Quasi-Birth-Death Processes, Tree-Like QBDs, Probabilistic 1-Counter Automata, and Pushdown Systems

We begin by observing that (discrete-time) QuasiBirth-Death Processes (QBDs) are equivalent, in a precise sense, to (discrete-time) probabilistic 1-Counter Automata (p1CAs), and both Tree-Like QBDs (TLQBDs) 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 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 �...

• Type: Conference
• Year: 2008
• Where: QEST
• Authors: Kousha Etessami, Dominik Wojtczak, Mihalis Yannakakis
• Keywords: Probabilistic 1-counter Automata, Probabilistic Pushdown Systems, QEST 2008, Quantitative Evaluation Of Systems, Termination Probabilities