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This papers studies the expressive and computational power of discreteOrdinary Differential Equations (ODEs). It presents a new framework usingdiscrete ODEs as a central tool for computation and provides several implicitcharacterizations of complexity and computability classes.  The proposed framework presents an original point of view on complexity andcomputability classes. It also unifies in an elegant settings variousconstructions that have been proposed for characterizing these classes. Thisincludes Cobham's and, Bellantoni and Cook's definition of polynomial time andlater extensions on the approach, as well as recent characterizations ofcomputability and complexity by classes of ordinary differential equations. Italso helps understanding the relationships between analog computations andclassical discrete models of computation theory.  At a more technical point of view, this paper points out the fundamental roleof linear (discrete) ordinary differential equations and classical ODE toolssuch as changes of variables to capture computability and complexity measures,or as a tool for programming various algorithms.

Recursion schemes, discrete differential equations and characterization of polynomial time computation