907 research outputs found
Complexity, BioComplexity, the Connectionist Conjecture and Ontology of Complexity\ud
This paper develops and integrates major ideas and concepts on complexity and biocomplexity - the connectionist conjecture, universal ontology of complexity, irreducible complexity of totality & inherent randomness, perpetual evolution of information, emergence of criticality and equivalence of symmetry & complexity. This paper introduces the Connectionist Conjecture which states that the one and only representation of Totality is the connectionist one i.e. in terms of nodes and edges. This paper also introduces an idea of Universal Ontology of Complexity and develops concepts in that direction. The paper also develops ideas and concepts on the perpetual evolution of information, irreducibility and computability of totality, all in the context of the Connectionist Conjecture. The paper indicates that the control and communication are the prime functionals that are responsible for the symmetry and complexity of complex phenomenon. The paper takes the stand that the phenomenon of life (including its evolution) is probably the nearest to what we can describe with the term “complexity”. The paper also assumes that signaling and communication within the living world and of the living world with the environment creates the connectionist structure of the biocomplexity. With life and its evolution as the substrate, the paper develops ideas towards the ontology of complexity. The paper introduces new complexity theoretic interpretations of fundamental biomolecular parameters. The paper also develops ideas on the methodology to determine the complexity of “true” complex phenomena.\u
Exhaustible sets in higher-type computation
We say that a set is exhaustible if it admits algorithmic universal
quantification for continuous predicates in finite time, and searchable if
there is an algorithm that, given any continuous predicate, either selects an
element for which the predicate holds or else tells there is no example. The
Cantor space of infinite sequences of binary digits is known to be searchable.
Searchable sets are exhaustible, and we show that the converse also holds for
sets of hereditarily total elements in the hierarchy of continuous functionals;
moreover, a selection functional can be constructed uniformly from a
quantification functional. We prove that searchable sets are closed under
intersections with decidable sets, and under the formation of computable images
and of finite and countably infinite products. This is related to the fact,
established here, that exhaustible sets are topologically compact. We obtain a
complete description of exhaustible total sets by developing a computational
version of a topological Arzela--Ascoli type characterization of compact
subsets of function spaces. We also show that, in the non-empty case, they are
precisely the computable images of the Cantor space. The emphasis of this paper
is on the theory of exhaustible and searchable sets, but we also briefly sketch
applications
Strong normalisation for applied lambda calculi
We consider the untyped lambda calculus with constructors and recursively
defined constants. We construct a domain-theoretic model such that any term not
denoting bottom is strongly normalising provided all its `stratified
approximations' are. From this we derive a general normalisation theorem for
applied typed lambda-calculi: If all constants have a total value, then all
typeable terms are strongly normalising. We apply this result to extensions of
G\"odel's system T and system F extended by various forms of bar recursion for
which strong normalisation was hitherto unknown.Comment: 14 pages, paper acceptet at electronic journal LMC
On the Executability of Interactive Computation
The model of interactive Turing machines (ITMs) has been proposed to
characterise which stream translations are interactively computable; the model
of reactive Turing machines (RTMs) has been proposed to characterise which
behaviours are reactively executable. In this article we provide a comparison
of the two models. We show, on the one hand, that the behaviour exhibited by
ITMs is reactively executable, and, on the other hand, that the stream
translations naturally associated with RTMs are interactively computable. We
conclude from these results that the theory of reactive executability subsumes
the theory of interactive computability. Inspired by the existing model of ITMs
with advice, which provides a model of evolving computation, we also consider
RTMs with advice and we establish that a facility of advice considerably
upgrades the behavioural expressiveness of RTMs: every countable transition
system can be simulated by some RTM with advice up to a fine notion of
behavioural equivalence.Comment: 15 pages, 0 figure
Effectivity and Density in Domains A Survey
AbstractThis article surveys the main results on effectivity and totality in domain theory and its applications. A more abstract and informative proof of Normann's generalized density theorem for total functionals of finite type over the reals is presented
Comparing hierarchies of total functionals
In this paper we consider two hierarchies of hereditarily total and
continuous functionals over the reals based on one extensional and one
intensional representation of real numbers, and we discuss under which
asumptions these hierarchies coincide. This coincidense problem is equivalent
to a statement about the topology of the Kleene-Kreisel continuous functionals.
As a tool of independent interest, we show that the Kleene-Kreisel functionals
may be embedded into both these hierarchies.Comment: 28 page
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