457 research outputs found
Many-one reductions and the category of multivalued functions
Multi-valued functions are common in computable analysis (built upon the Type
2 Theory of Effectivity), and have made an appearance in complexity theory
under the moniker search problems leading to complexity classes such as PPAD
and PLS being studied. However, a systematic investigation of the resulting
degree structures has only been initiated in the former situation so far (the
Weihrauch-degrees).
A more general understanding is possible, if the category-theoretic
properties of multi-valued functions are taken into account. In the present
paper, the category-theoretic framework is established, and it is demonstrated
that many-one degrees of multi-valued functions form a distributive lattice
under very general conditions, regardless of the actual reducibility notions
used (e.g. Cook, Karp, Weihrauch).
Beyond this, an abundance of open questions arises. Some classic results for
reductions between functions carry over to multi-valued functions, but others
do not. The basic theme here again depends on category-theoretic differences
between functions and multi-valued functions.Comment: an earlier version was titled "Many-one reductions between search
problems". in Mathematical Structures in Computer Science, 201
Selective Categories and Linear Canonical Relations
A construction of Wehrheim and Woodward circumvents the problem that
compositions of smooth canonical relations are not always smooth, building a
category suitable for functorial quantization. To apply their construction to
more examples, we introduce a notion of highly selective category, in which
only certain morphisms and certain pairs of these morphisms are "good". We then
apply this notion to the category of linear canonical
relations and the result of our version of the WW
construction, identifying the morphisms in the latter with pairs
consisting of a linear canonical relation and a nonnegative integer. We put a
topology on this category of indexed linear canonical relations for which
composition is continuous, unlike the composition in itself.
Subsequent papers will consider this category from the viewpoint of derived
geometry and will concern quantum counterparts
On Different Strategies for Eliminating Redundant Actions from Plans
Satisficing planning engines are often able to generate plans in a reasonable time, however, plans are often far from optimal. Such plans often contain a high number of redundant actions, that are actions, which can be removed without affecting the validity of the plans. Existing approaches for determining and eliminating redundant actions work in polynomial time, however, do not guarantee eliminating the "best" set of redundant actions, since such a problem is NP-complete. We introduce an approach which encodes the problem of determining the "best" set of redundant actions (i.e. having the maximum total-cost) as a weighted MaxSAT problem. Moreover, we adapt the existing polynomial technique which greedily tries to eliminate an action and its dependants from the plan in order to eliminate more expensive redundant actions. The proposed approaches are empirically compared to existing approaches on plans generated by state-of-the-art planning engines on standard planning benchmark
History and new possible research directions of hyperstructures
We present a summary of the origins and current developments of the theory of algebraic hyperstructures. We also sketch some possible lines of research
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