3,585 research outputs found
Counting Homomorphisms from Hypergraphs of Bounded Generalised Hypertree Width: A Logical Characterisation
We introduce the 2-sorted counting logic that expresses properties of
hypergraphs. This logic has available k variables to address hyperedges, an
unbounded number of variables to address vertices, and atomic formulas E(e,v)
to express that a vertex v is contained in a hyperedge e. We show that two
hypergraphs H, H' satisfy the same sentences of the logic if, and only
if, they are homomorphism indistinguishable over the class of hypergraphs of
generalised hypertree width at most k. Here, H, H' are called homomorphism
indistinguishable over a class C if for every hypergraph G in C the number of
homomorphisms from G to H equals the number of homomorphisms from G to H'. This
result can be viewed as a generalisation (from graphs to hypergraphs) of a
result by Dvorak (2010) stating that any two (undirected, simple, finite)
graphs H, H' are indistinguishable by the (k+1)-variable counting logic
if, and only if, they are homomorphism indistinguishable on the class
of graphs of tree width at most k
NiMo syntax: part 1
Many formalisms for the specification for concurrent and distributed systems have emerged. In particular considering boxes and strings approaches. Examples are action calculi, rewriting logic and graph rewriting, bigraphs. The boxes and string metaphor is addressed with different levels of granularity. One of the approaches is to consider a process network as an hypergraph. Based in this general framework, we encode NiMo nets as a class of Annotated hypergraphs. This class is defined by giving the alphabet and the operations used to construct such programs. Therefore we treat only editing operations on labelled hypergraphs and afterwards how this editing operation affects the graph. Graph transformation (execution rules) is not covered here.Postprint (published version
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
On structures in hypergraphs of models of a theory
We define and study structural properties of hypergraphs of models of a
theory including lattice ones. Characterizations for the lattice properties of
hypergraphs of models of a theory, as well as for structures on sets of
isomorphism types of models of a theory, are given
Achieving New Upper Bounds for the Hypergraph Duality Problem through Logic
The hypergraph duality problem DUAL is defined as follows: given two simple
hypergraphs and , decide whether
consists precisely of all minimal transversals of (in which case
we say that is the dual of ). This problem is
equivalent to deciding whether two given non-redundant monotone DNFs are dual.
It is known that non-DUAL, the complementary problem to DUAL, is in
, where
denotes the complexity class of all problems that after a nondeterministic
guess of bits can be decided (checked) within complexity class
. It was conjectured that non-DUAL is in . In this paper we prove this conjecture and actually
place the non-DUAL problem into the complexity class which is a subclass of . We here refer to the logtime-uniform version of
, which corresponds to , i.e., first order
logic augmented by counting quantifiers. We achieve the latter bound in two
steps. First, based on existing problem decomposition methods, we develop a new
nondeterministic algorithm for non-DUAL that requires to guess
bits. We then proceed by a logical analysis of this algorithm, allowing us to
formulate its deterministic part in . From this result, by
the well known inclusion , it follows
that DUAL belongs also to . Finally, by exploiting
the principles on which the proposed nondeterministic algorithm is based, we
devise a deterministic algorithm that, given two hypergraphs and
, computes in quadratic logspace a transversal of
missing in .Comment: Restructured the presentation in order to be the extended version of
a paper that will shortly appear in SIAM Journal on Computin
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