On Defining Expressions for Entropy and Cross-Entropy: The Entropic Transreals and Their Fracterm Calculus.

Classic formulae for entropy and cross-entropy contain operations x0 and log2x that are not defined on all inputs. This can lead to calculations with problematic subexpressions such as 0log20 and uncertainties in large scale calculations; partiality also introduces complications in logical analysis. Instead of adding conventions or splitting formulae into cases, we create a new algebra of real numbers with two symbols ±∞ for signed infinite values and a symbol named ⟂ for the undefined.

Axiomatizing physical experiments as oracles to algorithms.

We developed earlier a theory of combining algorithms with physical systems, on the basis of using physical experiments as oracles to algorithms. Although our concepts and methods are general, each physical oracle requires its own analysis, on the basis of some fragment of physical theory that specifies the equipment and its behaviour. For specific examples of physical systems (mechanical, optical, electrical), the computational power has been characterized using non-uniform complexity classes.