Jerzy Mycka

The article analyses the role of Church’s Thesis (hereinafter CT) in the context of the development of hypercomputation research. The text begins by presenting various views on the essence of computer science and the limitations of its methods. Then CT and its importance in determining the limits of methods used by computer science is presented. Basing on the above explanations, the work goes on to characterize various proposals of hypercomputation showing their relative power in relation to the arithmetic hierarchy. The general theme of the article is the analysis of mutual relations between the content of CT and the theories of hypercomputation. In the main part of the paper the arguments for abolition of CT caused by the introduction of hypercomputable methods in computer science are presented and critique of these views is presented. The role of the efficiency condition contained in the formulation of CT is stressed. The discussion ends with a summary defending the current status of Church’s thesis within the framework of philosophy and computer science as an important point of reference for determining what the notion of effective calculability really is. The considerations included in this article seem to be quite up-to-date relative to the current state of affairs in computer science.1.

The article presents several examples of different mathematical structures and interprets their properties related to the existence of universal functions. In this context, relations between the problem of totality of elements and possible forms of universal functions are analyzed. Furthermore, some global and local aspects of the mentioned functional systems are distinguished and compared. In addition, the paper attempts to link universality and totality with the dynamic and static properties of mathematical objects and to consider the problem of limitations in the construction of structures combining harmoniously the availability of information at the local and global level.

The article starts with various definitions of music and its components (taken es-pecially from medieval sources). Then the Statuit introit is presented as the main example useful for mathematical analysis in the paper. We discuss the distinc-tion between a musical work and its performance and give a review of possible representations/ notations of music with their short mathematical characteristics. The next point is devoted to the concept of modality, Gregorian scales and selected theories linking musical experience with mathematics. We use the Statuit introit as the illustration for the construction of its statistical analysis (for notes, intervals and progresses). The results of this analysis serve as a reference point to indicate the connection between the process of musical composition and the apparatus of mathematical linguistics. Based on the above considerations we suggest that the pattern analysis could be useful as a tool unifying some aspects of music and mathematics and explaining the perception of musical works

The class of recursive functions over the reals, denoted by, was introduced by Cristopher Moore in his seminal paper written in 1995. Since then many subsequent investigations brought new results: the class was put in relation with the class of functions generated by the General Purpose Analogue Computer of Claude Shannon; classical digital computation was embedded in several ways into the new model of computation; restrictions of were proved to represent different classes of recursive functions, e.g., recursive, primitive recursive and elementary functions, and structures such as the Ritchie and the Grzergorczyk hierarchies.The class of real recursive functions was then stratified in a natural way, and and the analytic hierarchy were recently recognised as two faces of the same mathematical concept.In this new article, we bring a strong foundational support to the Real Recursive Function Theory, rooted in Mathematical Analysis, in a way that the reader can easily recognise both its intrinsic mathematical beauty and its extreme simplicity. The new paradigm is now robust and smooth enough to be taught. To achieve such a result some concepts had to change and some new results were added.

This paper continues the line initiated in [Z. Grodzki, J. Mycka The equivalence of some classes of algorithms] of uniform formalization of some classes of formal algorithms. The successive new classes ${\cal RMA}_k$ and $\overline{{\cal RMA}_k}$ of right-hand side Markov-like $k$-algorithms are introduced. The classes ${\cal RMA}_k$ and $\overline{{\cal RMA}_k}$ of algorithms are "symmetric" to the classes ${\cal MA}_k$ and $\overline{{\cal MA}_k}$ of left-hand side Markov-like $k$-algorithms which have been introduced in [Z. Grodzki, J. Mycka The equivalence of some classes of algorithms]. The equivalence of the classes ${\cal RMA}_k$ and $\overline{{\cal RMA}_k}$ and the class ${\cal MNA}$ of Markov normal algorithms is shown here. This implies the closure properties of the above classes under the same operations as of ${\cal MNA}$