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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
Regular Expressions for Provenance
"... As noted by Green et al. several provenance analyses can be considered a special case of the general problem of computing formal polynomials resp. powerseries as solutions of an algebraic system. Specific provenance is then obtained by means of evaluating the formal polynomial under a suitable hom ..."
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As noted by Green et al. several provenance analyses can be considered a special case of the general problem of computing formal polynomials resp. powerseries as solutions of an algebraic system. Specific provenance is then obtained by means of evaluating the formal polynomial under a suitable homomorphism. Recently, we presented the idea of approximating the least solution of such algebraic systems by means of unfolding the system into a sequence of simpler algebraic systems. Similar ideas are at the heart of the seminaive evaluation algorithm for datalog. We apply these results to provenance problems: Seminaive evaluation can be seen as a particular implementation of fixed point iteration which can only be used to compute (finite) provenance polynomials. Other unfolding schemes, e.g. based on Newton’s method, allow us to compute a regular expression which yields a finite representation of (possibly infinite) provenance power series in the case of commutative and idempotent semirings. For specific semirings (e.g. Why(X)) we can then, in a second step, transform these regular expressions resp. power series into polynomials which capture the provenance. Using techniques like subterm sharing both the regular expressions and the polynomials can be succinctly represented.