75,682 research outputs found
Randomisation and Derandomisation in Descriptive Complexity Theory
We study probabilistic complexity classes and questions of derandomisation
from a logical point of view. For each logic L we introduce a new logic BPL,
bounded error probabilistic L, which is defined from L in a similar way as the
complexity class BPP, bounded error probabilistic polynomial time, is defined
from PTIME. Our main focus lies on questions of derandomisation, and we prove
that there is a query which is definable in BPFO, the probabilistic version of
first-order logic, but not in Cinf, finite variable infinitary logic with
counting. This implies that many of the standard logics of finite model theory,
like transitive closure logic and fixed-point logic, both with and without
counting, cannot be derandomised. Similarly, we present a query on ordered
structures which is definable in BPFO but not in monadic second-order logic,
and a query on additive structures which is definable in BPFO but not in FO.
The latter of these queries shows that certain uniform variants of AC0
(bounded-depth polynomial sized circuits) cannot be derandomised. These results
are in contrast to the general belief that most standard complexity classes can
be derandomised. Finally, we note that BPIFP+C, the probabilistic version of
fixed-point logic with counting, captures the complexity class BPP, even on
unordered structures
A Polynomial time Algorithm for Hamilton Cycle with maximum Degree 3
Based on the famous Rotation-Extension technique, by creating the new
concepts and methods: broad cycle, main segment, useful cut and insert,
destroying edges for a main segment, main goal Hamilton cycle, depth-first
search tree, we develop a polynomial time algorithm for a famous NPC: the
Hamilton cycle problem. Thus we proved that NP=P. The key points of this paper
are: 1) there are two ways to get a Hamilton cycle in exponential time: a full
permutation of n vertices; or, chose n edges from all k edges, and check all
possible combinations. The main problem is: how to avoid checking all
combinations of n edges from all edges. My algorithm can avoid this. Lemma 1
and lemma 2 are very important. They are the foundation that we always can get
a good branch in the depth-first search tree and can get a series of destroying
edges (all are bad edges) for this good branch in polynomial time. The
extraordinary insights are: destroying edges, a tree contains each main segment
at most one time at the same time, and dynamic combinations. The difficult part
is to understand how to construct a main segment's series of destroying edges
by dynamic combinations (see the proof of lemma 4). The proof logic is: if
there is at least on Hamilton cycle in the graph, we always can do useful cut
and inserts until a Hamilton cycle is got. The times of useful cut and inserts
are polynomial. So if at any step we cannot have a useful cut and insert, this
means that there are no Hamilton cycles in the graph.Comment: 49 pages. This time, I add a detailed polynomial time algorithm and
proof for 3S
Unification in the Description Logic EL
The Description Logic EL has recently drawn considerable attention since, on
the one hand, important inference problems such as the subsumption problem are
polynomial. On the other hand, EL is used to define large biomedical
ontologies. Unification in Description Logics has been proposed as a novel
inference service that can, for example, be used to detect redundancies in
ontologies. The main result of this paper is that unification in EL is
decidable. More precisely, EL-unification is NP-complete, and thus has the same
complexity as EL-matching. We also show that, w.r.t. the unification type, EL
is less well-behaved: it is of type zero, which in particular implies that
there are unification problems that have no finite complete set of unifiers.Comment: 31page
On Brambles, Grid-Like Minors, and Parameterized Intractability of Monadic Second-Order Logic
Brambles were introduced as the dual notion to treewidth, one of the most
central concepts of the graph minor theory of Robertson and Seymour. Recently,
Grohe and Marx showed that there are graphs G, in which every bramble of order
larger than the square root of the treewidth is of exponential size in |G|. On
the positive side, they show the existence of polynomial-sized brambles of the
order of the square root of the treewidth, up to log factors. We provide the
first polynomial time algorithm to construct a bramble in general graphs and
achieve this bound, up to log-factors. We use this algorithm to construct
grid-like minors, a replacement structure for grid-minors recently introduced
by Reed and Wood, in polynomial time. Using the grid-like minors, we introduce
the notion of a perfect bramble and an algorithm to find one in polynomial
time. Perfect brambles are brambles with a particularly simple structure and
they also provide us with a subgraph that has bounded degree and still large
treewidth; we use them to obtain a meta-theorem on deciding certain
parameterized subgraph-closed problems on general graphs in time singly
exponential in the parameter.
The second part of our work deals with providing a lower bound to Courcelle's
famous theorem, stating that every graph property that can be expressed by a
sentence in monadic second-order logic (MSO), can be decided by a linear time
algorithm on classes of graphs of bounded treewidth. Using our results from the
first part of our work we establish a strong lower bound for tractability of
MSO on classes of colored graphs
Primal logic of information
Primal logic arose in access control; it has a remarkably efficient (linear
time) decision procedure for its entailment problem. But primal logic is a
general logic of information. In the realm of arbitrary items of information
(infons), conjunction, disjunction, and implication may seem to correspond
(set-theoretically) to union, intersection, and relative complementation. But,
while infons are closed under union, they are not closed under intersection or
relative complementation.
It turns out that there is a systematic transformation of propositional
intuitionistic calculi to the original (propositional) primal calculi; we call
it Flatting. We extend Flatting to quantifier rules, obtaining arguably the
right quantified primal logic, QPL. The QPL entailment problem is
exponential-time complete, but it is polynomial-time complete in the case, of
importance to applications (at least to access control), where the number of
quantifiers is bounded
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