100 research outputs found
Algebraic foundations for qualitative calculi and networks
A qualitative representation is like an ordinary representation of a
relation algebra, but instead of requiring , as
we do for ordinary representations, we only require that , for each in the algebra. A constraint
network is qualitatively satisfiable if its nodes can be mapped to elements of
a qualitative representation, preserving the constraints. If a constraint
network is satisfiable then it is clearly qualitatively satisfiable, but the
converse can fail. However, for a wide range of relation algebras including the
point algebra, the Allen Interval Algebra, RCC8 and many others, a network is
satisfiable if and only if it is qualitatively satisfiable.
Unlike ordinary composition, the weak composition arising from qualitative
representations need not be associative, so we can generalise by considering
network satisfaction problems over non-associative algebras. We prove that
computationally, qualitative representations have many advantages over ordinary
representations: whereas many finite relation algebras have only infinite
representations, every finite qualitatively representable algebra has a finite
qualitative representation; the representability problem for (the atom
structures of) finite non-associative algebras is NP-complete; the network
satisfaction problem over a finite qualitatively representable algebra is
always in NP; the validity of equations over qualitative representations is
co-NP-complete. On the other hand we prove that there is no finite
axiomatisation of the class of qualitatively representable algebras.Comment: 22 page
An Integrated First-Order Theory of Points and Intervals over Linear Orders (Part II)
There are two natural and well-studied approaches to temporal ontology and
reasoning: point-based and interval-based. Usually, interval-based temporal
reasoning deals with points as a particular case of duration-less intervals. A
recent result by Balbiani, Goranko, and Sciavicco presented an explicit
two-sorted point-interval temporal framework in which time instants (points)
and time periods (intervals) are considered on a par, allowing the perspective
to shift between these within the formal discourse. We consider here two-sorted
first-order languages based on the same principle, and therefore including
relations, as first studied by Reich, among others, between points, between
intervals, and inter-sort. We give complete classifications of its
sub-languages in terms of relative expressive power, thus determining how many,
and which, are the intrinsically different extensions of two-sorted first-order
logic with one or more such relations. This approach roots out the classical
problem of whether or not points should be included in a interval-based
semantics. In this Part II, we deal with the cases of all dense and the case of
all unbounded linearly ordered sets.Comment: This is Part II of the paper `An Integrated First-Order Theory of
Points and Intervals over Linear Orders' arXiv:1805.08425v2. Therefore the
introduction, preliminaries and conclusions of the two papers are the same.
This version implements a few minor corrections and an update to the
affiliation of the second autho
Online Abelian Pattern Matching
Ejaz T, Rahmann S, Stoye J. Online Abelian Pattern Matching. Forschungsberichte der Technischen Fakultät, Abteilung Informationstechnik / Universität Bielefeld. Bielefeld: Technische Fakultät der Universität Bielefeld; 2008.An abelian pattern describes the set of strings that comprise of the same combination of characters. Given an abelian pattern P and a text T [Epsilon] [Sigma]^n, the task is to find all occurrences of P in T, i.e. all substrings S = T_i...T_j such that the frequency of each character in S matches the specified frequency of that character in P.
In this report we present simple online algorithms for abelian pattern matching, and give a lower bound for online algorithms which is [Omega](n)
An Integrated First-Order Theory of Points and Intervals over Linear Orders (Part I)
There are two natural and well-studied approaches to temporal ontology and
reasoning: point-based and interval-based. Usually, interval-based temporal
reasoning deals with points as a particular case of duration-less intervals. A
recent result by Balbiani, Goranko, and Sciavicco presented an explicit
two-sorted point-interval temporal framework in which time instants (points)
and time periods (intervals) are considered on a par, allowing the perspective
to shift between these within the formal discourse. We consider here two-sorted
first-order languages based on the same principle, and therefore including
relations, as first studied by Reich, among others, between points, between
intervals, and inter-sort. We give complete classifications of its
sub-languages in terms of relative expressive power, thus determining how many,
and which, are the intrinsically different extensions of two-sorted first-order
logic with one or more such relations. This approach roots out the classical
problem of whether or not points should be included in a interval-based
semantics
The Complexity of Reasoning about Spatial Congruence
In the recent literature of Artificial Intelligence, an intensive research
effort has been spent, for various algebras of qualitative relations used in
the representation of temporal and spatial knowledge, on the problem of
classifying the computational complexity of reasoning problems for subsets of
algebras. The main purpose of these researches is to describe a restricted set
of maximal tractable subalgebras, ideally in an exhaustive fashion with respect
to the hosting algebras. In this paper we introduce a novel algebra for
reasoning about Spatial Congruence, show that the satisfiability problem in the
spatial algebra MC-4 is NP-complete, and present a complete classification of
tractability in the algebra, based on the individuation of three maximal
tractable subclasses, one containing the basic relations. The three algebras
are formed by 14, 10 and 9 relations out of 16 which form the full algebra
Cores of Countably Categorical Structures
A relational structure is a core, if all its endomorphisms are embeddings.
This notion is important for computational complexity classification of
constraint satisfaction problems. It is a fundamental fact that every finite
structure has a core, i.e., has an endomorphism such that the structure induced
by its image is a core; moreover, the core is unique up to isomorphism. Weprove
that every \omega -categorical structure has a core. Moreover, every
\omega-categorical structure is homomorphically equivalent to a model-complete
core, which is unique up to isomorphism, and which is finite or \omega
-categorical. We discuss consequences for constraint satisfaction with \omega
-categorical templates
Reasoning about temporal relations : a maximal tractable subclass of Allen\u27s interval algebra
We introduce a new subclass of Allen\u27s interval algebra we call "ORD-Horn subclass", which is a strict superset of the "pointisable subclass". We prove that reasoning in the ORD-Horn subclass is a polynomial-time problem and show that the path-consistency method is sufficient for deciding satisfiability. Further, using an extensive machine-generated case analysis, we show that the ORD-Horn subclass is a maximal tractable subclass of the full algebra (assuming P neq NP). In fact, it is the unique greatest tractable subclass amongst the subclasses that contain all basic relations
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