27 research outputs found
Unambiguous Separators for Tropical Tree Automata
In this paper we show that given a max-plus automaton (over trees, and with real weights) computing a function f and a min-plus automaton (similar) computing a function g such that f ? g, there exists effectively an unambiguous tropical automaton computing h such that f ? h ? g.
This generalizes a result of Lombardy and Mairesse of 2006 stating that series which are both max-plus and min-plus rational are unambiguous. This generalization goes in two directions: trees are considered instead of words, and separation is established instead of characterization (separation implies characterization). The techniques in the two proofs are very different
Boundedness of Conjunctive Regular Path Queries
We study the boundedness problem for unions of conjunctive regular path queries with inverses (UC2RPQs). This is the problem of, given a UC2RPQ, checking whether it is equivalent to a union of conjunctive queries (UCQ). We show the problem to be ExpSpace-complete, thus coinciding with the complexity of containment for UC2RPQs. As a corollary, when a UC2RPQ is bounded, it is equivalent to a UCQ of at most triple-exponential size, and in fact we show that this bound is optimal. We also study better behaved classes of UC2RPQs, namely acyclic UC2RPQs of bounded thickness, and strongly connected UCRPQs, whose boundedness problem is, respectively, PSpace-complete and Pi_2^P-complete. Most upper bounds exploit results on limitedness for distance automata, in particular extending the model with alternation and two-wayness, which may be of independent interest
Recommended from our members
Approximate comparison of distance automata
Distance automata are automata weighted over the semiring (N∪ {∞}, min,+) (the tropical semiring). Such automata compute functions from words to N
∪{∞} 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-ε ) 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
Opérations polynomiales et hiérarchies de concaténation
RésuméSoit C une classe de langages. Notons Pol(C) la fermeture polynomiale de C. Pol(C) est la plus petite classe de langages contenant C et fermée par union finie et produit marqué LaL' où a est une lettre. Nous déterminons les clôtures polynomiales de diverses classes de langages rationnels puis nous étudions les propriétés des fermetures polynomiales. Par exemple, si C est fermée par quotients (resp. quotients et morphisme inverse), alors il en est de même de Pol(C). Notre résultat principal montre que si C est une algèbre de Boole fermée par résiduels alors Pol(C) est fermée par intersection. Comme application, nous affinons la hiérarchie de concaténation introduite par Straubing et nous prouvons la décidabilité des niveaux 12 et 32 de cette hiérarchie.AbstractGiven a class C of languages, let Pol(C) be the polynomial closure of C, that is, the smallest class of languages containing C and closed under the operations union and marked product LaL', where a is a letter. We determine the polynomial closure of various classes of rational languages and we study the properties of polynomial closures. For instance, if C is closed under quotients (resp. quotients and inverse morphism) then Pol(C) has the same property. Our main result shows that if C is a boolean algebra closed under quotients then Pol(C) is closed under intersection. As an application, we refine the concatenation hierarchy introduced by Straubing and we show that the levels 12 and 32 of this hierarcy are decidable
Sampled Semantics of Timed Automata
Sampled semantics of timed automata is a finite approximation of their dense
time behavior. While the former is closer to the actual software or hardware
systems with a fixed granularity of time, the abstract character of the latter
makes it appealing for system modeling and verification. We study one aspect of
the relation between these two semantics, namely checking whether the system
exhibits some qualitative (untimed) behaviors in the dense time which cannot be
reproduced by any implementation with a fixed sampling rate. More formally, the
\emph{sampling problem} is to decide whether there is a sampling rate such that
all qualitative behaviors (the untimed language) accepted by a given timed
automaton in dense time semantics can be also accepted in sampled semantics. We
show that this problem is decidable
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
Weak Cost Register Automata are Still Powerful
We consider one of the weakest variants of cost register automata over a
tropical semiring, namely copyless cost register automata over
with updates using and increments. We show that this model can simulate,
in some sense, the runs of counter machines with zero-tests. We deduce that a
number of problems pertaining to that model are undecidable, in particular
equivalence, disproving a conjecture of Alur et al. from 2012. To emphasize how
weak these machines are, we also show that they can be expressed as a
restricted form of linearly-ambiguous weighted automata.Comment: 16 page