On the Computational Complexity of Information Hiding

Abstract

In this work we study the intrinsic complexity of elimination algorithms in effective algebraic geometry and we focus our attention to elimination algorithms produced within the object–oriented paradigm. To this end, we describe a new computation model called quiz game (introduced in [1]) which models the notions of information hiding (due to Parnas, see [8]) and non–functional requirements (e.g. robustness) among other important concepts in software engineering. This characteristic distinguish our model from classical computation models such as the Turing machine or algebraic models.We illustrate our computation model with a non–trivial complexity lower bound for the identity function of polynomials. We show that any object–oriented (and robust) implementation of the identity function of polynomials is necessarily inefficient compared with a trivial implementation of this function. This result shows an existing synergy between Software Engineering and Algebraic Complexity Theory. Keywords: Abstract data type, abstraction function, data structure, information hiding, lower complexity bound, non–functional requirement, quantifier elimination, quiz game, robustness, scientific computing.