method. 2 2 Background In this section, we recall necessary definitions and notations for our presentation. Basically, we follow the notion of indexed ....

....(B # # A # ) induces Fischer style CPS translation [4] In #v , we evaluate the application from left to right(LR) Of course, the reverse order of evaluation should be studied in its own right. The CBV right to left(RL) evaluation method and its CPS transform was shown, for example, by Murthy[8]. In our formulation, we modify the evaluation context as follows: E : ME Ev. We also modify the translation as follows: MN : let y = in S = N : let x = in (M : let z = in (let y = zx in S) DJS s theory give a proof theoretical explanation to this phenomena; since implication is ....

Chetan R. Murthy. A computational analysis of Girard's translation and LC. In Proceedings, Seventh Annual IEEE Symposium on Logic in Computer Science, pages 90--101, Santa Cruz, California, 22--25 June 1992. IEEE Computer Society Press.

....and applying constructive systems to the formalization of algorithmic mathematics and to modeling and verifying complex computer systems. Our production system, Nuprl [7] has been used in several major applications [15, 5, 14, 1] and has been used to solve dicult problems in mathematics [17, 16, 13]. The mathematics formalized in the system have been used to teach computer science and logic courses [8] Many of the questions that arise in an e ort to build a global digital library of mathematics arise also in the formal setting. We see interesting possibilities for sharing ideas and methods ....

Chetan Murthy. A computational analysis of Girard's translation and LC. In Proceedings of Seventh Symposium on Logic in Comp. Sci., pages 90-101, 1992.

....fragment of the t protocol. Our term calculus for LKT can also be seen as an immediate successor of these works, although we do not use explicitly of these results. 3 As for the relation between constructive classical logic and CPS translation, Murthy s pioneering work is also noteworthy[18]. He shows that one can interpret Girard s LC[10] of which the negative fragment is LKT) by means of CPS calculi with intuitionistic extract method. However, he could not give an answer to the question whether this term extraction method is appropriate or not. This is because he doesn t ....

....decoration method is already mentioned in [5] though DJS does not put much stress on it. One can embed LKT into LJ by means of intuitionistic decoration. The introduced negations are implications; it is defined as OE : A j A oe OE, where OE is an arbitrary fixed formula. This Murthy s trick[18] enables us to embed LKT into minimal logic. If maps formulas to formulas, then if 1 = A 1 x 1 ; A n xn , we write 1 for the set (A 1 x1 ) A n xn ) For example, OE : 1 t for ( OE : A t 1 ) x1 ; OE : A t n ) xn , where t maps an LKT formula ....

Chetan R. Murthy. A computational analysis of Girard's translation and LC. In Proceedings, Seventh Annual IEEE Symposium on Logic in Computer Science, pages 90--101, Santa Cruz, California, 22--25 June 1992. IEEE Computer Society Press.

....which is an intuitionistic fragment of LKT. He interpreted LJT proofs as programs and cut elimination steps as reduction steps, just as we do here. We show that his system can be totally included in our CBN CPS calculus. Murthy s CPS calculus on LC: Murthy s pioneering work is also noteworthy. In [14], he shows that one can interpret Girard s LC (of which the negative fragment is LKT) by means of CPS calculi using the intuitionistic extract method. This is quite similar to our intuitionistic decoration method. However, he could not give an answer to the question whether this term extraction ....

....x g [ 0 by A x ; 0 . 5 denotes at most one unindexed formula. We say 5 is in the stoup [9] C denotes exactly one unindexed formula. The introduced negations are implications; it is defined as OE : A def = A OE, where OE is an arbitrary fixed formula. This method is introduced by Murthy[14]. This trick enables us to embed LKQ into minimal logic. If maps formulas to formulas, then if 1 = A 1 x1 ; An xn , we write 1 for the set (A 1 x1 ) An xn ) For example, OE : 1 q for ( OE : A q 1 ) x1 ; OE : A q n ) xn , where q maps an ....

C. R. Murthy, "A computational analysis of Girard's translation and LC," Proc. Seventh Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, California, Jun. 1992, pp. 90--101.

....language of an optimised version of Fischer s call byvalue CPS. We also did not nd in the literature the uniform CPS transform of Table 6. The relationship between types of terms and types of their CPS images was rst noticed by Meyer and Wand [MW85] Other important papers on types and CPS are [Mur92, HL93, HDM93] Theorem 10.2 is due to Meyer and Wand. The translation of arrow types is sometimes called idouble negation constructionj after Murthy [Mur92] The transformation of types for the callby name CPS in (36) and Theorem 10.9, are by Harper and Lillibridge [HL93] who follow what Meyer ....

....between types of terms and types of their CPS images was rst noticed by Meyer and Wand [MW85] Other important papers on types and CPS are [Mur92, HL93, HDM93] Theorem 10.2 is due to Meyer and Wand. The translation of arrow types is sometimes called idouble negation constructionj after Murthy [Mur92] The transformation of types for the callby name CPS in (36) and Theorem 10.9, are by Harper and Lillibridge [HL93] who follow what Meyer and Wand had done for call by value. Connections among functions, continuations and message passing are already well visible, though (as far as we know) ....

C. Murthy. A computational analysis of Girard's translation and LC. In 7th LICS Conf. IEEE Computer Society Press., 1992.

....our linear analysis. cf. page 19. The translation : A oe : B with sequents translated as : Gamma; Delta ) was independently considered by Parigot in [34] to prove strong normalization for his classical natural deduction by exhibiting a homomorphism to intuitionistic natural deduction. Murthy in [27, 28] applied similar ideas to control calculi (that is calculi extended with exception mechanisms, quite close to calculus) These calculi antedate the proof theoretic investigations of the normalization of classical logic and were recognized to be typable in propositional classical logic by ....

Murthy, C. (1992) A computational analysis of Girard's Translation and LC. In: Logic in Computer Science, pp. 90--101. IEEE Computer Society Press. LICS 1992.

....and fi reduction. In yet another direction, there is Girard s work on classical logic, based on the notion of polarity [10] Girard does not provide his system with expressions encoding proofs. Nevertheless, Murthy has shown how to extract terms from Girard s system, using Felleisen s operator [15]. More recently, Barbanera and Berardi have introduced an intriguing symmetric calculus based on a syntactic identi cation between a type and its double negation [4] All the above systems are based on computational interpretations of double negations. In the case of Delta and of the systems ....

C. R. Murthy. A computational analysis of Girard's translation and LC. In Proceedings of the seventh annual IEEE symposium on logic in computer science, pages 90101, 1992.

.... types and sorts, we can use a remark of Meyer and Wand [15] They noticed that there is a transformation on types that parallels the cps transform of call by value into call by name, which is: t ffl = t (oe ) ffl = oe ffl ( ffl o) o) We may also denote ( o) as : o (see [19]) Then, given our Proposition 4.2, we can reformulate Meyer and Wand s remark as follows, denoting by Gamma ffl the assumption that is the composition of Gamma with [ Delta] ffl : Lemma 6.1. Gamma M : C] Gamma ffl [ M ] ffl : o : o ffl [S] k : o ffl ; ....

....by Fournet and Gonthier in [11] where they give essentially the same encoding) More generally, P ] Pi k may be regarded as a term, up to innocuous j expansion. Before discussing the correctness of this cps transform, we first examine how types are correspondingly transformed (cf. [13,19]) Recall that we assumed given a type o, and that we denoted ( o) by : o . Then the ( Delta) Pi translation on types is: t Pi = o : o t (oe ) Pi = o : o (oe Pi Pi ) By definition, for any there exists a such that Pi = o : o . The ....

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C. Murthy, A computational analysis of Girard's translation and LC, LICS'92 (1992) 90-101.

....is an intuitionistic fragment of LKT. He interpret LJT proofs as programs ( calculus) and cut elimination steps as reduction steps, as we do. We show that calculus is completely included as an intuitionistic case of our CBN CPS calculus. We also have to mention to the pioneering work of Murthy[15]. He shows that one can interpret the proof of Girard s LC (of which negative fragment is LKT, positive fragment is LKQ) by CPS programs through the method called intuitionistic extract . This is quite similar to our intuitionistic decoration method. However he can t give an answer to the ....

Chetan R. Murthy. A computational analysis of Girard's translation and LC. In Proceedings, Seventh Annual IEEE Symposium on Logic in Computer Science, pages 90--101, Santa Cruz, California, 22--25 June 1992. IEEE Computer Society Press.

....LJT, which is an intuitionistic fragment of the LKT. He interpreted LJT proofs as programs and cut elimination steps as reduction steps, just as we do here. We show that his system can be totally included in the t . Murthy s CPS calculus on LC: Murthy s pioneering work is also noteworthy. In [12], he shows that one can interpret Girard s LC (of which the negative fragment is the LKT) by means of CPS calculi using the intuitionistic extract method. This is quite similar to our intuitionistic decoration method that appears in [13] However, he can t give an answer to the question ....

Chetan R. Murthy. A computational analysis of Girard's translation and LC. In Proceedings, Seventh Annual IEEE Symposium on Logic in Computer Science, pages 90--101, Santa Cruz, California, 22--25 June 1992. IEEE Computer Society Press.

....call byname (CBN) CPS translation on simply typed calculus induces a Godel s double negation translation on their types. Murthy described that one can interpret Girard LC (of which negative fragment is LKT, positive fragment is LKQ) by CBN CPS through the method of intuitionistic extract [16] which is similar to our intuitionistic decoration. However he can t give an answer to the question appropriateness of this term extraction method for LC . It is because he didn t consider the relation with the computation rules and cut elimination procedure. Incidentally, Murthy s paper also ....

....The origin of continuation variable is contexts of LKT while origin of observer variable is contexts. Notice that we exceptionally use the name m as head variable (instead of h) in (L )rule. This is because we follow standard usage of variable name on CPS translation. See for Murthy[16]. It remains to check that the fi contraction exactly corresponds to the cut elimination procedure on LKT LKQ. Propositon 3.1 (tight simulation) Let D and E be an equal intuitionistic decorated LKT derivation s.t. E is a result of a m cut elimination of D and of which end judgment is t : 0 ) 1, s ....

Chetan R. Murthy. A computational analysis of Girard's translation and LC. In Proceedings, Seventh Annual IEEE Symposium on Logic in Computer Science, pages 90--101, Santa Cruz, California, 22--25 June 1992. IEEE Computer Society Press.

....is an intuitionistic fragment of the LKT. He interpreted LJT proofs as programs and cut elimination steps as reduction steps, just as we do here. We show that his system can be totally included in our CBN CPS calculus. Murthy s CPS calculus on LC: Murthy s pioneering work is also noteworthy. In [16], he shows that one can interpret Girard s LC (of which the negative fragment is the LKT, but of which the positive fragment is NOT the LKQ) by means of CPS calculi using the intuitionistic extract method. This is quite similar to our intuitionistic decoration method. However, he can t give an ....

Chetan R. Murthy. A computational analysis of Girard's translation and LC. In Proceedings, Seventh Annual IEEE Symposium on Logic in Computer Science, pages 90--101, Santa Cruz, California, 22--25 June 1992. IEEE Computer Society Press.

....p. 67, Proposition 8.3] In 1990, however, T. Griffin opened a new research area by introducing a classical formulae astypes notion of control based on Felleisen s C operator [9] Since then, various authors have defined different systems that enlighten the constructive content of classical logic [1, 2, 7, 13, 14, 15, 16, 17]. Despite its originality, Griffin s work has been criticized by some logicians. In [15] for instance, M. Parigot writes that the system he (Griffin) obtains is not satisfactory from the logical point of view: the reduction is in fact a reduction strategy and the type assigned to C doesn t fit in ....

C. R. Murthy. A computational analysis of Girard's translation and LC. In Proceedings of the seventh annual IEEE symposium on logic in computer science, pages 90--101, 1992.

....works on classical logic, control operators and the Curry Howard isomorphism some initiated independently of his work. Most of these works study one typed calculus enriched with control operators; we call such calculi classical calculi. In the overwhelming majority of cases see for example [8, 9, 15, 22, 34, 48, 49, 50, 51, 52, 56, 57, 64] the calculus considered is essentially the simply typed or polymorphic calculus; other calculi considered include higher order calculus [31] ML [23] linear calculus [14] calculus with explicit substitutions [32, 68] or proof irrelevant logical pure type systems [79] In all cases, the ....

C.R. Murthy. A computational analysis of Girard's translation and LC. In Logic in Computer Science, 1992.

....V. Matiyasevich (Eds. Lecture Notes in Computer Science, Vol. 813, Springer Verlag (1994) pp. 142 152. In recent works, several authors have addressed the problem of extending the formulae astypes principle to classical logic, in order to express the computational content of classical proofs [2, 10, 12, 17, 18, 19]. This problem cannot have a unique solution because one knows that the technical content of the formulae as types principle, namely the Curry Howard isomorphism [3, 11, 14, 21] is strongly related to the constructive aspects of intuitionistic logic. Therefore, when dealing with classical logic, ....

C. R. Murthy. A computational analysis of Girard's translation and LC. In Proceedings of the seventh annual IEEE symposium on logic in computer science, pages 90101, 1992.

....Why can we say that this proof conversion on LK is CBN CPS translation We explain this by showing that we LKT is embeddable into LJ by means of intuitionistic negation decoration. This is analogue of linear decoration in DJS[4] Resulting formula(as types) is compatible with CBN CPS translation[14] on intuitionistic logic. Now we show the decoration. Translation on sequents are as follows: LKT 5 ; 0 ) 1 LJ 5 t ; 0 t ; 1 t ) OE where :A def = A i OE , OE is arbitrary. Translation on formulas are as follows: A t : A (for A atomic) A B) t : A t ) i ( B ....

....from LJ proofs, we get LKT proof. Moreover we don t need any correction cut during translation, thus cut elimination (computation) procedure proceeds in isomorphic manner, i.e. this is intuitionistic decoration. This decoration is the CPS CBN translation which is described, for example, by Murthy [14]. Classical linear decoration of LKT LKT 5 ; 0 ) 1 CLL 5 t ; 0 t ) 1 t A t : A (for A atomic) A B) t : A t 0ffi B t Proposition 5 (DJS) The above translation is a linear decoration for LKT. This means that we can map all derivation rule of LKT to CLL by adding ....

[Article contains additional citation context not shown here]

C. R. Murthy. A computational analysis of Girard's translation and LC. In Seventh Annual Symposium on Logic in Computer Science, pages 90--101, Santa Cruz, California, June 1992. IEEE Computer Society Press.

....call by name (CBN) CPS translation on simply typed calculus induces a Godel s double negation translation on their types. Murthy described that one can interpret Girard LC (of which negative fragment is LKT, positive fragment is LKQ) by CPS through the method of intuitionistic extract [16] which is quite similar to our method. However he can t give an answer to the question appropriateness of this term extraction method for LC . It is because he didn t consider the relation with the computation rules and cut elimination procedure. Murthy s paper includes good references for ....

....The origin of continuation variable is contexts of LKT while origin of observer variable is contexts. Notice that we exceptionally use the name m as head variable (instead of h) in (L ) rule. This is because we follow standard usage of variable name on CPS translation. See for Murthy[16]. It remains to check that the normalization exactly corresponds to the cut elimination procedure on LKT LKQ. Propositon 3.1 (simulation) Let D and E be an equal intuitionistic decorated LKT derivation s.t. E is a result of a m cut elimination of D and of which end judgment is t : 0 ) 1 , s : 0 ....

Chetan R. Murthy. A computational analysis of Girard's translation and LC. In Proceedings, Seventh Annual IEEE Symposium on Logic in Computer Science, pages 90--101, Santa Cruz, California, 22--25 June 1992. IEEE Computer Society Press.

....followed by several lines of work on classical logic, control operators, and the Curry Howard isomorphism some initiated independently of his work. It is not possible here to explain the aims and achievements of the individual lines of work; it must suffice simply to mention the work of Murthy [14, 59, 60, 61, 62], Barbanera and Berardi [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, ....

C.R. Murthy. A computational analysis of Girard's translation and LC. In Logic in Computer Science, 1992.

....Proceedings of the Colloquium on Trees in Algebra and Programming CAAP 94, Lecture Notes in Computer Science, Vol. 787 Springer Verlag (1994) pp. 85 99. During the last three years, several authors have introduced various systems that clarify the computational content of classical proofs [2, 3, 5, 6, 8, 9, 10]. In this paper, we investigate one of these systems, namely Parigot s calculus [10] Our investigation tool is merely syntactic: we propose a translation of the calculus into the well known calculus. This interpretation, which obey a continuation passing style, works for any term. It is ....

C. R. Murthy. A computational analysis of Girard's translation and LC. In Proceedings of the seventh annual IEEE symposium on logic in computer science, pages 90--101, 1992.

....LJT, which is an intuitionistic fragment of the LKT. He interpreted LJT proofs as programs and cut elimination steps as reduction steps, just as we do here. We show that his system can be totally included in the t . Murthy s CPS calculus on LC: Murthy s pioneering work is also noteworthy. In [11], he shows that one can interpret Girard s LC (of which the negative fragment is the LKT) by means of CPS calculi using the intuitionistic extract method. This is quite similar to our intuitionistic decoration method[12] However, he can t give an answer to the question appropriateness of ....

Chetan R. Murthy. A computational analysis of Girard's translation and LC. In Proceedings, Seventh Annual IEEE Symposium on Logic in Computer Science, pages 90--101, Santa Cruz, California, 22--25 June 1992. IEEE Computer Society Press.

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Chetan Murthy. A computational analysis of Girard's translation and LC. In Proceedings of Seventh Symposium on Logic in Comp. Sci., pages 90--101, 1992.

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Chet Murthy. A computational analysis of girard's translation and LC. In IEEE Annual Symposium on Logic in Computer Science. IEEE Computer Society Press, 1992.

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C.R. Murthy. A computational analysis of Girard's translation and LC. In Logic in Computer Science, 1992.

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