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The commutative ambiguity of a context-free grammar G assigns to each Parikh vector v the number of distinct leftmost derivations yielding a word with Parikh vector v. Based on the results on the generalization of Newtons method to omega-continuous semirings, we show how to approximate the commutative ambiguity by means of rational formal power series, and give a lower bound on the convergence speed of these approximations. From the latter result we deduce that the commutative ambiguity itself is rational modulo the generalized idempotence identity k=k+1 (for k some positive integer), and, subsequently, that it can be represented as a weighted sum of linear sets. This extends Parikhs well-known result that the commutative image of context-free languages is semilinear (k=1). Based on the well-known relationship between context-free grammars and algebraic systems over semirings, our results extend the work by Green et al. on the computation of the provenance of Datalog queries over commutative omega-continuous semirings.
Parikhs theorem states that the Parikh image of a context-free language is semilinear or, equivalently, that every context-free language has the same Parikh image as some regular language. We present a very simple construction that, given a context-f
We provide a direct proof of Agafonovs theorem which states that finite state selection preserves normality. We also extends this result to the more general setting of shifts of finite type by defining selections which are compatible the shift. A sli
Motivated by applications to stochastic differential equations, an extension of H{o}rmanders hypoellipticity theorem is proved for second-order degenerate elliptic operators with non-smooth coefficients. The main results are established using point-w
A classical theorem of Herglotz states that a function $nmapsto r(n)$ from $mathbb Z$ into $mathbb C^{stimes s}$ is positive definite if and only there exists a $mathbb C^{stimes s}$-valued positive measure $dmu$ on $[0,2pi]$ such that $r(n)=int_0^{2
We analyse conformal gauge, or isotropic, singularities in cosmological models in general relativity. Using the calculus of tractors, we find conditions in terms of tractor curvature for a local extension of the conformal structure through a cosmolog