5,373 research outputs found
Cohomology in Grothendieck Topologies and Lower Bounds in Boolean Complexity
This paper is motivated by questions such as P vs. NP and other questions in
Boolean complexity theory. We describe an approach to attacking such questions
with cohomology, and we show that using Grothendieck topologies and other ideas
from the Grothendieck school gives new hope for such an attack.
We focus on circuit depth complexity, and consider only finite topological
spaces or Grothendieck topologies based on finite categories; as such, we do
not use algebraic geometry or manifolds.
Given two sheaves on a Grothendieck topology, their "cohomological
complexity" is the sum of the dimensions of their Ext groups. We seek to model
the depth complexity of Boolean functions by the cohomological complexity of
sheaves on a Grothendieck topology. We propose that the logical AND of two
Boolean functions will have its corresponding cohomological complexity bounded
in terms of those of the two functions using ``virtual zero extensions.'' We
propose that the logical negation of a function will have its corresponding
cohomological complexity equal to that of the original function using duality
theory. We explain these approaches and show that they are stable under
pullbacks and base change. It is the subject of ongoing work to achieve AND and
negation bounds simultaneously in a way that yields an interesting depth lower
bound.Comment: 70 pages, abstract corrected and modifie
Stigmergic hyperlink's contributes to web search
Stigmergic hyperlinks are hyperlinks with a "heart beat": if used they stay healthy and online; if
neglected, they fade, eventually getting replaced. Their life attribute is a relative usage measure that
regular hyperlinks do not provide, hence PageRank-like measures have historically been well
informed about the structure of webs of documents, but unaware of what users effectively do with
the links.
This paper elaborates on how to input the users’ perspective into Google’s original, structure centric,
PageRank metric. The discussion then bridges to the Deep Web, some search challenges, and how
stigmergic hyperlinks could help decentralize the search experience, facilitating user generated
search solutions and supporting new related business models.info:eu-repo/semantics/publishedVersio
Computing with cells: membrane systems - some complexity issues.
Membrane computing is a branch of natural computing which abstracts computing models from the structure and the functioning of the living cell. The main ingredients of membrane systems, called P systems, are (i) the membrane structure, which consists of a hierarchical arrangements of membranes which delimit compartments where (ii) multisets of symbols, called objects, evolve according to (iii) sets of rules which are localised and associated with compartments. By using the rules in a nondeterministic/deterministic maximally parallel manner, transitions between the system configurations can be obtained. A sequence of transitions is a computation of how the system is evolving. Various ways of controlling the transfer of objects from one membrane to another and applying the rules, as well as possibilities to dissolve, divide or create membranes have been studied. Membrane systems have a great potential for implementing massively concurrent systems in an efficient way that would allow us to solve currently intractable problems once future biotechnology gives way to a practical bio-realization. In this paper we survey some interesting and fundamental complexity issues such as universality vs. nonuniversality, determinism vs. nondeterminism, membrane and alphabet size hierarchies, characterizations of context-sensitive languages and other language classes and various notions of parallelism
On the Relative Expressiveness of Argumentation Frameworks, Normal Logic Programs and Abstract Dialectical Frameworks
We analyse the expressiveness of the two-valued semantics of abstract
argumentation frameworks, normal logic programs and abstract dialectical
frameworks. By expressiveness we mean the ability to encode a desired set of
two-valued interpretations over a given propositional signature using only
atoms from that signature. While the computational complexity of the two-valued
model existence problem for all these languages is (almost) the same, we show
that the languages form a neat hierarchy with respect to their expressiveness.Comment: Proceedings of the 15th International Workshop on Non-Monotonic
Reasoning (NMR 2014
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