Journal of the ACM (JACM)
On probabilistic parallel programs with process creation and synchronisation
TACAS'11/ETAPS'11 Proceedings of the 17th international conference on Tools and algorithms for the construction and analysis of systems: part of the joint European conferences on theory and practice of software
Runtime analysis of probabilistic programs with unbounded recursion
ICALP'11 Proceedings of the 38th international conference on Automata, languages and programming - Volume Part II
Polynomial time algorithms for multi-type branching processesand stochastic context-free grammars
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Model checking stochastic branching processes
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
Stochastic context-free grammars, regular languages, and newton's method
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part II
CAV'13 Proceedings of the 25th international conference on Computer Aided Verification
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We consider equation systems of the form $X_1=f_1(X_1,\dots,X_n)$, $\dots$, $X_n = f_n(X_1,\dots,X_n)$, where $f_1,\dots,f_n$ are polynomials with positive real coefficients. In vector form we denote such an equation system by ${\bf X}={\bf f}({\bf X})$ and call ${\bf f}$ a system of positive polynomials (SPP). Equation systems of this kind appear naturally in the analysis of stochastic models like stochastic context-free grammars (with numerous applications to natural language processing and computational biology), probabilistic programs with procedures, web-surfing models with back buttons, and branching processes. The least nonnegative solution $\mu{\bf f}$ of an SPP equation ${\bf X}={\bf f}({\bf X})$ is of central interest for these models. Etessami and Yannakakis [J. ACM, 56 (2009), pp. 1-66] have suggested a particular version of Newton's method to approximate $\mu{\bf f}$. We extend a result of Etessami and Yannakakis and show that Newton's method starting at ${\bf 0}$ always converges to $\mu{\bf f}$. We obtain lower bounds on the convergence speed of the method. For so-called strongly connected SPPs we prove the existence of a threshold $k_{{\bf f}}\in\mathbb{N}$ such that for every $i\geq0$ the $(k_{{\bf f}}+i)$th iteration of Newton's method has at least $i$ valid bits of $\mu{\bf f}$. The proof yields an explicit bound for $k_{{\bf f}}$ depending only on syntactic parameters of ${\bf f}$. We further show that for arbitrary SPP equations, Newton's method still converges linearly: there exists a threshold $k_{{\bf f}}$ and an $\alpha_{{\bf f}}0$ such that for every $i\geq0$ the $(k_{{\bf f}}+\alpha_{{\bf f}}\cdot i)$th iteration of Newton's method has at least $i$ valid bits of $\mu{\bf f}$. The proof yields an explicit bound for $\alpha_{{\bf f}}$; the bound is exponential in the number of equations in ${\bf X}={\bf f}({\bf X})$, but we also show that it is essentially optimal. The proof does not yield any bound for $k_{{\bf f}}$, but only proves its existence. Constructing a bound for $k_{{\bf f}}$ is still an open problem. Finally, we also provide a geometric interpretation of Newton's method for SPPs.