The problem we consider in this paper is that of discovering formal rules which will enable us to decide when a question posed in English can be answered on the basis of one or more declarative English sentences. To illustrate how this may be done in very simple cases we give rules which translate certain declarative sentences and questions involving the quantifiers'some', 'every', 'any', and'no' into a modified first-order predicate calculus, and answer the questions by comparing their translated forms with those of the declaratives. We suggest that in order to capture the meanings of more complex sentences it will be necessary to go beyond the first-order predicate calculus, to a notation in which the scope of words other than quantifiers and negations is clearly indicated. We conclude by describing a notational form for connected sentences, which seems to be a natural extension of Chomsky's'deep structures'. INTRODUCTION In this paper we shall consider the problem of when an English sentence, or a series of sentences, provides enough information to answer a question, also posed in English.

K.A.PATON 411 24 Centromere finding: some shape descriptors for small chromosome outlines.

K.A.PATON 411 24 Centromere finding: some shape descriptors for small chromosome outlines.

G.R.KIss 155 11 Will seeing machines have illusions?

K.A.PATON 411 24 Centromere finding: some shape descriptors for small chromosome outlines.

Of course, it is not at all obvious that (5) and (1) describe the same function. In order to prove that they do, however, it is necessary only to check that the result of applying (5) to an arbitrary object z is the same as the result of applying (1) to z. In fact, we have (xx(sQRT((rimEs x)((pLus x)oNE)))z) (sQRT((TimEs z)((pLus z)oNE))) when (1) is applied to z.

Both the analytical philosopher and the designer of intelligent software are doing what might be called'mental engineering': constructing precise, formal models of some aspects of intelligent thought or behaviour. There is a wide gap between them, but it is narrowing.

The solution to this simple problem would then guide the solution of the original problem. Minsky (1961) has discussed complex planning of this'homomorphic model' type, and has stressed the potential reduction in total search effort to be won. In the same paper he has also considered the use of semantic models' as a form of complex planning in a mathematical context. The successful geometry theorem-proving program of Gelernter (1959), which used a diagram' to test the validity of propositions, is a wellknown example of this form of planning. Recently Sandewall (1969) has defined a Planning Problem Solver (P P This is an attempt to explore in detail complex planning of the homomorphic model' type as applied to the

Rather, the program exists in a state of continual revision and extension, and that part of it which is running and yielding results at the moment is intended to form the nucleus of a much larger and more extensive automatic deduction system in the future. The program is being developed with a number of different questions and applications in mind. First of all, some of the present generation of deduction programs are already capable of proving the sort of theorem of an elementary mathematical theory that appears in the textbooks either as a basic theorem or standard exercise (sometimes as a'starred' exercise), and might reasonably be classified as'somewhat' tricky. We have in mind the theorems of algebra and number theory obtained by programs discussed by Wos, Robinson and Carson (1965), Wos et al. (1967), and Luckham (1968). In addition, it is reported by Guard et al. (1969) that an open problem in modular lattice theory was solved with the aid of an on-line program (and it is interesting to note that, with reference to the initial set of axioms and hypotheses, this seems not to be the most difficult theorem that has been proved by a program so far). We are thus motivated to ask if, by adding a flexible interactive facility to a good deduction program, it is possible to construct a system that would be useful in investigating some fairly basic mathematics.

We will define the notions of abstract theorem-proving graph, abstract theorem-proving problem g and search strategy E for g. These concepts generalize the usual tree (or graph) searching problem and admit Hart, Nilsson and Raphael (1968) and Pohl (1969) theories of heuristic search. In particular the admissibility and optimality theorems of Hart, Nilsson and Raphael generalize for the classes 0 and 0" of diagonal search strategies for abstract theorem-proving problems. In addition the subclass au of 0 is shown to be optimal for 2. Implementation of diagonal search is treated in some detail for theorem-proving by resolution rules (Robinson 1965). SEARCH STRATEGIES, COMPLETENESS AND EFFICIENCY Completeness and efficiency of proof procedures can be studied only in the context of search strategies. A system T of inference rules and axioms can be complete or incomplete for a given class of intended interpretations. Similarly a search strategy E for T may or may not be complete for ...