In this Part II, we apply the general theory developed in Part I to a detailed analysis of the Constraint Satisfaction Problem (CSP). We show how specific types of resolution rules can be defined. In particular, we introduce the general notions of a chain and a braid. As in Part I, these notions are illustrated in detail with the Sudoku example - a problem known to be NP-complete and which is therefore typical of a broad class of hard problems. For Sudoku, we also show how far one can go in 'approximating' a CSP with a resolution theory and we give an empirical statistical analysis of how the various puzzles, corresponding to different sets of entries, can be classified along a natural scale of complexity. For any CSP, we also prove the confluence property of some Resolution Theories based on braids and we show how it can be used to define different resolution strategies. Finally, we prove that, in any CSP, braids have the same solving capacity as Trial-and-Error (T&E) with no guessing and we comment this result in the Sudoku case.

Many real world problems naturally appear as constraints satisfaction problems (CSP), for which very efficient algorithms are known. Most of these involve the combination of two techniques: some direct propagation of constraints between variables (with the goal of reducing their sets of possible values) and some kind of structured search (depth-first, breadth-first,...). But when such blind search is not possible or not allowed or when one wants a 'constructive' or a 'pattern-based' solution, one must devise more complex propagation rules instead. In this case, one can introduce the notion of a candidate (a 'still possible' value for a variable). Here, we give this intuitive notion a well defined logical status, from which we can define the concepts of a resolution rule and a resolution theory. In order to keep our analysis as concrete as possible, we illustrate each definition with the well known Sudoku example. Part I proposes a general conceptual framework based on first order logic; with the introduction of chains and braids, Part II will give much deeper results.

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.