## Change of Representation in Theorem Proving by Analogy (1993)

Citations: | 11 - 9 self |

### BibTeX

@INPROCEEDINGS{Melis93changeof,

author = {Erica Melis},

title = {Change of Representation in Theorem Proving by Analogy},

booktitle = {},

year = {1993}

}

### OpenURL

### Abstract

Constructing an analogy between a known and already proven theorem (the base case) and another yet to be proven theorem (the target case) often amounts to finding the appropriate representation at which the base and the target are similar. This is a well-known fact in mathematics, and it was corroborated by our empirical study of a mathematical textbook, which showed that a reformulation of the representation of a theorem and its proof is indeed more often than not a necessary prerequisite for an analogical inference. Thus machine supported reformulation becomes an important component of automated analogy-driven theorem proving too. The reformulation component proposed in this paper is embedded into a proof plan methodology based on methods and meta-methods, where the latter are used to change and appropriately adapt the methods. A theorem and its proof are both represented as a method and then reformulated by the set of metamethods presented in this paper. Our approach supports analog...

### Citations

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Citation Context ... plan construction. More precisely let us take a sorted higher order language which is extended by metavariables for formulae and terms as our object language 3 . In this paper Natural Deduction (ND) =-=[16]-=- is used as the proof calculus. Rules of this calculus are, for instance (! I); (8I), (I), and (! D) for the introduction of !;8, , and for the deletion of ! respectively. \Delta; \Sigma are used for ... |

267 | The use of explicit plans to guide inductive proofs
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Citation Context ...d control issues. Therefore methods and techniques have been suggested that more closely follow the reasoning patterns observed in humans, e.g., by Allen Newell [26] and, more recently, by Alan Bundy =-=[5]-=-, who proposed a framework for proof planning to improve the automation of theorem proving. As evidence for a mathematicians' strong reliance on proof planning we quote Gert Faltings, who discovered t... |

229 |
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Citation Context ... as homomorphism etc. The actual ratings are not properly developed currently. 3 A Model for Analogy-Driven Proof Plan Construction 3.1 Theorem Proving by Analogy Problem solving by analogy (see e.g. =-=[7, 18, 11]-=-) means to find a solution for a target problem on the basis of a given solution of a given base problem which is somehow similar -- or analogous -- to the target problem. The specific subject of theo... |

224 |
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Citation Context ... in the case study [23]. 2 Notions and Notation We assume the reader to be familiar with plan-based problem-solving (see e.g., [8]) as well as the main notions of automated theorem proving (see e.g., =-=[22]-=-). Analogy-driven theorem proving as outlined below presupposes a proof planning process that constructs a proof plan out of a reformulated originally given proof. Hence, our framework is essentially ... |

78 | Computational Approaches to Analogical Reasoning: A Comparative Analysis
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Citation Context ... as homomorphism etc. The actual ratings are not properly developed currently. 3 A Model for Analogy-Driven Proof Plan Construction 3.1 Theorem Proving by Analogy Problem solving by analogy (see e.g. =-=[7, 18, 11]-=-) means to find a solution for a target problem on the basis of a given solution of a given base problem which is somehow similar -- or analogous -- to the target problem. The specific subject of theo... |

74 | A science of reasoning
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Citation Context ...er. Thus in principle we become independent of the actual and often accidental representation of the given theorems. In particular we shall show how the proof plan framework of Alan Bundy (see, e.g., =-=[6]-=-) and the use of reformulations support analogy-driven proof construction. This paper focuses mainly on the formal description of meta-methods that reformulate problems and proofs that are represented... |

58 | An overview of machine learning - Carbonell, Michalski, et al. - 1983 |

52 | Mathematics and plausible reasoning - Polya |

50 | Little theories
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Citation Context ...Owen in [27], e.g., the identical symbols heuristic and the syntactic type heuristic. A good justification of Symbol Mapping is given by the so-called theory interpretations that are employed in IMPS =-=[15]-=-. PROCSYMBOL modifies M: ffl Replace all occurrences of the symbol s by f(s). ffl Change, if necessary, the corresponding sort declarations. Examples Examples for an application of Symbol Mapping are ... |

47 | A logical approach to reasoning by analogy
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Citation Context ... as homomorphism etc. The actual ratings are not properly developed currently. 3 A Model for Analogy-Driven Proof Plan Construction 3.1 Theorem Proving by Analogy Problem solving by analogy (see e.g. =-=[7, 18, 11]-=-) means to find a solution for a target problem on the basis of a given solution of a given base problem which is somehow similar -- or analogous -- to the target problem. The specific subject of theo... |

44 |
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Citation Context ...ace, but ill-understood. In particular, the role and the use of reformulating the base and the target problem for the analogy formation have found little attention although Indurkhya [20] and Russell =-=[29]-=- established the importance of of reformulation in analogical reasoning in general. Within mathematics, however, it is well known that constructing an analogy often involves finding the right represen... |

41 |
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Citation Context ...the proof was there already, and I only transferred this part appropriately. of problems and proofs that go beyond the usual symbol mapping techniques of established approaches to analogy (see, e.g., =-=[21, 25, 4, 27]-=-. In this paper we develop a methodology that, in contrast to former approaches to theorem proving by analogy, considers two problems to be analogous if there exists a chain of justified reformulation... |

35 |
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Citation Context ...the proof was there already, and I only transferred this part appropriately. of problems and proofs that go beyond the usual symbol mapping techniques of established approaches to analogy (see, e.g., =-=[21, 25, 4, 27]-=-. In this paper we develop a methodology that, in contrast to former approaches to theorem proving by analogy, considers two problems to be analogous if there exists a chain of justified reformulation... |

33 |
Metaphor and Cognition
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Citation Context ...oning is commonplace, but ill-understood. In particular, the role and the use of reformulating the base and the target problem for the analogy formation have found little attention although Indurkhya =-=[20]-=- and Russell [29] established the importance of of reformulation in analogical reasoning in general. Within mathematics, however, it is well known that constructing an analogy often involves finding t... |

21 |
Analogical reasoning and proof discovery
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Citation Context ...the proof was there already, and I only transferred this part appropriately. of problems and proofs that go beyond the usual symbol mapping techniques of established approaches to analogy (see, e.g., =-=[21, 25, 4, 27]-=-. In this paper we develop a methodology that, in contrast to former approaches to theorem proving by analogy, considers two problems to be analogous if there exists a chain of justified reformulation... |

18 | Difference matching
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Citation Context ...succeeding methods. Hence, to obtain a plan as complete as possible, additional methods have to be inserted which can be found, for instance, by searching bridge lemmas or by difference matching, see =-=[9, 1]-=-. The following gives the basic algorithm CONSTRUCT PROOF BY ANALOGY Input: problems P1, P2, and proof of P1 Output: proof plan for P2 1. INITIALIZE 2. MATCH if (M 1i , M 2i do not match) then go to 5... |

17 | Methods - The Basic Units for Planning and Verifying Proofs
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(Show Context)
Citation Context ... below presupposes a proof planning process that constructs a proof plan out of a reformulated originally given proof. Hence, our framework is essentially based on the concepts of proof planning from =-=[19]-=- and [6] which are, however, modified. The basic concepts in this framework are problems, methods, and metamethods. Problems consist of a theorem and the assumptions the theorem follows from. In the f... |

17 |
Analogy as a means of discovery in problem solving and learning
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Citation Context |

14 | A methodology for equational reasoning
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(Show Context)
Citation Context ...succeeding methods. Hence, to obtain a plan as complete as possible, additional methods have to be inserted which can be found, for instance, by searching bridge lemmas or by difference matching, see =-=[9, 1]-=-. The following gives the basic algorithm CONSTRUCT PROOF BY ANALOGY Input: problems P1, P2, and proof of P1 Output: proof plan for P2 1. INITIALIZE 2. MATCH if (M 1i , M 2i do not match) then go to 5... |

13 |
The heuristic of George Pólya and its relation to artificial intelligence
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- 1983
(Show Context)
Citation Context ...ning or any other such global search and control issues. Therefore methods and techniques have been suggested that more closely follow the reasoning patterns observed in humans, e.g., by Allen Newell =-=[26]-=- and, more recently, by Alan Bundy [5], who proposed a framework for proof planning to improve the automation of theorem proving. As evidence for a mathematicians' strong reliance on proof planning we... |

11 |
Tree Subsumption: Reasoning with Outlines
- Giunchiglia, Walsh
- 1992
(Show Context)
Citation Context ...T respectively. However, the defining formulae may differ from this standard case, e.g., if S and T are H-semimoduls as in many cases of HUA. Then the defining formula becomes 7 Giunchiglia and Walsh =-=[17]-=- speak of a different kind of abstraction (PI) if a proof of the abstracted problem exists and provides an outline of the proofs of the un-abstracted problems. Our abstracted proofs need not be proofs... |

8 |
Halbgruppen und Automaten, volume 99 of Heidelberger Taschenbucher
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- 1971
(Show Context)
Citation Context ...ere actually used for the necessary reformulations in the analogy-driven construction of proofs in an empirical case study [23]. Some examples from a mathematical textbook "Halbgruppen und Automa=-=ten" [13]-=- (abbreviated as HUA) and an example of Terry Boy de la Tour and Christoph Kreitz are given in section five. More proofs by analogy are investigated and presented in detail in the case study [23]. 2 N... |

5 |
Interview: Die Neugier, etwas ganz genau wissen zu wollen. bild der wissenschaft
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(Show Context)
Citation Context ...u�� man die Details einfugen und sieht, ob man es auch wirklich so machen kann. Es kommt aber auch durchaus vor, da�� man mal da sitzt, nicht mehr weiter wei�� und dann probiert, wohin der=-= Weg fuhrt" [14]-=- 1 . Another important problem solving strategy that can actually be combined with proof planning is the construction of proofs by analogy (see, e.g., Polya [28]), where the known proof of a theorem i... |

4 |
Some experiments with analogy in proof discovery (preliminary report
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- 1986
(Show Context)
Citation Context .... This meta-method was applied to the method of theorem 5.2. in HUA. Other suggestions for structuring Additional structuring heuristics have been investigated, inter alia by: ffl Bishop Brock et al. =-=[3]-=- propose to introduce so-called motivations for proof parts such that the range of one motivation represents a subproof. ffl Bishop Brock et al. [3] and Stephen Owen [27] suggest heuristics for identi... |

4 |
de la Tour and Ch. Kreitz. Building proofs by analogy via the curryhoward isomorphism
- Boy
- 1992
(Show Context)
Citation Context ... Example As an illustration of the previous algorithm consider the following example, which is presented in more technical detail later on in section 5. This example was used by de la Tour and Kreitz =-=[12]-=-, who tried to prove Theorem2 by analogy to the proof of Theorem1, where: Theorem1: 8x; y((Qxy ! 8z(Pxz ! Pyz))sQabs(PaasP bb) ! 9xPbx) and Theorem2: 8x; y((Qxy ! 8z(Pxz ! Pyz))sQabsQbcs(PaasP cc) ! 9... |

4 |
Analogies between proofs - a case study. SEKI-Report SR-93-12, Universitat des Saarlandes
- Melis
- 1993
(Show Context)
Citation Context ...e theorem, e.g., the right concepts or the right level of abstraction, such that the base and the target problem become similar. Thus, not surprisingly, our empirical study of a mathematical textbook =-=[23]-=- provided ample evidence that an appropriate reformulation of the theorem and its proof is often an important component of human theorem proving by analogy. Consequently, there is a need for a similar... |

1 |
A precondition prover for analogy. Memo ATP
- Bledsoe
- 1992
(Show Context)
Citation Context ... 3 1) 5. 3, 4; ` OE (! D 3 4) 6. ; ` the remaining lines of dec-cont(M) (same as in M) procedure schema interpreter history same as in M Change-Plan is our version of Bledsoe's precondition prover PC =-=[2]-=-. However, Bledsoe's procedure is more complicated and has additional features. This meta-method has been used to prove theorem 13.7. of HUA, analogously to the proof of theorem 6.9. in HUA. Reversion... |

1 |
Integrating derivational problem solving architecture
- Carbonell, Veloso
- 1988
(Show Context)
Citation Context ...ion five. More proofs by analogy are investigated and presented in detail in the case study [23]. 2 Notions and Notation We assume the reader to be familiar with plan-based problem-solving (see e.g., =-=[8]-=-) as well as the main notions of automated theorem proving (see e.g., [22]). Analogy-driven theorem proving as outlined below presupposes a proof planning process that constructs a proof plan out of a... |