Quantitative Analysis of Probabilistic Pushdown Automata: Expectations and Variances

Abstract

Probabilistic pushdown automata (pPDA) have been identified as a natural model for probabilistic programs with recursive procedure calls. Previous works considered the decidability and complexity of the model-checking problem for pPDA and various probabilistic temporal logics. In this paper we concentrate on computing the expected values and variances of various random variables defined over runs of a given probabilistic pushdown automaton. In particular, we show how to compute the expected accumulated reward and the expected gain for certain classes of reward functions. Using these results, we show how to analyze various quantitative properties of pPDA that are not expressible in conventional probabilistic temporal logics.

Analyzing probabilistic pushdown automata

“…Theorem 8 (see [23]) Let ∆ = (Q,Γ , → , Prob) be a pPDA, pα ∈ Q × Γ * a configuration of ∆ , C a regular set of configurations of ∆ represented by a DFA A such that P(Reach(pα, C )) > 0, and f a simple reward function. Further, let ρ > 0 and ε > 0 be rational constants.…”

“…Thus, we can also express the "expected contribution" of Z to the total reward accumulated along such a path. These considerations lead to a system of equations similar to Expect ∆ (we refer to [23] for details). Hence, Theorem 8 holds also for linear reward functions without any change.…”

“…In particular, one can approximate the conditional variance of Acc up to an arbitrarily small absolute error ε > 0 in polynomial space [23].…”

“…In particular, one such class has been extensively studied since the mid 00s: Markov chains generated by programs with (possibly recursive) procedures. This class is captured by two equivalent formalisms: probabilistic Pushdown Automata (pPDA) [22,23,14,16,8], and Recursive Markov Chains [29,27,28,30]. Intuitively, the equivalence of the two models derives from the well-known fact that a recursive program can be compiled into a "plain" program that manipulates a stack, i.e., into a pushdown automaton.…”

PReMo: An Analyzer for Probabilistic Recursive Models

“…They are formally equivalent to probabilistic Pushdown Systems (pPDSs) ( [2,3]), and they define a class of infinite-state Markov chains that generalize a number of well studied stochastic models such as Stochastic Context-Free Grammars (SCFGs) and Multi-Type Branching Processes. In a series of recent papers ( [4,5,6,7]), the second author and M. Yannakakis have developed algorithms for analysis and model checking of RMCs and their controlled and game extensions: 1-exit Recursive Markov Decision Processes (1-RMDPs) and 1-exit Recursive Simple Stochastic Games (1-RSSGs).…”