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An annotated corpus and a grammar model of theorem description

An annotated corpus and a grammar model of theorem description
An annotated corpus and a grammar model of theorem description

An Annotated Corpus and a Grammar Model of

Theorem Description

Yusuke Baba1and Masakazu Suzuki2

1Graduate School of Mathematics,Kyushu University

2Faculty of Mathematics,Kyushu University

6-10-1Hakozaki,Higashi-ku,Fukuoka,812-8581Japan

{ma201040,suzuki}@math.kyushu-u.ac.jp

Abstract.Digitizing documents is becoming increasingly popular in

various?elds,and training computers to understand the contents of dig-

itized documents is of growing interest.Since the early90’s,research of

natural language processing using large annotated corpora such as the

Penn TreeBank has developed.Applying the methods of corpus-based

research,we built a syntactically annotated corpus of theorem descrip-

tions,using a book of set theory,and extracted a grammar model of

theorems from the obtained corpus,as the?rst step to understanding

mathematical documents by computer.

1Introduction

In recent years,digitizing documents has become increasingly popular in various ?elds,for example,in mathematics[1][2][3][4].In connection with this movement, understanding the contents of digitized documents by computer is of growing interest[5][6].

The technology of understanding documents is applicable to useful systems such as machine translation,summarization and search.In mathematics,there are also particular application ideas of application e.g.the a system that trans-lates descriptions of a proof into a language of proof-checker.

The fundamental technology of understanding documents by computer is parsing.The e?ectiveness of the above-mentioned systems depends heavily on the accuracy of the parser.Generally,parsing is complicated by the characteristics of natural language such as ambiguity,omission and inversion.On the other hand, since mathematical documents are written in logical and precise expressions,we can expect that the typical methods of parsing for natural language give more accurate results for mathematical documents.

The availability of large,syntactically annotated corpora such as the Univer-sity of Pennsylvania Tree Bank(Penn TreeBank,[7])lead to rapid developments in the?eld of natural language processing.Sekine et al.[8]extracted rules of grammar from the Penn TreeBank and released a parser of English,the“Apple Pie Parser[9]”,using the grammar.

To extract a grammar from the Penn TreeBank,the?rst approach of Sekine et al.is based on the assumption that the corpus covers most of the possible

sentence structures in a domain.The outline of their idea follows.They assign parts-of-speech to an input sentence using a tagger,and then simply search for the same sequence of parts-of-speech in the corpus.The structure of the matched sequence is the output of their parser.

However,it turned out that this strategy was not practical.Only4.7%of sentences in the corpus had the same structure as another sentence in the corpus. As a result,they applied the idea to not only sentences but also noun phrases,and extracted rules of grammar about S(sentence)and NP(noun phrase).77.2%and 98.1%of S and NP structures have the same structure in the corpus,respectively [8].

If there were a large corpus covering most of the possible sentence structures in a domain,the method[8]would be more e?ective.While building a large corpus of natural language costs a great deal of labor and time,a corpus of theorem descriptions covering all the possible structures looks buildable with comparative ease because theorems have a limited vocabulary and consist of many idiomatic expressions.For this reason,we apply the corpus-based methods to process theorem descriptions.

We built a simply annotated corpus of theorem descriptions using a book of set theory,and extracted a context-free grammar with only three non-terminals from the corpus.The constructed corpus included about100instances,and the grammar that was extracted from the corpus had141generation rules.In order to evaluate the descriptive power of the obtained grammar,we performed an experiment as described in section4.

2Building a Corpus

2.1Preliminaries

We used a book of set theory[10]to collect samples of theorem descriptions as the source of the corpus.

To build an annotated corpus of theorems,we used28categories of Parts-Of-Speech(POS)symbols,and3categories of phrase and clause symbols.

We processed the theorem descriptions as a sequence of words.In our corpus,

a formula is simply a word.

2.2Sentence structure

To express the structure of theorem descriptions in the corpus,we de?ned a symbol of phrase(NP)and two symbols of clause(S,IFC),as follows.

Proposition-Clause(S)

S is a string of words which expresses a proposition.S can include any other propositions in itself.Naturally,a theorem description is described by S.

Examples of S

–If R is a strict order on X,then S is a non-strict order on X.

–(x1,x2)=(y1,y2)if and only if x1=x2and y1=y2.

–X is an open set

–Vα?Vβ

If-Clause(IFC)

IFC is a string of words which expresses the assumption of a proposition. Examples of IFC

–If|X|≤|Y|and|Y|≤|X|,then|X|=|Y|.

–In ZFC,every vector space has a basis.

Noun-Phrase(NP)

NP is a string of words which can be processed like a noun.

Examples of NP

–“any non-zero ordinal”

–“the set P={R(x):x∈X}of R-classes”

–“P={R(x):x∈X}”

–“X”

2.3Classi?cation of Parts-Of-Speech

Referring to the Penn TreeBank which has48categories of POS symbols[7],we use7original categories and21typical categories of POS on the grounds that the target domain of our corpus is only theorems.We de?ned original categories and typical categories of POS so that all the words that appeared in the samples would be classi?ed naturally.

(a)Original Parts-Of-Speech

Theorem descriptions have many idiomatic expressions.For example,out of the 96instances in the samples,50theorems(52.1%)have the structure:

“If[proposition],then[proposition]”or“Let[proposition],then[proposition].”11theorems(11.5%)have the structure:“[proposition]if and only if[proposi-tion].”The words‘if’and‘let’are generally classi?ed as a conjunction and a verb respectively,and‘if and only if’is generally partitioned into‘if’‘and’‘only’‘if’,in natural language processing.However,since these words have particular meaning in theorem descriptions,they should be given special treatment.In this way,we de?ned7original categories of POS referring to empirical knowledge of mathematics and rough characteristics obtained from the samples:

(1)Proposition-Conjunction(PPC),

(2)Assumption(IF),

(3)Existence(EX),

(4)Restriction(RST),

(5)Number(NUM),

(6)Explain-Conjunction(EXPC),

(7)Formula(FML).

We give detailed explanations for these categories below.

(1)Proposition-Conjunction(PPC)

PPC is used between two propositions,and expresses a relationship between the propositions.

Examples of PPC

–if and only if

–implies

Examples of Appearances in Samples

–A set is most countable if and only if it is?nite or countable.

–P(α)implies P(s(α))for any cardinalα.

(2)Assumption(IF)

IF is a word used in a description which expresses an assumption of the theorem (or a larger proposition which includes it).

Examples of IF

–if

–let

–suppose that

–assume that

Examples of Appearances in Samples

–If P is a partition of X,then R={(x,y):x,y∈p for some p∈P}is an equivalence relation on X.

–Let f be a function from X to Y.Then Im(f)is a subset of Y.

(3)Existence(EX)

EX is used to express the existence of something in a mathematical sense. Examples of EX

–there is

–there exists

–there is a bijective function from X to Y

–there exists a positive integer n

(4)Restriction(RST)

RST is used in order to impose restrictions on NP,in a mathematical sense. Examples of RST

–such that

–satisfying

Examples of Appearances in Samples

–There is no set S such that x∈S if and only if x∈x.

–there is a set Z∈X such that p(Z)=Z

(5)Number(NUM)

In mathematics there are various quantities,not only one,two,three,···,but also ‘only one’,‘all’,‘in?nite’···,which have particular meanings.NUM expresses such a numerical quantity in a mathematical sense.

Examples of NUM

–a

–unique

–any

–only one

–no

Examples of Appearances in Samples

–a well-ordered set

–For any formulaφ,···

–There is no set S such that x∈S if and only if x∈x.

(6)Explain-Conjunction(EXPC)

EXPC is used between propositions,and expresses an explanation or a trans-lation.

Examples of EXPC

–in other words

–that is

–For any set X ,there is an injection from X to P X but no bijection between these sets;that is ,|X |<|P X |.

(7)Formula(FML)

FML expresses a mathematical formula.We process a formula as a word.Examples of FML –p :P X →P X

–f [A ]={f (a ):a ∈A }–(x 1,x 2)=(y 1,y 2)–X –

n

(b)Typical Parts-Of-Speech

In addition to the Original POS,we de?ned 21categories of POS that are gen-erally used in natural language processing.Table 1is a list of these Typical POS.

Using S ,IFC ,NP and POS,we assigned the sentence structure for each member of the samples.The structure of a sentence can be expressed as a tree.An example of the expression of a sentence structure is shown in Fig.1.In the Figure,a box contains those parts of a sentence processed as a word.IF,EX,···are POS.Symbols of phrase and clause are assigned to nodes of the structure tree.

If

there is a

bijection

between n

and m ,

then

n =m .IF EX NUM NN IN FML CNJ FML COM RB FML SEP S

IFC

S

S

NP NP

NP

NP Fig.1.Example of expression of sentence structure

Table1.Typical POS

Symbol Category Examples of members NN Noun set,relation,function,···PR Pronoun it,we,following,···RPR Relative Pronoun what,which,that,whose,···JJ Adjective countable,empty,same,···BE be is,are,be,···

VB Verb have,lie,···

VBG Verb,gerund/present participle subsutituting,asserting,···VBN Verb,past participle based,denoted,···

MD Modal may,can,···

RB Adverb mutually,moreover,totally,···CNJ Conjunction and,but,or,as,···

IN Preposition on,of,in,from,to,···DT Determiner the,···

LPAR Left bracket character(,{,[

RPAR Right bracket character),},]

LOQ Left open quote‘-and-“

RCQ Right close quote’-and-”

SFP Sentence-?nal punctuation.

COM Comma,

COL Colon,Semi-colon:-and-;

SYM Other Symbols-,*,···

Our primary corpus has96instances of theorem descriptions which are all the theorems in[10].Table2shows some examples of members of the corpus.An instance of the corpus has4categories of data(a,b,c,d in the Table).Practically, we built the corpus in an XML?le using tag hierarchy.

Table2.Examples of Members of the Corpus

a:Character String of the Theorem Description

b:Partition of String into Words

c:Sequence of POS

d:Structure of Sentence

(the numbers‘n’appearing in the expression

represent the‘n-th’member of a sequence of POS)

a(x1,x2)=(y1,y2)if and only if x1=x2and y1=y2.

b[(x1,x2)=(y1,y2)][if and only if][x1=x2][and][y1=y2][.]

c FML PPC FML CNJ FML SFP

d(S(S(S1)2(S(S3)4(S5)))6)

a There is no set S such that x∈S if and only if x∈x.

b[There is][no][set][S][such that][x∈S][if and only if][x∈x][.]

c EX NUM NN FML RST FML PPC FML SFP

d(S(S1(NP23(NP4)5(S(S6)7(S8))))9)

a The union of at most countably many at most countable sets

is at most countable.

b[The][union][of][at most][countably][many][at most][countable][sets] [is][at most][countable][.]

c DT NN IN RB RB JJ RB JJ NN BE RB JJ SFP

d(S(S(NP123(NP456(NP789)))101112)13)

a If there is an injective function from X to Y,and X=φ,

then there is a surjective function from Y to X.

b[If][there is][an][injective][function][from][X][to][Y][,][and][X=φ]

[,][then][there is][a][surjective][function][from][Y][to][X][.]

c IF EX NUM JJ NN IN FML IN FML COM CNJ FML

COM RB EX NUM JJ NN IN FML IN FML SFP

d(S(IFC1(S2(NP3456(NP7)8(NP9))1011(S12))

1314(S15(NP16171819(NP20)21(NP22)))23)

3Extraction of Grammar

3.1Algorithm

The structure tree of the corpus that we built has three non-terminal symbols: S,IFC,NP.We applied the method[8]to the corpus and three symbols,and extracted generation rules of a grammar.The algorithm for extracting the rules is as follows.Let G be the set of rules of the grammar to be extracted. Repeat step1.and step2.for all the trees of the corpus.

The Algorithm of Extraction of Rules

Let G be the set of rules of the grammar to be extracted.

Repeat step1.and step2.for all the trees of the corpus.

step1.

For the root and all the nodes of the tree,

make a generation rule of the grammar such that

the Left-hand-side of the rule is

the symbol of the node or root,and

the Right-hand-side of the rule is

the left-to-right sequence of the symbols of the children of the node or root. step2.

Add the rules made in step1.to G

unless any rule is already in G.

Ex.The Rules obtained from Fig.1are as follows.

–S->IFC COM RB S SFP

–S->EX NP

–S->FML

–IFC->IF S

–NP->NUM NN IN NP

–NP->NP CNJ NP

–NP->FML

3.2The Obtained Grammar

Using Algorithm3.1,we extracted a grammar which has141generation rules, as the model of theorem description.The obtained rules consist of48S-rules,17 IFC-rules,and76NP-rules.Table3shows a selection of the obtained rules.Since the grammar has only3non-terminals,some rules have a long right-hand-side.

Table3.A selection of the obtained rules of grammar S-rules(Total48)

S->EX NP

S->FML

S->IFC SFP RB S

S->IN NUM NP COM S

S->NP BE JJ

S->NP BE NP

S->NP MD RB VB NP

S->NP VB

S->PR BE JJ CNJ JJ

S->PR BE NP LPAR PR BE COM JJ RPAR

S->S CNJ S

S->S COL EXPC COM NP S

···

IFC-rules(Total17)

IFC->IF S

IFC->IF S COM IF S

IFC->IF S SFP COM IF S

IFC->IN NP

IFC->IN NUM NP COM IF S

···

N-rules(Total76)

NP->DT JJ NN

NP->DT NN IN JJ NN

NP->DT NN IN NN IN NP IN NP

NP->DT NN IN NN IN NP RPR VB IN NUM DT NN NP

NP->NN NN

NP->NP CNJ NP

NP->NP CNJ PR NP

NP->NUM JJ CNJ JJ NN

NP->NUM JJ JJ NN IN NN RB NP

NP->NUM JJ NN

NP->RB JJ NN

NP->RB RB JJ NP

···

4Experiment

To evaluate the descriptive power of the grammar obtained in section3(let G be the grammar),we used100theorems collected from two books A[11]and B[12] which are written about Galois theory and model theory.

We assigned the correct structure of S,IFC,NP and POS to each theorem, and extracted generation rules using algorithm sec:al.If all the rules obtained from a theorem are elements of G,then the theorem can be described by G. We checked whether or not each theorem is described by G.Additionally,we expanded G into G by adding the rules which are not in G whenever they appear,and we then checked the descriptivity of G .

Firstly,we experimented using50theorems of book A(Step-1).Secondly we used50theorems of book B and G obtained from Step-1(Step-2).Table4 shows the results of the experiments.”G-describable”means that the theorem can be described by G.From the table,we can see that G describes only10% of new https://www.sodocs.net/doc/4210229567.html,ing the expanded grammar G did not produce a signi?cant increase in descriptive power.

We can consider that one factor in?uencing the results is a simplistic struc-ture model.Because our structure has only3non-terminals,obtained rules have greater diversity in their right-hand-sides.Moreover,the fact that the total num-ber of rules of G increased at a constant rate means that the test corpus is too small.

Therefore in order to improve the system,we have to introduce other struc-tures like verb-phrase or adjective-phrase,and expand our corpus.

Table4.Experimental Results

G-describable G -describable Total number of rules of G after Step Step-15/5011/50223

Step-25/5011/50316

5The Corpus Operation System

To build a large,syntactically-annotated corpus of mathematical text is trou-blesome work,and very time consuming.In order to make the work easier,we are designing a system that assists with many operations regarding using the corpus.

Fig.2is an image of the https://www.sodocs.net/doc/4210229567.html,ers need only assign the correct structure of a sentence via a user-interface,and they can see the sentence structure being edited in real time as a tree.Other operations:tagging,parsing,expanding the corpus,training the grammar,are all automatic.

Fig.2.The Corpus Operatrion System

6Conclusion

We collected about100theorem descriptions from a book of set theory[10],and built a model for an annotated corpus of theorems.

We extracted a grammar model of theorem description from the corpus, which has141generation rules.We also performed an experiment to evaluate the descriptive power of the grammar.

The results of the experiment indicated that the structure of our test corpus is too simplistic to generate an e?ective grammar,and the size of the corpus is too small.

As the next step in our work,we will introduce another structure and collect more instances for the corpus.Additionally,we plan to build the semi-automatic corpus operation system in the near future.

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定语从句关系副词/ 介词+关系代词/ 关系代词whose练习 一.填上适当的关系副词when/where/why,并把它写用两个简单句,注意介词的选择。1.It is a place _where_ an astronaut works and lives 句1:It is a place . 句2:An astronaut works and lives in the place. 2.It was the day _______ I was sent into the sky 句1: 句2: 3.Can you tell me the reason______________ you are taking pictures? 句1: 句2: 4.Do you remember the day______ we left you in charge? 句1: 句2: 5.I often think of the moment_________ I saw the UFO. 句1: 句2: 6.The police searched the house______the thief had stayed. 句1: 句2: 7.Is Shanghai the city ______he was born? 句1: 句2: 8.I don’t know the reason_________the house is so dirty. 句1: 句2: 9.The study is the place _________I often have talks with my father. 句1: 句2: 二.填上适当的关系代词which/whom,然后介词还原至原来位置,并找出先行词写在横线上,再写成两个简单句,如。 例: ①This is the question about _which_ we've had so much discussion. 还原:This is the question which we’ve had so much discussion about. 先行词:the question 句1:This is the question. 句2:we’v e had so much discussion about the question. ②He is the man about _________I told you yesterday. 还原: 先行词:句1: 句2: ③The woman to _________I spoke is a teacher. 还原: 先行词: 句1: 句2: ④Is this the house in _________ he was born? 还原: 先行词: 句1: 句2: ⑤The person to _________ you should write is Mr. Ball. 还原: 先行词: 句1: 句2: ⑥The games in _________ he took part were swimming and shooting. 还原: 先行词: 句1: 句2: 三.填上适当的介词。 1.Have you met the person _______whom he was speaking? 2.He is the man _______whom you should give your thanks. 3.The farm _______which we worked ten years ago isn’t what it used to be. 4.This is the tree _______which we used to play games. 5.His pen _____which he is writing now was bought yesterday. 6.The gas _______which we can not live is called oxygen. 7. The pencil ________ which he wrote was broken. 8. He built a telescope ________ which he could study the skies. 9. The woman, ________ whom I learned the news, is a nurse. 10. The wolf ________ which the sheep was killed was shot. 11. She has three children, all _______ whom are at school. 12. There is a tall tree outside, ________ which stands our teacher. 13. My glasses, _______ which I was like a blind man, fell to the ground and broke. 14. In the dark street, there wasn’t a person ________ whom she could turn for help. 15. I was surprised at the way ________ which he treated the old man. 16. The age _______ which children can go to school is seven. 17. His bike ________ which he went to work was stolen last night.

人教版新目标九年级英语课文翻译全一册

155******** 一单元 SECTION A 1a 我通过制作抽认卡来学习。通过和朋友一起学习。通过听磁带。通过做抽认卡。通过向老师求助。通过读课本。通过制作单词本。 1c A:你怎么为考试而学习。B:我通过参加学习小组来学习。 2a 1、你是通过看英文录像学英语的吗? 2、你曾和朋友们练习过对话吗? 3、听磁带怎么样? 4、大声朗读以练习发音怎么样?5、我曾经通过参加学习小组的方式学习过吗? 2b A是的,我通过那种学习方式学到了很多。B、哦,是的,它提高了我说英语的能力。C、有时那样做。我觉得他有用。D、不。(通过看英语录像学习)太难了,无法理解录像中的人所说的话。 2c A你曾经通过参加学习小组来学习吗?B、是的,我参加赤字,通过那种方式我学到了很多。 Grammer Focus 你怎么为准备一场考试而学习?我靠听磁带。你怎样学习英语?我通过参加学习小组来学习。你通过大声朗读来学习英语吗?是的,我是。你曾和朋友们练习过对话吗?哦,是的,他提高了我说英语的能力。你曾经通过参加学习?小组来学习吗?是的,我参加过。通过那种方式我学习到了很多。 3a如何才能学得最好 这星期我们询问了新星高中的同学关于学习更多英语的最佳方法的问题。许多同学说他们通过使用英语为学习它,一些还有很特别的建议。比如,李莉莲说学习新单词的最好的方法是阅读英语杂志。她说记忆浒音乐的歌词也有一些作用。当我们问及学习语法的问题时,她说:“我从不学习语法。它太枯燥了。” 魏明有不同的看法。他学习英语已经6年了,并且确实喜欢英语。他认为学习语法是学习一门语言的一种好方法。他还认为观看英语电影也不错,国灰他可以看到演员说话的情形。但是,有时候他发现看英语电影是件很头痛的事情,因为那些演员说话太快了。 刘畅说加入学校英语俱乐部是提高英语最好的方法。学生有很多练习的机会并且他们也有很多乐趣。她补充说和朋友练习会话一点用处也没有。“我们会因为某件事变得很激动,最后用汉语来讲,”她说。 3b A:我正在作一个关于学习英语的调查。我能问你一些问题吗?B:当然。A:太好了!你叫什么名字?B:魏明。A:那么你是怎样学习英语的,魏明?B:…… 4 A:你列词汇表吗?B:噢,是的。我常那样做。SECTION B 1a我不会发其中一些单词的音。我不会拼写一些英语单词。我听不懂英语口语。我在语法上犯错误。我读得很慢。1b我不知道怎么使用逗号。2a 1、不能正确发音。 2、忘记很多生词。 3、人们和我说话时我不能每次都听懂。 4、不能理解杂志中的单词。 5、没有获得很多写作训练。 2B A、你可以一直将生词写在你的笔记本里,并在家学习它们。B、你应该找一个笔友。C、听力能起作用。D、为什么不加入一个英语俱乐部来练习说英语呢? 2C A:我没有搭档来练习英语。B、也许我应该加入一个英语俱乐部。 3a我是怎样学习英语的 去年英语课对我来说很难。首先,对我来说听懂老师说话很难。开始,她说的太快,我不能听懂每个单词,后来,我意识到如果你听不懂每个单词并没有关系。而且我害怕在班上说话,因为我认为同学们可能会嘲笑我。我也不是总能造出完整的句子。然后我开始看英文电视。那很有用。我认为做大量听力练习是成为一个好的语言学习者的秘决之一。另一件我觉得很难的事是英语语法。所以我决定在每节课上记大量语法要点。然后我开始用我正在学的语法自己写新句子。这样作用处之大令人惊奇。现在我很喜欢学英语并且这学期我得了个A。我的老师对我印象很深。作者觉学英语很难是因为……1、老师发音差。2、她说话时人们总是嘲笑她。3、她在造完整的句子方面有困难。4、英语语法很难。当她开始…她的英语提高了。5、和说英语的朋友一起出去。6、大量的听力练习。7、在自己组织的句子里使用语法。 3b 亲爱的,我知道学英语不容易,但我有一些想法可能有用。你说你不能理解说话太快的人。那么,你可以尽量听最重要的单词,而不是每个单词。 4 1、关于学英语什么不容易。2、就这一点你作了什么? 3、你最喜欢的学习更多英语的方式是什么?韩文说如果人们语速太快听力有时就很难。 SELF CHECK 1 你应该在词汇表中写下新的英文单词。2、如果你不知道怎样拼写生词,就查词典。3、最好的提高你的英语(水平)的方法是加入英语俱乐部。4、另一件他觉得很困难的事是英语语法。5、这种纸摸上去非常柔软。 2 READING Section 2使用词典词典是有用的学习工具,但许多英语单词有不同的含义和用法。我们需要确定我们从词典中找到的含义与语境匹配。 我们该怎样解决我们的烦恼? 无论贫富、老少,我们都有烦恼。并且除非我们解决了问题,否则我们会轻易变得不开心。为我们的问题担忧会影响我们在学校的表现。它也会影响我们同家人相处的方式。所以我们该怎么解决我们的烦恼呢?有许多方法。它不是烦恼----它是挑战。享受面对它(的过程)。 通过学会忘记

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