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On the Notion of Interestingness in Automated Mathematical Discovery
- International Journal of Human Computer Studies
, 2000
"... We survey ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they've had. We focus on how they estimate the interestingness of concepts and conjectures and extract some common notions about interestingness in automated mathema ..."
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Cited by 70 (26 self)
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We survey ve mathematical discovery programs by looking in detail at the discovery processes they illustrate and the success they've had. We focus on how they estimate the interestingness of concepts and conjectures and extract some common notions about interestingness in automated mathematical discovery. We detail how empirical evidence is used to give plausibility to conjectures, and the dierent ways in which a result can be thought of as novel. We also look at the ways in which the programs assess how surprising and complex a conjecture statement is, and the dierent ways in which the applicability of a concept or conjecture is used. Finally, we note how a user can set tasks for the program to achieve and how this aects the calculation of interestingness. We conclude with some hints on the use of interestingness measures for future developers of discovery programs in mathematics.
Automatic Concept Formation in Pure Mathematics
"... The HR program forms concepts and makes conjectures in domains of pure mathematics andusestheoremproverOTTERandmodel generatorMACEtoproveordisprovetheconjectures. HRmeasurespropertiesofconcepts andassessesthetheoremsandproofsinvolving themtoestimatetheinterestingnessofeach concept and employ a best ..."
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Cited by 44 (31 self)
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The HR program forms concepts and makes conjectures in domains of pure mathematics andusestheoremproverOTTERandmodel generatorMACEtoproveordisprovetheconjectures. HRmeasurespropertiesofconcepts andassessesthetheoremsandproofsinvolving themtoestimatetheinterestingnessofeach concept and employ a best first search. This approachhasledHRtothediscoveryofinterestingnewmathematics and enables it to build theories from just the axioms of finite algebras.
Functional transformations in AI discovery systems
- Artificial Intelligence
, 1990
"... The power of scientilic discovery systems derives from two main sources: a set of heuristics that determine when to apply a creative operator (an operator for forming new operators and concepts) in a space that is being explored; and a set of creative operators that determine what new operators and ..."
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Cited by 20 (2 self)
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The power of scientilic discovery systems derives from two main sources: a set of heuristics that determine when to apply a creative operator (an operator for forming new operators and concepts) in a space that is being explored; and a set of creative operators that determine what new operators and concepts will be created for that exploration. This paper is mainly concerned with the second issue. A mechanism calledfuncfional transformations (lT) shows promising power in creating new and useful creative operators during exploration. The paper discusses the definition, creation, and application of functional transformations, and describes, as a demonstration of the power of ET, how the system ARE, starting with a small set of creative operations and a small set of heuristics, uses R’s to create all the concepts attained by Lenat’s AM system [4], and others as well. Besides showing an alternative way, of Lnat’s EURISKO [5], to meet the criticisms of too much pre-programmed knowledge [6] that have been leveled against AM, ARE provides a route to discovery systems that are capable of “refreshing ” themselves indefinitely by continually creating new operators. 1
HR - A System for Machine Discovery in Finite Algebras
- ECAI 98 Workshop Programme
, 1998
"... We describe the HR concept formation program which invents mathematical definitions and conjectures in finite algebras such as group theory and ring theory. We give the methods behind and the reasons for the concept formation in HR, an evaluation of its performance in its training domain, group theo ..."
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Cited by 8 (0 self)
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We describe the HR concept formation program which invents mathematical definitions and conjectures in finite algebras such as group theory and ring theory. We give the methods behind and the reasons for the concept formation in HR, an evaluation of its performance in its training domain, group theory, and a look at HR in domains other than group theory.
Automatic Theory Formation in Graph Theory
- IN ARGENTINE SYMPOSIUM ON ARTIFICIAL INTELLIGENCE
, 1999
"... This paper presents SCOT, a system for automatic theory construction in the domain of Graph Theory. Following on the footsteps of the programs ARE [9], HR [1] and Cyrano [6], concept discovery is modeled as search in a concept space. We propose a classification for discovery heuristics, which takes ..."
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Cited by 5 (0 self)
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This paper presents SCOT, a system for automatic theory construction in the domain of Graph Theory. Following on the footsteps of the programs ARE [9], HR [1] and Cyrano [6], concept discovery is modeled as search in a concept space. We propose a classification for discovery heuristics, which takes into account the main processes related to theory construction: concept construction, example production, example analysis, conjecture construction, and conjecture analysis.
Computational Discovery in Pure Mathematics
"... Abstract. We discuss what constitutes knowledge in pure mathematics and how new advances are made and communicated. We describe the impact of computer algebra systems, automated theorem provers, programs designed to generate examples, mathematical databases, and theory formation programs on the body ..."
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Abstract. We discuss what constitutes knowledge in pure mathematics and how new advances are made and communicated. We describe the impact of computer algebra systems, automated theorem provers, programs designed to generate examples, mathematical databases, and theory formation programs on the body of knowledge in pure mathematics. We discuss to what extent the output from certain programs can be considered a discovery in pure mathematics. This enables us to assess the state of the art with respect to Newell and Simon’s prediction that a computer would discover and prove an important mathematical theorem. 1
An Attempt to Automate NP-Hardness Reductions via SO∃ Logic
, 2004
"... We explore the possibility of automating NP-hardness reductions. We motivate the problem from an artificial intelligence perspective, then propose the use of second-order existential (SO#) logic as representation language for decision problems. Building upon the theoretical framework of J. Antonio ..."
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We explore the possibility of automating NP-hardness reductions. We motivate the problem from an artificial intelligence perspective, then propose the use of second-order existential (SO#) logic as representation language for decision problems. Building upon the theoretical framework of J. Antonio Medina, we explore the possibility of implementing seven syntactic operators. Each operator transforms SO# sentences in a way that preserves NP-completeness. We subsequently propose a program which implements these operators.
{simonco, bundy}<0dai. ed. ac. uk
"... The HR program forms concepts and makes conjectures in domains of pure mathematics and uses theorem prover OTTER and model generator MACE to prove or disprove the con-jectures. HR measures properties of concepts and assesses the theorems and proofs involving them to estimate the interestingness of e ..."
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The HR program forms concepts and makes conjectures in domains of pure mathematics and uses theorem prover OTTER and model generator MACE to prove or disprove the con-jectures. HR measures properties of concepts and assesses the theorems and proofs involving them to estimate the interestingness of each concept and employ a best first search. This approach has led HR to the discovery of inter-esting new mathematics and enables it to build theories from just the axioms of finite algebras. 1 In t roduc t ion The HR program invents definitions in finite algebras such as group and ring theory, and other areas of pure mathematics, such as graph and number theory. Using a set of production rules to derive a new concept from old ones and a set of measures for the interestingness of a concept, HR's best first search bases new concepts on the most interesting old ones. As it invents new defin-itions, HR uses empirical evidence to spot conjectures. Recently we have interfaced HR with the OTTER the-orem prover, [McCune, 1990], to prove some of the con-jectures HR makes. When OTTER fails, HR invokes the MACE model finder, [McCune, 1994], to find a counter-example. The proofs from OTTER help HR to assess the concepts involved in the conjectures, and the mod-els given by MACE provide further empirical evidence for future conjectures. This closes a cycle of mathemat-ical activity similar in nature to Buchberger's spiral of creativity, [Buchberger, 1993]. We detail how HR forms and assesses concepts and discuss how this has led to the introduction of new mathematics. We also show how a theory can be constructed from just the axioms of an al-gebra, and how the heuristic search improves the overall quality of the theory with respect to various measures. 1.1 Background Lenat, in [Lenat, 1976] chose pure mathematics as the domain for his AM program to demonstrate the use of heuristic search in concept formation. AM re-invented classically interesting definitions and conjectures, such
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"... In previous work we have designed an automatic bootstrapping algorithm to classify finite algebraic structures by generating properties that uniquely describe and discriminate different equivalence classes. One of the drawbacks of the approach was that during the classification a large number of dif ..."
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In previous work we have designed an automatic bootstrapping algorithm to classify finite algebraic structures by generating properties that uniquely describe and discriminate different equivalence classes. One of the drawbacks of the approach was that during the classification a large number of different discriminating properties were generated. This made it particularly difficult to compare classifying properties for different sizes of algebraic structures. To minimise the overall number of properties needed we have now experimented with parameterising structures by counting the number of elements with particular properties. Isomorphism classes are then discriminated by the different number of elements with the same property. With this new approach we can now classify large numbers of algebraic structures using only a small number of properties. We were able to construct and prove parameterisations for algebraic structures like loops of size 6 and groups of size 8. However, the approach is currently limited as translating counting arguments into pure first order or propositional logic, often makes for prohibitively long problem formulations. 1
Edinburgh Research Explorer On the Notion of Interestingness in Automated Mathematical
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