1,643 research outputs found
Optimal Spectral-Norm Approximate Minimization of Weighted Finite Automata
We address the approximate minimization problem for weighted finite automata
(WFAs) over a one-letter alphabet: to compute the best possible approximation
of a WFA given a bound on the number of states. This work is grounded in
Adamyan-Arov-Krein Approximation theory, a remarkable collection of results on
the approximation of Hankel operators. In addition to its intrinsic
mathematical relevance, this theory has proven to be very effective for model
reduction. We adapt these results to the framework of weighted automata over a
one-letter alphabet. We provide theoretical guarantees and bounds on the
quality of the approximation in the spectral and norm. We develop an
algorithm that, based on the properties of Hankel operators, returns the
optimal approximation in the spectral norm.Comment: 24 pages, authors appear in alphabetical order; minor correction in
Theorem 3.2 and consequently updated notation in Section 3, the validity of
the result is not affecte
Optimal Approximate Minimization of One-Letter Weighted Finite Automata
In this paper, we study the approximate minimization problem of weighted
finite automata (WFAs): to compute the best possible approximation of a WFA
given a bound on the number of states. By reformulating the problem in terms of
Hankel matrices, we leverage classical results on the approximation of Hankel
operators, namely the celebrated Adamyan-Arov-Krein (AAK) theory.
We solve the optimal spectral-norm approximate minimization problem for
irredundant WFAs with real weights, defined over a one-letter alphabet. We
present a theoretical analysis based on AAK theory, and bounds on the quality
of the approximation in the spectral norm and norm. Moreover, we
provide a closed-form solution, and an algorithm, to compute the optimal
approximation of a given size in polynomial time.Comment: 32 pages. arXiv admin note: substantial text overlap with
arXiv:2102.0686
Algebra, coalgebra, and minimization in polynomial differential equations
We consider reasoning and minimization in systems of polynomial ordinary
differential equations (ode's). The ring of multivariate polynomials is
employed as a syntax for denoting system behaviours. We endow this set with a
transition system structure based on the concept of Lie-derivative, thus
inducing a notion of L-bisimulation. We prove that two states (variables) are
L-bisimilar if and only if they correspond to the same solution in the ode's
system. We then characterize L-bisimilarity algebraically, in terms of certain
ideals in the polynomial ring that are invariant under Lie-derivation. This
characterization allows us to develop a complete algorithm, based on building
an ascending chain of ideals, for computing the largest L-bisimulation
containing all valid identities that are instances of a user-specified
template. A specific largest L-bisimulation can be used to build a reduced
system of ode's, equivalent to the original one, but minimal among all those
obtainable by linear aggregation of the original equations. A computationally
less demanding approximate reduction and linearization technique is also
proposed.Comment: 27 pages, extended and revised version of FOSSACS 2017 pape
Approximate Learning of Limit-Average Automata
Limit-average automata are weighted automata on infinite words that use average to aggregate the weights seen in infinite runs. We study approximate learning problems for limit-average automata in two settings: passive and active. In the passive learning case, we show that limit-average automata are not PAC-learnable as samples must be of exponential-size to provide (with good probability) enough details to learn an automaton. We also show that the problem of finding an automaton that fits a given sample is NP-complete. In the active learning case, we show that limit-average automata can be learned almost-exactly, i.e., we can learn in polynomial time an automaton that is consistent with the target automaton on almost all words. On the other hand, we show that the problem of learning an automaton that approximates the target automaton (with perhaps fewer states) is NP-complete. The abovementioned results are shown for the uniform distribution on words. We briefly discuss learning over different distributions
Bisimulation Metrics for Weighted Automata
We develop a new bisimulation (pseudo)metric for weighted finite automata (WFA) that generalizes Boreale\u27s linear bisimulation relation. Our metrics are induced by seminorms on the state space of WFA. Our development is based on spectral properties of sets of linear operators. In particular, the joint spectral radius of the transition matrices of WFA plays a central role. We also study continuity properties of the bisimulation pseudometric, establish an undecidability result for computing the metric, and give a preliminary account of applications to spectral learning of weighted automata
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