Abstract
In this paper, we analyze axiomatic and constructive issues of unconventional computations from a methodological and philosophical point of view. We explain how the new models of algorithms and unconventional computations change the algorithmic universe, making it open and allowing increased flexibility and expressive power that augment creativity. At the same time, the greater power of new types of algorithms also results in the greater complexity of the algorithmic universe, transforming it into the algorithmic multiverse and demanding new tools for its study. That is why we analyze new powerful tools brought forth by local mathematics, local logics, logical varieties and the axiomatic theory of algorithms, automata and computation. We demonstrate how these new tools allow efficient navigation in the algorithmic multiverse. Further work includes study of natural computation by unconventional algorithms and constructive approaches.
Highlights
The development of computer science and information technology brought forth a diversity of novel algorithms and algorithmic schemas, unconventional computations and nature-inspiredEntropy 2012, 14 processes, advanced functionality and conceptualizations
Local mathematics brings forth local logics because each local mathematical framework has its own logic and it is possible that different frameworks have different local logics
He wrote in [43]: “A main feature of Hilbert’s axiomatization of geometry is that the axiomatic method is presented and practiced in the spirit of the abstract conception of mathematics that arose at the end of the nineteenth century and which has generally been adopted in modern mathematics
Summary
The development of computer science and information technology brought forth a diversity of novel algorithms and algorithmic schemas, unconventional computations and nature-inspired. Assuming some simple basic conditions (in the form of postulates, axioms and conditions), many profound and far-reaching properties of algorithms are derived in this theory This allows one, when dealing with a specific model, not to prove this property, but only to check the conditions from the assumption, which is much easier than to prove the property under consideration. The projective approach in computer science has its counterpart in mathematics, where systems of unifying properties have been used for building new encompassing structures, proving indispensable properties in these new structures and projecting these properties on the encompassed domains Such projectivity has been explicitly utilized in category theory, which was developed and utilized with the goal of unification [11]. Each class of algorithmic model forms a local algorithmic universe, providing means for the development of local computer science in general and a local theory of algorithms in particular. Local mathematics brings forth local logics because each local mathematical framework has its own logic and it is possible that different frameworks have different local logics
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