Basic concepts in quantum computation by Ekert A., Hayden P., Inamori H.

By Ekert A., Hayden P., Inamori H.

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MATULA, Subtree isomorphism in 0(n5/2), Annals of Discrete Mathematics, 2 (1978), pp. 91-106. 12. G. L. MILLER AND J. H. I~EIF, Parallel tree contraction part 2: Further applications, SIAM Journal on Computing, 20 (1991), pp. 1128-1147. 13. W. L. Ruzzo, On uniform circuit complexity, J. Comput. , 22 (1981), pp. 365-383. Ultrafilter Logic and Generic Reasoning W. A. Carnielli* P . A . S . Veloso** Abstract We examine a new logical system, capturing the intuition of 'most' by means of generalised quantifiers over ultrafilters, with the aim of providing a basis for genetic reasoning.

The gate H[uw, vw*] outputs True iff it is not the case that vw* represents a subtree $2 which violates one of the conditions (3) or (4) of the definition of "S weakly ~--distinguishes U1 and U2". In view of Lemma 6, it is straightforward to verify that the circuit correctly performs tree comparison. One point that deserves further justification is the manner in which w* is chosen in F+[uw, v]: the string w* is chosen so that H[uw, vw*] can decide whether vw* represents a subtree $2 which violates condition (4) of the definition of "S weakly ~--distinguishes U1 and U2 ".

By a generic element we mean one that is generic for the set of all formulae of Lox0(~) v (with single free variable x). Generic elements are indiscernible among themselves: they cannot separated by formulae. In a broad intuitive sense, generic elements are somewhat reminiscent of Hilbert's ideal elements, or even of Platonic forms. So, it is not surprising that some ultrafilter structures fail to have generic elements. For instance, in the natural numbers with zero and successor and a non-principal ultrafilter, say one containing the cofinite subsets, any generic element turns out to be nonstandard.

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