678 research outputs found
First Order Theories of Some Lattices of Open Sets
We show that the first order theory of the lattice of open sets in some
natural topological spaces is -equivalent to second order arithmetic. We
also show that for many natural computable metric spaces and computable domains
the first order theory of the lattice of effectively open sets is undecidable.
Moreover, for several important spaces (e.g., , , and the
domain ) this theory is -equivalent to first order arithmetic
Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime
The synthesis of classical Computational Complexity Theory with Recursive
Analysis provides a quantitative foundation to reliable numerics. Here the
operators of maximization, integration, and solving ordinary differential
equations are known to map (even high-order differentiable) polynomial-time
computable functions to instances which are `hard' for classical complexity
classes NP, #P, and CH; but, restricted to analytic functions, map
polynomial-time computable ones to polynomial-time computable ones --
non-uniformly!
We investigate the uniform parameterized complexity of the above operators in
the setting of Weihrauch's TTE and its second-order extension due to
Kawamura&Cook (2010). That is, we explore which (both continuous and discrete,
first and second order) information and parameters on some given f is
sufficient to obtain similar data on Max(f) and int(f); and within what running
time, in terms of these parameters and the guaranteed output precision 2^(-n).
It turns out that Gevrey's hierarchy of functions climbing from analytic to
smooth corresponds to the computational complexity of maximization growing from
polytime to NP-hard. Proof techniques involve mainly the Theory of (discrete)
Computation, Hard Analysis, and Information-Based Complexity
Global semantic typing for inductive and coinductive computing
Inductive and coinductive types are commonly construed as ontological
(Church-style) types, denoting canonical data-sets such as natural numbers,
lists, and streams. For various purposes, notably the study of programs in the
context of global semantics, it is preferable to think of types as semantical
properties (Curry-style). Intrinsic theories were introduced in the late 1990s
to provide a purely logical framework for reasoning about programs and their
semantic types. We extend them here to data given by any combination of
inductive and coinductive definitions. This approach is of interest because it
fits tightly with syntactic, semantic, and proof theoretic fundamentals of
formal logic, with potential applications in implicit computational complexity
as well as extraction of programs from proofs. We prove a Canonicity Theorem,
showing that the global definition of program typing, via the usual (Tarskian)
semantics of first-order logic, agrees with their operational semantics in the
intended model. Finally, we show that every intrinsic theory is interpretable
in a conservative extension of first-order arithmetic. This means that
quantification over infinite data objects does not lead, on its own, to
proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories
are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were
used to characterize major computational complexity classes Their extensions
described here have similar potential which has already been applied
On the necessity of complexity
Wolfram's Principle of Computational Equivalence (PCE) implies that universal
complexity abounds in nature. This paper comprises three sections. In the first
section we consider the question why there are so many universal phenomena
around. So, in a sense, we week a driving force behind the PCE if any. We
postulate a principle GNS that we call the Generalized Natural Selection
Principle that together with the Church-Turing Thesis is seen to be equivalent
to a weak version of PCE. In the second section we ask the question why we do
not observe any phenomena that are complex but not-universal. We choose a
cognitive setting to embark on this question and make some analogies with
formal logic. In the third and final section we report on a case study where we
see rich structures arise everywhere.Comment: 17 pages, 3 figure
Feedback computability on Cantor space
We introduce the notion of feedback computable functions from to
, extending feedback Turing computation in analogy with the standard
notion of computability for functions from to . We then
show that the feedback computable functions are precisely the effectively Borel
functions. With this as motivation we define the notion of a feedback
computable function on a structure, independent of any coding of the structure
as a real. We show that this notion is absolute, and as an example characterize
those functions that are computable from a Gandy ordinal with some finite
subset distinguished
Basic notions of universal algebra for language theory and graph grammars
AbstractThis paper reviews the basic properties of the equational and recognizable subsets of general algebras; these sets can be seen as generalizations of the context-free and regular languages, respectively. This approach, based on Universal Algebra, facilitates the development of the theory of formal languages so as to include the description of sets of finite trees, finite graphs, finite hypergraphs, tuples of words, partially commutative words (also called traces) and other similar finite objects
- …