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Signifiable computability aims to separate what is theoretically computable from what is computable through performable processes on computers with finite amounts of memory. Real numbers and sequences thereof, data types, and instances are treated as finite texts, and memory limitations are made explicit through a requirement that the texts be stored in the available memory on the devices that manipulate them. In Part I of our investigation, we define the concepts of signification and reference of real numbers. We extend signification to number tuples, data types, and data instances and show that data structures representable as tuples of discretely finite numbers are signifiable. From the signification of real tuples, we proceed to the constructive signification of multidimensional matrices and show that any data structure representable as a multidimensional matrix of discretely finite numbers is signifiable. [ABSTRACT FROM AUTHOR]

We show that, for a primitive recursive function h (x , t) , where x is a n-tuple of natural numbers and t is a natural number, there exists a feedforward artificial neural network N (x , t) , such that for any n-tuple of natural numbers z and a positive natural number m, the first m + 1 terms of the sequence { h (z , t) } are the same as the terms of the tuple (N (z , 0) , ... , N (z , m)) . [ABSTRACT FROM AUTHOR]

Several characteristics of chess are investigated with methods of computability and number theories. It is shown that for an unfinished game it is primitive recursively decidable whether the game is winnable, drawable, or absolutely losable within a specified number of future moves for the player whose turn it is to play on the last board of the game. It is also shown that there exist primitive recursive procedures to compute optimal continuations of unfinished games within specified numbers of future moves and that the set of chess games is recursive. [ABSTRACT FROM AUTHOR]