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Solving FixedPoint Equations by Derivation Tree Analysis ⋆
"... Abstract. Systems of equations over ωcontinuous semirings can be mapped to contextfree 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. 1 ..."
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Abstract. Systems of equations over ωcontinuous semirings can be mapped to contextfree 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. 1
Derivation Tree Analysis for Accelerated FixedPoint Computation
"... Abstract. We show that for several classes of idempotent semirings the least fixedpoint of a polynomial system of equations X = f(X) is equal to the least fixedpoint of a linear system obtained by “linearizing ” the polynomials of f in a certain way. Our proofs rely on derivation tree analysis, a ..."
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Abstract. We show that for several classes of idempotent semirings the least fixedpoint of a polynomial system of equations X = f(X) is equal to the least fixedpoint 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 [5] 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 fixedpoint. We use these algorithms to derive several consequences, including an O(N 3) algorithm for computing the throughput of a contextfree grammar (obtained by speeding up the O(N 4) algorithm of [2]), and a generalization of Courcelle’s result stating that the downwardclosed image of a contextfree language is regular [3]. 1