Research Memorandum MIP-R-92. University of Edinburgh: Department of Machine Intelligence, School of Artificial Intelligence

Machine Intelligence 6

Ph.D. thesis, Edinburgh University. This thesis is concerned with algorithms for generating generalisations-from experience. These algorithms are viewed as examples of the general concept of a hypothesis discovery system which, in its turn, is placed in a framework in which it is seen as one component in a multi-stage process which includes stages of hypothesis criticism or justification, data gathering and analysis and prediction. Formal and informal criteria, which should be satisfied by the discovered hypotheses are given. In particular, they should explain experience and be simple. The formal work uses the first-order predicate calculus. These criteria are applied to the case of hypotheses which are generalisations from experience. A formal definition of generalisation from experience, relative to a body of knowledge is developed and several syntactical simplicity measures are defined. This work uses many concepts taken from resolution theory (Robinson, 1965). We develop a set of formal criteria that must be satisfied by any hypothesis generated by an algorithm for producing generalisation from experience. The mathematics of generalisation is developed. In particular, in the case when there is no body of knowledge, it is shown that there is always a least general generalisation of any two clauses, in the generalisation ordering. (In resolution theory, a clause is an abbreviation for a disjunction of literals.) This least general generalisation is effectively obtainable. Some lattices induced by the generalisation ordering, in the case where there is no body of knowledge, are investigated. The formal set of criteria is investigated. It is shown that for a certain simplicity measure, and under the assumption that there is no body of knowledge, there always exist hypotheses which satisfy them. Generally, however, there is no algorithm which, given the sentences describing experience, will produce as output a hypothesis satisfying the formal criteria. These results persist for a wide range of other simplicity measures. However several useful cases for which algorithms are available are described, as are some general properties of the set of hypotheses which satisfy the criteria. Some connections with philosophy are discussed. It is shown that, with sufficiently large experience, in some cases, any hypothesis which satisfies the formal criteria is acceptable in the sense of Hintikka and Hilpinen (1966). The role of simplicity is further discussed. Some practical difficulties which arise because of Goodman's (1965) "grue" paradox of confirmation theory are presented. A variant of the formal criteria suggested by the work of Meltzer (1970) is discussed. This allows an effective method to be developed when this was not possible before. However, the possibility is countenanced that inconsistent hypotheses might be proposed by the discovery algorithm. The positive results on the existence of hypotheses satisfying the formal criteria are extended to include some simple types of knowledge. It is shown that they cannot be extended much further without changing the underlying simplicity ordering. A program which implements one of the decidable cases is described. It is used to find definitions in the game of noughts and crosses and in family relationships. An abstract study is made of the progression of hypothesis discovery methods through time. Some possible and some impossible behaviours of such methods are demonstrated. This work is an extension of that of Gold (1967) and Feldman (1970). The results are applied to the case of machines that discover generalisations. They are found to be markedly sensitive to the underlying simplicity ordering employed.

Three approaches to symbolic integration in the 1960's are described. The first, from artificial intelligence, led to Slagle's SAINT and to a large degree to Moses' SIN. The second, from algebraic manipulation, led to Manove's implementation and to Horowitz' and Tobey's reexamination of the Hermite algorithm for integrating rational functions. The third, from mathematics, led to Richardson's proof of the unsolvability of the problem for a class of functions and for Risch's decision procedure for the elementary functions. Generalizations of Risch's algorithm to a class of special functions and programs for solving differential equations and for finding the definite integral are also described.

Ph.D. thesis, Stanford University.

"Heuristic DENDRAL is a computer program written to solve problems of inductive inference in organic chemistry. This paper will use the design of Heuristic DENDRAL and its performance on different problems for a discussion of the following topics: 1. the design for generality; 2. the performance problems attendant upon too much generality; 3. the coupling of expertise to the general problem solving processes; 4. the symbiotic relationship between generality and expertness, and the implications of this symbiosis for the study and design of problem solving systems. We conclude the paper with a view of the design for a general problem solver that is a variant of the "big switch" theory of generality."See also: Stanford University Report (ACM Citation)In Meltzer, B. and Michie, D. (Eds.), Machine Intelligence 6, pp. 165–190. Edinburgh University Press

At least we should be able to harness the power of the intelligent machines to tackle their threat to humanity, as now we use machines to counteract ill-effects of other machines. Although ethical notions change with time and across societies, no doubt some notions and restraints are shared by all societies. These could be programmed into the intelligent machines and in this way they could be made to'see' our morality. But as human morality changes, should the morality of the machines be made to follow? More generally, should we build human ethical and other inertias into intelligent machines? It would certainly be intolerable for us to be judged by values accepted in past times; so presumably we should wish the machines to follow our social mores as they change, rather than (like ideal clocks) be perfectly inertial, in the hope of representing ultimate values. Turning from the'seeing' of situations and events in terms of ethical values; there are deep problems over what it would be to say that machines'see' the common physical objects we take for granted. Here is the most difficult question of all: In what sense can a machine share our world? This question might be phrased in our terms as: How far can machines share our fiction?

"Problem-solving methods using some sort of heurstically guided search process have been the subject of much research in Artificial Intelligence. This paper groups these problem-solving methods under three major headings: the State-Space Approach, the Problem-Reduction Approach and the Formal-Logic Approach." New York: McGraw-Hill.

It is shown that any recognition problem solved by a polynomial time-bounded nondeterministic Turing machine can be "reduced" to the problem of determining whether a given propositional formula is a tautology. Here "reduced" means, roughly speaking, that the first problem can be solved deterministically in polynomial time provided an oracle is available for solving the second. From this notion of reducible, polynomial degrees of difficulty are defined, and it is shown that the problem of determining tautologyhood has the same polynomial degree as the problem of determining whether the first of two given graphs is isomorphic to a subgraph of the second. A method of measuring the complexity of proof procedures for the predicate calculus is introduced and discussed.

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'.