3,907 research outputs found
Undecidability and Irreducibility Conditions for Open-Ended Evolution and Emergence
Is undecidability a requirement for open-ended evolution (OEE)? Using methods
derived from algorithmic complexity theory, we propose robust computational
definitions of open-ended evolution and the adaptability of computable
dynamical systems. Within this framework, we show that decidability imposes
absolute limits to the stable growth of complexity in computable dynamical
systems. Conversely, systems that exhibit (strong) open-ended evolution must be
undecidable, establishing undecidability as a requirement for such systems.
Complexity is assessed in terms of three measures: sophistication, coarse
sophistication and busy beaver logical depth. These three complexity measures
assign low complexity values to random (incompressible) objects. As time grows,
the stated complexity measures allow for the existence of complex states during
the evolution of a computable dynamical system. We show, however, that finding
these states involves undecidable computations. We conjecture that for similar
complexity measures that assign low complexity values, decidability imposes
comparable limits to the stable growth of complexity, and that such behaviour
is necessary for non-trivial evolutionary systems. We show that the
undecidability of adapted states imposes novel and unpredictable behaviour on
the individuals or populations being modelled. Such behaviour is irreducible.
Finally, we offer an example of a system, first proposed by Chaitin, that
exhibits strong OEE.Comment: Reduced version of this article was submitted and accepted for oral
presentation at ALife XV (July 4-8, 2016, Cancun, Mexico
Definability as hypercomputational effect
The classical simulation of physical processes using standard models of
computation is fraught with problems. On the other hand, attempts at modelling
real-world computation with the aim of isolating its hypercomputational content
have struggled to convince. We argue that a better basic understanding can be
achieved through computability theoretic deconstruction of those physical
phenomena most resistant to classical simulation. From this we may be able to
better assess whether the hypercomputational enterprise is proleptic computer
science, or of mainly philosophical interest
Revisiting the Complexity of Stability of Continuous and Hybrid Systems
We develop a framework to give upper bounds on the "practical" computational
complexity of stability problems for a wide range of nonlinear continuous and
hybrid systems. To do so, we describe stability properties of dynamical systems
using first-order formulas over the real numbers, and reduce stability problems
to the delta-decision problems of these formulas. The framework allows us to
obtain a precise characterization of the complexity of different notions of
stability for nonlinear continuous and hybrid systems. We prove that bounded
versions of the stability problems are generally decidable, and give upper
bounds on their complexity. The unbounded versions are generally undecidable,
for which we give upper bounds on their degrees of unsolvability
Propositional computability logic I
In the same sense as classical logic is a formal theory of truth, the
recently initiated approach called computability logic is a formal theory of
computability. It understands (interactive) computational problems as games
played by a machine against the environment, their computability as existence
of a machine that always wins the game, logical operators as operations on
computational problems, and validity of a logical formula as being a scheme of
"always computable" problems. The present contribution gives a detailed
exposition of a soundness and completeness proof for an axiomatization of one
of the most basic fragments of computability logic. The logical vocabulary of
this fragment contains operators for the so called parallel and choice
operations, and its atoms represent elementary problems, i.e. predicates in the
standard sense. This article is self-contained as it explains all relevant
concepts. While not technically necessary, however, familiarity with the
foundational paper "Introduction to computability logic" [Annals of Pure and
Applied Logic 123 (2003), pp.1-99] would greatly help the reader in
understanding the philosophy, underlying motivations, potential and utility of
computability logic, -- the context that determines the value of the present
results. Online introduction to the subject is available at
http://www.cis.upenn.edu/~giorgi/cl.html and
http://www.csc.villanova.edu/~japaridz/CL/gsoll.html .Comment: To appear in ACM Transactions on Computational Logi
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
From truth to computability I
The recently initiated approach called computability logic is a formal theory
of interactive computation. See a comprehensive online source on the subject at
http://www.cis.upenn.edu/~giorgi/cl.html . The present paper contains a
soundness and completeness proof for the deductive system CL3 which axiomatizes
the most basic first-order fragment of computability logic called the
finite-depth, elementary-base fragment. Among the potential application areas
for this result are the theory of interactive computation, constructive applied
theories, knowledgebase systems, systems for resource-bound planning and
action. This paper is self-contained as it reintroduces all relevant
definitions as well as main motivations.Comment: To appear in Theoretical Computer Scienc
Peculiarities of Quantum Mechanics: Origins and Meaning
The most peculiar, specifically quantum, features of quantum mechanics ---
quantum nonlocality, indeterminism, interference of probabilities,
quantization, wave function collapse during measurement --- are explained on a
logical-geometrical basis. It is shown that truths of logical statements about
numerical values of quantum observables are quantum observables themselves and
are represented in quantum mechanics by density matrices of pure states.
Structurally, quantum mechanics is a result of applying non-Abelian symmetries
to truth operators and their eigenvectors --- wave functions. Wave functions
contain information about conditional truths of all possible logical statements
about physical observables and their correlations in a given physical system.
These correlations are logical, hence nonlocal, and exist when the system is
not observed. We analyze the physical conditions and logical and
decision-making operations involved in the phenomena of wave function collapse
and unpredictability of the results of measurements. Consistent explanations of
the Stern-Gerlach and EPR-Bohm experiments are presented."Comment: 51 pages. LaTeX document with 4 EPS figures. A 5th figure(figure 2)
can be obtained from
http://w4.lns.cornell.edu/public/CLNS/1996/CLNS96-1399/1399-fig2.gi
First-order queries on structures of bounded degree are computable with constant delay
A bounded degree structure is either a relational structure all of whose
relations are of bounded degree or a functional structure involving bijective
functions only. In this paper, we revisit the complexity of the evaluation
problem of not necessarily Boolean first-order queries over structures of
bounded degree. Query evaluation is considered here as a dynamical process. We
prove that any query on bounded degree structures is \constantdelaylin, i.e.,
can be computed by an algorithm that has two separate parts: it has a
precomputation step of linear time in the size of the structure and then, it
outputs all tuples one by one with a constant (i.e. depending on the size of
the formula only) delay between each. Seen as a global process, this implies
that queries on bounded structures can be evaluated in total time
O(f(|\phi|).(|\calS|+|\phi(\calS)|)) and space O(f(|\phi|).|\calS|) where
\calS is the structure, is the formula, \phi(\calS) is the result of
the query and is some function.
Among other things, our results generalize a result of \cite{Seese-96} on the
data complexity of the model-checking problem for bounded degree structures.
Besides, the originality of our approach compared to that \cite{Seese-96} and
comparable results is that it does not rely on the Hanf's model-theoretic
technic (see \cite{Hanf-65}) and is completely effective.Comment: 18 pages, 1 figur
Quantum Computation via Paraconsistent Computation
We present an original model of paraconsistent Turing machines (PTMs), a
generalization of the classical Turing machines model of computation using a
paraconsistent logic. Next, we briefl y describe the standard models of quantum
computation: quantum Turing machines and quantum circuits, and revise quantum
algorithms to solve the so-called Deutsch's problem and Deutsch-Jozsa problem.
Then, we show the potentialities of the PTMs model of computation simulating
the presented quantum algorithms via paraconsistent algorithms. This way, we
show that PTMs can resolve some problems in exponentially less time than any
classical deterministic Turing machine. Finally, We show that it is not
possible to simulate all characteristics (in particular entangled states) of
quantum computation by the particular model of PTMs here presented, therefore
we open the possibility of constructing a new model of PTMs by which it is
feasible to simulate such states
Well Quasiorders and Hierarchy Theory
We discuss some applications of WQOs to several fields were hierarchies and
reducibilities are the principal classification tools, notably to Descriptive
Set Theory, Computability theory and Automata Theory. While the classical
hierarchies of sets usually degenerate to structures very close to ordinals,
the extension of them to functions requires more complicated WQOs, and the same
applies to reducibilities. We survey some results obtained so far and discuss
open problems and possible research directions.Comment: 37 page
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