This paper describes some recent experiments with a computer program which is capable of useful, or at least interesting, application to a number of different problems. The program, the Graph Traverser, has been described in detail in a previous paper (Doran & Michie 1966). However, we shall here need to view the basic algorithm from a rather more general standpoint, corresponding to an actual extension in the flexibility of the program, so that a restatement of what the program can do is desirable. The Graph Traverser, which is written in Elliott 4100 Algol, is potentially applicable to problem situations which can be idealised in the following way (see for comparison Newell and Ernst 1965). There is given a set of'states', which are connected by a set of'transformations', or, as I shall call them, 'operators'. An operator will be applicable to some, but not necessarily all, of the states and two distinct operators applied to either the same or distinct states may each give the same state as end -product. Most of the concepts to be used here which are related to the use of operators were discussed in a paper by Michie (1967). This type of problem situation is represented in Figure 1 by a graph (in the mathematical sense) to which have been added various labels.