3 research outputs found
Fixed Points on Abstract Structures without the Equality Test
In this paper we present a study of definability properties of fixed points of effective operators on abstract structures without the equality test. In particular we prove that Gandy theorem holds for abstract structures. This provides a useful tool for dealing with recursive definitions using Sigma-formulas. One of the applications of Gandy theorem in the case of the reals without the equality test is that it allows us to define universal Sigma-predicates. It leads to a topological characterisation of Sigma-relations on |R
Recent Advances in Σ-definability over Continuous Data Types
The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Sigma-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the reals by the structure of hereditarily finite sets. In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals. We prove Engeler's Lemma for Sigma-definability over the reals without the equality test which relates Sigma-definability with definability in the constructive infinitary language L_{omega_1 omega}. Thus, a relation over the real numbers is Sigma-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas. This result reveals computational aspects of Sigma-definability and also gives topological characterisation of Sigma-definable relations over the reals without the equality test. We also illustrate how computability over the real numbers can be expressed in the language of Sigma-formulas
Hyperset Approach to Semi-structured Databases and the Experimental Implementation of the Query Language Delta
This thesis presents practical suggestions towards the implementation of the
hyperset approach to semi-structured databases and the associated query
language Delta. This work can be characterised as part of a top-down approach
to semi-structured databases, from theory to practice. The main original part
of this work consisted in implementation of the hyperset Delta query language
to semi-structured databases, including worked example queries. In fact, the
goal was to demonstrate the practical details of this approach and language.
The required development of an extended, practical version of the language
based on the existing theoretical version, and the corresponding operational
semantics. Here we present detailed description of the most essential steps of
the implementation. Another crucial problem for this approach was to
demonstrate how to deal in reality with the concept of the equality relation
between (hyper)sets, which is computationally realised by the bisimulation
relation. In fact, this expensive procedure, especially in the case of
distributed semi-structured data, required some additional theoretical
considerations and practical suggestions for efficient implementation. To this
end the 'local/global' strategy for computing the bisimulation relation over
distributed semi-structured data was developed and its efficiency was
experimentally confirmed.Comment: Technical Report (PhD thesis), University of Liverpool, Englan