7 research outputs found

    Quantum Certificate Complexity

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    Given a Boolean function f, we study two natural generalizations of the certificate complexity C(f): the randomized certificate complexity RC(f) and the quantum certificate complexity QC(f). Using Ambainis' adversary method, we exactly characterize QC(f) as the square root of RC(f). We then use this result to prove the new relation R0(f) = O(Q2(f)^2 Q0(f) log n) for total f, where R0, Q2, and Q0 are zero-error randomized, bounded-error quantum, and zero-error quantum query complexities respectively. Finally we give asymptotic gaps between the measures, including a total f for which C(f) is superquadratic in QC(f), and a symmetric partial f for which QC(f) = O(1) yet Q2(f) = Omega(n/log n).Comment: 9 page

    The complexity of the parity function in unbounded fan-in, unbounded depth circuits

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    AbstractAlmost everything is known on the complexity of the parity function in fan-in 2 circuits over various bases. Also the minimal depth of polynomial-size, unbounded fan-in {∧, ∨, ⌝\dl;} circuits for the parity function has been studied. Here the complexity without any depth restriction is considered. For the basis {∧, ∨, ⌝\dl;} almost optimal bounds, and for the basis of NOR gates and the basis of all threshold functions optimal bounds on the number of gates are obtained. For the basis {∧, ∨, ⌝\dl;} the minimal number of wires is determined. For threshold circuits an exponential gap between synchronous and asynchronous circuits is proved. The results not only answer open questions in complexity theory but also have implications for the real-life circuit design

    Properties of complexity measures for PRAMs and WRAMs

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    AbstractThe computation of Boolean functions by parallel computers with shared memory (PRAMs and WRAMs) is considered. In particular, complexity measures for parallel computers like critical and sensitive complexity are compared with other complexity measures for Boolean functions like branching program depth and length of prime implicants and clauses.The relations between these complexity measures and their asymptotic behaviour are investigated for the classes of Boolean functions, monotone functions and symmetric functions
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