Probabilistic Pushdown Automata Research Articles

Model Checking Temporal Properties of Recursive Probabilistic Programs

Probabilistic pushdown automata (pPDA) are a standard operational model for programming languages involving discrete random choices and recursive procedures. Temporal properties are useful for specifying the chronological order of events during program execution. Existing approaches for model checking pPDA against temporal properties have focused mostly on $\omega$-regular and LTL properties. In this paper, we give decidability and complexity results for the model checking problem of pPDA against $\omega$-visibly pushdown languages that can be described by specification logics such as CaRet. These logical formulae allow specifying properties that explicitly take the structured computations arising from procedural programs into account. For example, CaRet is able to match procedure calls with their corresponding future returns, and thus allows to express fundamental program properties such as total and partial correctness.

Exact Recursive Probabilistic Programming

Recursive calls over recursive data are useful for generating probability distributions, and probabilistic programming allows computations over these distributions to be expressed in a modular and intuitive way. Exact inference is also useful, but unfortunately, existing probabilistic programming languages do not perform exact inference on recursive calls over recursive data, forcing programmers to code many applications manually. We introduce a probabilistic language in which a wide variety of recursion can be expressed naturally, and inference carried out exactly. For instance, probabilistic pushdown automata and their generalizations are easy to express, and polynomial-time parsing algorithms for them are derived automatically. We eliminate recursive data types using program transformations related to defunctionalization and refunctionalization. These transformations are assured correct by a linear type system, and a successful choice of transformations, if there is one, is guaranteed to be found by a greedy algorithm.

Deciding probabilistic simulation between probabilistic pushdown automata and finite-state systems

Checking whether a pushdown automaton is simulated – in the sense of a simulation pre-order – by a finite-state automaton is EXPTIME-complete. This paper shows that the same computational complexity is obtained in a probabilistic setting. That is, the problem of deciding whether a probabilistic pushdown automaton (pPDA) is simulated by a finite Markov decision process (MDP) is EXPTIME-complete. The considered pPDA contain both probabilistic and non-deterministic branching. The EXPTIME-membership is shown by combining a partition-refinement algorithm with a tableaux system that is inspired by Stirling's technique for bisimilarity checking of ordinary pushdown automata. The hardness is obtained by exploiting the EXPTIME-hardness for the non-probabilistic case. Moreover, our decision problem is in PTIME when both the number of states of the pPDA and the number of states in the MDP are fixed.

Upper Bounds for Newton’s Method on Monotone Polynomial Systems, and P-Time Model Checking of Probabilistic One-Counter Automata

A central computational problem for analyzing and model checking various classes of infinite-state recursive probabilistic systems (including quasi-birth-death processes, multitype branching processes, stochastic context-free grammars, probabilistic pushdown automata and recursive Markov chains) is the computation of termination probabilities, and computing these probabilities in turn boils down to computing the least fixed point (LFP) solution of a corresponding monotone polynomial system (MPS) of equations, denoted x = P ( x ). It was shown in Etessami and Yannakakis [2009] that a decomposed variant of Newton’s method converges monotonically to the LFP solution for any MPS that has a nonnegative solution. Subsequently, Esparza et al. [2010] obtained upper bounds on the convergence rate of Newton’s method for certain classes of MPSs. More recently, better upper bounds have been obtained for special classes of MPSs [Etessami et al. 2010, 2012]. However, prior to this article, for arbitrary (not necessarily strongly connected) MPSs, no upper bounds at all were known on the convergence rate of Newton’s method as a function of the encoding size |P| of the input MPS, x = P ( x ). In this article, we provide worst-case upper bounds, as a function of both the input encoding size |P|, and ε > 0, on the number of iterations required for decomposed Newton’s method (even with rounding) to converge to within additive error ε > 0 of q*, for an arbitrary MPS with LFP solution q*. Our upper bounds are essentially optimal in terms of several important parameters of the problem. Using our upper bounds, and building on prior work, we obtain the first P-time algorithm (in the standard Turing model of computation) for quantitative model checking, to within arbitrary desired precision, of discrete-time QBDs and (equivalently) probabilistic 1-counter automata, with respect to any (fixed) ω -regular or LTL property.

Runtime analysis of probabilistic programs with unbounded recursion

We study the runtime in probabilistic programs with unbounded recursion. As underlying formal model for such programs we use probabilistic pushdown automata (pPDAs) which exactly correspond to recursive Markov chains. We show that every pPDA can be transformed into a stateless pPDA (called “pBPA”) whose runtime and further properties are closely related to those of the original pPDA. This result substantially simplifies the analysis of runtime and other pPDA properties. We prove that for every pPDA the probability of performing a long run decreases exponentially in the length of the run, if and only if the expected runtime in the pPDA is finite. If the expectation is infinite, then the probability decreases “polynomially”. We show that these bounds are asymptotically tight. Our tail bounds on the runtime are generic, i.e., applicable to any probabilistic program with unbounded recursion.

Analyzing probabilistic pushdown automata

The paper gives a summary of the existing results about algorithmic analysis of probabilistic pushdown automata and their subclasses.

Probabilistic regular graphs

Deterministic graph grammars generate regular graphs, that form a structural extension of configuration graphs of pushdown systems. In this paper, we study a probabilistic extension of regular graphs obtained by labelling the terminal arcs of the graph grammars by probabilities. Stochastic properties of these graphs are expressed using PCTL, a probabilistic extension of computation tree logic. We present here an algorithm to perform approximate verification of PCTL formulae. Moreover, we prove that the exact model-checking problem for PCTL on probabilistic regular graphs is undecidable, unless restricting to qualitative properties. Our results generalise those of EKM06, on probabilistic pushdown automata, using similar methods combined with graph grammars techniques.

Branching-Time Model-Checking of Probabilistic Pushdown Automata

In this paper we study complexity of the model-checking problem for probabilistic pushdown automata (pPDA) and qualitative fragments of two branching-time logics PCTL* and PECTL*. We prove tha this problem is in 2-EXPTIME for pPDA and qualitative PCTL*. Consequently, we prove that model-checking of stateless pPDA (pBPA) and both qualitative PCTL* and qualitative PECTL* is 2-EXPTIME-hard. These results combined with results of several other papers give us that the model-checking problem for pPDA (and also for pBPA) and both qualitative PCTL* and qualitative PECTL* is 2-EXPTIME-complete. Finally, we survey known results on model-checking of pPDA and branching-time logics.

Deciding probabilistic bisimilarity over infinite-state probabilistic systems

We prove that probabilistic bisimilarity is decidable over probabilistic extensions of BPA and BPP processes. For normed subclasses of probabilistic BPA and BPP processes we obtain polynomial-time algorithms. Further, we show that probabilistic bisimilarity between probabilistic pushdown automata and finite-state systems is decidable in exponential time. If the number of control states in PDA is bounded by a fixed constant, then the algorithm needs only polynomial time.

Methods for Quantitative Analysis of Probabilistic Pushdown Automata

We present several constructions and techniques which have recently been used to tackle the prob- lems of qualitative/quantitative analysis of probabilistic pushdown automata.

PROBABILISTIC PUSHDOWN AUTOMATA

The properties of Probabilistic Pushdown Automata (PPA) are examined. First PPA is defined, various basic properties of it are derived, and several more restrictive types are considered. Then, the families of random languages acceptable and generable by various types of PPA, as well as the families of languages acceptable and generable by various types of PPA with cut-point, are studied. The relationships among these families are also examined, and many interesting results are obtained. Among them, it is shown that the family of languages generable by PPA having uniform output length is the homomorphic closure of the family of languages acceptable by PPA.