26 research outputs found
Finite-Degree Predicates and Two-Variable First-Order Logic
We consider two-variable first-order logic on finite words with a fixed
number of quantifier alternations. We show that all languages with a neutral
letter definable using the order and finite-degree predicates are also
definable with the order predicate only. From this result we derive the
separation of the alternation hierarchy of two-variable logic on this
signature
A Crevice on the Crane Beach: Finite-Degree Predicates
First-order logic (FO) over words is shown to be equiexpressive with FO
equipped with a restricted set of numerical predicates, namely the order, a
binary predicate MSB, and the finite-degree predicates: FO[Arb] = FO[<,
MSB, Fin].
The Crane Beach Property (CBP), introduced more than a decade ago, is true of
a logic if all the expressible languages admitting a neutral letter are
regular.
Although it is known that FO[Arb] does not have the CBP, it is shown here
that the (strong form of the) CBP holds for both FO[<, Fin] and FO[<, MSB].
Thus FO[<, Fin] exhibits a form of locality and the CBP, and can still express
a wide variety of languages, while being one simple predicate away from the
expressive power of FO[Arb]. The counting ability of FO[<, Fin] is studied as
an application.Comment: Submitte
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
Monadic Second-Order Logic with Arbitrary Monadic Predicates
We study Monadic Second-Order Logic (MSO) over finite words, extended with
(non-uniform arbitrary) monadic predicates. We show that it defines a class of
languages that has algebraic, automata-theoretic and machine-independent
characterizations. We consider the regularity question: given a language in
this class, when is it regular? To answer this, we show a substitution property
and the existence of a syntactical predicate.
We give three applications. The first two are to give very simple proofs that
the Straubing Conjecture holds for all fragments of MSO with monadic
predicates, and that the Crane Beach Conjecture holds for MSO with monadic
predicates. The third is to show that it is decidable whether a language
defined by an MSO formula with morphic predicates is regular.Comment: Conference version: MFCS'14, Mathematical Foundations of Computer
Science Journal version: ToCL'17, Transactions on Computational Logi
Continuity of Functional Transducers: A Profinite Study of Rational Functions
A word-to-word function is continuous for a class of languages~
if its inverse maps _languages to~. This notion
provides a basis for an algebraic study of transducers, and was integral to the
characterization of the sequential transducers computable in some circuit
complexity classes.
Here, we report on the decidability of continuity for functional transducers
and some standard classes of regular languages. To this end, we develop a
robust theory rooted in the standard profinite analysis of regular languages.
Since previous algebraic studies of transducers have focused on the sole
structure of the underlying input automaton, we also compare the two algebraic
approaches. We focus on two questions: When are the automaton structure and the
continuity properties related, and when does continuity propagate to
superclasses
Continuity and Rational Functions
A word-to-word function is continuous for a class of languages V if its inverse maps V languages to V. This notion provides a basis for an algebraic study of transducers, and was integral to the characterization of the sequential transducers computable in some circuit complexity classes.
Here, we report on the decidability of continuity for functional transducers and some standard classes of regular languages. Previous algebraic studies of transducers have focused on the structure of the underlying input automaton, disregarding the output. We propose a comparison of the two algebraic approaches through two questions: When are the automaton structure and the continuity properties related, and when does continuity propagate to superclasses
Two-variable first order logic with modular predicates over words
We consider first order formulae over the signature consisting of the symbols of the alphabet, the symbol < (interpreted as a linear order) and the set MOD of modular numerical predicates. We study the expressive power of FO 2 [<, MOD], the two-variable first order logic over this signature, interpreted over finite words. We give an algebraic characterization of the corresponding regular languages in terms of their syntactic morphisms and we also give simple unambiguous regular expressions for them. It follows that one can decide whether a given regular language is captured by FO 2 [<, MOD]. Our proofs rely on a combination of arguments from semigroup theory (stamps), model theory (Ehrenfeucht-Fraïssé games) and combinatorics
Topological Sorting with Regular Constraints
We introduce the constrained topological sorting problem (CTS): given a regular language K and a directed acyclic graph G with labeled vertices, determine if G has a topological sort that forms a word in K. This natural problem applies to several settings, e.g., scheduling with costs or verifying concurrent programs. We consider the problem CTS[K] where the target language K is fixed, and study its complexity depending on K. We show that CTS[K] is tractable when K falls in several language families, e.g., unions of monomials, which can be used for pattern matching. However, we show that CTS[K] is NP-hard for K = (ab)^* and introduce a shuffle reduction technique to show hardness for more languages. We also study the special case of the constrained shuffle problem (CSh), where the input graph is a disjoint union of strings, and show that CSh[K] is additionally tractable when K is a group language or a union of district group monomials. We conjecture that a dichotomy should hold on the complexity of CTS[K] or CSh[K] depending on K, and substantiate this by proving a coarser dichotomy under a different problem phrasing which ensures that tractable languages are closed under common operators
Locality and Centrality: The Variety ZG
We study the variety ZG of monoids where the elements that belong to a group
are central, i.e., commute with all other elements. We show that ZG is local,
that is, the semidirect product ZG * D of ZG by definite semigroups is equal to
LZG, the variety of semigroups where all local monoids are in ZG. Our main
result is thus: ZG * D = LZG. We prove this result using Straubing's delay
theorem, by considering paths in the category of idempotents. In the process,
we obtain the characterization ZG = MNil \vee Com, and also characterize the ZG
languages, i.e., the languages whose syntactic monoid is in ZG: they are
precisely the languages that are finite unions of disjoint shuffles of
singleton languages and regular commutative languages.Comment: 31 pages. Corrected small errors and improved the presentation.
Submitte
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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 u*, 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