J. I. Zucker. Cut-elimination and normalization. Annals of Mathematical Logic, 1(1):1--112, 1974.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....theorem proving and proof checking for these inference systems by providing simple operational interpretations for the connectives of hh 2 . The correspondence between cut free sequential proofs and normal natural deduction proofs for first order intuitionistic logic has been formalized in [16, 13]. This paper appears in Proceedings of the 1989 International Workshop on Extensions of Logic Programming, Peter Schroeder Heister, editor, Springer Verlag Lecture Notes in AI, 1991. There, the formal relation between cut elimination and proof normalization is also explored in detail. In ....

J. I. Zucker. Cut-elimination and normalization. Annals of Mathematical Logic, 1(1):1--112, 1974.

....can also be seen as a refinement of Murthy s idea from the point of view of cut elimination as computation . Cut elimination and normalization: The idea of assigning typed terms to a Gentzen style logical system is not new. There are early proof theoretical works on the LJ by Prawitz[22] Zucker[23], Pottinger[21] and Mints[11] Our term assignment can also be seen as an extension to classical logic of their investigation. Specifically, our results introduce two different term assignments and reductions (i.e. CBN and CBN) even if we restrict the LK to the LJ. Moreover, our system has ....

....Rules for the LKT which appears for the first time in whole derivation. We assume all initial indices are distinct unless they are truly related (i.e. subject of further implicit contraction) This is possible, by introducing the concatenation to indices on every binary rules. See Zucker[23]. We introduce a new notion for t terms. In a word, it is a bi directional form of application. They correspond to the two orientation of cut in Gentzen style classical logic. Accordingly we have two kind of fi contractions. The raw t terms, ranged over s; t; u; etc. are defined as ....

J. I. Zucker. Correspondence between cutelimination and normalization, part i and ii. Annals of Mathematical Logic, 7:1--156,

....steps as reduction steps, as we do. We show that calculus is completely included as an intuitionistic case of our CBN CPS calculus. The idea of assigning typed terms to Gentzen style logical system, itself, is not new. As there are an early proof theoretical works on LJ by Prawitz[27] Zucker[28] and Pottinger[26] Pottinger s main concern is to prove normalization as a homomorphic image of cut elimination i.e. normalization is simulated by cut elimination. However, our term assignment is by far precise, besides we consider classical case. As first step, we simulate SN and CR ....

....proofs are equal if they differ only by the index of formula in the proved sequent. Initial index is an index which appears for the first time in whole proof. We assume all initial index are distinct when it appears. This is possible, by introducing the concatenation to indexes, see for Zucker[28]. In our logical system, structural rules are implicit. As we interpret contexts as sets, occurrences of a formula with the same index are automatically contracted. One can interpret this that binary rules are always followed by appropriate explicit contractions which is renaming of index to the ....

J. I. Zucker. Correspondence between cutelimination and normalization, part i and ii. Annals of Mathematical Logic, 7:1--156, 1974.

....also be seen as a refinement of Murthy s idea from the point of view of cut elimination as computation . Cut elimination and normalization: The idea of assigning typed terms to a Gentzen style logical system is not new. There are early proof theoretical works on the LJ by Prawitz[18] Zucker[19], Pottinger[17] and Mints[10] Our term assignment can also be seen as an extension to classical logic of their investigation. Specifically, our results introduce two different term assignments and reductions (i.e. CBN and CBV) even if we restrict the LK to the LJ. Moreover, our system has ....

....Initial index is an index which appears for the first time in whole derivation. We assume all initial indices are distinct unless they are truly related (i.e. subject of further implicit contraction) This is possible, by introducing the concatenation to indices on every binary rules. See Zucker[19]. 2.2 Raw t terms We introduce a new notion for t terms. In a word, it has a bi directional form of application. They correspond to the two orientation of cut in Gentzen style classical logic. We explain this in the next subsection. Accordingly we have two kind of fi contractions. ....

J. I. Zucker. Correspondence between cutelimination and normalization, part i and ii. Annals of Mathematical Logic, 7:1--156,

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