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- Marek Karpinski, Angus Macintyre
- STOC
- 1994

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 first time, the VC Dimension of analog neural networks with the sigmoid activation function o(y) = 1/1 + e-y to be bounded by a quadratic polynomial in the number of programmable parameters.

- Marek Karpinski, Angus Macintyre
- J. Comput. Syst. Sci.
- 1997

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)

- Angus Macintyre
- J. Symb. Log.
- 1976

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 give axiomatizations and prove quantifier elimination theorems for first-order theories of unramified valued fields with an automorphism having a close interaction with the valuation. We achieve an analogue of the classical Ostrowski theory of pseudoconvergence. In the outstanding case of Witt vectors with their Frobenius map, we use the ∂-ring formalism… (More)

- Angus Macintyre
- Ann. Pure Appl. Logic
- 1990

- Angus Macintyre
- Ann. Pure Appl. Logic
- 1997

- Lou van den Dries, Angus Macintyre, David Marker
- Ann. Pure Appl. Logic
- 2001

- Angus Macintyre
- Bulletin of Symbolic Logic
- 2003

- Marek Karpinski, Angus Macintyre
- EuroCOLT
- 1995

Techniques from diierential topology are used to give polynomial bounds for the VC-dimension of sigmoidal neural networks. The bounds are quadratic in w, the dimension of the space of weights. Similar results are obtained for a wide class of Pfaaan activation functions. The obstruction (in diierential topology) to improving the bound to an optimal bound O… (More)