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19
Computing a glimpse of randomness
 Experimental Mathematics
"... A Chaitin Omega number is the halting probability of a universal Chaitin (selfdelimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In ..."
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Cited by 18 (9 self)
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A Chaitin Omega number is the halting probability of a universal Chaitin (selfdelimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly noncomputable. The aim of this paper is to describe a procedure, which combines Java programming and mathematical proofs, for computing the exact values of the first 63 bits of a Chaitin Omega: 000000100000010000100000100001110111001100100111100010010011100. Full description of programs and proofs will be given elsewhere. 1
Algorithmic randomness of closed sets
 J. LOGIC AND COMPUTATION
, 2007
"... We investigate notions of randomness in the space C[2 N] of nonempty closed subsets of {0, 1} N. A probability measure is given and a version of the MartinLöf test for randomness is defined. Π 0 2 random closed sets exist but there are no random Π 0 1 closed sets. It is shown that any random 4 clos ..."
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Cited by 11 (8 self)
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We investigate notions of randomness in the space C[2 N] of nonempty closed subsets of {0, 1} N. A probability measure is given and a version of the MartinLöf test for randomness is defined. Π 0 2 random closed sets exist but there are no random Π 0 1 closed sets. It is shown that any random 4 closed set is perfect, has measure 0, and has box dimension log2. A 3 random closed set has no nc.e. elements. A closed subset of 2 N may be defined as the set of infinite paths through a tree and so the problem of compressibility of trees is explored. If Tn = T ∩ {0, 1} n, then for any random closed set [T] where T has no dead ends, K(Tn) ≥ n − O(1) but for any k, K(Tn) ≤ 2 n−k + O(1), where K(σ) is the prefixfree complexity of σ ∈ {0, 1} ∗.
Algorithmically Coding the Universe
 Developments in Language Theory, World Scientific
, 1994
"... All science is founded on the assumption that the physical universe is ordered. Our aim is to challenge this hypothesis using arguments from the algorithmic information theory. 1 Introduction Algorithmic information theory opens new vistas that extend far beyond the traditional boundaries of mathem ..."
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Cited by 10 (7 self)
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All science is founded on the assumption that the physical universe is ordered. Our aim is to challenge this hypothesis using arguments from the algorithmic information theory. 1 Introduction Algorithmic information theory opens new vistas that extend far beyond the traditional boundaries of mathematics and computer science. How can we describe the seemingly random processes in nature and reconcile them with the supposed order? How much can a given piece of information be compressed? These are matters of fundamental scientific importance that will be discussed below, mainly from an informal or semiformal point of view. The descriptional complexity of a sequence of bits, finite or infinite, is the length of the shortest sequence of bits defining the originally given sequence. A given sequence being random means, roughly, that its descriptional complexity equals its length. In other words the simplest way to define the sequence is to write it down. This seems to be the case for the seq...
Is Complexity a Source of Incompleteness?
 IS COMPLEXITY A SOURCE OF INCOMPLETENESS
, 2004
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Extending the Loop Language with HigherOrder Procedural Variables
 Special issue of ACM TOCL on Implicit Computational Complexity
, 2010
"... We extend Meyer and Ritchie’s Loop language with higherorder procedures and procedural variables and we show that the resulting programming language (called Loop ω) is a natural imperative counterpart of Gödel System T. The argument is twofold: 1. we define a translation of the Loop ω language int ..."
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Cited by 9 (6 self)
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We extend Meyer and Ritchie’s Loop language with higherorder procedures and procedural variables and we show that the resulting programming language (called Loop ω) is a natural imperative counterpart of Gödel System T. The argument is twofold: 1. we define a translation of the Loop ω language into System T and we prove that this translation actually provides a lockstep simulation, 2. using a converse translation, we show that Loop ω is expressive enough to encode any term of System T. Moreover, we define the “iteration rank ” of a Loop ω program, which corresponds to the classical notion of “recursion rank ” in System T, and we show that both translations preserve ranks. Two applications of these results in the area of implicit complexity are described. 1
LanguageTheoretic Complexity of Disjunctive Sequences
 DISCRETE APPL. MATH
, 1995
"... A sequence over an alphabet # is called disjunctive [13] if it contains all possible finite strings over # as its substrings. Disjunctive sequences have been recently studied in various contexts, e.g. [12, 9]. They abound in both category and measure senses [5]. In this paper we measure the complexi ..."
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Cited by 5 (1 self)
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A sequence over an alphabet # is called disjunctive [13] if it contains all possible finite strings over # as its substrings. Disjunctive sequences have been recently studied in various contexts, e.g. [12, 9]. They abound in both category and measure senses [5]. In this paper we measure the complexity of a sequence x by the complexity of the language P (x) consisting of all prefixes of x. The languages P (x) associated to disjunctive sequences can be arbitrarily complex. We show that for some disjunctive numbers x the language P (x) is contextsensitive, but no language P (x) associate to a disjunctive number can be contextfree. We also show that computing a disjunctive number x by rationals corresponding to an infinite subset of P (x)does not decrease the complexity of the procedure, i.e. if x is disjunctive, then P (x) contains no infinite contextfree language. This result reinforces, in a way, Chaitin's thesis [6] according to which perfect sets, i.e. sets for which there is no w...
Regressive Ramsey numbers are Ackermannian
 J. Combin. Theory Ser. A
, 1999
"... Abstract. We give an elementary proof of the fact that regressive Ramsey numbers are Ackermannian. This fact was first proved by Kanamori and McAloon with mathematical logic techniques. Nous vivons encore sous le règne de la logique, voilà, bien entendu, à quoi je voulais en venir. Mais les procédés ..."
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Cited by 5 (2 self)
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Abstract. We give an elementary proof of the fact that regressive Ramsey numbers are Ackermannian. This fact was first proved by Kanamori and McAloon with mathematical logic techniques. Nous vivons encore sous le règne de la logique, voilà, bien entendu, à quoi je voulais en venir. Mais les procédés logiques, de nos jours, ne s’appliquent plus qu’à la résolution de problèmes d’intérêt secondaire. [1, 1924, p. 13] 649 revision:19980508 modified:20020227 1.
Strong Determinism vs. Computability
 The Foundational Debate, Complexity and Constructivity in Mathematics and
, 1995
"... Are minds subject to laws of physics? Are the laws of physics computable? Are conscious thought processes computable? Currently there is little agreement as to what are the right answers to these questions. Penrose ([41], p. 644) goes one step further and asserts that: a radical new theory is indeed ..."
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Cited by 3 (1 self)
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Are minds subject to laws of physics? Are the laws of physics computable? Are conscious thought processes computable? Currently there is little agreement as to what are the right answers to these questions. Penrose ([41], p. 644) goes one step further and asserts that: a radical new theory is indeed needed, and I am suggesting, moreover, that this theory, when it is found, will be of an essentially noncomputational character. The aim of this paper is three fold: 1) to examine the incompatibility between the hypothesis of strong determinism and computability, 2) to give new examples of uncomputable physical laws, and 3) to discuss the relevance of Gödel’s Incompleteness Theorem in refuting the claim that an algorithmic theory—like strong AI—can provide an adequate theory of mind. Finally, we question the adequacy of the theory of computation to discuss physical laws and thought processes. 1
Reflections on Quantum Computing
, 2000
"... In this rather speculative note three problems pertaining to the power and limits of quantum computing are posed and partially answered: a) when are quantum speedups possible?, b) is fixedpoint computing a better model for quantum computing?, c) can quantum computing trespass the Turing barrier? 1 ..."
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Cited by 2 (0 self)
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In this rather speculative note three problems pertaining to the power and limits of quantum computing are posed and partially answered: a) when are quantum speedups possible?, b) is fixedpoint computing a better model for quantum computing?, c) can quantum computing trespass the Turing barrier? 1 When are quantum speedups possible? This section discusses the possibility that speedups in quantum computing can be achieved only for problems which have a few or even unique solutions [12]. For instance, this includes the computational complexity class UP [15]. Typical examples are Shor's quantum algorithm for prime factoring [18] and Grover's database search algorithm [13] for a single item satisfying a given condition in an unsorted database (see also Gruska [14]). In quantum complexity, one popular class of problems is BQP,whichisthe set of decision problems that can be solved in polynomial time (on a quantum computer) so that the correct answer is obtained with probability at l...
Effective Category and Measure in Abstract Complexity Theory (Extended Abstract)
, 1995
"... Complexity TheoryExtended Abstract #+ Cristian Calude # and Marius Zimand Abstract Strong variants of the Operator Speedup Theorem, Operator Gap Theorem and Compression Theorem are obtained using an e#ective version of Baire Category Theorem. It is also shown that all complexity classes of re ..."
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Cited by 1 (0 self)
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Complexity TheoryExtended Abstract #+ Cristian Calude # and Marius Zimand Abstract Strong variants of the Operator Speedup Theorem, Operator Gap Theorem and Compression Theorem are obtained using an e#ective version of Baire Category Theorem. It is also shown that all complexity classes of recursive predicates have e#ective measure zero in the space of recursive predicates and, on the other hand, the class of predicates with almost everywhere complexity above an arbitrary recursive threshold has recursive measure one in the class of recursive predicates. Keywords: Complexity measure, Operator Speedup Theorem, Operator Gap Theorem, Compression Theorem, e#ective Baire classification, e#ective measure. 1 Introduction The abstract complexity theory initiated by Blum [2] (see also Bridges [5], Calude [8], Hartmanis and Hopcroft [17], Machtey and Young [23], Seiferas [34]) has revealed fundamental properties of complexity measures. The striking importance of this theory relies in i...