9 research outputs found
Capturing Logarithmic Space and Polynomial Time on Chordal Claw-Free Graphs
We show that the class of chordal claw-free graphs admits LREC-definable
canonization. LREC is a logic that extends first-order logic with counting
by an operator that allows it to formalize a limited form of recursion. This
operator can be evaluated in logarithmic space. It follows that there exists a
logarithmic-space canonization algorithm, and therefore a logarithmic-space
isomorphism test, for the class of chordal claw-free graphs. As a further
consequence, LREC captures logarithmic space on this graph class. Since
LREC is contained in fixed-point logic with counting, we also obtain that
fixed-point logic with counting captures polynomial time on the class of
chordal claw-free graphs.Comment: 34 pages, 13 figure
Isoperimetric Inequalities on Hexagonal Grids
We consider the edge- and vertex-isoperimetric probem on finite and infinite
hexagonal grids: For a subset W of the hexagonal grid of given cardinality, we
give a lower bound for the number of edges between W and its complement, and
lower bounds for the number of vertices in the neighborhood of W and for the
number of vertices in the boundary of W. For the infinite hexagonal grid the
given bounds are tight
Capturing Polynomial Time using Modular Decomposition
The question of whether there is a logic that captures polynomial time is one
of the main open problems in descriptive complexity theory and database theory.
In 2010 Grohe showed that fixed point logic with counting captures polynomial
time on all classes of graphs with excluded minors. We now consider classes of
graphs with excluded induced subgraphs. For such graph classes, an effective
graph decomposition, called modular decomposition, was introduced by Gallai in
1976. The graphs that are non-decomposable with respect to modular
decomposition are called prime. We present a tool, the Modular Decomposition
Theorem, that reduces (definable) canonization of a graph class C to
(definable) canonization of the class of prime graphs of C that are colored
with binary relations on a linearly ordered set. By an application of the
Modular Decomposition Theorem, we show that fixed point logic with counting
captures polynomial time on the class of permutation graphs. Within the proof
of the Modular Decomposition Theorem, we show that the modular decomposition of
a graph is definable in symmetric transitive closure logic with counting. We
obtain that the modular decomposition tree is computable in logarithmic space.
It follows that cograph recognition and cograph canonization is computable in
logarithmic space.Comment: 38 pages, 10 Figures. A preliminary version of this article appeared
in the Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer
Science (LICS '17
Capturing Polynomial Time and Logarithmic Space using Modular Decompositions and Limited Recursion
Diese Arbeit leistet BeitrĂ€ge im Bereich der deskriptiven KomplexitĂ€tstheorie. ZunĂ€chst beschĂ€ftigen wir uns mit der ungelösten Frage, ob es eine Logik gibt, welche die Klasse der Polynomialzeit-Eigenschaften (PTIME) charakterisiert. Wir betrachten Graphklassen, die unter induzierten Teilgraphen abgeschlossen sind. Auf solchen Graphklassen lĂ€sst sich die 1976 von Gallai eingefĂŒhrte modulare Zerlegung anwenden. Graphen, die durch modulare Zerlegung nicht zerlegbar sind, heiĂen prim. Wir stellen ein neues Werkzeug vor: das Modulare Zerlegungstheorem. Es reduziert (definierbare) Kanonisierung einer Graphklasse C auf (definierbare) Kanonisierung der Klasse aller primen Graphen aus C, die mit binĂ€ren Relationen auf einer linear geordneten Menge gefĂ€rbt sind. Mit Hilfe des Modularen Zerlegungstheorems zeigen wir, dass Fixpunktlogik mit ZĂ€hlen (FP+C) PTIME auf der Klasse aller Permutationsgraphen und auf der Klasse aller chordalen KomparabilitĂ€tsgraphen charakterisiert. Wir beweisen zudem, dass modulare ZerlegungsbĂ€ume in Symmetrisch-Transitive-HĂŒllen-Logik mit ZĂ€hlen (STC+C) definierbar und damit in logarithmischem Platz berechenbar sind.
Weiterhin definieren wir eine neue Logik fĂŒr die KomplexitĂ€tsklasse Logarithmischer Platz (LOGSPACE). Wir erweitern die Logik erster Stufe mit ZĂ€hlen um einen Operator, der eine in logarithmischem Platz berechenbare Form der Rekursion erlaubt. Die resultierende Logik LREC ist ausdrucksstĂ€rker als die Deterministisch-Transitive-HĂŒllen-Logik mit ZĂ€hlen (DTC+C) und echt in FP+C enthalten. Wir zeigen, dass LREC LOGSPACE auf gerichteten BĂ€umen charakterisiert. Zudem betrachten wir eine Erweiterung LREC= von LREC, die sich gegenĂŒber LREC durch bessere Abschlusseigenschaften auszeichnet und im Gegensatz zu LREC ausdrucksstĂ€rker als die Symmetrisch-Transitive-HĂŒllen-Logik (STC) ist. Wir beweisen, dass LREC= LOGSPACE sowohl auf der Klasse der Intervallgraphen als auch auf der Klasse der chordalen klauenfreien Graphen charakterisiert.This theses is making contributions to the field of descriptive complexity theory. First, we look at the main open problem in this area: the question of whether there exists a logic that captures polynomial time (PTIME). We consider classes of graphs that are closed under taking induced subgraphs. For such graph classes, an effective graph decomposition, called modular decomposition, was introduced by Gallai in 1976. The graphs that are non-decomposable with respect to modular decomposition are called prime. We present a tool, the Modular Decomposition Theorem, that reduces (definable) canonization of a graph class C to (definable) canonization of the class of prime graphs of C that are colored with binary relations on a linearly ordered set. By an application of the Modular Decomposition Theorem, we show that fixed-point logic with counting (FP+C) captures PTIME on the class of permutation graphs and the class of chordal comparability graphs. We also prove that the modular decomposition tree is definable in symmetric transitive closure logic with counting (STC+C), and therefore, computable in logarithmic space.
Further, we introduce a new logic for the complexity class logarithmic space (LOGSPACE). We extend first-order logic with counting by a new operator that allows it to formalize a limited form of recursion which can be evaluated in logarithmic space. We prove that the resulting logic LREC is strictly more expressive than deterministic transitive closure logic with counting (DTC+C) and that it is strictly contained in FP+C. We show that LREC captures LOGSPACE on the class of directed trees. We also study an extension LREC= of LREC that has nicer closure properties and that, unlike LREC, is more expressive than symmetric transitive closure logic (STC). We prove that LREC= captures LOGSPACE on the class of interval graphs and on the class of chordal claw-free graphs
Capturing Logarithmic Space and Polynomial Time on Chordal Claw-Free Graphs
We show that the class of chordal claw-free graphs admits LREC-definable
canonization. LREC is a logic that extends first-order logic with counting
by an operator that allows it to formalize a limited form of recursion. This
operator can be evaluated in logarithmic space. It follows that there exists a
logarithmic-space canonization algorithm, and therefore a logarithmic-space
isomorphism test, for the class of chordal claw-free graphs. As a further
consequence, LREC captures logarithmic space on this graph class. Since
LREC is contained in fixed-point logic with counting, we also obtain that
fixed-point logic with counting captures polynomial time on the class of
chordal claw-free graphs
Capturing Polynomial Time using Modular Decomposition
The question of whether there is a logic that captures polynomial time is one
of the main open problems in descriptive complexity theory and database theory.
In 2010 Grohe showed that fixed point logic with counting captures polynomial
time on all classes of graphs with excluded minors. We now consider classes of
graphs with excluded induced subgraphs. For such graph classes, an effective
graph decomposition, called modular decomposition, was introduced by Gallai in
1976. The graphs that are non-decomposable with respect to modular
decomposition are called prime. We present a tool, the Modular Decomposition
Theorem, that reduces (definable) canonization of a graph class C to
(definable) canonization of the class of prime graphs of C that are colored
with binary relations on a linearly ordered set. By an application of the
Modular Decomposition Theorem, we show that fixed point logic with counting
captures polynomial time on the class of permutation graphs. Within the proof
of the Modular Decomposition Theorem, we show that the modular decomposition of
a graph is definable in symmetric transitive closure logic with counting. We
obtain that the modular decomposition tree is computable in logarithmic space.
It follows that cograph recognition and cograph canonization is computable in
logarithmic space
L-Recursion and a new Logic for Logarithmic Space
We extend first-order logic with counting by a new operator that allows it to formalise a limited form of recursion which can be evaluated in logarithmic space. The resulting logic LREC has a data complexity in LOGSPACE, and it defines LOGSPACE-complete problems like deterministic reachability and Boolean formula evaluation. We prove that LREC is strictly more expressive than deterministic transitive closure logic with counting and incomparable in expressive power with symmetric transitive closure logic STC and transitive closure logic (with or without counting). LREC is strictly contained in fixed-point logic with counting FP+C. We also study an extension LREC = of LREC that has nicer closure properties and is more expressive than both LREC and STC, but is still contained in FP+C and has a data complexity in LOGSPACE. Our main results are that LREC captures LOGSPACE on the class of directed trees and that LREC = captures LOGSPACE on the class of interval graphs
L-Recursion and a new Logic for Logarithmic Space
We extend first-order logic with counting by a new operator that allows it to
formalise a limited form of recursion which can be evaluated in logarithmic
space. The resulting logic LREC has a data complexity in LOGSPACE, and it
defines LOGSPACE-complete problems like deterministic reachability and Boolean
formula evaluation. We prove that LREC is strictly more expressive than
deterministic transitive closure logic with counting and incomparable in
expressive power with symmetric transitive closure logic STC and transitive
closure logic (with or without counting). LREC is strictly contained in
fixed-point logic with counting FPC. We also study an extension LREC= of LREC
that has nicer closure properties and is more expressive than both LREC and
STC, but is still contained in FPC and has a data complexity in LOGSPACE. Our
main results are that LREC captures LOGSPACE on the class of directed trees and
that LREC= captures LOGSPACE on the class of interval graphs