Applications of quantifier elimination to the proofs of algebraic inequalities in engineering problems related to Todinov's method for risk reduction

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Ioakimidis, Nikolaos
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An interesting and efficient method based on algebraic inequalities for the reduction of risk and uncertainty as well as for the generation of new knowledge and the optimization of systems and processes has been recently proposed by Todinov. This method requires the proof of the related inequality and several classical approaches were successfully used by Todinov for this task. Here the use of the well-known method of quantifier elimination for the same task, that is for the proof of algebraic inequalities related to Todinov's method, as an additional approach appropriate for use with respect to some algebraic inequalities is proposed and actually used in some problems already proposed and successfully solved by Todinov on the basis of his own method of algebraic inequalities. The present approach is applied to some algebraic inequalities of engineering interest already proved by Todinov, more explicitly (i) to the inequality for the equivalent resistances of two resistors in an electrical circuit, (ii) to the inequality for the equivalent spring constants of two, three or four elastic springs, (iii) to inequalities concerning the supply of high-reliability components, (iv) to inequalities concerning ranking systems, (v) to the construction of a system with superior reliability and (vi) to the accumulated strain energy in bars under tension or in cantilevers under bending. Naturally, because of the well-known doubly-exponential computational complexity of quantifier elimination the present approach is applicable only when the related universally quantified formula contains a small total number of variables (free and quantified). Yet, the method of quantifier elimination can also be combined with the method of proof by induction and this additional possibility is also illustrated in two inequalities related to Todinov's method for their proof in the general case.
Risk reduction, Uncertainty reduction, Algebraic inequalities, Todinov's method, Equivalent resistances, Resistors, Equivalent spring constants, High-reliability components, Ranking systems, Superior reliability, Strain energy, Bars, Tension, Cantilevers, Bending, Proof by induction, Universal quantifier, Quantifier elimination, Quantifier-free formulae