2,124 research outputs found
Trace Complexity of Chaotic Reversible Cellular Automata
Delvenne, K\r{u}rka and Blondel have defined new notions of computational
complexity for arbitrary symbolic systems, and shown examples of effective
systems that are computationally universal in this sense. The notion is defined
in terms of the trace function of the system, and aims to capture its dynamics.
We present a Devaney-chaotic reversible cellular automaton that is universal in
their sense, answering a question that they explicitly left open. We also
discuss some implications and limitations of the construction.Comment: 12 pages + 1 page appendix, 4 figures. Accepted to Reversible
Computation 2014 (proceedings published by Springer
Generalized Effective Reducibility
We introduce two notions of effective reducibility for set-theoretical
statements, based on computability with Ordinal Turing Machines (OTMs), one of
which resembles Turing reducibility while the other is modelled after Weihrauch
reducibility. We give sample applications by showing that certain (algebraic)
constructions are not effective in the OTM-sense and considerung the effective
equivalence of various versions of the axiom of choice
The semaphore codes attached to a Turing machine via resets and their various limits
We introduce semaphore codes associated to a Turing machine via resets.
Semaphore codes provide an approximation theory for resets. In this paper we
generalize the set-up of our previous paper "Random walks on semaphore codes
and delay de Bruijn semigroups" to the infinite case by taking the profinite
limit of -resets to obtain -resets. We mention how this opens new
avenues to attack the P versus NP problem.Comment: 28 pages; Sections 3-6 appeared in a previous version of
arXiv:1509.03383 as Sections 9-12 (the split of the previous paper was
suggested by the journal); Sections 1-2 and 7 are ne
Intuitionistic computability logic
Computability logic (CL) is a systematic formal theory of computational tasks
and resources, which, in a sense, can be seen as a semantics-based alternative
to (the syntactically introduced) linear logic. With its expressive and
flexible language, where formulas represent computational problems and "truth"
is understood as algorithmic solvability, CL potentially offers a comprehensive
logical basis for constructive applied theories and computing systems
inherently requiring constructive and computationally meaningful underlying
logics.
Among the best known constructivistic logics is Heyting's intuitionistic
calculus INT, whose language can be seen as a special fragment of that of CL.
The constructivistic philosophy of INT, however, has never really found an
intuitively convincing and mathematically strict semantical justification. CL
has good claims to provide such a justification and hence a materialization of
Kolmogorov's known thesis "INT = logic of problems". The present paper contains
a soundness proof for INT with respect to the CL semantics. A comprehensive
online source on CL is available at http://www.cis.upenn.edu/~giorgi/cl.htm
-Generic Computability, Turing Reducibility and Asymptotic Density
Generic computability has been studied in group theory and we now study it in
the context of classical computability theory. A set A of natural numbers is
generically computable if there is a partial computable function f whose domain
has density 1 and which agrees with the characteristic function of A on its
domain. A set A is coarsely computable if there is a computable set C such that
the symmetric difference of A and C has density 0. We prove that there is a
c.e. set which is generically computable but not coarsely computable and vice
versa. We show that every nonzero Turing degree contains a set which is not
coarsely computable. We prove that there is a c.e. set of density 1 which has
no computable subset of density 1. As a corollary, there is a generically
computable set A such that no generic algorithm for A has computable domain. We
define a general notion of generic reducibility in the spirt of Turing
reducibility and show that there is a natural order-preserving embedding of the
Turing degrees into the generic degrees which is not surjective
Non-Transitive Self-Knowledge: Luminosity via Modal μ-Automata
This essay provides a novel account of iterated epistemic states. The essay argues that states of epistemic determinacy might be secured by countenancing self-knowledge on the model of fixed points in monadic second-order modal logic, i.e. the modal -calculus. Despite the epistemic indeterminacy witnessed by the invalidation of modal axiom 4 in the sorites paradox -- i.e. the KK principle: -- an epistemic interpretation of a -automaton permits fixed points to entrain a principled means by which to account for necessary conditions on self-knowledge
- …