75 research outputs found
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
Profinite Techniques for Probabilistic Automata and the Markov Monoid Algorithm
We consider the value 1 problem for probabilistic automata over finite words:
it asks whether a given probabilistic automaton accepts words with probability
arbitrarily close to 1. This problem is known to be undecidable. However,
different algorithms have been proposed to partially solve it; it has been
recently shown that the Markov Monoid algorithm, based on algebra, is the most
correct algorithm so far. The first contribution of this paper is to give a
characterisation of the Markov Monoid algorithm. The second contribution is to
develop a profinite theory for probabilistic automata, called the prostochastic
theory. This new framework gives a topological account of the value 1 problem,
which in this context is cast as an emptiness problem. The above
characterisation is reformulated using the prostochastic theory, allowing us to
give a simple and modular proof.Comment: Conference version: STACS'2016, Symposium on Theoretical Aspects of
Computer Science Journal version: TCS'2017, Theoretical Computer Scienc
Lower Bounds for Alternating Online State Complexity
The notion of Online State Complexity, introduced by Karp in 1967, quantifies
the amount of states required to solve a given problem using an online
algorithm, which is represented by a deterministic machine scanning the input
from left to right in one pass. In this paper, we extend the setting to
alternating machines as introduced by Chandra, Kozen and Stockmeyer in 1976:
such machines run independent passes scanning the input from left to right and
gather their answers through boolean combinations. We devise a lower bound
technique relying on boundedly generated lattices of languages, and give two
applications of this technique. The first is a hierarchy theorem , stating that
the polynomial hierarchy of alternating online state complexity is infinite,
and the second is a linear lower bound on the alternating online state
complexity of the prime numbers written in binary. This second result
strengthens a result of Hartmanis and Shank from 1968, which implies an
exponentially worse lower bound for the same model
Finitary languages
The class of omega-regular languages provides a robust specification language
in verification. Every omega-regular condition can be decomposed into a safety
part and a liveness part. The liveness part ensures that something good happens
"eventually". Finitary liveness was proposed by Alur and Henzinger as a
stronger formulation of liveness. It requires that there exists an unknown,
fixed bound b such that something good happens within b transitions. In this
work we consider automata with finitary acceptance conditions defined by
finitary Buchi, parity and Streett languages. We study languages expressible by
such automata: we give their topological complexity and present a
regular-expression characterization. We compare the expressive power of
finitary automata and give optimal algorithms for classical decisions
questions. We show that the finitary languages are Sigma 2-complete; we present
a complete picture of the expressive power of various classes of automata with
finitary and infinitary acceptance conditions; we show that the languages
defined by finitary parity automata exactly characterize the star-free fragment
of omega B-regular languages; and we show that emptiness is NLOGSPACE-complete
and universality as well as language inclusion are PSPACE-complete for finitary
parity and Streett automata
Monadic Second-Order Logic with Arbitrary Monadic Predicates
We study Monadic Second-Order Logic (MSO) over finite words, extended with
(non-uniform arbitrary) monadic predicates. We show that it defines a class of
languages that has algebraic, automata-theoretic and machine-independent
characterizations. We consider the regularity question: given a language in
this class, when is it regular? To answer this, we show a substitution property
and the existence of a syntactical predicate.
We give three applications. The first two are to give very simple proofs that
the Straubing Conjecture holds for all fragments of MSO with monadic
predicates, and that the Crane Beach Conjecture holds for MSO with monadic
predicates. The third is to show that it is decidable whether a language
defined by an MSO formula with morphic predicates is regular.Comment: Conference version: MFCS'14, Mathematical Foundations of Computer
Science Journal version: ToCL'17, Transactions on Computational Logi
Lower bounds for the state complexity of probabilistic languages and the language of prime numbers
This paper studies the complexity of languages of finite words using automata
theory. To go beyond the class of regular languages, we consider infinite
automata and the notion of state complexity defined by Karp. Motivated by the
seminal paper of Rabin from 1963 introducing probabilistic automata, we study
the (deterministic) state complexity of probabilistic languages and prove that
probabilistic languages can have arbitrarily high deterministic state
complexity. We then look at alternating automata as introduced by Chandra,
Kozen and Stockmeyer: such machines run independent computations on the word
and gather their answers through boolean combinations. We devise a lower bound
technique relying on boundedly generated lattices of languages, and give two
applications of this technique. The first is a hierarchy theorem, stating that
there are languages of arbitrarily high polynomial alternating state
complexity, and the second is a linear lower bound on the alternating state
complexity of the prime numbers written in binary. This second result
strengthens a result of Hartmanis and Shank from 1968, which implies an
exponentially worse lower bound for the same model.Comment: Submitted to the Journal of Logic and Computation, Special Issue on
LFCS'2016) (Logical Foundations of Computer Science). Guest Editors: S.
Artemov and A. Nerode. This journal version extends two conference papers:
the first published in the proceedings of LFCS'2016 and the second in the
proceedings of LICS'2018. arXiv admin note: substantial text overlap with
arXiv:1607.0025
Pushing undecidability of the isolation problem for probabilistic automata
This short note aims at proving that the isolation problem is undecidable for
probabilistic automata with only one probabilistic transition. This problem is
known to be undecidable for general probabilistic automata, without restriction
on the number of probabilistic transitions. In this note, we develop a
simulation technique that allows to simulate any probabilistic automaton with
one having only one probabilistic transition
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