40 research outputs found
On FO2 quantifier alternation over words
We show that each level of the quantifier alternation hierarchy within
FO^2[<] -- the 2-variable fragment of the first order logic of order on words
-- is a variety of languages. We then use the notion of condensed rankers, a
refinement of the rankers defined by Weis and Immerman, to produce a decidable
hierarchy of varieties which is interwoven with the quantifier alternation
hierarchy -- and conjecturally equal to it. It follows that the latter
hierarchy is decidable within one unit: given a formula alpha in FO^2[<], one
can effectively compute an integer m such that alpha is equivalent to a formula
with at most m+1 alternating blocks of quantifiers, but not to a formula with
only m-1 blocks. This is a much more precise result than what is known about
the quantifier alternation hierarchy within FO[<], where no decidability result
is known beyond the very first levels
Going higher in the First-order Quantifier Alternation Hierarchy on Words
We investigate the quantifier alternation hierarchy in first-order logic on
finite words. Levels in this hierarchy are defined by counting the number of
quantifier alternations in formulas. We prove that one can decide membership of
a regular language to the levels (boolean combination of
formulas having only 1 alternation) and (formulas having only 2
alternations beginning with an existential block). Our proof works by
considering a deeper problem, called separation, which, once solved for lower
levels, allows us to solve membership for higher levels
The Complexity of Finding Reset Words in Finite Automata
We study several problems related to finding reset words in deterministic
finite automata. In particular, we establish that the problem of deciding
whether a shortest reset word has length k is complete for the complexity class
DP. This result answers a question posed by Volkov. For the search problems of
finding a shortest reset word and the length of a shortest reset word, we
establish membership in the complexity classes FP^NP and FP^NP[log],
respectively. Moreover, we show that both these problems are hard for
FP^NP[log]. Finally, we observe that computing a reset word of a given length
is FNP-complete.Comment: 16 pages, revised versio
Semigroups with operation-compatible Green’s quasiorders
We call a semigroup on which the Green’s quasiorder ≤ J (≤ L, ≤ R) is operation-compatible, a ≤ J-compatible (≤ L-compatible, ≤ R-compatible) semigroup. We study the classes of ≤ J-compatible, ≤ L-compatible and ≤ R-compatible semigroups, using the smallest operation-compatible quasiorders containing Green’s quasiorders as a tool. We prove a number of results, including the following. The class of ≤ L-compatible (≤ R-compatible) semigroups is closed under taking homomorphic images. A regular periodic semigroup is ≤ J-compatible if and only if it is a semilattice of simple semigroups. Every negatively orderable semigroup can be embedded into a negatively orderable ≤ J-compatible semigroup
Newton series, coinductively
We present a comparative study of four product operators on weighted languages: (i) the convolution, (ii) the shue, (iii) the inltration, and (iv) the Hadamard product. Exploiting the fact that the set of weighted languages is a nal coalgebra, we use coinduction to prove that a classical operator from dierence calculus in mathematics: the Newton transform, generalises (from innite sequences) to weighted lan- guages. We show that the Newton transform is an isomorphism of rings that transforms the Hadamard product of two weighted languages into an inltration product, and we develop various representations for the Newton transform of a language, together with concrete calculation rules for computing them
Partially Ordered Two-way B\"uchi Automata
We introduce partially ordered two-way B\"uchi automata and characterize
their expressive power in terms of fragments of first-order logic FO[<].
Partially ordered two-way B\"uchi automata are B\"uchi automata which can
change the direction in which the input is processed with the constraint that
whenever a state is left, it is never re-entered again. Nondeterministic
partially ordered two-way B\"uchi automata coincide with the first-order
fragment Sigma2. Our main contribution is that deterministic partially ordered
two-way B\"uchi automata are expressively complete for the first-order fragment
Delta2. As an intermediate step, we show that deterministic partially ordered
two-way B\"uchi automata are effectively closed under Boolean operations.
A small model property yields coNP-completeness of the emptiness problem and
the inclusion problem for deterministic partially ordered two-way B\"uchi
automata.Comment: The results of this paper were presented at CIAA 2010; University of
Stuttgart, Computer Scienc
Church-Rosser Systems, Codes with Bounded Synchronization Delay and Local Rees Extensions
What is the common link, if there is any, between Church-Rosser systems,
prefix codes with bounded synchronization delay, and local Rees extensions? The
first obvious answer is that each of these notions relates to topics of
interest for WORDS: Church-Rosser systems are certain rewriting systems over
words, codes are given by sets of words which form a basis of a free submonoid
in the free monoid of all words (over a given alphabet) and local Rees
extensions provide structural insight into regular languages over words. So, it
seems to be a legitimate title for an extended abstract presented at the
conference WORDS 2017. However, this work is more ambitious, it outlines some
less obvious but much more interesting link between these topics. This link is
based on a structure theory of finite monoids with varieties of groups and the
concept of local divisors playing a prominent role. Parts of this work appeared
in a similar form in conference proceedings where proofs and further material
can be found.Comment: Extended abstract of an invited talk given at WORDS 201
Closures of regular languages for profinite topologies
The Pin-Reutenauer algorithm gives a method, that can be viewed as a descriptive procedure, to compute the closure in the free group of a regular language with respect to the Hall topology. A similar descriptive procedure is shown to hold for the
pseudovariety A of aperiodic semigroups, where the closure is taken in the free aperiodic omega-semigroup. It is inherited by a
subpseudovariety of a given pseudovariety if both of them enjoy the property of being full. The pseudovariety A, as well as some of
its subpseudovarieties are shown to be full. The interest in such descriptions stems from the fact that, for each of the main pseudovarieties V in our examples, the closures of two regular languages are disjoint if and only if the languages can be separated by a language whose syntactic semigroup lies in V. In the cases of A and of the pseudovariety DA of semigroups in which all regular elements are idempotents, this is a new result.PESSOA French-Portuguese project
Egide-Grices 11113YM, "Automata, profinite semigroups and symbolic dynamics".FCT -- Fundação para a Ciência e a Tecnologia, respectively under the projects
PEst-C/MAT/UI0144/2011 and PEst-C/MAT/UI0013/2011.ANR 2010 BLAN 0202 01 FREC.AutoMathA programme of the European Science Foundation.FCT and the project PTDC/MAT/65481/2006 which was partly funded by the European Community Fund FEDER