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This paper studies the expressive and computational power of discreteOrdinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations.It presents a new framework using these equations as a central tool forcomputation and algorithm design. We present the general theory of discreteODEs for computation theory, we illustrate this with various examples ofalgorithms, and we provide several implicit characterizations of complexity andcomputability classes.  The proposed framework presents an original point of view on complexity andcomputation classes. It unifies several constructions that have been proposedfor characterizing these classes including classical approaches in implicitcomplexity using restricted recursion schemes, as well as recentcharacterizations of computability and complexity by classes of continuousordinary differential equations. It also helps understanding the relationshipsbetween analog computations and classical discrete models of computationtheory.  At a more technical point of view, this paper points out the fundamental roleof linear (discrete) ODEs and classical ODE tools such as changes of variablesto capture computability and complexity measures, or as a tool for programmingmany algorithms.

A characterization of functions over the integers computable in polynomial time using discrete differential equations