7,347 research outputs found
Proof equivalence in MLL is PSPACE-complete
MLL proof equivalence is the problem of deciding whether two proofs in
multiplicative linear logic are related by a series of inference permutations.
It is also known as the word problem for star-autonomous categories. Previous
work has shown the problem to be equivalent to a rewiring problem on proof
nets, which are not canonical for full MLL due to the presence of the two
units. Drawing from recent work on reconfiguration problems, in this paper it
is shown that MLL proof equivalence is PSPACE-complete, using a reduction from
Nondeterministic Constraint Logic. An important consequence of the result is
that the existence of a satisfactory notion of proof nets for MLL with units is
ruled out (under current complexity assumptions). The PSPACE-hardness result
extends to equivalence of normal forms in MELL without units, where the
weakening rule for the exponentials induces a similar rewiring problem.Comment: Journal version of: Willem Heijltjes and Robin Houston. No proof nets
for MLL with units: Proof equivalence in MLL is PSPACE-complete. In Proc.
Joint Meeting of the 23rd EACSL Annual Conference on Computer Science Logic
and the 29th Annual ACM/IEEE Symposium on Logic in Computer Science, 201
Rainbow Graphs and Switching Classes
A rainbow graph is a graph that admits a vertex-coloring such that every
color appears exactly once in the neighborhood of each vertex. We investigate
some properties of rainbow graphs. In particular, we show that there is a
bijection between the isomorphism classes of n-rainbow graphs on 2n vertices
and the switching classes of graphs on n vertices.Comment: Added more reference, fixed some typos (revision for journal
submission
On the hardness of switching to a small number of edges
Seidel's switching is a graph operation which makes a given vertex adjacent
to precisely those vertices to which it was non-adjacent before, while keeping
the rest of the graph unchanged. Two graphs are called switching-equivalent if
one can be made isomorphic to the other one by a sequence of switches.
Jel\'inkov\'a et al. [DMTCS 13, no. 2, 2011] presented a proof that it is
NP-complete to decide if the input graph can be switched to contain at most a
given number of edges. There turns out to be a flaw in their proof. We present
a correct proof.
Furthermore, we prove that the problem remains NP-complete even when
restricted to graphs whose density is bounded from above by an arbitrary fixed
constant. This partially answers a question of Matou\v{s}ek and Wagner
[Discrete Comput. Geom. 52, no. 1, 2014].Comment: 19 pages, 7 figures. An extended abstract submitted to COCOON 201
On the Minimum Degree up to Local Complementation: Bounds and Complexity
The local minimum degree of a graph is the minimum degree reached by means of
a series of local complementations. In this paper, we investigate on this
quantity which plays an important role in quantum computation and quantum error
correcting codes. First, we show that the local minimum degree of the Paley
graph of order p is greater than sqrt{p} - 3/2, which is, up to our knowledge,
the highest known bound on an explicit family of graphs. Probabilistic methods
allows us to derive the existence of an infinite number of graphs whose local
minimum degree is linear in their order with constant 0.189 for graphs in
general and 0.110 for bipartite graphs. As regards the computational complexity
of the decision problem associated with the local minimum degree, we show that
it is NP-complete and that there exists no k-approximation algorithm for this
problem for any constant k unless P = NP.Comment: 11 page
Equiangular lines in Euclidean spaces
We obtain several new results contributing to the theory of real equiangular
line systems. Among other things, we present a new general lower bound on the
maximum number of equiangular lines in d dimensional Euclidean space; we
describe the two-graphs on 12 vertices; and we investigate Seidel matrices with
exactly three distinct eigenvalues. As a result, we improve on two
long-standing upper bounds regarding the maximum number of equiangular lines in
dimensions d=14, and d=16. Additionally, we prove the nonexistence of certain
regular graphs with four eigenvalues, and correct some tables from the
literature.Comment: 24 pages, to appear in JCTA. Corrected an entry in Table
- …