5,948 research outputs found
On the Expressive Power of Query Languages for Matrices
We investigate the expressive power of MATLANG, a formal language for matrix manipulation based on common matrix operations and linear algebra. The language can be extended with the operation inv of inverting a matrix. In MATLANG + inv we can compute the transitive closure of directed graphs, whereas we show that this is not possible without inversion. Indeed we show that the basic language can be simulated in the relational algebra with arithmetic operations, grouping, and summation. We also consider an operation eigen for diagonalizing a matrix, which is defined so that different eigenvectors returned for a same eigenvalue are orthogonal. We show that inv can be expressed in MATLANG + eigen. We put forward the open question whether there are boolean queries about matrices, or generic queries about graphs, expressible in MATLANG + eigen but not in MATLANG + inv. The evaluation problem for MATLANG + eigen is shown to be complete for the complexity class Exists R
On the Expressiveness of LARA: A Unified Language for Linear and Relational Algebra
We study the expressive power of the Lara language - a recently proposed unified model for expressing relational and linear algebra operations - both in terms of traditional database query languages and some analytic tasks often performed in machine learning pipelines. We start by showing Lara to be expressive complete with respect to first-order logic with aggregation. Since Lara is parameterized by a set of user-defined functions which allow to transform values in tables, the exact expressive power of the language depends on how these functions are defined. We distinguish two main cases depending on the level of genericity queries are enforced to satisfy. Under strong genericity assumptions the language cannot express matrix convolution, a very important operation in current machine learning operations. This language is also local, and thus cannot express operations such as matrix inverse that exhibit a recursive behavior. For expressing convolution, one can relax the genericity requirement by adding an underlying linear order on the domain. This, however, destroys locality and turns the expressive power of the language much more difficult to understand. In particular, although under complexity assumptions the resulting language can still not express matrix inverse, a proof of this fact without such assumptions seems challenging to obtain
When Can Matrix Query Languages Discern Matrices?
We investigate when two graphs, represented by their adjacency matrices, can be distinguished by means of sentences formed in MATLANG, a matrix query language which supports a number of elementary linear algebra operators. When undirected graphs are concerned, and hence the adjacency matrices are real and symmetric, precise characterisations are in place when two graphs (i.e., their adjacency matrices) can be distinguished. Turning to directed graphs, one has to deal with asymmetric adjacency matrices. This complicates matters. Indeed, it requires to understand the more general problem of when two arbitrary matrices can be distinguished in MATLANG. We provide characterisations of the distinguishing power of MATLANG on real and complex matrices, and on adjacency matrices of directed graphs in particular. The proof techniques are a combination of insights from the symmetric matrix case and results from linear algebra and linear control theory
Context-Free Path Querying by Matrix Multiplication
Graph data models are widely used in many areas, for example, bioinformatics,
graph databases. In these areas, it is often required to process queries for
large graphs. Some of the most common graph queries are navigational queries.
The result of query evaluation is a set of implicit relations between nodes of
the graph, i.e. paths in the graph. A natural way to specify these relations is
by specifying paths using formal grammars over the alphabet of edge labels. An
answer to a context-free path query in this approach is usually a set of
triples (A, m, n) such that there is a path from the node m to the node n,
whose labeling is derived from a non-terminal A of the given context-free
grammar. This type of queries is evaluated using the relational query
semantics. Another example of path query semantics is the single-path query
semantics which requires presenting a single path from the node m to the node
n, whose labeling is derived from a non-terminal A for all triples (A, m, n)
evaluated using the relational query semantics. There is a number of algorithms
for query evaluation which use these semantics but all of them perform poorly
on large graphs. One of the most common technique for efficient big data
processing is the use of a graphics processing unit (GPU) to perform
computations, but these algorithms do not allow to use this technique
efficiently. In this paper, we show how the context-free path query evaluation
using these query semantics can be reduced to the calculation of the matrix
transitive closure. Also, we propose an algorithm for context-free path query
evaluation which uses relational query semantics and is based on matrix
operations that make it possible to speed up computations by using a GPU.Comment: 9 pages, 11 figures, 2 table
Semantics, Modelling, and the Problem of Representation of Meaning -- a Brief Survey of Recent Literature
Over the past 50 years many have debated what representation should be used
to capture the meaning of natural language utterances. Recently new needs of
such representations have been raised in research. Here I survey some of the
interesting representations suggested to answer for these new needs.Comment: 15 pages, no figure
On the Expressive Power of Linear Algebra on Graphs
Most graph query languages are rooted in logic. By contrast, in this paper we consider graph query languages rooted in linear algebra. More specifically, we consider MATLANG, a matrix query language recently introduced, in which some basic linear algebra functionality is supported. We investigate the problem of characterising equivalence of graphs, represented by their adjacency matrices, for various fragments of MATLANG. A complete picture is painted of the impact of the linear algebra operations in MATLANG on their ability to distinguish graphs
Two-Way Visibly Pushdown Automata and Transducers
Automata-logic connections are pillars of the theory of regular languages.
Such connections are harder to obtain for transducers, but important results
have been obtained recently for word-to-word transformations, showing that the
three following models are equivalent: deterministic two-way transducers,
monadic second-order (MSO) transducers, and deterministic one-way automata
equipped with a finite number of registers. Nested words are words with a
nesting structure, allowing to model unranked trees as their depth-first-search
linearisations. In this paper, we consider transformations from nested words to
words, allowing in particular to produce unranked trees if output words have a
nesting structure. The model of visibly pushdown transducers allows to describe
such transformations, and we propose a simple deterministic extension of this
model with two-way moves that has the following properties: i) it is a simple
computational model, that naturally has a good evaluation complexity; ii) it is
expressive: it subsumes nested word-to-word MSO transducers, and the exact
expressiveness of MSO transducers is recovered using a simple syntactic
restriction; iii) it has good algorithmic/closure properties: the model is
closed under composition with a unambiguous one-way letter-to-letter transducer
which gives closure under regular look-around, and has a decidable equivalence
problem
Logic-Based Decision Support for Strategic Environmental Assessment
Strategic Environmental Assessment is a procedure aimed at introducing
systematic assessment of the environmental effects of plans and programs. This
procedure is based on the so-called coaxial matrices that define dependencies
between plan activities (infrastructures, plants, resource extractions,
buildings, etc.) and positive and negative environmental impacts, and
dependencies between these impacts and environmental receptors. Up to now, this
procedure is manually implemented by environmental experts for checking the
environmental effects of a given plan or program, but it is never applied
during the plan/program construction. A decision support system, based on a
clear logic semantics, would be an invaluable tool not only in assessing a
single, already defined plan, but also during the planning process in order to
produce an optimized, environmentally assessed plan and to study possible
alternative scenarios. We propose two logic-based approaches to the problem,
one based on Constraint Logic Programming and one on Probabilistic Logic
Programming that could be, in the future, conveniently merged to exploit the
advantages of both. We test the proposed approaches on a real energy plan and
we discuss their limitations and advantages.Comment: 17 pages, 1 figure, 26th Int'l. Conference on Logic Programming
(ICLP'10
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