Missing Proofs Found

For close to a century, despite the efforts of fine minds that include Hilbert and Ackermann, Tarski and Bernays, ukasiewicz, and Rose and Rosser, various proofs of a number of significant theorems have remained missing – at least not reported in the literature – amply demonstrating the depth of the corresponding problems. The types of such missing proofs are indeed diverse. For one example, a result may be guaranteed provable because of being valid, and yet no proof has been found. For a second example, a theorem may have been proved via metaargument, but the desired axiomatic proof based solely on the use of a given inference rule may have eluded the experts. For a third example, a theorem may have been announced by a master, but no proof was supplied. The finding of missing proofs of the cited types, as well as of other types, is the focus of this article. The means to finding such proofs rests with heavy use of McCune's automated reasoning program OTTER, reliance on a variety of powerful strategies this program offers, and employment of diverse methodologies. Here we present some of our successes and, because it may prove useful for circuit design and program synthesis as well as in the context of mathematics and logic, detail our approach to finding missing proofs. Well-defined and unmet challenges are included.     

Automatically Assessing MPEG Coding Fidelity

As the market and number of products for multimedia applications grow, the need for testing methods and fidelity metrics becomes critical. Testing devices are important for the development and evaluation of multimedia products. This article describes a testing method and prototype that can be used for assessing MPEG coding fidelity     

Vanquishing the XCB Question: The Methodological Discovery of the Last Shortest Single Axiom for the Equivalential Calculus

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 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 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. To show that XCB 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-five 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.     

Dietary docosahexaenoic acid has little effect on peroxisomes in healthy mice

NMRI mice were fed diets supplemented with 0.05, 0.2, or 2% (w/w) docosahexaenoic acid (DHA), a polyunsaturated fatty acid present in fish oil, for 3 d, 3 wk, or 3 mon. The doses of DHA were chosen to supply the mice with concentrations of DHA which approximate those that have been reported to be beneficial to patients with peroxisomal disease. Diets containing 0.05 or 0.2% DHA did not change hepatic, myocardial, and renal catalase (EC 1.11.1.6) activity except for a slight but significant increase (to 120%) in myocardial catalase activity in mice treated with the 0.05% DHA diet for 3 mon. A diet with 2% DHA induced myocardial catalase activity to 150% after both 3 d and 3 wk of administration. In the liver of mice fed this diet for 3 wk, hepatic catalase activity was increased to 140% while no induction of palmitoyl-CoA oxidase (EC 1.3.99.3), urate oxidase (EC 1.7.3.3), andl-α-hydroxyisovalerate oxidase (EC 1.1.3.a) was observed. With the light microscope, no changes in peroxisomal morphology were visually evaluated in catalase stained sections of liver, myocardium, and kidney of mice fed either diet. Our results show that in healthy mice a low dietary DHA dose (<0.2%; this corresponds to a dose prescribed to peroxisomal patients) has no effect on several hepatic peroxisomal H₂O₂-producing enzymes, including the rate-limiting enzyme of the peroxisomal fatty acid β-oxidation. This may indicate that such a DHA dose will not add a strong load on the often disturbed fatty acid metabolism in the liver of patients with peroxisomal disorders.     

Distributivity in Łℵ0 and Other Sentential Logics

Certain distributivity results for ukasiewicz's infinite-valued logic ₀ are proved axiomatically (for the first time) with the help of the automated reasoning program OTTER. In addition, nondistributivity results are established for a wide variety of positive substructural logics by the use of logical matrices discovered with the automated model-finding programs MACE and MAGIC.     

Analyzing pulmonary capillary pressure

A new mutual wavelet packets scheme is proposed for subband adaptive filtering. Its relation to the existing literature on the subject is outlined. The proposed scheme performs a decomposition of the two signals used in the adaptation process onto the same optimal basis. That basis is chosen so that it favors the adaptation performance. A new criterion, based on a functional of the two signals, is introduced. This proposed criterion consists in the maximization of the magnitude of the cross-correlation samples of the two signals considered. This maximizes the cross-correlation vector of the well known normal equations. At the expense of a slightly higher computational complexity, the mutual wavelet packets scheme reduces the aliasing drawback of regular subband adaptive filtering, preserving its main advantages. Simulations have been performed on pulmonary capillary pressure transients for respiratory interference cancelling. The method proved to be more efficient than classical methods