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A semantic construction of two-ary integers

100%
Ricci G.
Discussiones Mathematicae - General Algebra and Applications
|
2005
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tom 25
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nr 2
165-219
EN
To binary trees, two-ary integers are what usual integers are to natural numbers, seen as unary trees. We can represent two-ary integers as binary trees too, yet with leaves labelled by binary words and with a structural restriction. In a sense, they are simpler than the binary trees, they relativize. Hence, contrary to the extensions known from Arithmetic and Algebra, this integer extension does not make the starting objects more complex. We use a semantic construction to get this extension. This method differs from the algebraic ones, mainly because it is able to find equational features of the extended objects. Two-ary integers turn out to form the free algebra corresponding to the Jónsson-Tarski's "paradoxical" equations. This entails that they have a "sum" operation as well as other operations of higher dimensions. Two-ary integers can provide LISP memories with convenient direct access jumps and the above low complexity hints at feasible hardware implementations.
2
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Notes on a linguistic description as the basis for automatic image understanding

100%
Tadeusiewicz R., Ogiela M., Szczepaniak P.
International Journal of Applied Mathematics and Computer Science
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2009
|
tom 19
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nr 1
143-150
EN
The main paradigm of image understanding and a concept for its practical machine realisation are presented. The crucial elements of the presented approach are the formalisation of human knowledge about the class of images that are to be automatically interpreted, a linguistic description and the realization of cognitive resonance.
3
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Universality of Logic

72%
Woleński J.
Bulletin of the Section of Logic
|
2017
|
tom 46
|
nr 1/2
EN
This paper deals with the problem of universality property of logic. At first, this property is analyzed in the context of first-order logic. Three senses of the universality property are distinguished: universal applicability, topical neutrality and validity (truth in all models). All theses senses can be proved to be justified. The fourth understanding, namely the amount of expressive power, is connected with the criticism of the first-order thesis: first-order logic is the logic. The categorical approach to logic is presented as associated with the last understanding of universality. The author concludes that two senses of universality should be sharply discriminated and defends the first-order thesis.
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