163,622 research outputs found
The word problem for omega-terms over the Trotter-Weil hierarchy [extended abstract]
© Springer International Publishing Switzerland 2016. Over finitewords, there is a tight connection between the quantifier alternation hierarchy inside two-variable first-order logic FO 2 and a hierarchy of finite monoids: theTrotter-Weil Hierarchy. The variousways of climbing up this hierarchy include Mal’cev products, deterministic and codeterministic concatenation as well as identities of ω-terms.We show that the word problem for ω-terms over each level of the Trotter-Weil Hierarchy is decidable; this means, for every variety V of the hierarchy and every identity u = v of ω-terms, one can decide whether all monoids in V satisfy u = v. More precisely, for every fixed variety V, our approach yields nondeterministic logarithmic space (NL) and deterministic polynomial time algorithms, which are more efficient than straightforward translations of the NL-algorithms. From a language perspective, the word problem for ω- terms is the following: for every language variety V in theTrotter-Weil Hierarchy and every language varietyWgivenbyan identity of ω-terms, one can decide whether V ⊆ W. This includes the case where V is some level of the FO 2 quantifier alternation hierarchy. As an application of our results, we show that the separation problems for the so-called corners of the Trotter- Weil Hierarchy are decidable
Tight lower bounds for certain parameterized NP-hard problems
Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of well-known NP-hard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on depth-t circuits cannot be solved in time no(k) poly(m), where n is the circuit input length, m is the circuit size, and k is the parameter, unless the (t − 1)-st level W [t − 1] of the W-hierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NP-hard problems, including weighted sat, dominating set, hitting set, set cover, and feature set, cannot be solved in time no(k) poly(m), where n is the size of the universal set from which the k elements are to be selected and m is the instance size, unless the first level W [1] of the W-hierarchy collapses to FPT. We also prove that another group of parameterized problems which includes weighted q-sat (for any fixed q ≥ 2), clique, and independent set, cannot be solved in time no(k) unless all search problems in the syntactic class SNP, introduced by Papadimitriou and Yannakakis, are solvable in subexponential time. Note that all these parameterized problems have trivial algorithms of running time either n k poly(m) or O(n k).
Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages
The purpose of this paper is to provide efficient algorithms that decide
membership for classes of several Boolean hierarchies for which efficiency (or
even decidability) were previously not known. We develop new forbidden-chain
characterizations for the single levels of these hierarchies and obtain the
following results: - The classes of the Boolean hierarchy over level
of the dot-depth hierarchy are decidable in (previously only the
decidability was known). The same remains true if predicates mod for fixed
are allowed. - If modular predicates for arbitrary are allowed, then
the classes of the Boolean hierarchy over level are decidable. - For
the restricted case of a two-letter alphabet, the classes of the Boolean
hierarchy over level of the Straubing-Th\'erien hierarchy are
decidable in . This is the first decidability result for this hierarchy. -
The membership problems for all mentioned Boolean-hierarchy classes are
logspace many-one hard for . - The membership problems for quasi-aperiodic
languages and for -quasi-aperiodic languages are logspace many-one complete
for
Hierarchical Path Search with Partial Materialization of Costs for a Smart Wheelchair
In this paper, the off-line path planner module of a smart wheelchair aided navigation
system is described. Environmental information is structured into a hierarchical graph (H-graph) and
used either by the user interface or the path planner module. This information structure facilitates
efficient path search and easier information access and retrieval. Special path planning issues like
planning between floors of a building (vertical path planning) are also viewed. The H-graph proposed
is modelled by a tree. The hierarchy of abstractions contained in the tree has several levels of detail.
Each abstraction level is a graph whose nodes can represent other graphs in a deeper level of the
hierarchy. Path planning is performed using a path skeleton which is built from the deepest
abstraction levels of the hierarchy to the most upper levels and completed in the last step of the
algorithm. In order not to lose accuracy in the path skeleton generation and speed up the search, a set
of optimal subpaths are previously stored in some nodes of the H-graph (path costs are partially
materialized). Finally, some experimental results are showed and compared to traditional heuristic
search algorithms used in robot path planning.Comisión Interministerial de Ciencia y TecnologÃa TER96-2056-C02-0
Extracting tag hierarchies
Tagging items with descriptive annotations or keywords is a very natural way
to compress and highlight information about the properties of the given entity.
Over the years several methods have been proposed for extracting a hierarchy
between the tags for systems with a "flat", egalitarian organization of the
tags, which is very common when the tags correspond to free words given by
numerous independent people. Here we present a complete framework for automated
tag hierarchy extraction based on tag occurrence statistics. Along with
proposing new algorithms, we are also introducing different quality measures
enabling the detailed comparison of competing approaches from different
aspects. Furthermore, we set up a synthetic, computer generated benchmark
providing a versatile tool for testing, with a couple of tunable parameters
capable of generating a wide range of test beds. Beside the computer generated
input we also use real data in our studies, including a biological example with
a pre-defined hierarchy between the tags. The encouraging similarity between
the pre-defined and reconstructed hierarchy, as well as the seemingly
meaningful hierarchies obtained for other real systems indicate that tag
hierarchy extraction is a very promising direction for further research with a
great potential for practical applications.Comment: 25 pages with 21 pages of supporting information, 25 figure
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