## Reconstructing Proofs at the Assertion Level (1994)

Citations: | 65 - 9 self |

### BibTeX

@INPROCEEDINGS{Huang94reconstructingproofs,

author = {Xiaorong Huang},

title = {Reconstructing Proofs at the Assertion Level},

booktitle = {},

year = {1994},

pages = {738--752},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

Most automated theorem provers suffer from the problem that they can produce proofs only in formalisms difficult to understand even for experienced mathematicians. Effort has been made to reconstruct natural deduction (ND) proofs from such machine generated proofs. Although the single steps in ND proofs are easy to understand, the entire proof is usually at a low level of abstraction, containing too many tedious steps. To obtain proofs similar to those found in mathematical textbooks, we propose a new formalism, called ND style proofs at the assertion level , where derivations are mostly justified by the application of a definition or a theorem. After characterizing the structure of compound ND proof segments allowing assertion level justification, we show that the same derivations can be achieved by domain-specific inference rules as well. Furthermore, these rules can be represented compactly in a tre structure. Finally, we describe a system called PROVERB , which substantially sh...

### Citations

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Citation Context ...rs, we believe that patterns of repeated applications of an assertion may be remembered as new rules. Similar phenomena is called in other systems the learning of macro-operators [FHN72], or chunking =-=[New90]-=-. On account of this, domain-specific rules are also referred to as compound rules or macro-rules. We continue with our subset example to illustrate this. Example 1 (Continued): Suppose that a reasone... |

829 |
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Citation Context ...tisfying this constraint will be referred to as the natural expansion of corresponding assertion level justification. This constraint is closely related to one of Johnson-Laird's effective procedures =-=[JL83]-=-, aimed at accounting for spontaneous daily reasoning. Unfortunately, the psychological explanations provided by him can not be extended to predicate logic straightforwardly. 3 Assertion Level Inferen... |

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Citation Context ...om machine oriented formalisms into more natural formalisms [And80, Mil83, Pfe87, Lin90]. As the target formalism, usually a variation of the natural deduction (ND) proof first proposed by G. Gentzen =-=[Gen35]-=- is chosen. Heuristics of various kinds are developed to improve the quality of the target ND proof. For instance, C. Lingenfelder utilizes the topological structures of the refutation graph both to p... |

457 |
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Citation Context ...mbered as new operators, we believe that patterns of repeated applications of an assertion may be remembered as new rules. Similar phenomena is called in other systems the learning of macro-operators =-=[FHN72]-=-, or chunking [New90]. On account of this, domain-specific rules are also referred to as compound rules or macro-rules. We continue with our subset example to illustrate this. Example 1 (Continued): S... |

275 | The use of explicit plans to guide inductive proofs
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Citation Context ...ertion Level The existence of a hierarchy of proof units in proofs constructed by mathematicians can be accounted for by a computational model of human deductive reasoning [Hua93]. Following A. Bundy =-=[Bun88]-=-, this theory cast theorem proving as a planing process, where a planner constructs a proof by applying methods (called tactics in some earlier systems [GMW79, CAB + 86]) on open goals. The proof unde... |

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Citation Context ...ly in a template driven way. Equipped with more advanced techniques developed in the field of natural language generation, a more coherent translation was obtained by the MUMBLE system of D. McDonald =-=[McD83]-=-, where emphasis was laid on the generation of utterances highlighting important global structures of the proofs, as well as utterances mediating between subproofs. A more recent attempt can be found ... |

43 | Transforming Matings into Natural Deduction Proofs - Andrews - 1980 |

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Citation Context ... developers or knowledge engineers, they do not meet the requirement of a typical end-user. To solve this problem, a new, so called reconstructive paradigm for explanation has emerged in recent years =-=[WT92]-=-. The central idea of this approach is that a distinct knowledge base should be used to reconstruct a new solution based on the original one found by the expert system. The reconstructive approach for... |

32 |
Human Oriented Proof Presentation: A Reconstructive Approach
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- 1994
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Citation Context ... to natural expansions, but includes other logically equivalent compound segments. This procedure is the preprocessor of PROVERB , a system transforming natural deduction proofs into natural language =-=[Hua94]-=-. PROVERB is the explanation component for\Omega --MKRP, an interactive proof development environment [HKK + 93]. As argued above, in order to produce natural language proofs comparable with proofs fo... |

29 |
The translation of formal proofs into English
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Citation Context ...g lemmas is reported in [PN90]. Until now the reconstruction stops here and ND proofs are used as inputs by systems producing proofs in natural language. The first such attempt was made by D. Chester =-=[Che76]-=-. His system EXPOUND is usually characterized as an example of direct translation. Although a sophisticated linearization is applied on the input ND proofs, the steps are translated locally in a templ... |

23 |
Transformation and structuring of computer generated proofs
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- 1996
(Show Context)
Citation Context ...of the target ND proof. For instance, C. Lingenfelder utilizes the topological structures of the refutation graph both to produce more direct proofs as well as to avoid redundancy by inserting lemmas =-=[Lin90]-=-. Another technique for inserting lemmas is reported in [PN90]. Until now the reconstruction stops here and ND proofs are used as inputs by systems producing proofs in natural language. The first such... |

23 | Proof Transformations in Higher-Order Logic - PFENNING - 1987 |

17 | Learning plan schemata from observation: Explanation-based learning for plan recognition
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Citation Context ... symbols not originally occurring in A, the assertion being applied, by new metavariables. A similar variablization is a standard technique employed in the context of explanation based generalization =-=[Moo90]-=-. Obviously, the replaced constant symbols must occur in formulas serving as premises, such as a 1 , U 1 and F 1 in a 1 2 U 1 and U 1 ae F 1 in our example. Contraposition The second way of acquiring ... |

15 | Presenting Intuitive Deductions via Symmetric Simplification
- Pfenning, Nesmith
- 1990
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Citation Context ...the topological structures of the refutation graph both to produce more direct proofs as well as to avoid redundancy by inserting lemmas [Lin90]. Another technique for inserting lemmas is reported in =-=[PN90]-=-. Until now the reconstruction stops here and ND proofs are used as inputs by systems producing proofs in natural language. The first such attempt was made by D. Chester [Che76]. His system EXPOUND is... |

13 | C.P.: Edinburgh LCF: A Mechanized Logic of Computation. LNCS 78 - Gordon, Milner, et al. - 1979 |

10 |
Natural Language Explanation of Natural Deduction Proofs
- Edgar, Pelletier
- 1993
(Show Context)
Citation Context ...hasis was laid on the generation of utterances highlighting important global structures of the proofs, as well as utterances mediating between subproofs. A more recent attempt can be found in THINKER =-=[EP93]-=-, where different styles for explaining ND proofs are exploited. In short, it was believed that ND proofs can be adequately presented by resorting solely to ordering , pruning , and augmentation. All ... |

7 | An explanation facility for today's expert systems - Wick, Slagle - 1989 |

4 |
Halbgruppen und Automaten
- Deussen
- 1971
(Show Context)
Citation Context ... proofs, which allow atomic justifications at a higher level of abstraction. To gain more reliable experience with the levels of justifications, we have analyzed proofs in mathematical textbooks like =-=[Deu71]-=-. Based on our preliminary empirical study, justifications are provided at three levels. -- Logic level justifications are simply verbalizations of the ND inference rules, such as the rule of Modus Po... |

4 |
An Extensible Natural Calculus for Argument Presentation. SEKI-Report SR-91-03, Fachbereich Informatik, Universitat
- Huang
- 1991
(Show Context)
Citation Context ...c2 can be constructed, using the corresponding contrapositions of the logic level rules, as depicted in Fig. 3(b). The following property makes a more succinct representation of equation (4) possible =-=[Hua91]-=-: R(A; Contra(B) [ B) = R(A; B) [Contra(R(A; B)) where B is an arbitrary set of logic level inference rules. A natural corollary is: Contra(R(A; Contra(B) [ B)) ae R(A; Contra(B) [ B) Intuitively, thi... |

4 |
Applications of assertions as elementary tactics in proof planning
- Huang
- 1992
(Show Context)
Citation Context ...ertions by constructing a compound proof segment is specified in terms of a so called decomposition-composition constraint imposed on such proof segments identified in our preliminary empirical study =-=[Hua92]-=-. The following two definitions are necessary for the discussion of this constraint. Definition: An inference rule of the form 4`F;4`P1 ;:::;4`Pn 4`Q is a decomposition rule with respect to the formul... |

4 |
An explanatory framework for human theorem proving
- Huang
- 1992
(Show Context)
Citation Context ...und Proof Segment at the Assertion Level The existence of a hierarchy of proof units in proofs constructed by mathematicians can be accounted for by a computational model of human deductive reasoning =-=[Hua93]-=-. Following A. Bundy [Bun88], this theory cast theorem proving as a planing process, where a planner constructs a proof by applying methods (called tactics in some earlier systems [GMW79, CAB + 86]) o... |