Recognizing Strong Random Reals

Abstract

Abstract. The class of strong random reals can be defined via a natural conception of effective null set. We show that the same class is also characterized by a learning-theoretic criterion of 'recognizability'.1. Characterizing randomness. Consider a physical process that, if suitably idealized, generates an indefinite sequence of independent random bits. One such process might be radioactive decay of a lump of uranium whose mass is kept at just the level needed to ensure that the probability is one-half that no alpha particle is emitted in the nth microsecond of the experiment. Let us think of the bits as drawn from {0, 1} and denote the resulting sequence by x with coordinates x 0 , x 1 , . . .. Now wouldn't it be odd if there were a computer program P with the following property? The program will not, in general, allow prediction of x i inasmuch as there is no requirement that the ultimate bit b m written by P(i) be marked as final. Nonetheless, shouldn't randomness exclude any computational process from having the kind of intimate knowledge of x i described in 1? For any input i, P enters a nonterminating routine that writes a nonemptyThe tension engendered by 1 afflicts a celebrated theory of randomness developed over the last half century. 2 The theory offers diverse criteria, each well motivated, for the concept 'infinite sequence of random bits'. Remarkably, the criteria yield the same collection of sequences -a collection, moreover, of measure 1 with respect to the 'coin flip' measure on the collection of infinite binary sequences. Despite this evidence for theoretical adequacy, some of the sequences labeled 'random' can be associated with a program P satisfying 1.One response to this state of affairs has been to modify (in a simple and satisfying way) the randomness criteria originally proposed by Martin-Löf (1966). The resulting collection Received xxxxx, 200x. 1 The sequences satisfying 1 correspond to the limit recursive sets of Gold (1965) and trial and error predicates of Putnam (1965); such limits were introduced earlier by Shoenfield (1959) to characterize the sets recursive in 0 .

On Analogues of the Church–turing Thesis in Algorithmic Randomness

“…However, there are some Martin-Löf random sequences that are not weakly 2-random, particularly all 0 2 Martin-Löf random sequences. 25 In fact, some have held that the existence of 0 2 Martin-Löf random sequences provides grounds for rejecting the Martin-Löf-Chaitin thesis (see, for instance, Raatikainen (2000) and Osherson and Weinstein (2008)). The objection here is that 0 2 random sequences, though not computable, are nonetheless decidable in the limit.…”

“…Second, the search for a definition of randomness that captures the so-called intuitive conception was a central impetus in the development of algorithmic randomness; 8 here I account for why this search could not be successfully carried out. Lastly, although theses such as the Martin-Löf-Chaitin thesis do not currently play a role in the theory of algorithmic randomness (unlike the case of the Church-Turing thesis in computability theory 9 ), in some recent work, both the Martin-Löf-Chaitin thesis ([Del11], [Das11]) and the Weak 2-Randomness thesis ( [OW08]) have been defended. Thus, the view against which I am arguing is not without its adherents.…”

“…As discussed above, the measures of the critical regions given by a Martin-Löf test (U i ) i∈ω are effectively bounded: 2 Martin-Löf random sequences. 25 In fact, that some Martin-Löf random sequences are ∆ 0 2 has been taken by some to be grounds for rejecting the Martin-Löf-Chaitin thesis (see, for instance, [Raa00] and [OW08]). The objection here is that ∆ 0 2 random sequences, though not computable, are nonetheless decidable in the limit.…”

Initial segment complexities of randomness notions

“…Strong randomness is by some authors considered to be the next counterpart of Kurtz randomness, although it is not the relativised version; therefore they call Kurtz random also "weakly random" and strongly random also "weakly 2-random" [13]. Strong randomness [8,17] has various nice characterisations, in particular the following: A is strongly random iff A is Martin-Löf random and forms a minimal pair with K with respect to Turing reducibility [4,Footnote 2]. For these notions, in order to quantify the degree of non-randomness of a sequence, one studies from which value f (m) onwards all initial segments can be compressed by m bits.…”

Relativizations of randomness and genericity notions

“…Furthermore, a set is called "weakly 2-random" [20] or "strongly random" [24] if and only if it is Martin-Löf random and forms a minimal pair with the halting problem.…”