We discuss combining physical experiments with machine computations and introduce a form of analogue-digital (AD) Turing machine. We examine in detail a case study where an experimental procedure based on Newtonian kinematics is combined with a class of Turing machines. Three forms of AD machine are studied, in which physical parameters can be set exactly and approximately. Using non-uniform complexity theory, and some probability, we prove theorems that show that these machines can compute more than classical Turing machines.

Earlier, to explore the idea of combining physical experiments with algorithms, we introduced a new form of analogue-digital (AD) Turing machine. We examined in detail a case study where an experimental procedure, based on Newtonian kinematics, is used as an oracle with classes of Turing machines. The physical cost of oracle calls was counted and three forms of AD queries were studied, in which physical parameters can be set exactly and approximately. Here, in this sequel, we complete the classification of the computational power of these AD Turing machines and determine precisely what they can compute, using non-uniform complexity classes and probabilities.

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We define the notion of a catalytic-space computation. This is a computation that has a small amount of clean space available and is equipped with additional auxiliary space, with the caveat that the additional space is initially in an arbitrary, possibly incompressible, state and must be returned to this state when the computation is finished. We show that the extra space can be used in a nontrivial way, to compute uniform TC 1 -circuits with just a logarithmic amount of clean space. The extra space thus works analogously to a catalyst in a chemical reaction. TC 1 -circuits can compute for example the determinant of a matrix, which is not known to be computable in logspace.In order to obtain our results we study an algebraic model of computation, a variant of straight-line programs. We employ register machines with input registers x1, .

In this paper we show that BPP is truth-table reducible to the set of Kolmogorov random strings R K . It was previously known that PSPACE, and hence BPP is Turing-reducible to R K . The earlier proof relied on the adaptivity of the Turingreduction to find a Kolmogorov-random string of polynomial length using the set R K as oracle. Our new non-adaptive result relies on a new fundamental fact about the set R K , namely each initial segment of the characteristic sequence of R K is not compressible by recursive means. As a partial converse to our claim we show that strings of high Kolmogorov-complexity when used as advice are not much more useful than randomly chosen strings.

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Abstract. This paper is motivated by a conjecture [All12, ADF + 13] that BPP can be characterized in terms of polynomial-time nonadaptive reductions to the set of Kolmogorovrandom strings. In this paper we show that an approach laid out in [ADF + 13] to settle this conjecture cannot succeed without significant alteration, but that it does bear fruit if we consider time-bounded Kolmogorov complexity instead.We show that if a set A is reducible in polynomial time to the set of time-t-bounded Kolmogorov-random strings (for all large enough time bounds t), then A is in P/poly, and that if in addition such a reduction exists for any universal Turing machine one uses in the definition of Kolmogorov complexity, then A is in PSPACE.