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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.

In an attempt to overcome the great inefficiency of theorem proving methods, several existing methods are studied, and several now ones are proposed. A concentrated attempt is made to devise a unified proof procedure whose inference rules are designed for the efficient use by a search strategy. For unsatisfiable sets of Horn clauses, it is shown that p1-resolution and selective linear negative (SLN) resolution can be alternated heuristically to conduct a bi-directional search. This bi-directional search is shown to be more efficient than either of P.-resolution and SLN-resolution. The extreme sparseness of the SLN-search spaces lead to the extension of SLN-resolution to a more general and more powerful resolution rule, selective linear (SL) resolution, which resembles Loveland's model elimination strategy. With SL-resolution, all immediate descendants of a clause are obtained by resolving upon a single selected literal of that clause. Linear resolution, s-linear resolution and t-linear resolution are shown to be as powerful as the most powerful resolution systems. By slightly decreasing the power, considerable increase in the sparseness of search spaces is obtained by using SL-resolution. The amenability of SL-resolution to applications of heuristic methods suggest that, on these grounds alone, it is at least competitive with theorem-proving procedures designed solely from heuristic considerations. Considerable attention is devoted to various anticipation procedures which allow an estimate of the sparseness of search trees before their generation. Unlimited anticipation takes the form of pseudo-search trees which construct outlines of possible proofs. Anticipation procedures together with a number of heuristic measures are suggested for the implementation of an exhaustive search strategy for SL-resolution.

This thesis was motivated by a case study involving the formalisation of arguments that simplify the verification of tree-oriented multicast protocols. As well as covering the case study itself, it discusses our solution to problems we encountered concerning expressivity and automation. The expressivity problems related to the need for theory interpretation. We found the existing Locale and axiomatic type class mechanisms provided by the Isabelle theorem prover we were using to be inadequate. This led us to develop a new prototype implementation of theory interpretation. To support this implementation, we developed a novel system of proof terms for the HOL logic that we also describe in this thesis. We found existing automation to perform poorly, which led us to experiment with additional kinds of automation. We describe our approach, focusing on features that make automation suitable for interactive use. Our presentation of the case study starts with our formalisation of an abstract theory of distributed systems, covering state transition systems, forward and backward simulation relations, and related properties of LTL (linear temporal logic). We then summarise proofs of simulation relations holding for particular abstract multicast protocols. We discuss the mechanisation styles we experimented with in the case study. We also discuss the methodology behind our proofs. We cover aspects such as how to discover and construct proofs, and how to explore the space of proofs, how to make good definitions and lemmas, how to increase modularity, reuse, stability and malleability of proofs, and reduce maintenance of proofs, and the gap between intuitively understood proofs and their formalisation.

This project looks at using diagrammatic reasoning to prove mathematical theorems. The work is motivated by a need for theorem provers whose reasoning is readily intelligible to human beings. It should also have practical applications in mathematics teaching. We focus on the continuous domain of analysis - a geometric subject, but one which is taught using a dry algebraic formalism which many students find hard. The geometric nature of the domain makes it suitable for a diagram-based approach. However it is a difficult domain, and there are several problems, including handling alternating quantifiers, sequences and generalisation. We developed representations and reasoning methods to solve these. Our diagram logic isn't complete, but does cover a reasonable range of theorems. It utilises computers to extend diagrammatic reasoning in new directions – including using animation. This work is tested for soundness, and evaluated empirically for ease of use. We demonstrate that computerised diagrammatic theorem proving is not only possible in the domain of real analysis, but that students perform better using it than with an equivalent algebraic computer system.

Developing logic in machines has always been an area of concern for scientists. Automated Theorem Proving is a field that has implemented the concept of logical consequence to a certain level. However, if the number of available axioms is very large then the probability of getting a proof for a conjecture in a reasonable time limit can be very small. This is where the ability to learn from previously proved theorems comes into play. If we see in our own lives, whenever a new situation S(NEW) is encountered we try to recollect all old scenarios S(OLD) in our neural system similar to the new one. Based on them we then try to find a solution for S(NEW) with the help of all related facts F(OLD) to S(OLD). Similar is the concept in this research. The thesis deals with developing a solution and finally implementing it in a tool that tries to prove a failed conjecture (a problem that the ATP system failed to prove) by extracting a sufficient set of axioms (we call it Refined Axiom Set (RAS)) from a large pool of available axioms. The process is carried out by measuring the similarity of a failed conjecture with solved theorems (already proved) of the same domain. We call it "process1", which is based on syntactic selection of axioms. After process1, RAS may still have irrelevant axioms, which motivated us to apply semantic selection approach on RAS so as to refine it to a much finer level. We call this approach as "process2". We then try to prove failed conjecture either from the output of process1 or process2, depending upon whichever approach is selected by the user. As for our testing result domain, we picked all FOF problems from the TPTP problem domain called SWC, which consisted of 24 broken conjectures (problems for which the ATP system is able to show that proof exists but not able to find it because of limited resources), 124 failed conjectures and 274 solved theorems. The results are produced by keeping in account both the broken and failed problems. The percentage of broken conjectures being solved with respect to the failed conjectures is obviously higher and the tool has shown a success of 100 % on the broken set and 19.5 % on the failed ones.

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 230-235). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 141-155). Available also in a digital version from Dissertation Abstracts.

Thesis (M.S.)--Mathematics, Georgia Institute of Technology, 2005. / Belinfante, Johan, Committee Chair ; Green, William, Committee Member ; Manolios, Panagiotis, Committee Member. Includes bibliographical references.

Thesi supervisor: Cesare Tinelli. Includes bibliographic references (p. 214-220).