58 research outputs found
Establishing a Connection Between Graph Structure, Logic, and Language Theory
The field of graph structure theory was given life by the Graph Minors Project of Robertson and Seymour, which developed many tools for understanding the way graphs relate to each other and culminated in the proof of the Graph Minors Theorem. One area of ongoing research in the field is attempting to strengthen the Graph Minors Theorem to sets of graphs, and sets of sets of graphs, and so on.
At the same time, there is growing interest in the applications of logic and formal languages to graph theory, and a significant amount of work in this field has recently been consolidated in the publication of a book by Courcelle and Engelfriet.
We investigate the potential applications of logic and formal languages to the field of graph structure theory, suggesting a new area of research which may provide fruitful
Logic Meets Algebra: the Case of Regular Languages
The study of finite automata and regular languages is a privileged meeting
point of algebra and logic. Since the work of Buchi, regular languages have
been classified according to their descriptive complexity, i.e. the type of
logical formalism required to define them. The algebraic point of view on
automata is an essential complement of this classification: by providing
alternative, algebraic characterizations for the classes, it often yields the
only opportunity for the design of algorithms that decide expressibility in
some logical fragment.
We survey the existing results relating the expressibility of regular
languages in logical fragments of MSO[S] with algebraic properties of their
minimal automata. In particular, we show that many of the best known results in
this area share the same underlying mechanics and rely on a very strong
relation between logical substitutions and block-products of pseudovarieties of
monoid. We also explain the impact of these connections on circuit complexity
theory.Comment: 37 page
On the Monadic Second-Order Transduction Hierarchy
We compare classes of finite relational structures via monadic second-order
transductions. More precisely, we study the preorder where we set C \subseteq K
if, and only if, there exists a transduction {\tau} such that
C\subseteq{\tau}(K). If we only consider classes of incidence structures we can
completely describe the resulting hierarchy. It is linear of order type
{\omega}+3. Each level can be characterised in terms of a suitable variant of
tree-width. Canonical representatives of the various levels are: the class of
all trees of height n, for each n \in N, of all paths, of all trees, and of all
grids
Logical and Algebraic Characterizations of Rational Transductions
Rational word languages can be defined by several equivalent means: finite
state automata, rational expressions, finite congruences, or monadic
second-order (MSO) logic. The robust subclass of aperiodic languages is defined
by: counter-free automata, star-free expressions, aperiodic (finite)
congruences, or first-order (FO) logic. In particular, their algebraic
characterization by aperiodic congruences allows to decide whether a regular
language is aperiodic.
We lift this decidability result to rational transductions, i.e.,
word-to-word functions defined by finite state transducers. In this context,
logical and algebraic characterizations have also been proposed. Our main
result is that one can decide if a rational transduction (given as a
transducer) is in a given decidable congruence class. We also establish a
transfer result from logic-algebra equivalences over languages to equivalences
over transductions. As a consequence, it is decidable if a rational
transduction is first-order definable, and we show that this problem is
PSPACE-complete
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