Bind every sequence

Every Sequence Is Decompressible from a Random One

Kučera and Gács independently showed that every infinite sequence is Turing reducible to a Martin-Löf random sequence. This result is extended by showing that every infinite sequence S is Turing reducible to a Martin-Löf random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S. It is shown that this ...

A Variant of the Euclid-Mullin Sequence Containing Every Prime

We consider a generalization of Euclid’s proof of the infinitude of primes and show that it leads to variants of the Euclid-Mullin sequence that provably contain every prime number.

Every Sequence Is Reducible to a Random One

Charles Bennett asked whether every infinite binary sequence can be obtained from an "incompressible" one by a Turing machine, He proved that this is the case for arithmetical sequences. The question has some philosophical interest because it permits us to view even very pathological sequences as the result of the combination of two relatively wellunderstood processes: the completely chaotic ou...

Every sequence is compressible to a random one

Kučera and Gaćs independently showed that every infinite sequence is Turing reducible to a Martin-Löf random sequence. We extend this result to show that every infinite sequence S is Turing reducible to a Martin-Löf random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S. We show that this is the op...

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