23 research outputs found

    When is Containment Decidable for Probabilistic Automata?

    Get PDF
    The containment problem for quantitative automata is the natural quantitative generalisation of the classical language inclusion problem for Boolean automata. We study it for probabilistic automata, where it is known to be undecidable in general. We restrict our study to the class of probabilistic automata with bounded ambiguity. There, we show decidability (subject to Schanuel's conjecture) when one of the automata is assumed to be unambiguous while the other one is allowed to be finitely ambiguous. Furthermore, we show that this is close to the most general decidable fragment of this problem by proving that it is already undecidable if one of the automata is allowed to be linearly ambiguous

    The Decidability Frontier for Probabilistic Automata on Infinite Words

    Get PDF
    We consider probabilistic automata on infinite words with acceptance defined by safety, reachability, B\"uchi, coB\"uchi, and limit-average conditions. We consider quantitative and qualitative decision problems. We present extensions and adaptations of proofs for probabilistic finite automata and present a complete characterization of the decidability and undecidability frontier of the quantitative and qualitative decision problems for probabilistic automata on infinite words

    POMDPs under Probabilistic Semantics

    Full text link
    We consider partially observable Markov decision processes (POMDPs) with limit-average payoff, where a reward value in the interval [0,1] is associated to every transition, and the payoff of an infinite path is the long-run average of the rewards. We consider two types of path constraints: (i) quantitative constraint defines the set of paths where the payoff is at least a given threshold lambda_1 in (0,1]; and (ii) qualitative constraint which is a special case of quantitative constraint with lambda_1=1. We consider the computation of the almost-sure winning set, where the controller needs to ensure that the path constraint is satisfied with probability 1. Our main results for qualitative path constraint are as follows: (i) the problem of deciding the existence of a finite-memory controller is EXPTIME-complete; and (ii) the problem of deciding the existence of an infinite-memory controller is undecidable. For quantitative path constraint we show that the problem of deciding the existence of a finite-memory controller is undecidable.Comment: Appears in Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence (UAI2013

    IST Austria Technical Report

    Get PDF
    We consider probabilistic automata on infinite words with acceptance defined by safety, reachability, Büchi, coBüchi and limit-average conditions. We consider quantitative and qualitative decision problems. We present extensions and adaptations of proofs of [GO09] and present a precise characterization of the decidability and undecidability frontier of the quantitative and qualitative decision problems

    Nonnegativity Problems for Matrix Semigroups

    Full text link
    The matrix semigroup membership problem asks, given square matrices M,M1,…,MkM,M_1,\ldots,M_k of the same dimension, whether MM lies in the semigroup generated by M1,…,MkM_1,\ldots,M_k. It is classical that this problem is undecidable in general but decidable in case M1,…,MkM_1,\ldots,M_k commute. In this paper we consider the problem of whether, given M1,…,MkM_1,\ldots,M_k, the semigroup generated by M1,…,MkM_1,\ldots,M_k contains a non-negative matrix. We show that in case M1,…,MkM_1,\ldots,M_k commute, this problem is decidable subject to Schanuel's Conjecture. We show also that the problem is undecidable if the commutativity assumption is dropped. A key lemma in our decidability result is a procedure to determine, given a matrix MM, whether the sequence of matrices (Mn)n≥0(M^n)_{n\geq 0} is ultimately nonnegative. This answers a problem posed by S. Akshay (arXiv:2205.09190). The latter result is in stark contrast to the notorious fact that it is not known how to determine effectively whether for any specific matrix index (i,j)(i,j) the sequence (Mn)i,j(M^n)_{i,j} is ultimately nonnegative (which is a formulation of the Ultimate Positivity Problem for linear recurrence sequences)

    Nonnegativity problems for matrix semigroups

    Get PDF
    The matrix semigroup membership problem asks, given square matrices M, M1, ..., Mk of the same dimension, whether M lies in the semigroup generated by M1, ..., Mk. It is classical that this problem is undecidable in general, but decidable in case M1, ..., Mk commute. In this paper we consider the problem of whether, given M1, ..., Mk, the semigroup generated by M1, ..., Mk contains a non-negative matrix. We show that in case M1, ..., Mk commute, this problem is decidable subject to Schanuel's Conjecture. We show also that the problem is undecidable if the commutativity assumption is dropped. A key lemma in our decidability proof is a procedure to determine, given a matrix M, whether the sequence of matrices (Mn)∞n=0 is ultimately nonnegative. This answers a problem posed by S. Akshay [1]. The latter result is in stark contrast to the notorious fact that it is not known how to determine, for any specific matrix index (i, j), whether the sequence (Mn)i,j is ultimately nonnegative. Indeed the latter is equivalent to the Ultimate Positivity Problem for linear recurrence sequences, a longstanding open problem

    A Probabilistic Higher-Order Fixpoint Logic

    Get PDF
    We introduce PHFL, a probabilistic extension of higher-order fixpoint logic, which can also be regarded as a higher-order extension of probabilistic temporal logics such as PCTL and the μp\mu^p-calculus. We show that PHFL is strictly more expressive than the μp\mu^p-calculus, and that the PHFL model-checking problem for finite Markov chains is undecidable even for the μ\mu-only, order-1 fragment of PHFL. Furthermore the full PHFL is far more expressive: we give a translation from Lubarsky's μ\mu-arithmetic to PHFL, which implies that PHFL model checking is Π11\Pi^1_1-hard and Σ11\Sigma^1_1-hard. As a positive result, we characterize a decidable fragment of the PHFL model-checking problems using a novel type system

    The big-O problem for labelled markov chains and weighted automata

    Get PDF
    Given two weighted automata, we consider the problem of whether one is big-O of the other, i.e., if the weight of every finite word in the first is not greater than some constant multiple of the weight in the second. We show that the problem is undecidable, even for the instantiation of weighted automata as labelled Markov chains. Moreover, even when it is known that one weighted automaton is big-O of another, the problem of finding or approximating the associated constant is also undecidable. Our positive results show that the big-O problem is polynomial-time solvable for unambiguous automata, coNP-complete for unlabelled weighted automata (i.e., when the alphabet is a single character) and decidable, subject to Schanuel’s conjecture, when the language is bounded (i.e., a subset of w_1^* … w_m^* for some finite words w_1,… ,w_m). On labelled Markov chains, the problem can be restated as a ratio total variation distance, which, instead of finding the maximum difference between the probabilities of any two events, finds the maximum ratio between the probabilities of any two events. The problem is related to ε-differential privacy, for which the optimal constant of the big-O notation is exactly exp(ε)

    Alternating Tree Automata with Qualitative Semantics

    Get PDF
    We study alternating automata with qualitative semantics over infinite binary trees: Alternation means that two opposing players construct a decoration of the input tree called a run, and the qualitative semantics says that a run of the automaton is accepting if almost all branches of the run are accepting. In this article, we prove a positive and a negative result for the emptiness problem of alternating automata with qualitative semantics. The positive result is the decidability of the emptiness problem for the case of Büchi acceptance condition. An interesting aspect of our approach is that we do not extend the classical solution for solving the emptiness problem of alternating automata, which first constructs an equivalent non-deterministic automaton. Instead, we directly construct an emptiness game making use of imperfect information. The negative result is the undecidability of the emptiness problem for the case of co-Büchi acceptance condition. This result has two direct consequences: The undecidability of monadic second-order logic extended with the qualitative path-measure quantifier and the undecidability of the emptiness problem for alternating tree automata with non-zero semantics, a recently introduced probabilistic model of alternating tree automata
    corecore