Results 1  10
of
18
Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations
 IN STACS
, 2005
"... We define Recursive Markov Chains (RMCs), a class of finitely presented denumerable Markov chains, and we study algorithms for their analysis. Informally, an RMC consists of a collection of finitestate Markov chains with the ability to invoke each other in a potentially recursive manner. RMCs offer ..."
Abstract

Cited by 95 (13 self)
 Add to MetaCart
We define Recursive Markov Chains (RMCs), a class of finitely presented denumerable Markov chains, and we study algorithms for their analysis. Informally, an RMC consists of a collection of finitestate Markov chains with the ability to invoke each other in a potentially recursive manner. RMCs offer a natural abstract model for probabilistic programs with procedures. They generalize, in a precise sense, a number of well studied stochastic models, including Stochastic ContextFree Grammars (SCFG) and MultiType Branching Processes (MTBP). We focus on algorithms for reachability and termination analysis for RMCs: what is the probability that an RMC started from a given state reaches another target state, or that it terminates? These probabilities are in general irrational, and they arise as (least) fixed point solutions to certain (monotone) systems of nonlinear equations associated with RMCs. We address both the qualitative problem of determining whether the probabilities are 0, 1 or inbetween, and
Quasibirthdeath processes, TreeLike QBDs, probabilistic 1counter automata, and pushdown systems
, 2008
"... We begin by observing that (discretetime) QuasiBirthDeath Processes (QBDs) are equivalent, in a precise sense, to (discretetime) probabilistic 1Counter Automata (p1CAs), and both TreeLike QBDs (TLQBDs) and TreeStructured QBDs (TSQBDs) are equivalent to both probabilistic Pushdown Systems ..."
Abstract

Cited by 23 (8 self)
 Add to MetaCart
(Show Context)
We begin by observing that (discretetime) QuasiBirthDeath Processes (QBDs) are equivalent, in a precise sense, to (discretetime) probabilistic 1Counter Automata (p1CAs), and both TreeLike QBDs (TLQBDs) and TreeStructured QBDs (TSQBDs) are equivalent to both probabilistic Pushdown Systems
Newtonian Program Analysis
, 2010
"... This article presents a novel generic technique for solving dataflow equations in interprocedural dataflow analysis. The technique is obtained by generalizing Newton’s method for computing a zero of a differentiable function to ωcontinuous semirings. Complete semilattices, the common program analy ..."
Abstract

Cited by 15 (5 self)
 Add to MetaCart
This article presents a novel generic technique for solving dataflow equations in interprocedural dataflow analysis. The technique is obtained by generalizing Newton’s method for computing a zero of a differentiable function to ωcontinuous semirings. Complete semilattices, the common program analysis framework, are a special class of ωcontinuous semirings. We show that our generalized method always converges to the solution, and requires at most as many iterations as current methods based on Kleene’s fixedpoint theorem. We also show that, contrary to Kleene’s method, Newton’s method always terminates for arbitrary idempotent and commutative semirings. More precisely, in the latter setting the number of iterations required to solve a system of n equations is at most n.
Verification of Probabilistic Recursive Sequential Programs
, 2007
"... This work studies algorithmic verification of infinitestate probabilistic systems generated by probabilistic pushdown automata (pPDA). Probabilistic pushdown automata are obtained as a probabilistic variant of pushdown automata that proved to be a successful abstract model of recursive sequential p ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
This work studies algorithmic verification of infinitestate probabilistic systems generated by probabilistic pushdown automata (pPDA). Probabilistic pushdown automata are obtained as a probabilistic variant of pushdown automata that proved to be a successful abstract model of recursive sequential programs. The main aim of this work is to study decidability and complexity of the problem whether a given probabilistic system generated by a pPDA satisfies a given property expressed in a suitable formalism. There are plenty of formalisms available for specifying properties of probabilistic systems. In this work we consider various temporal properties expressed by finitestate automata on infinite words and formulae of temporal logics, longrun average properties, and properties connected with expected behavior. Concerning temporal logics, we consider both linear and branching time ones. Among others we consider linear temporal logic (LTL) and probabilistic computation tree logic (PCTL), which is a probabilistic variant of the wellknown logic CTL. We also consider a general logic PECTL ∗ , which combines automata based
Efficient analysis of probabilistic programs with an unbounded counter
 CoRR
"... Abstract. We show that a subclass of infinitestate probabilistic programs that can be modeled by probabilistic onecounter automata (pOC) admits an efficient quantitative analysis. In particular, we show that the expected termination time can be approximated up to an arbitrarily small relative erro ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
(Show Context)
Abstract. We show that a subclass of infinitestate probabilistic programs that can be modeled by probabilistic onecounter automata (pOC) admits an efficient quantitative analysis. In particular, we show that the expected termination time can be approximated up to an arbitrarily small relative error with polynomially many arithmetic operations, and the same holds for the probability of all runs that satisfy a given ωregular property. Further, our results establish a powerful link between pOC and martingale theory, which leads to fundamental observations about quantitative properties of runs in pOC. In particular, we provide a “divergence gap theorem”, which bounds a positive nontermination probability in pOC away from zero. 1
On the Memory Consumption of Probabilistic Pushdown Automata
 LIPICS LEIBNIZ INTERNATIONAL PROCEEDINGS IN INFORMATICS
, 2009
"... ..."
Analyzing Probabilistic Pushdown Automata
 FORMAL METHODS IN SYSTEM DESIGN
, 2012
"... The paper gives a summary of the existing results about algorithmic analysis of probabilistic pushdown automata and their subclasses. ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The paper gives a summary of the existing results about algorithmic analysis of probabilistic pushdown automata and their subclasses.
QuasiBirthDeath Processes, TreeLike QBDs, Probabilistic 1Counter Automata, and Pushdown
"... We begin by observing that (discretetime) QuasiBirthDeath Processes (QBDs) are equivalent, in a precise sense, to probabilistic 1Counter Automata (p1CAs), and both TreeLike QBDs (TLQBDs) and TreeStructured QBDs (TSQBDs) are equivalent to both probabilistic Pushdown Systems (pPDSs) and Recurs ..."
Abstract
 Add to MetaCart
(Show Context)
We begin by observing that (discretetime) QuasiBirthDeath Processes (QBDs) are equivalent, in a precise sense, to probabilistic 1Counter Automata (p1CAs), and both TreeLike QBDs (TLQBDs) and TreeStructured QBDs (TSQBDs) 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, 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 unitcost 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 wellknown “linear/quadratic convergence ” of numerical analysis, known for QBDs and TLQBDs, which typically gives no constructive bound in terms of the encoding size of the system being solved. In fact, we observe (based on recent re