Median constrained bucket order rank aggregation
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In [She82], it is shown that four basic functional properties are enough to characterize plain Kolmogorov complexity, hence obtaining an axiomatic characterization of this notion. In this paper, we try to extend this work, both by looking at alternative axiomatic systems for plain complexity and by considering potential axiomatic systems for other types of complexity. First we show that the axiomatic system given by Shen cannot be weakened (at least in any natural way). We then give an analogue of Shen's axiomatic system for conditional complexity. In a the second part of the paper, we look at prefix-free complexity and try to construct an axiomatic system for it. We show however that the natural analogues of Shen's axiomatic systems fails to characterize prefix-free complexity.
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In [6], it is shown that four of its basic functional properties are enough to characterize plain Kolmogorov complexity, hence obtaining an axiomatic characterization of this notion. In this paper, we try to extend this work, both by looking at alternative axiomatic systems for plain complexity and by considering potential axiomatic systems for other types of complexity. First we show that the axiomatic system given by Shen cannot be weakened (at least in any natural way). We then give an analogue of Shen's axiomatic system for conditional complexity. In the second part of the paper, we look at prefix-free complexity and try to construct an axiomatic system for it. We show however that the natural analogues of Shen's axiomatic systems fail to characterize prefix-free complexity.
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In [She82], it is shown that four basic functional properties are enough to characterize plain Kolmogorov complexity, hence obtaining an axiomatic characterization of this notion. In this paper, we try to extend this work, both by looking at alternative axiomatic systems for plain complexity and by considering potential axiomatic systems for other types of complexity. First we show that the axiomatic system given by Shen cannot be weakened (at least in any natural way). We then give an analogue of Shen's axiomatic system for conditional complexity. In a the second part of the paper, we look at prefix-free complexity and try to construct an axiomatic system for it. We show however that the natural analogues of Shen's axiomatic systems fails to characterize prefix-free complexity.
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The main goal of "Naive Axiomatic Mengenlehre" (NAM) is to find a more or less adequately explicit criterion that precisely formalizes the intuitive notion of a "normal set". NAM is mainly a construction procedure for building several formal systems NAMix, each of which can turn out to be an adequate codification of the contentual naive set theory. ("i" is a natural number which enumerates the used "normality" condition, and "x" is a letter which points to the variants of the used axioms.) Parallel to NAM, the Naive Axiomatic Class Theory NACT is constructed as a system of systems too.
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Recent work of Qi et al. arXiv:2004.11240v7 proposes a set of axioms for tensor rank functions. The current paper presents examples showing that their axioms allow rank functions to have some undesirable properties, and a stronger set of axioms is suggested that eliminates these properties. Two questions raised by Qi et al. involving the submax rank function are also answered.
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In Chapters 11 and 12, we have asserted the important fact that decision machines cannot be constructed for PL and PL(=). This has an obvious consequence for axiomatic systems such as those discussed in Chapter 10. Let an axiomatic system be given by citing axioms involving certain non-logical constants, and by taking PL, PL(=), or PL(=) with symbols like "+" as the associated logic. We are now interested in discussing such axiomatic systems in general. It can be assumed that the only axiomatic systems of interest to us are (simply) consistent. We may then investigate the (simple) completeness of such axiomatic systems, and the question of whether or not decision machines can be constructed for them. Clearly, some (simply) consistent axiomatic systems are (simply) incomplete. We had some examples in Chapter 10. It is also clear that decision machines are not available for at least some axiomatic systems. To show this, we can imagine that each axiomatic system is re-abstracted to forms of the logical system with which it is associated. Then the question of whether a particular sentence is a theo?rem of the axiomatic system becomes equivalent to a question of whether a particular logical form is derivable from a set of other logical forms.
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Axiomatic method is of great importance in formulating legal theories,the reason of which lies in that it can enhance the comprehensiveness of the theories and result in the theories better-knit and more convincing and developing quickly.Thus axiomatic method is often adopted by people while writing law works or compiling codes.But still some issues should be noted while we are doing our work:the fewer the axiomatic method is,the better;the axiomatic method chosen should be of prima facie rationale;truth is self-evident,etc.
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