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    Applications of quantifier elimination to the proofs of algebraic inequalities in engineering problems related to Todinov's method for risk reduction
    (Κανένας, ) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    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.
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    Problems under uncertainty : quantifier elimination to universally–existentially (AE) quantified formulae related to two or more horizons of uncertainty
    (Κανένας, ) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    Problems under uncertainty appear frequently in practical applications. Ben-Haim's IGDT (info-gap decision theory) constitutes a very efficient method for the study of such problems. The three components (or elements) of Ben-Haim's IGDT are (i) the system model, (ii) the info-gap uncertainty model and (iii) the performance requirement(s). Appropriate (mainly positivity) assumptions can also be made. Here we use the IGDT only partially by restricting our attention to its first component, the system model, and to its second component, the info-gap uncertainty model, but paying no attention to the performance requirement(s) also very important in the IGDT. Here an emphasis is put on the use of (mixed) universally–existentially (AE) quantified formulae assuring the validity of the system model (under the assumptions made) for all values of the universally quantified uncertain variable(s) and for at least one value (or a set of values) of the existentially quantified uncertain variable(s) of course provided that these quantified variables satisfy the adopted info-gap uncertainty model here the popular fractional-error model. On the other hand, here we also assume that each uncertain variable (either universally or existentially quantified) has its own uncertainty parameter (or horizon of uncertainty). Next, by using the method of quantifier elimination in its powerful implementation in the computer algebra system Mathematica we transform the quantified formula to an equivalent QFF (quantifier-free formula) free from the quantifiers and the quantified variables, but, evidently, including the horizons of uncertainty. Two simple applications concerning (i) a product/quotient and (ii) the buckling load of a fixed–free column illustrate the present approach with the derivation of the related QFFs, some of which can also be verified manually.
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    Quantifier elimination and quantifier-free formulae for universally–existentially (AE) quantified formulae in Ben-Haim's info-gap model of uncertainty
    (Κανένας, ) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    The method of quantifier elimination with implementations in some computer algebra systems already proved useful for the computation of both the robustness and the opportuneness (or opportunity) functions in Ben-Haim's info-gap (or information-gap) model of uncertainty. As is well known, this model constitutes an interesting and practical tool in decision theory. Moreover, quantifier elimination concerning the robustness/opportuneness functions can be performed to the related universally/existentially quantified formulae. Here we proceed to the consideration of the additional mixed (AE) case, where both the universal and the existential quantifiers are present in the quantified formula related to Ben-Haim's info-gap model of uncertainty. In this mixed (AE) case, evidently now with more than one uncertain variable, the universal quantifier concerns one (or more than one) uncertain variable and similar is the case with the existential quantifier. After performing quantifier elimination to this quantified formula (here by using the computer algebra system Mathematica), we derive the related QFF (quantifier-free formula) that concerns the horizon of uncertainty. The case of more than one horizon of uncertainty can also be similarly studied. In this way, an expression for the horizon of uncertainty in a logical form with the appropriate inequalities is derived. From this form it is observed that additional immunity functions (beyond the classical robustness and opportuneness functions) appear in the mixed universal–existential (AE) case. The present approach is applied to four uncertainty problems which are based on info-gap models and concern (i) the area of a rectangle, (ii) the buckling load of a fixed–free column, (iii) the volume of a rectangular cuboid and (iv) the reactions at the ends of a fixed beam loaded by a concentrated load.
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    Robust reliability under uncertainty conditions by using modified info-gap models with two to four horizons of uncertainty and quantifier elimination
    (Κανένας, ) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    Quantifier elimination for real variables constitutes an interesting computational tool with efficient implementations in some popular computer algebra systems and many applications in several disciplines. On the other hand, many practical problems concern situations under uncertainty, where uncertainty intervals and, more generally, reliability regions of uncertain quantities have to be computed. Here the interest is in the popular Ben-Haim's IGDT (info-gap or information-gap decision theory) for problems under severe uncertainty based on info-gap models, where quantifier elimination already proved to constitute a possible tool for the computation of the related reliability regions and robustness functions. Here Ben-Haim's IGDT is considered again, but now in a modified form, where more than one horizon of uncertainty is present (here two, three or four). More explicitly, here each uncertain quantity is assumed to have its own horizon of uncertainty contrary to the usual case in the IGDT, where only one horizon of uncertainty is present in the related info-gap model. Six applications are presented showing the usefulness of the present computational approach. These applications (mainly based on fractional-error info-gap models) concern (i) a linear system, (ii) a sum, (iii) the area of a rectangle, (iv) the volume of a rectangular cuboid, (v) the buckling load of a fixed-free column and (vi) the von Mises yield criterion in two-dimensional elasticity. Beyond the uncertain quantities (here two, three or four) one, two or three parameters may also be present and appear in the derived QFFs (quantifier-free formulae). Of course, it is noted that quantifier elimination generally has a doubly-exponential computational complexity and this restricts its applicability to problems with a small total number of variables (quantified and free).
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    Application of quantifier elimination to robust reliability under severe uncertainty conditions by using the info-gap decision theory (IGDT)
    (Αυτο-έκδοση, ) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    Ben-Haim's info-gap (or information-gap) decision theory (IGDT) constitutes a very interesting and popular method for the study of problems in engineering and in many other scientific disciplines under severe uncertainty conditions. On the other hand, quantifier elimination constitutes an equally interesting approach implemented in some computer algebra systems and aiming at the transformation of quantified formulae (i.e. formulae including the universal and/or the existential quantifiers) to logically equivalent formulae but free from these quantifiers and the related quantified variables. Here we apply the method of quantifier elimination (by using its implementation in Mathematica) to the info-gap decision theory and we compute the related reliability regions and, next, the related robustness functions. The computation of the opportuneness (or opportunity) functions is also considered in brief. More explicitly, the four problems studied here concern: (i) the Hertzian contact of two isotropic elastic spheres, (ii) a spring with a linear stiffness but also with an uncertain cubic non-linearity in its stiffness, (iii) the robust reliability of a project with uncertain activity (task) durations and (iv) a gap-closing electrostatic actuator. In all these problems here under uncertainty conditions, the present results are seen to be in complete agreement with the results already derived for the same problems by Ben-Haim and his collaborators (who used appropriate more elementary methods) with respect to the robustness and/or opportuneness functions, but here the reliability regions are also directly computed. Moreover, the present approach permits the study of some difficult parametric cases (e.g. in the problem of the gap-closing electrostatic actuator with a non-linearity in its stiffness), where the help of a computer algebra system seems to be necessary.