Methods of Approach At the outset it might be well to distinguish sharply between two general approaches to the problem of machine learning.

We have divided this area into categories and cross-indexed the references accordingly.

The editors have provided him with special introductory sections as a ing guide to a thorough understand of the reports.

An early expression of this difficulty is the socalled might be stated that Socrates's birthplace was Athens.

Step 6 is a goal-assertion the input, another algorithm might result. Thus one could resolution that functions similarly to the goal-goal resolution break a into a[1],..., a [length(a)/2] and a [length(a)/ above. The final synthesized program is: 2 1],..., a[length(a)] and find an algorithm that recursively calls f on both the first and second halves of its f(x) if x NIL then 0 else car(x) f(cdr(x)).

Printed in the United States of America PREFACE Among the most challenging scientific questions of our time are the corresponding analytic and synthetic problems: How does the brain function? Can we design a machine which will simulate a brain? Speculation on these problems, which can be traced back many centuries, usually reflects in any period the characteristics of machines then in use. Descartes, in DeBomine, sees the lower animals and, in many of his functions, man as automata. Using analogies drawn from water-clocks, fountains and mechanical devices common to the seventeenth century, he imagined that the nerves transmitted signals by tiny mechanical motions.

IDijkstra 19591 Dijkstra, E., NA Note on Two Problems in Connection with Graphs," Numeri.scht'

It is no pure coincidence that an organic chemist rather than a computer specialist is writing this foreword.

\emph{Canonical (logic) programs} (CP) refer to normal logic programs augmented with connective $not\ not$. In this paper we address the question of whether CP are \emph{succinctly incomparable} with \emph{propositional formulas} (PF). Our main result shows that the PARITY problem, which can be polynomially represented in PF but \emph{only} has exponential representations in CP. In other words, PARITY \emph{separates} PF from CP. Simply speaking, this means that exponential size blowup is generally inevitable when translating a set of formulas in PF into an equivalent program in CP (without introducing new variables). Furthermore, since it has been shown by Lifschitz and Razborov that there is also a problem that separates CP from PF (assuming $\mathsf{P}\nsubseteq \mathsf{NC^1/poly}$), it follows that CP and PF are indeed succinctly incomparable. From the view of the theory of computation, the above result may also be considered as the separation of two \emph{models of computation}, i.e., we identify a language in $\mathsf{NC^1/poly}$ which is not in the set of languages computable by polynomial size CP programs.

In this paper we explore how machine learning techniques can be applied to the discovery of efficient mathematical identities. We introduce an attribute grammar framework for representing symbolic expressions. Given a grammar of math operators, we build trees that combine them in different ways, looking for compositions that are analytically equivalent to a target expression but of lower computational complexity. However, as the space of trees grows exponentially with the complexity of the target expression, brute force search is impractical for all but the simplest of expressions. Consequently, we introduce two novel learning approaches that are able to learn from simpler expressions to guide the tree search. The first of these is a simple n-gram model, the other being a recursive neural-network. We show how these approaches enable us to derive complex identities, beyond reach of brute-force search, or human derivation.