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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 ..."
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)...
Vanquishing the XCB question: The methodological discovery of the last shortest single axiom for the equivalential calculus
 J. Automated Reasoning
"... Abstract. With the inclusion of an e ective methodology, this article answers in detail a question that, for a quarter of a century, remained open despite intense study by various researchers. Is the formula XCB = e(x � e(e(e(x � y)�e(z�y))�z)) a single axiom for the classical equivalential calculus ..."
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Abstract. With the inclusion of an e ective methodology, this article answers in detail a question that, for a quarter of a century, remained open despite intense study by various 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 reexivity, symmetry, and transitivity, respectively. (Wenote 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 twenty ve 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.
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
On the Complexity of Hilbert’s 17th Problem
"... Hilbert posed the following problem as the 17th in the list of 23 problems in his famous 1900 lecture: Given a multivariate polynomial that takes only nonnegative values over the reals, can it be represented as a sum of squares of rational functions? In 1927, E. Artin gave an affirmative answer to ..."
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Hilbert posed the following problem as the 17th in the list of 23 problems in his famous 1900 lecture: Given a multivariate polynomial that takes only nonnegative values over the reals, can it be represented as a sum of squares of rational functions? In 1927, E. Artin gave an affirmative answer to this question. His result guaranteed the existence of such a finite representation and raised the following important question: What is the minimum number of rational functions needed to represent any nonnegative nvariate, degree d polynomial? In 1967, Pfister proved that any nvariate nonnegative polynomial over the reals can be written as sum of squares of at most 2 n rational functions. In spite of a considerable effort by mathematicians for over 75 years, it is not known whether n + 2 rational functions are sufficient! In lieu of the lack of progress towards the resolution of this question, we initiate the study of Hilbert’s 17th problem from the point of view of Computational Complexity. In this setting, the following question is a natural relaxation: What is the descriptive complexity of the sum of squares representation (as rational functions) of a nonnegative, nvariate, degree d polynomial? We consider arithmetic circuits as a natural representation of rational functions. We are able to show, assuming a standard conjecture in complexity theory, that it is impossible that every nonnegative, nvariate, degree four polynomial can be represented as a sum of squares of a small (polynomial in n) number of rational functions, each of which has a small size arithmetic circuit (over the rationals) computing it. Our result points to the direction that it is unlikely that every nonnegative, nvariate polynomial over the reals can be written as a sum of squares of a polynomial (in n) number of rational functions. Further, relating to standard (and believed to be hard to prove) complexitytheoretic conjectures sheds some light on why it has been difficult for mathematicians to close the n + 2 and 2 n gap. We hope that our line of work will play an important role in the resolution of this question. 1 1
A Spectrum of Applications of Automated Reasoning
, 2002
"... The likelihood of an automated reasoning program being of substantial assistance for a wide spectrum of applications rests with the nature of the options and parameters it o ers on which to base needed strategies and methodologies. This article focuses on such a spectrum, featuring W. McCune's ..."
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The likelihood of an automated reasoning program being of substantial assistance for a wide spectrum of applications rests with the nature of the options and parameters it o ers on which to base needed strategies and methodologies. This article focuses on such a spectrum, featuring W. McCune's program OTTER, discussing widely varied successes in answering open questions, and touching on some of the strategies and methodologies that played a key role. The applications include nding a rst proof, discovering single axioms, locating improved axiom systems, and simplifying existing proofs. The last application is directly pertinent to the recently found (by R. Thiele) Hilbert's twentyfourth problemwhich is extremely amenable to attack with the appropriate automated reasoning programa problem concerned with proof simpli cation. The methodologies include those for seeking shorter proofs and for nding proofs that avoid unwanted lemmas or classes of term, a speci c option for seeking proofs with smaller equational or formula complexity, and a di erent option to address the variable richness of a proof. The type of proof one obtains with the use of OTTER is Hilbertstyle
Proof Nets and the Identity of Proofs
, 2006
"... These are the notes for a 5lecturecourse given at ESSLLI 2006 in Malaga, Spain. The URL ..."
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These are the notes for a 5lecturecourse given at ESSLLI 2006 in Malaga, Spain. The URL
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.