## Towards the Mechanical Verification of Textbook Proofs

Citations: | 1 - 1 self |

### BibTeX

@MISC{Zinn_towardsthe,

author = {Claus Zinn},

title = {Towards the Mechanical Verification of Textbook Proofs},

year = {}

}

### OpenURL

### Abstract

Our goal is to implement a program for the machine verification of textbook proofs. We study the task from both the linguistics and automated reasoning perspective and give an in-depth analysis for a sample textbook proof. We propose a framework for natural language proof understanding that extends and integrates state-of-the-art technologies from Natural Language Processing (Discourse Representation Theory) and Automated Reasoning (Proof Planning) in a novel and promising way, having the potential to initiate progress in both of these disciplines.

### Citations

787 |
From Discourse to Logic
- Kamp, Reyle
- 1993
(Show Context)
Citation Context ...itten a definite clause grammar that covers the first proofs of [11], and the proof we discussed above. For semantics construction, our starting point was Kamp’s Discourse Representation Theory (DRT) =-=[13]-=-. To deal with particular syntactic6 Towards the Mechanical Verification of Textbook Proofs assertion object introduced by <a = p ′ 1 p′ 2 . . . p′ s p′′ 1 . . . p′′ t constructions that occur in mat... |

592 |
Untersuchungen über das logische Schließen. Mathematische Zeitschrift, 39:176–210 and 405–431
- Gentzen
- 1934
(Show Context)
Citation Context ...pply Modus Ponens using (3.1) and (3.2). To prove (3.2), we can assume n being an arbitrary number and assume ∀k ∈ N : k < n → P(k). These steps are on the inference level of a Gentzen-style calculus =-=[9]-=-. 4 RELATED AND FUTURE WORK The automated verification of textbook proofs has long been ignored. The only citable reference is Simon’s Phd thesis [22]. However, Simon fails to seriously address both l... |

331 |
Reference to Abstract Objects in Discourse
- Asher
- 1993
(Show Context)
Citation Context ...still from this great reservoir whatever is mechanizable” [28]. This is a promising approach to make proof machines more usable. Our work also relates to extensions of DRT, in particular Asher’s SDRT =-=[2]-=-. SDRT discourse update mechanism allows to effectively interface semantics with pragmatics. The discourse update procedure consists of relating a sentence to its prior discourse (not just adding it).... |

165 | A Problem for RST: The Need for Multi-Level Discourse Analysis
- Moore, Pollack
- 1992
(Show Context)
Citation Context ...ithin a discourse segment contribute to a common discourse purpose) and the informational structure (describing how sentences within a segment are related to each other by some relation) of the proof =-=[19]-=-. From the automated reasoning point of view, verifying textbook proofs is a deductive task where one must identify the logical structure of the proof: identifying assumptions and conclusions, the sco... |

160 | Implementing mathematics with the Nuprl proof development system
- Constable, Allen, et al.
- 1986
(Show Context)
Citation Context ...d and asks the system to apply them. The system keeps track of the proof obligations and guarantees that the constructed proof is correct. A well-known example of a proof system of this kind is Nuprl =-=[8]-=-. For both approaches, proof planning technology could be used to allow the generation of high-level proofs [4]. During proof construction, instead of carrying out elementary steps at the inference le... |

128 |
The Nervous System
- Robertson
- 1975
(Show Context)
Citation Context ...ribes the application of a sequence of inference rules. Tactics are formally described by methods which specify the preconditions that must hold for a tactic to be carried out. The Oyster/Clam system =-=[5]-=- implements an AI plan formation algorithm which seeks to construct a proof plan by searching a tree of methods that combine to prove a given conjecture. These proof plans lead to formal proofs if eac... |

93 |
Discourse Processing and Commonsense Plans
- Litman, Allen
- 1990
(Show Context)
Citation Context ...way of handling discourse structure compares with existing discourse models like [10]. Another related line of research is to view the problem of discourse understanding as a plan recognition problem =-=[17]-=-,[6], [21]. The proof planner we described certainly generates expectations and tries to match them later. But proof recognition goes hand in hand with constructive planning. The relation between proo... |

92 | Discourse Relations and Defeasible Knowledge
- Lascarides, Asher
- 1991
(Show Context)
Citation Context ...rior discourse (not just adding it). To compute such rhetorical relations (e.g., background, narration, elaboration), Asher and Lascarides make use of domain knowledge and their DICE inference engine =-=[15, 3]-=-. Given the enormous complexity of the entire problem, much implementation work is to be done to complete vip. However, our definite clause grammar covers already a considerable amount of mathematical... |

88 | How to write a proof
- Lamport
- 1993
(Show Context)
Citation Context ... by a numbering scheme. For example, take the condition labelled 2.2.4. It contains the terms b and a. Their quantification is 1 PRSs are similar in many respects to Lamport-style proof presentations =-=[14]-=-.Towards the Mechanical Verification of Textbook Proofs 7 determined by the closest accessible let constructs that introduce these names. let a ∈ N be universally quantified ∗ 1 theorem a > 1 → prod ... |

68 | Modeling the User’s Plans and Goals
- Carberry
- 1988
(Show Context)
Citation Context ...f handling discourse structure compares with existing discourse models like [10]. Another related line of research is to view the problem of discourse understanding as a plan recognition problem [17],=-=[6]-=-, [21]. The proof planner we described certainly generates expectations and tries to match them later. But proof recognition goes hand in hand with constructive planning. The relation between proof co... |

64 |
The plan recognition problem: an intersection of psychology and artificial intelligence[J],Aritificial Intelligence,1978(11):45~83
- Schmidt
(Show Context)
Citation Context ...dling discourse structure compares with existing discourse models like [10]. Another related line of research is to view the problem of discourse understanding as a plan recognition problem [17],[6], =-=[21]-=-. The proof planner we described certainly generates expectations and tries to match them later. But proof recognition goes hand in hand with constructive planning. The relation between proof construc... |

53 | The semantics and pragmatics of presupposition
- Asher, Lascarides
- 1998
(Show Context)
Citation Context ...rior discourse (not just adding it). To compute such rhetorical relations (e.g., background, narration, elaboration), Asher and Lascarides make use of domain knowledge and their DICE inference engine =-=[15, 3]-=-. Given the enormous complexity of the entire problem, much implementation work is to be done to complete vip. However, our definite clause grammar covers already a considerable amount of mathematical... |

48 |
Benthem Jutting. Checking Landau’s “Grundlagen” in the Automath system
- van
- 1977
(Show Context)
Citation Context ... approach). A well-known example of a large formalisation is van Benthem-Jutting’s translation of Landau’s ‘Grundlagen der Analysis’ into aut-qe, one of the formalisms of the Automath-language family =-=[26]-=-. The Mizar system offers, besides the rich formal language and the proof checker component, a large library of formalised mathematics which allows one to formalise proofs without excessive preparator... |

45 | Proof verbalization as an application of NLG
- Huang, Fiedler
- 1997
(Show Context)
Citation Context ...oof author’s argumentation line and for subsequent formalisation. The inverse of our task, translating formal proofs into natural language proofs, has first been described in [7]. More recent work is =-=[12]-=-. Facing the problem that proofs generated by conventional theorem provers are unstructured, tediously long and therefore unreadable, a readable, structured and short proof (omitting all the low-level... |

40 |
Toward mechanical mathematics
- Wang
- 1960
(Show Context)
Citation Context ...great wealth of material and constitutes the major source of our understanding of mathematical reasoning. The reasonable course would be to distill from this great reservoir whatever is mechanizable” =-=[28]-=-. This is a promising approach to make proof machines more usable. Our work also relates to extensions of DRT, in particular Asher’s SDRT [2]. SDRT discourse update mechanism allows to effectively int... |

33 |
An introduction to the theory of numbers, Oxford at the Clarendon
- Hardy, Wright
- 1938
(Show Context)
Citation Context ... returned. Each of these formal proofs constitutes one possible semantics for the given informal proof. 3.1 The Parser Module We have written a definite clause grammar that covers the first proofs of =-=[11]-=-, and the proof we discussed above. For semantics construction, our starting point was Kamp’s Discourse Representation Theory (DRT) [13]. To deal with particular syntactic6 Towards the Mechanical Ver... |

28 |
The translation of formal proofs into English
- Chester
- 1976
(Show Context)
Citation Context ...ment for following the proof author’s argumentation line and for subsequent formalisation. The inverse of our task, translating formal proofs into natural language proofs, has first been described in =-=[7]-=-. More recent work is [12]. Facing the problem that proofs generated by conventional theorem provers are unstructured, tediously long and therefore unreadable, a readable, structured and short proof (... |

22 |
On the Shape of Mathematical Arguments
- Gasteren
- 1988
(Show Context)
Citation Context ..., however, has a number of characteristics that makes this task feasible. The art of writing good mathematical texts focuses on clearness and conciseness and not on an embellished style of expression =-=[27]-=-. The expert language used by mathematicians is stylised, that is, characterised by the use of standard phrases and keywords [25]. The use of terms and formulae introduce formal parts into mathematica... |

18 | Benthem Jutting, Checking Landau's "Grundlagen - van - 1977 |

17 |
Computer programs for checking mathematical proofs
- McCarthy
- 1961
(Show Context)
Citation Context ...romising way, having the potential to initiate progress in both of these disciplines. Keywords: text understanding, proof verification, discourse representation theory, proof planning 1 MOTIVATION In =-=[18]-=-, John McCarthy notes that “Checking mathematical proofs is potentially one of the most interesting and useful applications of automatic computers”. In the first half of the 1960s, one of his students... |

15 |
The Mizar-QC/6000 Logic Information Language
- Trybulec
- 1978
(Show Context)
Citation Context .... A human user is required to perform the formalisation task. Basically, two main approaches towards the formalisation and verification of proofs were taken. In the first, the Automath/Mizar approach =-=[20, 24]-=-, the user is required to give a full and explicit construction of a formal proof. The proof component then checks the proof for correctness (the compiler approach). A well-known example of a large fo... |

14 | How to read and do proofs - Solow - 1982 |

9 |
Checking Number Theory Proofs in Natural Language
- Simon
- 1990
(Show Context)
Citation Context ...on the inference level of a Gentzen-style calculus [9]. 4 RELATED AND FUTURE WORK The automated verification of textbook proofs has long been ignored. The only citable reference is Simon’s Phd thesis =-=[22]-=-. However, Simon fails to seriously address both linguistic and mathematical issues. It remains unclear how Simon handles linguistic phenomena – he does not propose an adequate theory for doing so. It... |

8 |
The Use of Explicit Proof Plans to Guide Inductive Proofs
- Bundy
- 1988
(Show Context)
Citation Context ...nstructed proof is correct. A well-known example of a proof system of this kind is Nuprl [8]. For both approaches, proof planning technology could be used to allow the generation of high-level proofs =-=[4]-=-. During proof construction, instead of carrying out elementary steps at the inference level, tactics are applied. A tactic is a program that describes the application of a sequence of inference rules... |

8 |
Elementary Theory of Numbers
- LeVeque
- 1960
(Show Context)
Citation Context ...s of the proof. Gaps and flaws must be recognised.2 EXAMPLE Towards the Mechanical Verification of Textbook Proofs 3 Fig. 1 depicts LeVeque’s existence proof of the fundamental theorem of arithmetic =-=[16]-=-. We analyse this proof from both the linguistic and mathematical point of view. Theorem 2-2. Every integer a > 1 can be represented as a product of one or more primes. Proof: The theorem is true for ... |

7 | Understanding Mathematical Discourse
- Zinn
- 1999
(Show Context)
Citation Context ...′ 2 . . . p′ sp′′ 1 . . . p′′ t . 2.1 Linguistic analysis Fig. 1. A textbook proof taken from [16] All kinds of linguistic phenomena which can occur in other text genres also occur in textbook proofs =-=[29]-=-. We discuss a selection of problems for the proof at hand. A prerequisite for parsing textbook proofs is to be able to parse formulae that occur in these proofs. Parsing formulae in isolation is triv... |

7 |
Machine verification of mathematical proofs
- ABRAHAMS
- 1963
(Show Context)
Citation Context ...ing and useful applications of automatic computers”. In the first half of the 1960s, one of his students, Paul Abrahams, implemented a Lisp program for the machine verification of mathematical proofs =-=[1]-=-. The program, named Proofchecker, “was primarily directed towards the verification of textbook proofs, i.e., proofs resembling those that normally appear in mathematical textbooks and journals”. Abra... |

6 | The plan recognition problem - Schmidt, Sridharan, et al. - 1978 |

4 |
Writing mathematical papers in english
- TRZECIAK
- 1993
(Show Context)
Citation Context ...learness and conciseness and not on an embellished style of expression [27]. The expert language used by mathematicians is stylised, that is, characterised by the use of standard phrases and keywords =-=[25]-=-. The use of terms and formulae introduce formal parts into mathematical discourse. In addition, textbook proofs are, in general, a highly structured form of discourse. A crucial prerequisite to under... |

1 | Machine veri of mathematical proofs - Abrahams - 1963 |