Stefan Kiefer

Parikh's theorem: A simple and direct automaton construction

Parikhs theorem states that the Parikh image of a context-free language is semilinear or, equivalently, that every context-free language has the same Parikh image as some regular language. We present a very simple construction that, given a context-free grammar, produces a finite automaton recognizing such a regular language.

Language equivalence for probabilistic automata

In this paper, we propose a new randomised algorithm for deciding language equivalence for probabilistic automata. This algorithm is based on polynomial identity testing and thus returns an answer with an error probability that can be made arbitrarily small. We implemented our algorithm, as well as deterministic algorithms of Tzeng and Doyen et al., optimised for running time whilst adequately handling issues of numerical stability. We conducted extensive benchmarking experiments, including the verification of randomised anonymity protocols, the outcome of which establishes that the randomised algorithm significantly outperforms the deterministic ones in a majority of our test cases. Finally, we also provide fine-grained analytical bounds on the complexity of these algorithms, accounting for the differences in performance.

BPA bisimilarity is EXPTIME-hard

Given a basic process algebra (BPA) and two stack symbols, the BPA bisimilarity problem asks whether the two stack symbols are bisimilar. We show that this problem is EXPTIME-hard.

The Odds of Staying on Budget

Given Markov chains and Markov decision processes MDPs whose transitions are labelled with non-negative integer costs, we study the computational complexity of deciding whether the probability of paths whose accumulated cost satisfies a Boolean combination of inequalities exceeds a given threshold. For acyclic Markov chains, we show that this problem is PP-complete, whereas it is hard for the PosSLP problem and in PSpace for general Markov chains. Moreover, for acyclic and general MDPs, we prove PSpace- and EXP-completeness, respectively. Our results have direct implications on the complexity of computing reward quantiles in succinctly represented stochastic systems.

Efficient Analysis of Probabilistic Programs with an Unbounded Counter

We show that a subclass of infinite-state probabilistic programs that can be modeled by probabilistic one-counter automata (pOC) admits an efficient quantitative analysis. We start by establishing 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 non-termination probability in pOC away from zero. Using these observations, we show that the expected termination time can be approximated up to an arbitrarily small relative error in polynomial time, and the same holds for the probability of all runs that satisfy a given ω-regular property encoded by a deterministic Rabin automaton.

Analyzing probabilistic pushdown automata

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

On the complexity of equivalence and minimisation for Q-weighted automata

This paper is concerned with the computational complexity of equivalence and minimisation for automata with transition weights in the field Q of rational numbers. We use polynomial identity testing and the Isolation Lemma to obtain complexity bounds, focussing on the class NC of problems within P solvable in polylogarithmic parallel time. For finite Q-weighted automata, we give a randomised NC procedure that either outputs that two automata are equivalent or returns a word on which they differ. We also give an NC procedure for deciding whether a given automaton is minimal, as well as a randomised NC procedure that minimises an automaton. We consider probabilistic automata with rewards, similar to Markov Decision Processes. For these automata we consider two notions of equivalence: expectation equivalence and distribution equivalence. The former requires that two automata have the same expected reward on each input word, while the latter requires that each input word induce the same distribution on rewards in each automaton. For both notions we give algorithms for deciding equivalence by reduction to equivalence of Q-weighted automata. Finally we show that the equivalence problem for Q-weighted visibly pushdown automata is logspace equivalent to the polynomial identity testing problem.

APEX: an analyzer for open probabilistic programs

We present APEX, a tool for analysing probabilistic programs that are open, i.e. where variables or even functions can be left unspecified. APEX transforms a program into an automaton that captures the programs probabilistic behaviour under all instantiations of the unspecified components. The translation is compositional and effectively leverages state reduction techniques. APEX can then further analyse the produced automata; in particular, it can check two automata for equivalence which translates to equivalence of the corresponding programs under all environments. In this way, APEX can verify a broad range of anonymity and termination properties of randomised protocols and other open programs, sometimes with an exponential speed-up over competing state-of-the-art approaches.

Bisimilarity of Pushdown Automata is Nonelementary

Given two pushdown automata, the bisimilarity problem asks whether the infinite transition systems they induce are bisimilar. While this problem is known to be decidable our main result states that it is nonelementary, improving EXPTIME-hardness, which was the best previously known lower bound for this problem. Our lower bound result holds for normed pushdown automata as well.

On the complexity of the equivalence problem for probabilistic automata

Deciding equivalence of probabilistic automata is a key problem for establishing various behavioural and anonymity properties of probabilistic systems. In recent experiments a randomised equivalence test based on polynomial identity testing outperformed deterministic algorithms. In this paper we show that polynomial identity testing yields efficient algorithms for various generalisations of the equivalence problem. First, we provide a randomized NC procedure that also outputs a counterexample trace in case of inequivalence. Second, we consider equivalence of probabilistic cost automata. In these automata transitions are labelled with integer costs and each word is associated with a distribution on costs, corresponding to the cumulative costs of the accepting runs on that word. Two automata are equivalent if they induce the same cost distributions on each input word. We show that equivalence can be checked in randomised polynomial time. Finally we show that the equivalence problem for probabilistic visibly pushdown automata is logspace equivalent to the problem of whether a polynomial represented by an arithmetic circuit is identically zero.