Convergence of Newton's Method over Commutative Semirings

We give a lower bound on the speed at which Newton's method (as defined in [11]) converges over arbitrary omega-continuous commutative semirings. From this result, we deduce that Newton's method converges within a finite number of iterations over any semiring which is "collapsed at some k is an element of N" (i.e. k = k + 1 holds) in the sense of Bloom and Esik [2]. We apply these results to (1) obtain a generalization of Parikh's theorem, (2) compute the provenance of Datalog queries, and (3) analyze weighted pushdown systems. We further show how to compute Newton's method over any omega-continuous semiring by constructing a grammar unfolding w.r.t. "tree dimension". We review several concepts equivalent to tree dimension and prove a new relation to pathwidth. (C) 2015 Elsevier Inc. All rights reserved.