H. Nakano. The Logical Structures of the Catch and Throw Mechanism. PhD thesis, The University of Tokyo, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....formulas as types paradigm to classical logic has been widely investigated by T. G. Griffin [10] C. R. Murthy [14] F. Barbanera and S. Berardi [1] N. J. Rehof and M. H. Srensen [23] P. De Groote [6] J. L. Krivine [13] and M. Parigot [18, 19] H. Nakano, Y. Kameyama and M. Sato [16, 15, 17, 11, 12, 26] have proposed various logical frameworks that are intended to provide a type system for a lexical variant of the catch throw mechanism used in functional languages such as Lisp. Moreover, H. Nakano has shown that it is possible to restrict the catch throw mechanism in order to stay in an ....

....which is confluent and strongly normalizing in the second order framework [20] ffl We deal with the first and second order frameworks. ffl We consider a type system a la Curry, which allows us to rephrase the above restriction on pure (i.e. untyped) terms, and not only on typed terms as in [17]. ffl We propose a computational interpretation of our restriction. ffl On the other hand, we do not consider tag abstraction in this paper. However, our restriction is symmetrical and thus easily extends to duality. Our purpose is to type tag abstraction with the subtraction (the connector dual ....

H. Nakano. The Logical Structures of the Catch and Throw Mechanism. PhD thesis, The University of Tokyo, 1995.

....of the members in the list. If 0 is found in the list, then the result must be 0 without computing any further, so it returns 0 by the throw expression. The catch throw mechanism is useful if one wants to escape from nested function calls at a time, especially in run time errors. Nakano [11 14] proposed calculi with inference rules which give logical interpretations of the catch throw constructs in Lisp. His calculi di er from the actual catch throwconstructs in Common Lisp in the following two ways. 1) He changed the scope rule of the catch construct from a dynamic one to a lexical ....

....four calculi for the catch throw mechanism: Author Intuitionistic Logic Classical Logic Nakano L c=t L K c=t Sato NJ c=t NK c=t In this paper, we investigate the strong normalizability (SN) of the above four calculi, in particular, L K c=t and NK c=t . The SN of L c=t was proved by Nakano [14], but his proof was based on complex model theoretic arguments. In our previous works, we proved the SN of NJ c=t in [8] and the SN of a large fragment of L K c=t in [9] but the SN of the full fragments of classical calculi L K c=t and NK c=t was an open problem. This paper solves this problem ....

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H. Nakano, \Logical Structures of the Catch and Throw Mechanism", Ph. D Dissertation, University of Tokyo, (1995).

....members in the list. If 0 is found in the list, then the result must be 0 without computing any further, so it returns 0 by the throw expression. The catch throw mechanism is useful if one wants to escape from nested function calls at a time, especially in run time errors. Nakano [10] 11] 12][13] proposed calculi with inference rules which give logical interpretations of the catch throw constructs in Lisp. His calculi differ from the actual catch throw constructs in Common Lisp in two ways. 1) He changed the scope rule of the catch construct from a dynamic one to a lexical one. In the ....

....four calculi for the catch throw mechanism: Author Intuitionistic Logic Classical Logic Nakano L c=t L K c=t Sato NJ c=t NK c=t In this paper, we investigate the strong normalizability (SN) of the above four calculi, in particular, L K c=t and NK c=t . The SN of L c=t was proved by Nakano [13], but his proof was based on complex model theoretic arguments. In our previous works, we proved the SN of NJ c=t in [6] and the SN of a large fragment of L K c=t in [7] but the SN of the full fragments of classical calculi L K c=t and NK c=t was an open problem. This paper solves this ....

[Article contains additional citation context not shown here]

Nakano, H.: "Logical Structures of the Catch and Throw Mechanism", Ph. D Dissertation, University of Tokyo, 1995. 23

....the strong normalizability any reduction sequence is finite . This result is contrast to the real programming languages such as Common Lisp and Standard ML where tags (exception names) have dynamic scope and there are non terminating programs. The strong normalizability of L c=t was proved in [11] 2 by a quite elaborate proof. We simplified the proof in our draft[4] and applied it to the second author s stronger calculus[13] but it still needs a tricky technique, and works only for the calculi with the restriction on the introduction rule. In this paper, we develop a quite natural ....

....B (A E) By applying the formation rule to a, we obtain a term x B :a of type B A. Since FV (x B :a) fg and FTV (x B :a) fu E g, intuitively we have (B A)E. Hence we have deduced 1 Here we do not give the precise meaning of essential occurrence in L c=t . Refer to [8] and [11] for details. 5 (B A) E from B (A E) But this is valid only in a classical calculus, and is not valid in an intuitionistic calculus. Nakano put a restriction on this rule to obtain an intuitionistic calculus L c=t . As an example of type inference, the following figure is a proof of the ....

Nakano, H.: "Logical Structures of the Catch and Throw Mechanism", Ph. D Dissertation, University of Tokyo, 1995.

....On the other hand, the term x: throw u pair(x; z) is not well typed, since the thrown term pair(x; z) depends on the variable x. The dependency on variables is defined in a slightly complex manner due to the existence of the case construct. For the precise definition of the condition, refer to [10] or [5] Nakano s system L c=t is constructed with this side condition, and has a nice correspondence to the intuitionistic propositional calculus. Also it enjoys several desirable properties such as 1 subject reduction and strong normalization[10] If we add a primitive recursion operator on the ....

....the precise definition of the condition, refer to [10] or [5] Nakano s system L c=t is constructed with this side condition, and has a nice correspondence to the intuitionistic propositional calculus. Also it enjoys several desirable properties such as 1 subject reduction and strong normalization[10]. If we add a primitive recursion operator on the list type to L c=t , we can type the following program in Common Lisp: defun multiply (xl) catch zero (multiply2 xl) defun multiply2 (xl) if (null xl) 1 (if (zerop (car xl) throw zero 0) car xl) multiply2 (cdr xl) This ....

Nakano, H. Logical Structures of the Catch and Throw Mechanism. Ph. D Dissertation, University of Tokyo, 1995.

....science. Any advance in the understanding of the intensional properties of proofs will enrich computer science and strengthen its links with logic. Connections between classical proofs and control features in functional computations have already been investigated by various people, see e.g. [14, 22, 23]. Outline of the programme In this extended abstract we sketch an approach to understand classical proofs semantically and from a computational perspective. It is our hope that this work will shed light on some of the conceptual questions identified above. We take as our first task the ....

....of Felleisen et al. 10] and relating them to classical proofs, there has been a great deal of interest in classical proofs. The following is a tentative (and incomplete) classification: ffl Algorithm extraction, control operators: Griffin [14] Murthy [22] Krivine [21] de Groote [9] Nakano [23], Hirokawa [16] Schwichtenberg and Berger [4] Coquand [6] etc. ffl Formal systems and calculi: Girard [11, 12] Parigot [24] Berardi and Barbanera [2] Danos, Joinet and Schellinx [8] etc. ffl Proofs and semantics of cut elimination: Girard [11] Hofmann [17] Coquand [5] Pfenning [25] ....

H. Nakano. Logical Structure of Catch and Throw Mechanism. PhD thesis, University of Tokyo, 1995.

....in [28] he stated that and (see Example 2.2) have behaviour close to Felleisen s C operator and Scheme callcc respectively but offered no explanation. de Groote was the first to study calculus from a computational viewpoint: in [9] he considered a cps translation. Various researchers e.g. [26, 36] have studied control operators in relation to classical proofs. More recently Bierman [5] has sketched an explanation of in terms of a kind of abstract machine with control environment . In this section we demonstrate the expressive richness of pcf v by using it to interpret Felleisen s C v ....

H. Nakano. Logical Structure of Catch and Throw Mechanism. PhD thesis, University of Tokyo, 1995.

....symmetric lambda calculus [1] are related to our NK c=t . The exact comparison of these systems also remains to be done in the future. Lastly, we should comment on the points where Nakano s L c=t is better than our NJ c=t . All the typable terms of Nakano s L c=t are strongly normalizeing (see [6]) while it is not known whether it is the case in our reduction system. Also, the condition under which implication introduction rule may be applied is more liberal in L c=t than in NJ c=t . After submitting the first version of this paper, we could reformulate NJ c=t so that it has the same ....

Nakano, H., Logical Structures of the Catch and Throw Mechanism, Ph.D. Dissertation, The University of Tokyo, 1995.

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