Solving Fixed-Point Equations by Derivation Tree Analysis

Abstract

Abstract. Systems of equations over ω-continuous semirings can be mapped to context-free grammars in a natural way. We show how an analysis of the derivation trees of the grammar yields new algorithms for approximating and even computing exactly the least solution of the system.

A Brief History of Strahler Numbers

“…It is perhaps less known that µf can be given a "language-theoretic" interpretation. We explain this by means of an example (see [14] for more details). Consider the equation…”

FPSOLVE: A Generic Solver for Fixpoint Equations Over Semirings

“…The two main generic schemes implemented in FPsolve for the approximation (and sometimes exact computation) of the least solution of an algebraic system are classical fixpoint iteration and Newton's method. Following [15,10], we introduce them as procedures that "unfold" the algebraic system up to a certain depth which allows both to unify and at the same time simplify their presentation.…”

“…Newton's method for arbitrary ω-continuous semirings, as described in [9], can be much faster than Kleene iteration. It is shown in [10] that the method can also be presented as an unfolding of the algebraic system: This time, the system is unfolded w.r.t. the Strahler number or dimension of its associated derivation trees (see [10,15]).…”

“…It is shown in [10] that the method can also be presented as an unfolding of the algebraic system: This time, the system is unfolded w.r.t. the Strahler number or dimension of its associated derivation trees (see [10,15]). The dimension of a rooted tree t is defined as the height of the largest perfect binary tree that is a minor of t.…”

“…Straight-forward induction now shows that over the nonnegative reals the values of X d and X <d resp. Y d and X =d coincide [10]. In particular, the defining equation of X =d can be seen as the generalization of the derivative of aX 2 in the noncommutative case.…”