6,581 research outputs found
On the Computability of Solomonoff Induction and Knowledge-Seeking
Solomonoff induction is held as a gold standard for learning, but it is known
to be incomputable. We quantify its incomputability by placing various flavors
of Solomonoff's prior M in the arithmetical hierarchy. We also derive
computability bounds for knowledge-seeking agents, and give a limit-computable
weakly asymptotically optimal reinforcement learning agent.Comment: ALT 201
Poly-infix operators and operator families
Poly-infix operators and operator families are introduced as an alternative
for working modulo associativity and the corresponding bracket deletion
convention. Poly-infix operators represent the basic intuition of repetitively
connecting an ordered sequence of entities with the same connecting primitive.Comment: 8 page
Highly Undecidable Problems For Infinite Computations
We show that many classical decision problems about 1-counter
omega-languages, context free omega-languages, or infinitary rational
relations, are -complete, hence located at the second level of the
analytical hierarchy, and "highly undecidable". In particular, the universality
problem, the inclusion problem, the equivalence problem, the determinizability
problem, the complementability problem, and the unambiguity problem are all
-complete for context-free omega-languages or for infinitary rational
relations. Topological and arithmetical properties of 1-counter
omega-languages, context free omega-languages, or infinitary rational
relations, are also highly undecidable. These very surprising results provide
the first examples of highly undecidable problems about the behaviour of very
simple finite machines like 1-counter automata or 2-tape automata.Comment: to appear in RAIRO-Theoretical Informatics and Application
Computational reverse mathematics and foundational analysis
Reverse mathematics studies which subsystems of second order arithmetic are
equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main
philosophical application of reverse mathematics proposed thus far is
foundational analysis, which explores the limits of different foundations for
mathematics in a formally precise manner. This paper gives a detailed account
of the motivations and methodology of foundational analysis, which have
heretofore been largely left implicit in the practice. It then shows how this
account can be fruitfully applied in the evaluation of major foundational
approaches by a careful examination of two case studies: a partial realization
of Hilbert's program due to Simpson [1988], and predicativism in the extended
form due to Feferman and Sch\"{u}tte.
Shore [2010, 2013] proposes that equivalences in reverse mathematics be
proved in the same way as inequivalences, namely by considering only
-models of the systems in question. Shore refers to this approach as
computational reverse mathematics. This paper shows that despite some
attractive features, computational reverse mathematics is inappropriate for
foundational analysis, for two major reasons. Firstly, the computable
entailment relation employed in computational reverse mathematics does not
preserve justification for the foundational programs above. Secondly,
computable entailment is a complete relation, and hence employing it
commits one to theoretical resources which outstrip those available within any
foundational approach that is proof-theoretically weaker than
.Comment: Submitted. 41 page
Revising Type-2 Computation and Degrees of Discontinuity
By the sometimes so-called MAIN THEOREM of Recursive Analysis, every
computable real function is necessarily continuous. Weihrauch and Zheng
(TCS'2000), Brattka (MLQ'2005), and Ziegler (ToCS'2006) have considered
different relaxed notions of computability to cover also discontinuous
functions. The present work compares and unifies these approaches. This is
based on the concept of the JUMP of a representation: both a TTE-counterpart to
the well known recursion-theoretic jump on Kleene's Arithmetical Hierarchy of
hypercomputation: and a formalization of revising computation in the sense of
Shoenfield.
We also consider Markov and Banach/Mazur oracle-computation of discontinuous
fu nctions and characterize the computational power of Type-2 nondeterminism to
coincide with the first level of the Analytical Hierarchy.Comment: to appear in Proc. CCA'0
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