The initial plan for LOGLISP [1] was simply that it would offer, within LISP, a Horn-clause relational programming facility akin to PROLOG. This it does, but with some differences from PROLOG, notably the use of a breadth-first, rather than depth-first, elaboration of the underlying tree of alternative linear proofs, and the consequent avoidance of explicit backtracking as a control mechanism. It was because of these differences that the facility was called LOGIC rather than PROLOG, which would have been misleading. The name LOGLISP then refers to the combined system: LOGIC LISP. It soon became apparent, however, that the main interest of LOGLISP lay rather in its (relatively crude, but genuine) attempt to merge the functional programming style of LISP with the relational programming style of LOGIC and PROLOG. This was done by introducing the notion of'Lisp-transforms' into LOGIC.

The particular task in question is the interpretation of the electrical signals generated by the heart muscle, known as the electrocardiographic (Eco) interpretation.

A thunderstorm is considered severe if any one of the following phenomena accompanies the thunderstorm (and is reported): - tornadoes (intense, small-scale cyclones); - hailstones cm (in.) in diameter; - surface wind gusts in excess of 93 km h-1 (50 knots) and/or significant wind damage.

It is given by them without supporting analysis but with the statement that the bishops'cannot win if the weaker side can obtain a position similar to the above, but they win in most cases'. The second position, a win, is then given with a solution and a number of supporting variations extending to 14 moves. One or other of both positions is repeated in the subsequent literature up to 1983 (e.g.

The traditional approach involving protracted interaction between a knowledge engineer and a domain expert is viable only to the extent that both these resources are available; this approach will not meet the apparently exponential growth in demand for expert systems. A solution to this dilemma requires rethinking the way knowledge-based products are built. An example of this reappraisal of methodology appears in Michie (1983), and is based on the principle of formalizing and refining the knowledge implicit in collections of examples or data bases. Dietterich and Michalski (1983) give an overview of methods for learning from examples. There are many such, all based on the idea of inductive generalization. One of the simplest of these methods dates back to work by Hunt in the late fifties (Hunt et al., 1966). Each given example, described by measuring certain fixed properties, belongs to a known class and the'learning' takes the form of developing a classification rule that can then be applied to new objects. Simple though it may be, derivatives of this method have achieved useful results; Kononenko et al. (1984), for example, have managed to generate five medical diagnosis systems with minimal reference to diagnosticians.

It is assumed that Si are sorted in increasing order of s(Si). Non-linearities of four trees are shown in Figure 6. Ti is absolutely linear; thus its non-linearity measure is zero. T2 is very close to being a balanced tree: non-linearity one. T3 is preferred to T4, i.e. this function is sensitive to the location of non-linearity within a tree (the lower a non-linearity occurs in a tree the lower (better) its measure).

Such methods can effectively and efficiently induce good descriptions from a given set of examples and, optionally, induce counter-examples (for example Michalski, 1975, 1980a; Quinlan, 1979; Langley et al., 1983). These methods cannot modify concept descriptions which are contradicted by new examples, but must re-learn the descriptions from scratch. In contrast, incremental learning methods modify concept descriptions to accommodate new learning events (Winston, 1975; Michalski and Larson, 1978). When we observe human learning we clearly see that it is incremental.

It is a rich source of difficult and challenging problems which involve issues of knowledge representation, the analysis of natural language, and the automation of practical and common-sense reasoning.

In this paper we will be concerned with such reasoning in its most general form, that is, in inferences that are defeasible: given more information, we may retract them. The purpose of this paper is to introduce a form of non-monotonic inference based on the notion of a partial model of the world. We take partial models to reflect our partial knowledge of the true state of affairs. We then define non-monotonic inference as the process of filling in unknown parts of the model with conjectures: statements that could turn out to be false, given more complete knowledge. To take a standard example from default reasoning: since most birds can fly, if Tweety is a bird it is reasonable to assume that she can fly, at least in the absence of any information to the contrary. We thus have some justification for filling in our partial picture of the world with this conjecture. If our knowledge includes the fact that Tweety is an ostrich, then no such justification exists, and the conjecture must be retracted.

This paper describes some work on automatically generating finite counterexamples in topology, and the use of counterexamples to speed up proof discovery in intermediate analysis, and gives some examples theorems where human provers are aided in proof discovery by the use of examples.