4,231 research outputs found
Graphical Reasoning in Compact Closed Categories for Quantum Computation
Compact closed categories provide a foundational formalism for a variety of
important domains, including quantum computation. These categories have a
natural visualisation as a form of graphs. We present a formalism for
equational reasoning about such graphs and develop this into a generic proof
system with a fixed logical kernel for equational reasoning about compact
closed categories. Automating this reasoning process is motivated by the slow
and error prone nature of manual graph manipulation. A salient feature of our
system is that it provides a formal and declarative account of derived results
that can include `ellipses'-style notation. We illustrate the framework by
instantiating it for a graphical language of quantum computation and show how
this can be used to perform symbolic computation.Comment: 21 pages, 9 figures. This is the journal version of the paper
published at AIS
LiteMat: a scalable, cost-efficient inference encoding scheme for large RDF graphs
The number of linked data sources and the size of the linked open data graph
keep growing every day. As a consequence, semantic RDF services are more and
more confronted with various "big data" problems. Query processing in the
presence of inferences is one them. For instance, to complete the answer set of
SPARQL queries, RDF database systems evaluate semantic RDFS relationships
(subPropertyOf, subClassOf) through time-consuming query rewriting algorithms
or space-consuming data materialization solutions. To reduce the memory
footprint and ease the exchange of large datasets, these systems generally
apply a dictionary approach for compressing triple data sizes by replacing
resource identifiers (IRIs), blank nodes and literals with integer values. In
this article, we present a structured resource identification scheme using a
clever encoding of concepts and property hierarchies for efficiently evaluating
the main common RDFS entailment rules while minimizing triple materialization
and query rewriting. We will show how this encoding can be computed by a
scalable parallel algorithm and directly be implemented over the Apache Spark
framework. The efficiency of our encoding scheme is emphasized by an evaluation
conducted over both synthetic and real world datasets.Comment: 8 pages, 1 figur
Initial Algebra Semantics for Cyclic Sharing Tree Structures
Terms are a concise representation of tree structures. Since they can be
naturally defined by an inductive type, they offer data structures in
functional programming and mechanised reasoning with useful principles such as
structural induction and structural recursion. However, for graphs or
"tree-like" structures - trees involving cycles and sharing - it remains
unclear what kind of inductive structures exists and how we can faithfully
assign a term representation of them. In this paper we propose a simple term
syntax for cyclic sharing structures that admits structural induction and
recursion principles. We show that the obtained syntax is directly usable in
the functional language Haskell and the proof assistant Agda, as well as
ordinary data structures such as lists and trees. To achieve this goal, we use
a categorical approach to initial algebra semantics in a presheaf category.
That approach follows the line of Fiore, Plotkin and Turi's models of abstract
syntax with variable binding
Homotopy Type Theory in Lean
We discuss the homotopy type theory library in the Lean proof assistant. The
library is especially geared toward synthetic homotopy theory. Of particular
interest is the use of just a few primitive notions of higher inductive types,
namely quotients and truncations, and the use of cubical methods.Comment: 17 pages, accepted for ITP 201
Intuitionistic fuzzy XML query matching and rewriting
With the emergence of XML as a standard for data representation, particularly on the web, the need for intelligent query languages that can operate on XML documents with structural heterogeneity has recently gained a lot of popularity. Traditional Information Retrieval and Database approaches have limitations when dealing with such scenarios. Therefore, fuzzy (flexible) approaches have become the predominant. In this thesis, we propose a new approach for approximate XML query matching and rewriting which aims at achieving soft matching of XML queries with XML data sources following different schemas. Unlike traditional querying approaches, which require exact matching, the proposed approach makes use of Intuitionistic Fuzzy Trees to achieve approximate (soft) query matching. Through this new approach, not only the exact answer of a query, but also approximate answers are retrieved. Furthermore, partial results can be obtained from multiple data sources and merged together to produce a single answer to a query. The proposed approach introduced a new tree similarity measure that considers the minimum and maximum degrees of similarity/inclusion of trees that are based on arc matching. New techniques for soft node and arc matching were presented for matching queries against data sources with highly varied structures. A prototype was developed to test the proposed ideas and it proved the ability to achieve approximate matching for pattern queries with a number of XML schemas and rewrite the original query so that it obtain results from the underlying data sources. This has been achieved through several novel algorithms which were tested and proved efficiency and low CPU/Memory cost even for big number of data sources
Termination of rewrite relations on -terms based on Girard's notion of reducibility
In this paper, we show how to extend the notion of reducibility introduced by
Girard for proving the termination of -reduction in the polymorphic
-calculus, to prove the termination of various kinds of rewrite
relations on -terms, including rewriting modulo some equational theory
and rewriting with matching modulo , by using the notion of
computability closure. This provides a powerful termination criterion for
various higher-order rewriting frameworks, including Klop's Combinatory
Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems
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