We describe a method to perform functional operations on probability distributions of random variables. The method uses reproducing kernel Hilbert space representations of probability distributions, and it is applicable to all operations which can be applied to points drawn from the respective distributions. We refer to our approach as {\em kernel probabilistic programming}. We illustrate it on synthetic data, and show how it can be used for nonparametric structural equation models, with an application to causal inference.

Pointwise-competitive classifier from class F is required to classify identically to the best classifier in hindsight from F. For noisy, agnostic settings we present a strategy for learning pointwise-competitive classifiers from a finite training sample provided that the classifier can abstain from prediction at a certain region of its choice. For some interesting hypothesis classes and families of distributions, the measure of this rejected region is shown to be diminishing at a fast rate, with high probability. Exact implementation of the proposed learning strategy is dependent on an ERM oracle that can be hard to compute in the agnostic case. We thus consider a heuristic approximation procedure that is based on SVMs, and show empirically that this algorithm consistently outperforms a traditional rejection mechanism based on distance from decision boundary.

R. L. GREGORY, Psychological Laboratory, Cambridge Discussion on paper 5 683 6 Some questions concerning the explanation of learning 691 in animals MR. A. J. WATS01,4 Psychological Laboratory, Cambridge Discussion on paper 6 721 7 Information, redundancy and decay of the memory trace 729 DR. Y. BAR-HILLEL, The Hebrew University, Jerusalem Discussion on paper 2 801 3 To what extent can administration be mechanized?

If ability to perform complex calculations were a sufficient criterion, then even a conventional digital computor could lay claim to more intelligence than any of usand perhaps we had better let it make away with the word and be done with it.

A. YONEZAWA and C. HEWITT 41 REPRESENTATIONS FOR ABSTRACT REASONING 4 A maximal method for set variables in automatic theorem-proving W. W. BLEDSOE

J. Do RAN 433 22 Chromosome classification and segmentation as exercises in knowing what to expect.

B.RANDELL 3 PROGRAM PROOF AND MANIPULATION 2 Some techniques for proving correctness of programs which alter data structures.

The two outstanding figures in the history of computer science are Alan Turing and John von Neumann, and they shared the view that logic was the key to understanding and automating computation. In particular, it was Turing who gave us in the mid-1930s the fundamental analysis, and the logical definition, of the concept of'computability by machine' and who discovered the surprising and beautiful basic fact that there exist universal machines which by suitable programming can be made to t This essay is an expanded and revised version of one entitled The Role of Logic in Computer Science and Artificial Intelligence, which was completed in January 1992 (and was later published in the Proceedings of the Fifth Generation computer Systems 1992 Conference). Since completing that essay I have had the benefit of extremely helpful discussions on many of the details with Professor Donald Michie and Professor I. J. Good, both of whom knew Turing well during the war years at Bletchley Park. Professor J. A. N. Lee, whose knowledge of the literature and archives of the history of computing is encyclopedic, also provided additional information, some of which is still unpublished. Further light has very recently been shed on the von Neumann side of the story by Norman Macrae's excellent biography John von Neumann (Macrae 1992). Accordingly, it seemed appropriate to undertake a more complete and thorough version of the FGCS'92 essay, focussing somewhat more on the interesting historical and biographical issues. I am grateful to Donald Michie and Stephen Muggleton for inviting me to contribute such a'second edition' to the present volume, and I would also like to thank the Institute for New Computer Technology (ICOT) for kind permission to make use of the FGCS'92 essay in this way. 1 LOGIC, COMPUTERS, TURING, AND VON NEUMANN

OXFORD 1994 Oxford University Press, Walton Street, Oxford 0X2 6DP Oxford New York Athens Auckland Bangkok Bombay Calcutta Cape Town Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kuala Lumpur Madras Madrid Melbourne Mexico City Nairobi Paris Singapore Taipei Tokyo Toronto and associated companies in Berlin lbadan Published in the United States by Oxford University Press Inc., New York 0 E. K. Furukawa, D. Michie, and S. Muggleton, 1994 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press. Enquiries concerning reproduction outside those terms and in other countries should be sent to the Rights Department, Oxford University Press, at the address above. This book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. The founder of modern computational logic, J.A. Robinson, opens this volume with a chapter on the field's great forefathers John von Neumann and Alan Turing.

When you lose a game of chess to a computer then don't pretend that you didn't think at the game.