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From proof nets to the free * autonomous category
 Logical Methods in Computer Science, 2(4:3):1–44, 2006. Available from: http://arxiv.org/abs/cs/0605054. [McK05] Richard McKinley. Classical categories and deep inference. In Structures and Deduction 2005 (Satellite Workshop of ICALP’05
, 2005
"... Vol. 2 (4:3) 2006, pp. 1–44 www.lmcsonline.org ..."
Vanquishing the XCB Question: The Methodological Discovery of the Last Shortest Single Axiom for the Equivalential Calculus
 J. Automated Reasoning
, 2002
"... With the inclusion of an effective methodology, this article answers in detail a question that, for a quarter of a century, remained open despite intense study byvarious researchers. Is the formula XCB = e(x# e(e(e(x# y)#e(z#y))#z)) a single axiom for the classical equivalential calculus when the ru ..."
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With the inclusion of an effective methodology, this article answers in detail a question that, for a quarter of a century, remained open despite intense study byvarious researchers. Is the formula XCB = e(x# e(e(e(x# y)#e(z#y))#z)) a single axiom for the classical equivalential calculus when the rules of inference consist of detachment(modus ponens) and substitution? Where the function e represents equivalence, this calculus can be axiomatized quite naturally with the formulas e(x# x), e(e(x# y)#e(y#x)), and e(e(x# y)#e(e(y#z)#e(x# z))), which correspond to reflexivity, symmetry, and transitivity, respectively.(We note that e(x# x) is dependent on the other two axioms.) Heretofore, thirteen shortest single axioms for classical equivalence of length eleven had been discovered, and XCB was the only remaining formula of that length whose status was undetermined. ToshowthatXCB is indeed such a single axiom, we focus on the rule of condensed detachment, a rule that captures detachment together with an appropriately general, but restricted, form of substitution. The proof we present in this paper consists of twentyfi e applications of condensed detachment, completing with the deduction of transitivity followed by a deduction of symmetry.We also discuss some factors that may explain in part why XCB resisted relinquishing its treasure for so long. Our approach relied on diverse strategies applied by the automated reasoning program OTTER. Thus ends the search for shortest single axioms for the equivalential calculus.
The Flowering of Automated Reasoning
, 2001
"... This article celebrates with obvious joy the role automated reasoning now plays for mathematics and logic. Simultaneously, this article evidences the realization of a dream thought impossible just four decades ago by almost all. But there were believers, including Joerg Siekmann to whom this article ..."
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This article celebrates with obvious joy the role automated reasoning now plays for mathematics and logic. Simultaneously, this article evidences the realization of a dream thought impossible just four decades ago by almost all. But there were believers, including Joerg Siekmann to whom this article is dedicated in honor of his sixtieth birthday. Indeed, today (in the year 2001)...
Canonical proof nets for classical logic
"... Abstract. Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proofnet, they are in essence the same proof. Providing a convincing proofnet counterpart to proofs in the classical sequent calculus is thus an ..."
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Abstract. Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proofnet, they are in essence the same proof. Providing a convincing proofnet counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs. By convincing, we mean that (a) there should be a canonical function from sequent proofs to proof nets, (b) it should be possible to check the correctness of a net in polynomial time, (c) every correct net should be obtainable from a sequent calculus proof, and (d) there should be a cutelimination procedure which preserves correctness. Previous attempts to give proofnetlike objects for propositional classical logic have failed at least one of the above conditions. In [23], the author presented a calculus of proof nets (expansion nets) satisfying (a) and (b); the paper defined a sequent calculus corresponding to expansion nets but gave no explicit demonstration of (c). That sequent calculus, called LK ∗ in this paper, is a novel onesided sequent calculus with both additively and multiplicatively formulated disjunction rules. In this paper (a selfcontained extended version of [23]) , we give a full proof of (c) for expansion nets with respect to LK ∗, and in addition give a cutelimination procedure internal to expansion nets – this makes expansion nets the first notion of proofnet for classical logic satisfying all four criteria. 1
ISSN 02496399 ISRN INRIA/RR????FR+ENGProof Nets and the Identity of Proofs
, 2006
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The Arrival of Automated Reasoning
"... For some, the object of automated reasoning is the design and implementation of a program that offers sufficient power to enable one to contribute new and significant results to mathematics and to logic, as well as elsewhere. One measure of success rests with the number and quality of the results ob ..."
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For some, the object of automated reasoning is the design and implementation of a program that offers sufficient power to enable one to contribute new and significant results to mathematics and to logic, as well as elsewhere. One measure of success rests with the number and quality of the results obtained with the assistance of the program in focus. A less obvious measure (heavily in focus here) rests with the ability of a novice, in the domain under investigation, to make significant contributions to one or more fields of science by relying heavily on a given reasoning program. For example, if one who is totally unfamiliar with the area of study but skilled in automated reasoning can discover with an automated reasoning program impressive proofs, previously unknown axiom dependencies, and far more, then the field of automated reasoning has indeed arrived. This article details such—how one novice, with much experience with W. McCune’s program OTTER but no knowledge of the domains under investigation, obtained startling results in the study of areas of logic that include the BCSK logic and various extensions of that logic.