Derivation tree analysis for accelerated fixed-point computation

We show that for several classes of idempotent semirings the least fixed-point of a polynomial system of equations X = f ( X ) is equal to the least fixed-point of a linear system obtained by “linearizing” the polynomials of f in a certain way. Our proofs rely on derivation tree analysis, a proof principle that combines methods from algebra, calculus, and formal language theory, and was first used in Esparza et al. (2007) [10], to show that Newton’s method over commutative and idempotent semirings converges in a linear number of steps. Our results lead to efficient generic algorithms for computing the least fixed-point. We use these algorithms to derive several consequences, including an O ( N 3 ) algorithm for computing the throughput of a context-free grammar (obtained by speeding up the O ( N 4 ) algorithm of Caucal et al. (2007) [7]), and a generalization of Courcelle’s result stating that the downward-closed image of a context-free language is regular (Courcelle, 1991) [8].

Authors

• Javier Esparza
• Stefan Kiefer
• Michael Luttenberger

Keywords

• Fixed-point equations
• Semirings
• Derivation trees