3,559 research outputs found
An effective characterization of the alternation hierarchy in two-variable logic
We characterize the languages in the individual levels of the quantifier
alternation hierarchy of first-order logic with two variables by identities.
This implies decidability of the individual levels. More generally we show that
the two-sided semidirect product of a decidable variety with the variety J is
decidable
The half-levels of the FO2 alternation hierarchy
© 2016, Springer Science+Business Media New York. The alternation hierarchy in two-variable first-order logic FO 2 [ < ] over words was shown to be decidable by Kufleitner and Weil, and independently by Krebs and Straubing. We consider a similar hierarchy, reminiscent of the half levels of the dot-depth hierarchy or the Straubing-Thérien hierarchy. The fragment Σm2 of FO 2 is defined by disallowing universal quantifiers and having at most m−1 nested negations. The Boolean closure of Σm2 yields the m th level of the FO 2 -alternation hierarchy. We give an effective characterization of Σm2, i.e., for every integer m one can decide whether a given regular language is definable in Σm2. Among other techniques, the proof relies on an extension of block products to ordered monoids
Block products and nesting negations in FO2
The alternation hierarchy in two-variable first-order logic FO 2 [∈ < ∈] over words was recently shown to be decidable by Kufleitner and Weil, and independently by Krebs and Straubing. In this paper we consider a similar hierarchy, reminiscent of the half levels of the dot-depth hierarchy or the Straubing-Thérien hierarchy. The fragment of FO 2 is defined by disallowing universal quantifiers and having at most m∈-∈1 nested negations. One can view as the formulas in FO 2 which have at most m blocks of quantifiers on every path of their parse tree, and the first block is existential. Thus, the m th level of the FO 2 -alternation hierarchy is the Boolean closure of. We give an effective characterization of, i.e., for every integer m one can decide whether a given regular language is definable by a two-variable first-order formula with negation nesting depth at most m. More precisely, for every m we give ω-terms U m and V m such that an FO 2 -definable language is in if and only if its ordered syntactic monoid satisfies the identity U m ∈V m. Among other techniques, the proof relies on an extension of block products to ordered monoids. © 2014 Springer International Publishing Switzerland
From algebra to logic: there and back again -- the story of a hierarchy
This is an extended survey of the results concerning a hierarchy of languages
that is tightly connected with the quantifier alternation hierarchy within the
two-variable fragment of first order logic of the linear order.Comment: Developments in Language Theory 2014, Ekaterinburg : Russian
Federation (2014
Tree Languages Defined in First-Order Logic with One Quantifier Alternation
We study tree languages that can be defined in \Delta_2 . These are tree
languages definable by a first-order formula whose quantifier prefix is forall
exists, and simultaneously by a first-order formula whose quantifier prefix is
. For the quantifier free part we consider two signatures, either the
descendant relation alone or together with the lexicographical order relation
on nodes. We provide an effective characterization of tree and forest languages
definable in \Delta_2 . This characterization is in terms of algebraic
equations. Over words, the class of word languages definable in \Delta_2 forms
a robust class, which was given an effective algebraic characterization by Pin
and Weil
Adding modular predicates to first-order fragments
We investigate the decidability of the definability problem for fragments of
first order logic over finite words enriched with modular predicates. Our
approach aims toward the most generic statements that we could achieve, which
successfully covers the quantifier alternation hierarchy of first order logic
and some of its fragments. We obtain that deciding this problem for each level
of the alternation hierarchy of both first order logic and its two-variable
fragment when equipped with all regular numerical predicates is not harder than
deciding it for the corresponding level equipped with only the linear order and
the successor. For two-variable fragments we also treat the case of the
signature containing only the order and modular predicates.Relying on some
recent results, this proves the decidability for each level of the alternation
hierarchy of the two-variable first order fragmentwhile in the case of the
first order logic the question remains open for levels greater than two.The
main ingredients of the proofs are syntactic transformations of first order
formulas as well as the algebraic framework of finite categories
Languages of Dot-depth One over Infinite Words
Over finite words, languages of dot-depth one are expressively complete for
alternation-free first-order logic. This fragment is also known as the Boolean
closure of existential first-order logic. Here, the atomic formulas comprise
order, successor, minimum, and maximum predicates. Knast (1983) has shown that
it is decidable whether a language has dot-depth one. We extend Knast's result
to infinite words. In particular, we describe the class of languages definable
in alternation-free first-order logic over infinite words, and we give an
effective characterization of this fragment. This characterization has two
components. The first component is identical to Knast's algebraic property for
finite words and the second component is a topological property, namely being a
Boolean combination of Cantor sets.
As an intermediate step we consider finite and infinite words simultaneously.
We then obtain the results for infinite words as well as for finite words as
special cases. In particular, we give a new proof of Knast's Theorem on
languages of dot-depth one over finite words.Comment: Presented at LICS 201
One Quantifier Alternation in First-Order Logic with Modular Predicates
Adding modular predicates yields a generalization of first-order logic FO
over words. The expressive power of FO[<,MOD] with order comparison and
predicates for has been investigated by Barrington,
Compton, Straubing and Therien. The study of FO[<,MOD]-fragments was initiated
by Chaubard, Pin and Straubing. More recently, Dartois and Paperman showed that
definability in the two-variable fragment FO2[<,MOD] is decidable. In this
paper we continue this line of work.
We give an effective algebraic characterization of the word languages in
Sigma2[<,MOD]. The fragment Sigma2 consists of first-order formulas in prenex
normal form with two blocks of quantifiers starting with an existential block.
In addition we show that Delta2[<,MOD], the largest subclass of Sigma2[<,MOD]
which is closed under negation, has the same expressive power as two-variable
logic FO2[<,MOD]. This generalizes the result FO2[<] = Delta2[<] of Therien and
Wilke to modular predicates. As a byproduct, we obtain another decidable
characterization of FO2[<,MOD]
Two-variable logics with some betweenness relations: Expressiveness, satisfiability and membership
We study two extensions of FO2[<], first-order logic interpreted in finite
words, in which formulas are restricted to use only two variables. We adjoin to
this language two-variable atomic formulas that say, "the letter appears
between positions and " and "the factor appears between positions
and ". These are, in a sense, the simplest properties that are not
expressible using only two variables.
We present several logics, both first-order and temporal, that have the same
expressive power, and find matching lower and upper bounds for the complexity
of satisfiability for each of these formulations. We give effective conditions,
in terms of the syntactic monoid of a regular language, for a property to be
expressible in these logics. This algebraic analysis allows us to prove, among
other things, that our new logics have strictly less expressive power than full
first-order logic FO[<]. Our proofs required the development of novel
techniques concerning factorizations of words
The Covering Problem
An important endeavor in computer science is to understand the expressive
power of logical formalisms over discrete structures, such as words. Naturally,
"understanding" is not a mathematical notion. This investigation requires
therefore a concrete objective to capture this understanding. In the
literature, the standard choice for this objective is the membership problem,
whose aim is to find a procedure deciding whether an input regular language can
be defined in the logic under investigation. This approach was cemented as the
right one by the seminal work of Sch\"utzenberger, McNaughton and Papert on
first-order logic and has been in use since then. However, membership questions
are hard: for several important fragments, researchers have failed in this
endeavor despite decades of investigation. In view of recent results on one of
the most famous open questions, namely the quantifier alternation hierarchy of
first-order logic, an explanation may be that membership is too restrictive as
a setting. These new results were indeed obtained by considering more general
problems than membership, taking advantage of the increased flexibility of the
enriched mathematical setting. This opens a promising research avenue and
efforts have been devoted at identifying and solving such problems for natural
fragments. Until now however, these problems have been ad hoc, most fragments
relying on a specific one. A unique new problem replacing membership as the
right one is still missing. The main contribution of this paper is a suitable
candidate to play this role: the Covering Problem. We motivate this problem
with 3 arguments. First, it admits an elementary set theoretic formulation,
similar to membership. Second, we are able to reexplain or generalize all known
results with this problem. Third, we develop a mathematical framework and a
methodology tailored to the investigation of this problem
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