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We investigate the truth-table degrees of (co-)c.e.\ sets, in particular,sets of random strings. It is known that the set of random strings with respectto any universal prefix-free machine is Turing complete, but that truth-tablecompleteness depends on the choice of universal machine. We show that for suchsets of random strings, any finite set of their truth-table degrees do not meetto the degree~0, even within the c.e. truth-table degrees, but when taking themeet over all such truth-table degrees, the infinite meet is indeed~0. Thelatter result proves a conjecture of Allender, Friedman and Gasarch. We alsoshow that there are two Turing complete c.e. sets whose truth-table degreesform a minimal pair.

Random strings and tt-degrees of Turing complete C.E. sets