213 research outputs found
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Approximate Comparison of Functions Computed by Distance Automata
Distance automata are automata weighted over the semiring (NâȘ{â},min,+) (the tropical semiring). Such automata compute functions from words to NâȘ{â}. It is known from Krob that the problems of deciding â fâ€gâ or â f=gâ for f and g computed by distance automata is an undecidable problem. The main contribution of this paper is to show that an approximation of this problem is decidable. We present an algorithm which, given Δ>0 and two functions f,g computed by distance automata, answers âyesâ if fâ€(1âΔ)g, ânoâ if fâŠÌžg, and may answer âyesâ or ânoâ in all other cases. The core argument behind this quasi-decision procedure is an algorithm which is able to provide an approximated finite presentation of the closure under products of sets of matrices over the tropical semiring. Lastly, our theorem of affine domination gives better bounds on the precision of known decision procedures for cost automata, when restricted to distance automata
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Approximate comparison of distance automata
Distance automata are automata weighted over the semiring (NâȘ {â}, min,+) (the tropical semiring). Such automata compute functions from words to N
âȘ{â} such as the number of occurrences of a given letter. It is known that testing f 0 and two functions f,g computed by distance automata, answers "yes" if f <= (1-Δ ) g, "no" if f \not\leq g, and may answer "yes" or "no" in all other cases. This result highly refines previously known decidability results of the same type. The core argument behind this quasi-decision procedure is an algorithm which is able to provide an approximated finite presentation to the closure under products of sets of matrices over the tropical semiring. We also provide another theorem, of affine domination, which shows that previously known decision procedures for cost-automata have an improved precision when used over distance automata
The Strahler number of a parity game
The Strahler number of a rooted tree is the largest height of a perfect binary tree that is its minor. The Strahler number of a parity game is proposed to be defined as the smallest Strahler number of the tree of any of its attractor decompositions. It is proved that parity games can be solved in quasi-linear space and in time that is polynomial in the number of vertices~n and linear in (d/2k)k, where d is the number of priorities and k is the Strahler number. This complexity is quasi-polynomial because the Strahler number is at most logarithmic in the number of vertices. The proof is based on a new construction of small Strahler-universal trees.
It is shown that the Strahler number of a parity game is a robust parameter: it coincides with its alternative version based on trees of progress measures and with the register number defined by Lehtinen~(2018). It follows that parity games can be solved in quasi-linear space and in time that is polynomial in the number of vertices and linear in (d/2k)k, where k is the register number. This significantly improves the running times and space achieved for parity games of bounded register number by Lehtinen (2018) and by Parys (2020).
The running time of the algorithm based on small Strahler-universal trees yields a novel trade-off kâ
lg(d/k)=O(logn) between the two natural parameters that measure the structural complexity of a parity game, which allows solving parity games in polynomial time. This includes as special cases the asymptotic settings of those parameters covered by the results of Calude, Jain Khoussainov, Li, and Stephan (2017), of JurdziĆski and LaziÄ (2017), and of Lehtinen (2018), and it significantly extends the range of such settings, for example to d=2O(lgnâ) and k=O(lgnââ)
Varieties of Cost Functions.
Regular cost functions were introduced as a quantitative generalisation of regular languages, retaining many of their equivalent characterisations and decidability properties. For instance, stabilisation monoids play the same role for cost functions as monoids do for regular languages. The purpose of this article is to further extend this algebraic approach by generalising two results on regular languages to cost functions: Eilenberg's varieties theorem and profinite equational characterisations of lattices of regular languages. This opens interesting new perspectives, but the specificities of cost functions introduce difficulties that prevent these generalisations to be straightforward. In contrast, although syntactic algebras can be defined for formal power series over a commutative ring, no such notion is known for series over semirings and in particular over the tropical semiring
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Register complexity and determinisation of max-plus automata
We survey some results about the sequentiality problem for max-plus automata and its generalisation, the register complexity problem for cost register automata. We compare classes of functions computed by maxplus automata and by cost register automata with respect to the notion of ambiguity. The two models are introduced gently, so the novice reader is welcome
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Comparison of max-plus automata and joint spectral radius of tropical matrices
Weighted automata over the max-plus semiring S are closely related to finitely generated semigroups of matrices over S. In this paper, we use results in automata theory to study two quantities associated with sets of matrices: the joint spectral radius and the ultimate rank. We prove that these two quantities are not computable over the tropical semiring, i.e. there is no algorithm that takes as input a finite set of matrices M and provides as output the joint spectral radius (resp. the ultimate rank) of M. On the other hand, we prove that the joint spectral radius is nevertheless approximable and we exhibit restricted cases in which the joint spectral radius and the ultimate rank are computable. To reach this aim, we study the problem of comparing functions computed by weighted automata over the tropical semiring. This problem is known to be undecidable and we prove that it remains undecidable in some specific subclasses of automata
Numerical study of homogeneous dynamo based on experimental von Karman type flows
A numerical study of the magnetic induction equation has been performed on
von Karman type flows. These flows are generated by two co-axial
counter-rotating propellers in cylindrical containers. Such devices are
currently used in the von Karman sodium (VKS) experiment designed to study
dynamo action in an unconstrained flow. The mean velocity fields have been
measured for different configurations and are introduced in a periodic
cylindrical kinematic dynamo code. Depending on the driving configuration, on
the poloidal to toroidal flow ratio and on the conductivity of boundaries, some
flows are observed to sustain growing magnetic fields for magnetic Reynolds
numbers accessible to a sodium experiment. The response of the flow to an
external magnetic field has also been studied: The results are in excellent
agreement with experimental results in the single propeller case but can differ
in the two propellers case.Comment: 20 pages, 32 figure
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Which Classes of Origin Graphs Are Generated by Transducers.
We study various models of transducers equipped with origin information. We consider the semantics of these models as particular graphs, called origin graphs, and we characterise the families of such graphs recognised by streaming string transducers
Electrical conductance of a 2D packing of metallic beads under thermal perturbation
Electrical conductivity measurements on a 2D packing of metallic beads have
been performed to study internal rearrangements in weakly pertubed granular
materials. Small thermal perturbations lead to large non gaussian conductance
fluctuations. These fluctuations are found to be intermittent and gathered in
bursts. The distributions of the waiting time between to peaks is found to be a
power law inside bursts. The exponent is independent of the bead network, the
intensity of the perturbation and external stress. these bursts are interpreted
as the signature of individual bead creep rather than collective vaults
reorganisations. We propose a simple model linking the exponent of the waiting
time distribution to the roughness exponent of the surface of the beads.Comment: 7 pages, 6 figure
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Classes of languages generated by the Kleene star of a word
In this paper, we study the lattice and the Boolean algebra, possibly closed under quotient, generated by the languages of the form uâ, where u is a word. We provide effective equational characterisations of these classes, i.e. one can decide using our descriptions whether a given regular language belongs or not to each of them
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