Angus Macintyre

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We introduce a new method for proving explicit upper bounds on the VC Dimension of general functional basis networks, and prove as an application, for the rst time, that the VC Dimension of analog neural networks with the sigmoidal activation function (y) = 1=1+e ?y is bounded by a quadratic polynomial O((lm) 2) in both the number l of programmable(More)
Using relativizations of results of Goncharov and Peretyat'kin on decidable homogeneous models, we prove that if M is S-saturated for some Scott set S, and F is an enumeration of S, then M has a presentation recursive in F. Applying this result we are able to classify degrees coding (i) the reducts of models of PA to addition or multiplication, (ii)(More)
We introduce a new powerful method of approximating the volume (and integrals) for a vast number of geometric body classes defined by boolean combinations of Pfaffian conditions. The method depends on the VC Dimension of the underlying classes of bodies. The resulting approximation algorithms are quite different in spirit from the other up to now known(More)