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forthcoming)). Applying lakatosstyle reasoning to ai problems
 Thinking Machines and the philosophy of computer science: Concepts and principles. IGI Global
, 2010
"... One current direction in AI research is to focus on combining different reasoning styles such as deduction, induction, abduction, analogical reasoning, nonmonotonic reasoning, vague and uncertain reasoning. The philosopher Imre Lakatos produced one such theory of how people with different reasoning ..."
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One current direction in AI research is to focus on combining different reasoning styles such as deduction, induction, abduction, analogical reasoning, nonmonotonic reasoning, vague and uncertain reasoning. The philosopher Imre Lakatos produced one such theory of how people with different reasoning styles collaborate to develop mathematical ideas. Lakatos argued that mathematics is a quasiempirical, flexible, fallible, human endeavour, involving negotiations, mistakes, vague concept definitions and disagreements, and he outlined a heuristic approach towards the subject. In this chapter we apply these heuristics to the AI domains of evolving requirement specifications, planning and constraint satisfaction problems. In drawing analogies between Lakatos’s theory and these three domains we identify areas of work which correspond to each heuristic, and suggest extensions and further ways in which Lakatos’s philosophy can inform AI problem solving. Thus, we show how we might begin to produce a philosophicallyinspired AI theory of combined reasoning. 1
Schemebased Definition and Conjecture Synthesis for Inductive Theories
"... Human mathematical discovery processes include the invention of definitions, conjectures, theorems, examples, problems and algorithms for solving these problems. Automating these processes is an exciting area for research which is now recognised by the automated reasoning community ..."
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Human mathematical discovery processes include the invention of definitions, conjectures, theorems, examples, problems and algorithms for solving these problems. Automating these processes is an exciting area for research which is now recognised by the automated reasoning community
J Autom Reasoning (2007) 39:109–139 DOI 10.1007/s1081700790705 User Interaction with the Matita Proof Assistant
"... Abstract Matita is a new, documentcentric, tacticbased interactive theorem prover. This paper focuses on some of the distinctive features of the user interaction with Matita, characterized mostly by the organization of the library as a searchable knowledge base, the emphasis on a highquality not ..."
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Abstract Matita is a new, documentcentric, tacticbased interactive theorem prover. This paper focuses on some of the distinctive features of the user interaction with Matita, characterized mostly by the organization of the library as a searchable knowledge base, the emphasis on a highquality notational rendering, and the complex interplay between syntax, presentation, and semantics.
STUDIES IN LOGIC, GRAMMAR AND RHETORIC 10 (23) 2007 Automated Discovery of Inductive Theorems ⋆
"... Abstract. Inductive mathematical theorems have, as a rule, historically been quite difficult to prove – both for mathematics students and for automated theorem provers. That said, there has been considerable progress over the past several years, within the automated reasoning community, towards prov ..."
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Abstract. Inductive mathematical theorems have, as a rule, historically been quite difficult to prove – both for mathematics students and for automated theorem provers. That said, there has been considerable progress over the past several years, within the automated reasoning community, towards proving some of these theorems. However, little work has been done thus far towards automatically discovering them. In this paper we present our methods of discovering (as well as proving) inductive theorems, within an automated system. These methods have been tested over the natural numbers, with regards to addition and multiplication, as well as to exponents of group elements. 1
Decompositions of Natural Numbers: From A Case Study in Mathematical Theory Exploration ∗
"... In the context of a scheme based exploration model proposed by Bruno Buchberger, we investigate the idea of decomposition, applied in the exploration of natural numbers. The free decomposition problem (i.e. whether an element can always be decomposed with respect to an operation) can be arbitrarily ..."
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In the context of a scheme based exploration model proposed by Bruno Buchberger, we investigate the idea of decomposition, applied in the exploration of natural numbers. The free decomposition problem (i.e. whether an element can always be decomposed with respect to an operation) can be arbitrarily difficult, and we illustrate this in the theory of natural numbers. We consider a restriction, the decomposition in domains with a wellfounded partial ordering: we introduce the notions of irreducible elements, reducible elements w.r.t. a composition operation, decomposition of domain elements into irreducible ones, and also the problem of irreducible decomposition which we then solve. Natural numbers can be classified as a decomposition domain, in which we know how to solve the decomposition problem. This leads to the prime decomposition theorem.
Scheme–Based Systematic Exploration of Natural Numbers ∗
"... In this paper, we report a case study of computer supported exploration of the theory of natural numbers, using a theory exploration model based on knowledge schemes, proposed by Bruno Buchberger. We illustrate with examples from the exploration: (i) the invention of new concepts (functions, relatio ..."
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In this paper, we report a case study of computer supported exploration of the theory of natural numbers, using a theory exploration model based on knowledge schemes, proposed by Bruno Buchberger. We illustrate with examples from the exploration: (i) the invention of new concepts (functions, relations) in the theory, using knowledge schemes, (ii) the invention of new propositions, using proposition schemes, (iii) the invention of problems, using knowledge schemes, (iv) the introduction of new reasoning rules, by lifting knowledge to the inference level, after their correctness was proved. 1
Applying Lakatosstyle reasoning to AI problems
"... Lakatos (1976) argued that mathematics develops in a much more organic way than its rigid textbook presentation of definitiontheoremproof would suggest. He outlined a heuristic approach which holds that mathematics progresses by a series of primitive conjectures, proofs, counterexamples, proofgen ..."
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Lakatos (1976) argued that mathematics develops in a much more organic way than its rigid textbook presentation of definitiontheoremproof would suggest. He outlined a heuristic approach which holds that mathematics progresses by a series of primitive conjectures, proofs, counterexamples, proofgenerated concepts, modified conjectures and modified proofs. The purpose of this chapter is to apply these heuristics to the AI domains of evolving requirement specifications, planning and constraint satisfaction problems. In drawing analogies between Lakatos’s theory and these three domains, we identify areas of work which correspond to each heuristic, and suggest extensions and further ways in which Lakatos’s philosophy can inform AI problem solving. As well as the implications for added flexibility in AI, these analogies extend both the power and our understanding of Lakatos’s theory, thus this work contributes to the philosophy of mathematics. In addition, we start to build a bridge between these largely disparate AI domains. 1