On the completeness of quantum computation models

The notion of computability is stable (i.e. independent of the choice of an indexing) over infinite-dimensional vector spaces provided they have a finite "tensorial dimension". Such vector spaces with a finite tensorial dimension permit to define an absolute notion of completeness for quantum computation models and give a precise meaning to the Church-Turing thesis in the framework of quantum theory. (Extra keywords: quantum programming languages, denotational semantics, universality.)Comment: 15 pages, LaTe

On the solution of trivalent decision problems by quantum state identification

The trivalent functions of a trit can be grouped into equipartitions of three elements. We discuss the separation of the corresponding functional classes by quantum state identifications

Constructive Dimension and Turing Degrees

This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dim_H(S) and constructive packing dimension dim_P(S) is Turing equivalent to a sequence R with dim_H(R) <= (dim_H(S) / dim_P(S)) - epsilon, for arbitrary epsilon > 0. Furthermore, if dim_P(S) > 0, then dim_P(R) >= 1 - epsilon. The reduction thus serves as a *randomness extractor* that increases the algorithmic randomness of S, as measured by constructive dimension. A number of applications of this result shed new light on the constructive dimensions of Turing degrees. A lower bound of dim_H(S) / dim_P(S) is shown to hold for the Turing degree of any sequence S. A new proof is given of a previously-known zero-one law for the constructive packing dimension of Turing degrees. It is also shown that, for any regular sequence S (that is, dim_H(S) = dim_P(S)) such that dim_H(S) > 0, the Turing degree of S has constructive Hausdorff and packing dimension equal to 1. Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor, and that bounded Turing reductions cannot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truth-table and bounded Turing reductions differ in their ability to extract dimension.Comment: The version of this paper appearing in Theory of Computing Systems, 45(4):740-755, 2009, had an error in the proof of Theorem 2.4, due to insufficient care with the choice of delta. This version modifies that proof to fix the error

The diagonalization method in quantum recursion theory

As quantum parallelism allows the effective co-representation of classical mutually exclusive states, the diagonalization method of classical recursion theory has to be modified. Quantum diagonalization involves unitary operators whose eigenvalues are different from one.Comment: 15 pages, completely rewritte

Program analysis is harder than verification: A computability perspective

We study from a computability perspective static program analysis, namely detecting sound program assertions, and verification, namely sound checking of program assertions. We first design a general computability model for domains of program assertions and correspond- ing program analysers and verifiers. Next, we formalize and prove an instantiation of Rice\u2019s theorem for static program analysis and verifica- tion. Then, within this general model, we provide and show a precise statement of the popular belief that program analysis is a harder prob- lem than program verification: we prove that for finite domains of pro- gram assertions, program analysis and verification are equivalent prob- lems, while for infinite domains, program analysis is strictly harder than verification

Depth, Highness and DNR Degrees

A sequence is Bennett deep [5] if every recursive approximation of the Kolmogorov complexity of its initial segments from above satisfies that the difference between the approximation and the actual value of the Kolmogorov complexity of the initial segments dominates every constant function. We study for different lower bounds r of this difference between approximation and actual value of the initial segment complexity, which properties the corresponding r(n)-deep sets have. We prove that for r(n) = εn, depth coincides with highness on the Turing degrees. For smaller choices of r, i.e., r is any recursive order function, we show that depth implies either highness or diagonally-non-recursiveness (DNR). In particular, for left-r.e. sets, order depth already implies highness. As a corollary, we obtain that weakly-useful sets are either high or DNR. We prove that not all deep sets are high by constructing a low order-deep set. Bennett's depth is defined using prefix-free Kolmogorov complexity. We show that if one replaces prefix-free by plain Kolmogorov complexity in Bennett's depth definition, one obtains a notion which no longer satisfies the slow growth law (which stipulates that no shallow set truth-table computes a deep set); however, under this notion, random sets are not deep (at the unbounded recursive order magnitude). We improve Bennett's result that recursive sets are shallow by proving all K-trivial sets are shallow; our result is close to optimal. For Bennett's depth, the magnitude of compression improvement has to be achieved almost everywhere on the set. Bennett observed that relaxing to infinitely often is meaningless because every recursive set is infinitely often deep. We propose an alternative infinitely often depth notion that doesn't suffer this limitation (called i.o. depth).We show that every hyperimmune degree contains a i.o. deep set of magnitude εn, and construct a π01- class where every member is an i.o. deep set of magnitude εn. We prove that every non-recursive, non-DNR hyperimmune-free set is i.o. deep of constant magnitude, and that every nonrecursive many-one degree contains such a set

Quantum value indefiniteness

The indeterministic outcome of a measurement of an individual quantum is certified by the impossibility of the simultaneous, definite, deterministic pre-existence of all conceivable observables from physical conditions of that quantum alone. We discuss possible interpretations and consequences for quantum oracles.Comment: 19 pages, 2 tables, 2 figures; contribution to PC0