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We present abstraction techniques that transform a given non-linear dynamicalsystem into a linear system or an algebraic system described by polynomials ofbounded degree, such that, invariant properties of the resulting abstractioncan be used to infer invariants for the original system. The abstractiontechniques rely on a change-of-basis transformation that associates each statevariable of the abstract system with a function involving the state variablesof the original system. We present conditions under which a given change ofbasis transformation for a non-linear system can define an abstraction.Furthermore, the techniques developed here apply to continuous systems definedby Ordinary Differential Equations (ODEs), discrete systems defined bytransition systems and hybrid systems that combine continuous as well asdiscrete subsystems. The techniques presented here allow us to discover, givena non-linear system, if a change of bases transformation involvingdegree-bounded polynomials yielding an algebraic abstraction exists. If so, ourtechnique yields the resulting abstract system, as well. This approach isfurther extended to search for a change of bases transformation that abstractsa given non-linear system into a system of linear differential inclusions. Ourtechniques enable the use of analysis techniques for linear systems to inferinvariants for non-linear systems. We present preliminary evidence of thepractical feasibility of our ideas using a prototype implementation.

Change-Of-Bases Abstractions for Non-Linear Systems