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

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

Minimisation of Multiplicity Tree Automata

We consider the problem of minimising the number of states in a multiplicity tree automaton over the field of rational numbers. We give a minimisation algorithm that runs in polynomial time assuming unit-cost arithmetic. We also show that a polynomial bound in the standard Turing model would require a breakthrough in the complexity of polynomial identity testing by proving that the latter problem is logspace equivalent to the decision version of minimisation. The developed techniques also improve the state of the art in multiplicity word automata: we give an NC algorithm for minimising multiplicity word automata. Finally, we consider the minimal consistency problem: does there exist an automaton with $n$ states that is consistent with a given finite sample of weight-labelled words or trees? We show that this decision problem is complete for the existential theory of the rationals, both for words and for trees of a fixed alphabet rank.Comment: Paper to be published in Logical Methods in Computer Science. Minor editing changes from previous versio

Probabilistic Model-Based Safety Analysis

Model-based safety analysis approaches aim at finding critical failure combinations by analysis of models of the whole system (i.e. software, hardware, failure modes and environment). The advantage of these methods compared to traditional approaches is that the analysis of the whole system gives more precise results. Only few model-based approaches have been applied to answer quantitative questions in safety analysis, often limited to analysis of specific failure propagation models, limited types of failure modes or without system dynamics and behavior, as direct quantitative analysis is uses large amounts of computing resources. New achievements in the domain of (probabilistic) model-checking now allow for overcoming this problem. This paper shows how functional models based on synchronous parallel semantics, which can be used for system design, implementation and qualitative safety analysis, can be directly re-used for (model-based) quantitative safety analysis. Accurate modeling of different types of probabilistic failure occurrence is shown as well as accurate interpretation of the results of the analysis. This allows for reliable and expressive assessment of the safety of a system in early design stages

An Individual-based Probabilistic Model for Fish Stock Simulation

We define an individual-based probabilistic model of a sole (Solea solea) behaviour. The individual model is given in terms of an Extended Probabilistic Discrete Timed Automaton (EPDTA), a new formalism that is introduced in the paper and that is shown to be interpretable as a Markov decision process. A given EPDTA model can be probabilistically model-checked by giving a suitable translation into syntax accepted by existing model-checkers. In order to simulate the dynamics of a given population of soles in different environmental scenarios, an agent-based simulation environment is defined in which each agent implements the behaviour of the given EPDTA model. By varying the probabilities and the characteristic functions embedded in the EPDTA model it is possible to represent different scenarios and to tune the model itself by comparing the results of the simulations with real data about the sole stock in the North Adriatic sea, available from the recent project SoleMon. The simulator is presented and made available for its adaptation to other species.Comment: In Proceedings AMCA-POP 2010, arXiv:1008.314