By Solomon Marcus (auth.), Michael J. Dinneen, Bakhadyr Khoussainov, André Nies (eds.)
This Festschrift quantity has been released in honor of Cristian Calude at the party of his sixtieth birthday and includes contributions from invited audio system and typical papers provided on the foreign Workshop on Theoretical computing device technology, WTCS 2012, held in Auckland, New Zealand, in February 2012.
Cristian Calude has made an important contribution to analyze in machine technological know-how idea. besides early paintings via Chaitin, Kučera, Kurtz, Solovay, and Terwijn his papers released within the mid-1990s together with Khoussainov, Hertling, and Wang laid the root for the improvement of recent conception of algorithmic randomness. His paintings used to be crucial for constructing the best function of latest Zealand during this zone.
The examine pursuits of Cristian Calude are mirrored within the issues lined via the 32 papers incorporated during this ebook, particularly: algorithmic details conception, algorithms, automata and formal languages, computing and typical sciences, computability and purposes, common sense and functions, philosophy of computation, physics and computation, and unconventional types of computation. they've been geared up into 4 components. the 1st half contains papers discussing his existence achievements. this is often through papers within the 3 basic parts of complexity, computability, and randomness; physics, philosophy (and logic), and computation; and algorithms, automata, and formal types (including unconventional computing).
Read or Download Computation, Physics and Beyond: International Workshop on Theoretical Computer Science, WTCS 2012, Dedicated to Cristian S. Calude on the Occasion of His 60th Birthday, Auckland, New Zealand, February 21-24, 2012, Revised Selected and Invited Papers PDF
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Additional resources for Computation, Physics and Beyond: International Workshop on Theoretical Computer Science, WTCS 2012, Dedicated to Cristian S. Calude on the Occasion of His 60th Birthday, Auckland, New Zealand, February 21-24, 2012, Revised Selected and Invited Papers
In other terms, the sum is non-random if and only if the ratio ri /mi tends to 0. Proof. Assume that ri /mi → 0. Then for every ε we can let hi = ε mi and get a lower semicomputable sequence that satisfies the conditions of Theorem 7. Therefore α is not random. We can also prove that α is not complete (thus providing an alternative proof of its non-randomness). Recall the argument used in the proof of Theorem 2: if ri mi , then ∑ ri 1 ∑ mi . And if ri cmi , then ∑ ri c ∑ mi . This remains true if the inequality ri cmi is true for all sufficiently large i.
We provide the proof of this result below, starting with one direction in the next section 3 and finishing the other direction in section 5. 3 Complete Lower Semicomputable Reals Are Random The fact that lower semicomputable reals are random, is Chaitin’s theorem (randomness of Ω ). It is usually proved by using complexity characterization of randomness. ). First, we prove that there exists a lower semicomputable random real. For that we consider an effectively open set U of measure less than (say) 1/2 that covers all nonrandom reals in [0, 1].
A weaker property is lower semicomputability. A real number α is lower semicomputable if it is a limit of a computable increasing sequence of rational numbers. Such a sequence is called approximation of α from below in the sequel. Equivalent definition: α is lower semicomputable if the set of all rational numbers less than α is enumerable. One more reformulation: if α = ∑i 0 di where di is computable series of rational numbers, and all di with i > 0 are non-negative. ) It is easy to see that α is computable if and only if α and −α are lower semicomputable.