A programming language based on sets; motivation and examples.

D.B.VIGOR 33 MECHANISED MATHEMATICS 4 An approach to analytic integration using ordered algebraic expressions. L.I.HonGsox 47 5 Some theorem-proving strategies based on the resolution principle.

K3 Th is a statement of facts and theories relevant to all the ei.

Spectra 70 / 46 and the IBM 360/50, that performs many second-order inferences in addition to carrying out first-order'resolutions' on Skolemized disjunctive formulae. The second-order aspect of the program is represented by an extended matching procedure, which operates in conjunction with rules for lambda abstraction and application, and for existential generalization and instantiation. These rules abstract properties from first-order formulae and apply axioms such as mathematical induction to these properties, thereby generating new second-order formulae as well as first-order formulae that could not have been produced by the resolution method alone. The mechanization of second-order logic is of potential usefulness in the areas of deductive question answering, illustrated by an example, and to the proving of formal properties of programs. Among the more interesting recent developments in the theory and practice of automatic theorem proving are the incorporation of formal techniques such as J. A. Robinson's resolution method into the'deductive sections' of question-answering and problem-solving systems (Chadwick et al. 1969, Green 1969), the solution of previously'open' problems (Guard et a/.

Given P {P1,.. P„} as input, set j 0, 00 8 (the identity substitution), and go to step 2. Step 2. (for j 0): if Pi0j is a singleton for each i, 1 1,.. n,

If the original formula included this function, we must also allow for its negation. This can either be eliminated or the following formula complicated to allow for this situation. However, in our examples this negation cannot appear and so we do not include it.

The distinction between analytic and empirical statements obviously has some potential philosophical overtones. We hope to avoid most of them by formulating the distinction in terms of an assumption on the verbs believe, know, and so on, rather than in terms of philosophical considerations. The predicate'Holds' The connectedness' of our set of functions and relations requires that there should be some unary relation'Holds' such that Holds(m

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.