34,042 research outputs found
Linearly bounded infinite graphs
Linearly bounded Turing machines have been mainly studied as acceptors for
context-sensitive languages. We define a natural class of infinite automata
representing their observable computational behavior, called linearly bounded
graphs. These automata naturally accept the same languages as the linearly
bounded machines defining them. We present some of their structural properties
as well as alternative characterizations in terms of rewriting systems and
context-sensitive transductions. Finally, we compare these graphs to rational
graphs, which are another class of automata accepting the context-sensitive
languages, and prove that in the bounded-degree case, rational graphs are a
strict sub-class of linearly bounded graphs
Guarded Second-Order Logic, Spanning Trees, and Network Flows
According to a theorem of Courcelle monadic second-order logic and guarded
second-order logic (where one can also quantify over sets of edges) have the
same expressive power over the class of all countable -sparse hypergraphs.
In the first part of the present paper we extend this result to hypergraphs of
arbitrary cardinality. In the second part, we present a generalisation dealing
with methods to encode sets of vertices by single vertices
Finiteness conditions for graph algebras over tropical semirings
Connection matrices for graph parameters with values in a field have been
introduced by M. Freedman, L. Lov{\'a}sz and A. Schrijver (2007). Graph
parameters with connection matrices of finite rank can be computed in
polynomial time on graph classes of bounded tree-width. We introduce join
matrices, a generalization of connection matrices, and allow graph parameters
to take values in the tropical rings (max-plus algebras) over the real numbers.
We show that rank-finiteness of join matrices implies that these graph
parameters can be computed in polynomial time on graph classes of bounded
clique-width. In the case of graph parameters with values in arbitrary
commutative semirings, this remains true for graph classes of bounded linear
clique-width. B. Godlin, T. Kotek and J.A. Makowsky (2008) showed that
definability of a graph parameter in Monadic Second Order Logic implies rank
finiteness. We also show that there are uncountably many integer valued graph
parameters with connection matrices or join matrices of fixed finite rank. This
shows that rank finiteness is a much weaker assumption than any definability
assumption.Comment: 12 pages, accepted for presentation at FPSAC 2014 (Chicago, June 29
-July 3, 2014), to appear in Discrete Mathematics and Theoretical Computer
Scienc
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