Solving Polynomial Systems on Semirings: A Generalization of Newton’s Method

Systems of polynomials on semirings arise in several branches of computer science, like static analysis of procedural programs or formal language theory. We propose a new technique for calculating the least fixed points of such polynomial systems. This technique is a generalization of Newton’s method, the well-known method for approximating a zero of a nonlinear function on the reals. We show that our generalization of Newton’s method converges at least as fast as the standard fixed point iteration, and identify classes of semirings on which Newton’s method even reaches the least fixed point after a finite number of steps in contrast to the standard fixed point iteration. We further obtain from these convergence results interesting links to other topics of formal language theory, for instance, languages of finite index and the Parikh theorem. Motivated by our results on the convergence of Newton’s method, we then identify three more classes of semirings which allow for an even faster calculation of the least fixed point. Two of these semirings allow for reducing a nonlinear polynomial system to a linear system in such a way that the least fixed point is preserved. In the third case already a finite number of standard fixed point iterations suffice to calculate the least fixed point, although this class of semirings does not satisfy the ascending chain condition. We then turn to min-max-systems, a natural generalization of polynomial systems on semirings whose natural order is total. We identify a class of semirings which allow to solve nonlinear min-max-systems by the established approach of strategy iteration. In particular, we consider strategy iteration using nondeterministic strategies and show that these strategies allow for choosing an optimal successor.