to appear in QEST 2008We begin by observing that (discrete-time) Quasi-Birth-Death Processes
(QBDs) are equivalent, in a precise sense, to (discrete-time)
probabilistic 1-Counter Automata (p1CAs), and both Tree-Like QBDs
(TL-QBDs) and Tree-Structured QBDs (TS-QBDs) are equivalent to both
probabilistic Pushdown Systems (pPDSs) and Recursive Markov Chains
(RMCs).
We then proceed to exploit these connections to obtain a number of new
algorithmic upper and lower bounds for central computational problems
about these models. Our main result is this: for an arbitrary QBD
(even a null-recurrent one), we can approximate its termination
probabilities (i.e., its G matrix) to within i bits of precision
(i.e., within additive error 1/2i), in time polynomial in
\underline{both} the encoding size of the QBD and in i, in the
unit-cost rational arithmetic RAM model of computation. Specifically,
we show that a decomposed Newton's method can be used to achieve this.
We emphasize that this bound is very different from the well-known
``linear/quadratic convergence'' of numerical analysis, known for QBDs
and TL-QBDs, which typically gives no constructive bound in terms of
the encoding size of the system being solved. In fact, we observe
(based on recent results for pPDSs) that for the more general TL-QBDs
this bound fails badly. Specifically, in the worst case Newton's
method ``converges linearly'' to the termination probabilities for
TL-QBDs, but requires exponentially many iterations in the encoding
size of the TL-QBD to approximate these probabilities within any
non-trivial constant error c<1.
Our upper bound proof for QBDs combines several ingredients: a
detailed analysis of the structure of 1-counter automata, an iterative
application of a classic condition number bound for errors in linear
systems, and a very recent constructive bound on the performance of
Newton's method for monotone systems of polynomial equations