113 research outputs found
Kleene Algebra with Converse
International audienceThe equational theory generated by all algebras of binary relations with operations of union, composition, converse and reflexive transitive closure was studied by Bernátsky, Bloom, Ésik, and Stefanescu in 1995. We reformulate some of their proofs in syntactic and elementary terms, and we provide a new algorithm to decide the corresponding theory. This algorithm is both simpler and more efficient; it relies on an alternative automata construction, that allows us to prove that the considered equational theory lies in the complexity class PSPACE. Specific regular languages appear at various places in the proofs. Those proofs were made tractable by considering appropriate automata recognising those languages, and exploiting symmetries in those automata
Uniform generation in trace monoids
We consider the problem of random uniform generation of traces (the elements
of a free partially commutative monoid) in light of the uniform measure on the
boundary at infinity of the associated monoid. We obtain a product
decomposition of the uniform measure at infinity if the trace monoid has
several irreducible components-a case where other notions such as Parry
measures, are not defined. Random generation algorithms are then examined.Comment: Full version of the paper in MFCS 2015 with the same titl
Reachability problems for products of matrices in semirings
We consider the following matrix reachability problem: given square
matrices with entries in a semiring, is there a product of these matrices which
attains a prescribed matrix? We define similarly the vector (resp. scalar)
reachability problem, by requiring that the matrix product, acting by right
multiplication on a prescribed row vector, gives another prescribed row vector
(resp. when multiplied at left and right by prescribed row and column vectors,
gives a prescribed scalar). We show that over any semiring, scalar reachability
reduces to vector reachability which is equivalent to matrix reachability, and
that for any of these problems, the specialization to any is
equivalent to the specialization to . As an application of this result and
of a theorem of Krob, we show that when , the vector and matrix
reachability problems are undecidable over the max-plus semiring
. We also show that the matrix, vector, and scalar
reachability problems are decidable over semirings whose elements are
``positive'', like the tropical semiring .Comment: 21 page
Discounting in LTL
In recent years, there is growing need and interest in formalizing and
reasoning about the quality of software and hardware systems. As opposed to
traditional verification, where one handles the question of whether a system
satisfies, or not, a given specification, reasoning about quality addresses the
question of \emph{how well} the system satisfies the specification. One
direction in this effort is to refine the "eventually" operators of temporal
logic to {\em discounting operators}: the satisfaction value of a specification
is a value in , where the longer it takes to fulfill eventuality
requirements, the smaller the satisfaction value is.
In this paper we introduce an augmentation by discounting of Linear Temporal
Logic (LTL), and study it, as well as its combination with propositional
quality operators. We show that one can augment LTL with an arbitrary set of
discounting functions, while preserving the decidability of the model-checking
problem. Further augmenting the logic with unary propositional quality
operators preserves decidability, whereas adding an average-operator makes some
problems undecidable. We also discuss the complexity of the problem, as well as
various extensions
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Containment and equivalence of weighted automata: Probabilistic and max-plus cases
This paper surveys some results regarding decision problems for probabilistic and max-plus automata, such as containment and equivalence. Probabilistic and max-plus automata are part of the general family of weighted automata, whose semantics are maps from words to real values. Given two weighted automata, the equivalence problem asks whether their semantics are the same, and the containment problem whether one is point-wise smaller than the other one. These problems have been studied intensively and this paper will review some techniques used to show (un)decidability and state a list of open questions that still remain
Completeness and Incompleteness of Synchronous Kleene Algebra
Synchronous Kleene algebra (SKA), an extension of Kleene algebra (KA), was
proposed by Prisacariu as a tool for reasoning about programs that may execute
synchronously, i.e., in lock-step. We provide a countermodel witnessing that
the axioms of SKA are incomplete w.r.t. its language semantics, by exploiting a
lack of interaction between the synchronous product operator and the Kleene
star. We then propose an alternative set of axioms for SKA, based on Salomaa's
axiomatisation of regular languages, and show that these provide a sound and
complete characterisation w.r.t. the original language semantics.Comment: Accepted at MPC 201
Combinatorial Hopf algebras and Towers of Algebras
Bergeron and Li have introduced a set of axioms which guarantee that the
Grothendieck groups of a tower of algebras can be
endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap,
and independently Lam and Shimozono constructed dual graded graphs from
primitive elements in Hopf algebras. In this paper we apply the composition of
these constructions to towers of algebras. We show that if a tower
gives rise to graded dual Hopf algebras then we must
have where .Comment: 7 page
Spectral extension of the quantum group cotangent bundle
The structure of a cotangent bundle is investigated for quantum linear groups
GLq(n) and SLq(n). Using a q-version of the Cayley-Hamilton theorem we
construct an extension of the algebra of differential operators on SLq(n)
(otherwise called the Heisenberg double) by spectral values of the matrix of
right invariant vector fields. We consider two applications for the spectral
extension. First, we describe the extended Heisenberg double in terms of a new
set of generators -- the Weyl partners of the spectral variables. Calculating
defining relations in terms of these generators allows us to derive SLq(n) type
dynamical R-matrices in a surprisingly simple way. Second, we calculate an
evolution operator for the model of q-deformed isotropic top introduced by
A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we
present two possible expressions for it. The first one is a Riemann theta
function in the spectral variables. The second one is an almost free motion
evolution operator in terms of logarithms of the spectral variables. Relation
between the two operators is given by a modular functional equation for Riemann
theta function.Comment: 38 pages, no figure
A quantum analogue of the dihedral action on Grassmannians
In recent work, Launois and Lenagan have shown how to construct a cocycle
twisting of the quantum Grassmannian and an isomorphism of the twisted and
untwisted algebras that sends a given quantum minor to the minor whose index
set is permuted according to the -cycle , up to a power of
. This twisting is needed because does not naturally induce an
automorphism of the quantum Grassmannian, as it does classically and
semi-classically. We extend this construction to give a quantum analogue of the
action on the Grassmannian of the dihedral subgroup of generated by
and , the longest element, and this analogue takes the form of a
groupoid. We show that there is an induced action of this subgroup on the
torus-invariant prime ideals of the quantum Grassmannian and also show that
this subgroup acts on the totally nonnegative and totally positive
Grassmannians. Then we see that this dihedral subgroup action exists
classically, semi-classically (by Poisson automorphisms and anti-automorphisms,
a result of Yakimov) and in the quantum and nonnegative settings.Comment: 22 pages, 2 figures. Version 2: substantially revised organisation
and presentatio
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