63 research outputs found
An introduction to finite automata and their connection to logic
This is a tutorial on finite automata. We present the standard material on
determinization and minimization, as well as an account of the equivalence of
finite automata and monadic second-order logic. We conclude with an
introduction to the syntactic monoid, and as an application give a proof of the
equivalence of first-order definability and aperiodicity
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
Algebraic Characterization of the Alternation Hierarchy in FO^2[<] on Finite Words
We give an algebraic characterization of the quantifier alternation hierarchy in first-order two-variable logic on finite words. As a result, we obtain a new proof that this hierarchy is strict. We also show that the first two levels of the hierarchy have decidable membership problems, and conjecture an algebraic decision procedure for the other levels
Wreath Products of Forest Algebras, with Applications to Tree Logics
We use the recently developed theory of forest algebras to find algebraic
characterizations of the languages of unranked trees and forests definable in
various logics. These include the temporal logics CTL and EF, and first-order
logic over the ancestor relation. While the characterizations are in general
non-effective, we are able to use them to formulate necessary conditions for
definability and provide new proofs that a number of languages are not
definable in these logics
Piecewise testable tree languages
This paper presents a decidable characterization of tree languages that can
be defined by a boolean combination of Sigma_1 sentences. This is a tree
extension of the Simon theorem, which says that a string language can be
defined by a boolean combination of Sigma_1 sentences if and only if its
syntactic monoid is J-trivial
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
Applications of the theory of automata in enumeration
AbstractA general method for applying techniques from the theory of automata to enumeration problems is developed, and several applications are given. These include the enumeration of reduced words in a finite group action and a new proof of the MacMahon Master Theorem
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