On the Complexity of the Equivalence Problem for Probabilistic Automata

Checking two probabilistic automata for equivalence has been shown to be a key problem for efficiently 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 show how to check for equivalence two probabilistic automata with (cumulative) rewards. Our algorithm runs in deterministic polynomial time, if the number of reward counters is fixed. Finally we show that the equivalence problem for probabilistic visibly pushdown automata is logspace equivalent to the Arithmetic Circuit Identity Testing problem, which is to decide whether a polynomial represented by an arithmetic circuit is identically zero.Comment: technical report for a FoSSaCS'12 pape

Game Characterization of Probabilistic Bisimilarity, and Applications to Pushdown Automata

We study the bisimilarity problem for probabilistic pushdown automata (pPDA) and subclasses thereof. Our definition of pPDA allows both probabilistic and non-deterministic branching, generalising the classical notion of pushdown automata (without epsilon-transitions). We first show a general characterization of probabilistic bisimilarity in terms of two-player games, which naturally reduces checking bisimilarity of probabilistic labelled transition systems to checking bisimilarity of standard (non-deterministic) labelled transition systems. This reduction can be easily implemented in the framework of pPDA, allowing to use known results for standard (non-probabilistic) PDA and their subclasses. A direct use of the reduction incurs an exponential increase of complexity, which does not matter in deriving decidability of bisimilarity for pPDA due to the non-elementary complexity of the problem. In the cases of probabilistic one-counter automata (pOCA), of probabilistic visibly pushdown automata (pvPDA), and of probabilistic basic process algebras (i.e., single-state pPDA) we show that an implicit use of the reduction can avoid the complexity increase; we thus get PSPACE, EXPTIME, and 2-EXPTIME upper bounds, respectively, like for the respective non-probabilistic versions. The bisimilarity problems for OCA and vPDA are known to have matching lower bounds (thus being PSPACE-complete and EXPTIME-complete, respectively); we show that these lower bounds also hold for fully probabilistic versions that do not use non-determinism

Containment and equivalence of weighted automata: Probabilistic and max-plus cases

This paper surveys some results regarding decision problems for probabilistic and max-plus automata, such as containment and equivalence. Probabilistic and max-plus automata are part of the general family of weighted automata, whose semantics are maps from words to real values. Given two weighted automata, the equivalence problem asks whether their semantics are the same, and the containment problem whether one is point-wise smaller than the other one. These problems have been studied intensively and this paper will review some techniques used to show (un)decidability and state a list of open questions that still remain

Complexity of Equivalence and Learning for Multiplicity Tree Automata

We consider the complexity of equivalence and learning for multiplicity tree automata, i.e., weighted tree automata over a field. We first show that the equivalence problem is logspace equivalent to polynomial identity testing, the complexity of which is a longstanding open problem. Secondly, we derive lower bounds on the number of queries needed to learn multiplicity tree automata in Angluin's exact learning model, over both arbitrary and fixed fields. Habrard and Oncina (2006) give an exact learning algorithm for multiplicity tree automata, in which the number of queries is proportional to the size of the target automaton and the size of a largest counterexample, represented as a tree, that is returned by the Teacher. However, the smallest tree-counterexample may be exponential in the size of the target automaton. Thus the above algorithm does not run in time polynomial in the size of the target automaton, and has query complexity exponential in the lower bound. Assuming a Teacher that returns minimal DAG representations of counterexamples, we give a new exact learning algorithm whose query complexity is quadratic in the target automaton size, almost matching the lower bound, and improving the best previously-known algorithm by an exponential factor

What is known about the Value 1 Problem for Probabilistic Automata?

The value 1 problem is a decision problem for probabilistic automata over finite words: are there words accepted by the automaton with arbitrarily high probability? Although undecidable, this problem attracted a lot of attention over the last few years. The aim of this paper is to review and relate the results pertaining to the value 1 problem. In particular, several algorithms have been proposed to partially solve this problem. We show the relations between them, leading to the following conclusion: the Markov Monoid Algorithm is the most correct algorithm known to (partially) solve the value 1 problem