20,489 research outputs found
Definability Equals Recognizability for -Outerplanar Graphs
One of the most famous algorithmic meta-theorems states that every graph
property that can be defined by a sentence in counting monadic second order
logic (CMSOL) can be checked in linear time for graphs of bounded treewidth,
which is known as Courcelle's Theorem. These algorithms are constructed as
finite state tree automata, and hence every CMSOL-definable graph property is
recognizable. Courcelle also conjectured that the converse holds, i.e. every
recognizable graph property is definable in CMSOL for graphs of bounded
treewidth. We prove this conjecture for -outerplanar graphs, which are known
to have treewidth at most .Comment: 40 pages, 8 figure
The Complexity of Enriched Mu-Calculi
The fully enriched μ-calculus is the extension of the propositional
μ-calculus with inverse programs, graded modalities, and nominals. While
satisfiability in several expressive fragments of the fully enriched
μ-calculus is known to be decidable and ExpTime-complete, it has recently
been proved that the full calculus is undecidable. In this paper, we study the
fragments of the fully enriched μ-calculus that are obtained by dropping at
least one of the additional constructs. We show that, in all fragments obtained
in this way, satisfiability is decidable and ExpTime-complete. Thus, we
identify a family of decidable logics that are maximal (and incomparable) in
expressive power. Our results are obtained by introducing two new automata
models, showing that their emptiness problems are ExpTime-complete, and then
reducing satisfiability in the relevant logics to these problems. The automata
models we introduce are two-way graded alternating parity automata over
infinite trees (2GAPTs) and fully enriched automata (FEAs) over infinite
forests. The former are a common generalization of two incomparable automata
models from the literature. The latter extend alternating automata in a similar
way as the fully enriched μ-calculus extends the standard μ-calculus.Comment: A preliminary version of this paper appears in the Proceedings of the
33rd International Colloquium on Automata, Languages and Programming (ICALP),
2006. This paper has been selected for a special issue in LMC
Enumeration of -Fibonacci Paths using Infinite Weighted Automata
In this paper, we introduce a new family of generalized colored Motzkin
paths, where horizontal steps are colored by means of colors, where
is the th -Fibonacci number. We study the enumeration of this
family according to the length. For this, we use infinite weighted automata.Comment: arXiv admin note: substantial text overlap with arXiv:1310.244
Graph Spectral Properties of Deterministic Finite Automata
We prove that a minimal automaton has a minimal adjacency matrix rank and a
minimal adjacency matrix nullity using equitable partition (from graph spectra
theory) and Nerode partition (from automata theory). This result naturally
introduces the notion of matrix rank into a regular language L, the minimal
adjacency matrix rank of a deterministic automaton that recognises L. We then
define and focus on rank-one languages: the class of languages for which the
rank of minimal automaton is one. We also define the expanded canonical
automaton of a rank-one language.Comment: This paper has been accepted at the following conference: 18th
International Conference on Developments in Language Theory (DLT 2014),
August 26 - 29, 2014, Ekaterinburg, Russi
Computing Aggregate Properties of Preimages for 2D Cellular Automata
Computing properties of the set of precursors of a given configuration is a
common problem underlying many important questions about cellular automata.
Unfortunately, such computations quickly become intractable in dimension
greater than one. This paper presents an algorithm --- incremental aggregation
--- that can compute aggregate properties of the set of precursors
exponentially faster than na{\"i}ve approaches. The incremental aggregation
algorithm is demonstrated on two problems from the two-dimensional binary Game
of Life cellular automaton: precursor count distributions and higher-order mean
field theory coefficients. In both cases, incremental aggregation allows us to
obtain new results that were previously beyond reach
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