A note on the polynomial form of boolean functions and related topics

A Note on the

Polynomial Form of Boolean Functions

and Related Topics

Bogdan J.Falkowski,Senior Member,IEEE

AbstractDThis note relates to a recently published partly tutorial article that presents some discussion of the polynomial form of Boolean functions and its applications based on the literature published in English and German.We show that a lot of the research in this area has also been done in Eastern Europe,and this note aims to present these unknown developments.The most recent work in this area is also described.

Index TermsDBoolean functions,multiple-valued functions,polynomial forms, arithmetic transform,mixed arithmetic transform,arithmetic derivatives.

?

1I NTRODUCTION

S CHNEEWEISS recently discussed a few derivations and applica-tions of the polynomial form of a Boolean function in a partly tutorial report[1].By writing his article,the author of[1]did a respectable job since he provided some useful suggestions for further work toward extension and possibly optimization of methods used in the theory of arithmetic expansions,as well as mentioned some earlier and recent related papers available in English and German.Unfortunately,the author,who is a German expert on arithmetic transform,is unaware of many developments in this area done in Eastern Europe and only one book from this geographical area was mentioned by him[2].

The purpose of this note is to recall some earlier work representing the origins of the theory and give a short and full overview of the research in this important area.It should be mentioned that some of the discussed work was published in Russian and whenever this author knows of its English translation, then the translation is mentioned;otherwise,the note refers to the originals in Russian.Finally,this note discusses different terminol-ogy used in the above area by various authors and makes a short overview of the most recent research done since the original article [1]was submitted for review.It is believed that all these topics are important and useful for further work in this and related areas.

2A RITHMETIC T RANSFORM

The polynomial form of Boolean functions discussed in[1]is nothing more than the orthogonal expansions of the corresponding Boolean functions by the set of basis functions for an arithmetic transform.It should be noted that,in the classical literature on the subject[3],the termapolynomial representationousually refers to the representation of Boolean functions based on the operations in the field of integers modulo2,constants0and1,and the unary operation of the complementation that is known under the names of Zhegalkin[4],EXCLUSIVE OR canonical form[5],or Reed-Muller expansions[6],[7].However,the same term was frequently used to denote polynomial expansions of Boolean functions based on Walsh transform[8],[9]known also under the name of orthogonal Walsh series expansion[10]or abstract Fourier trans-form expansion[5].As there is a direct relation between polynomial expansions of Boolean functions based on Reed-Muller,Arithmetic and Walsh functions[3],[5],[11],[12],[13],it is quite understandable to use this term for arithmetic expansions, as was the case of[1].To differentiate between arithmetic and Reed-Muller transforms and expansions,reference[14],similarly to other authors from the Soviet Union,uses the names apolynomial arithmeticaloandapolynomial logicalotransforms and expansions for the former and latter,accordingly.It should also be noticed that the arithmetic,Walsh,and Reed-Muller transform are examples of abstract harmonic analysis of Boolean functions on finite Abelian groups[5],[15].The arithmetic transform can be also found in the literature under different names ofaprobabilistic transformo[16],[17],aalgebraic transformo[18]and,in some early Russian literature[19],as well as in the recent publication[14],is known asainverse conjunctive trans-form.oAt this point,it should be mentioned that the sets of basis functions for the arithmetic transform obtained from its inverse transform matrix and known under the name ofaconjunctive transformo[14],[19],oraadding transformo[20],is the same as the sets of basis functions for the better known Reed-Muller transform [12],[14].Sometimes,the Reed-Muller transform is also called the aconjunctive logic transformowith addition of the adjectivealogicoto differentiate it fromaconjunctive arithmetic transformo[14].As the conjunctive matrix can be easily constructed via Pascal triangle, which was shown for the first time in[11],this transform is sometimes called aaPascal transformo[14].The transform matrix of the arithmetic transform is nonsingular and its inverse is formed of all the elements having entries equal to1with calculations performed over the complex field g or its subfield of integers . For the Reed-Muller transform,its transform matrix is self-inverse since,in GF(2),the addition and subtraction coincide.The arithmetic transform of a matrix order x P n for an n-variable Boolean function can be defined as[14]:

e x

e x

P

H

àe x

P

e x

P

P

R

Q

S Y e

I I Y x P Y Q Y F F F

Also,

e x e P e x

P

for x P Y Q Y F F F Y

where the symbol is a Kronecker product.

The corresponding arithmetic polynomial form is given as equation(1)in[1].

In Section2.2in[1],the author describes the method of finding the coefficients for the polynomial form via linear equations.This method requires solution of P n equations for an n-variable Boolean function.There are a number of more efficient techniques to perform the same tasks that should be mentioned.As the arithmetic transform matrix is Kronecker-based,then it may be easily computed by means of fast transform[14],[16].While more efficient than the standard transform matrix multiplication,this method requires the full vector of the original Boolean function to be operated on.However,frequently in modern CAD VLSI systems,these functions are represented in reduced forms,such as array of cubes or decision diagrams[21],which would have to be expanded to full minterm truth vectors in order to apply fast algorithms.Hence,methods have been developed that allow calculating the coefficients directly from arrays of cubes or different decision diagrams[22],[23],[24],[25],[26].Some of the methods for decision diagrams are based on various variants of the Shannon decompositionDsee equation(3)in[1].Moreover,these methods also allow calculating only some chosen arithmetic coefficient without generation of the whole spectrum what is useful in some applications of arithmetic spectra,for example,in

.The author is with the School of Electrical and Electronic Engineering, Nanyang Technological University,Block S1,Nanyang Avenue,Singapore 639798.

E-mail:efalkowski@https://www.wendangku.net/doc/f616378933.html,.sg.

Manuscript received18Nov.1998;revised18May1999.

For information on obtaining reprints of this article,please send e-mail to: tc@https://www.wendangku.net/doc/f616378933.html,,and reference IEEECS Log Number108281.

0018-9340/99/$10.00?1999IEEE

testing[27].Link of arithmetic transforms with different compu-tationally efficient graphical representations of Boolean functions, such as OBDDs,FBDDs,EVBDDs,and MTBDDs,have also been shown[18],[23],[24],[25],[28],[29].In Section3of[1],the Shannon decomposition in GF(2)and arithmetic form are dis-cussed.It should be mentioned that there are three basic types of expansion using EXOR operators:Shannon expansion,and positive and negative Davio expansions[12],[29].Similarly,there are three types of corresponding arithmetic expansions discussed with proofs in details in[3],[30].

3A PPLICATIONS OF A RITHMETIC T RANSFORM AND R ECENT D EVELOPMENTS

In[1],applications of arithmetic expansion in probabilistic equivalence verification and reliability analysis were discussed and the author referred mainly to the literature in English and German.In this review,we will stress the lesser known research done in the areas of arithmetic transform in Eastern Europe.The detailed discussion of status and achievements for arithmetic transform was done in a recent tutorial paper and the interested reader is referred to this work[31].Another good reference is the recent DSc thesis of Yanushkevich[32],with a review of arithmetic transform and a long list of references.Some review of the research work in this area was also done in[33]and this note uses this information as well.As mentioned in[1],the first attempts to present logic operations by arithmetical ones were done by the founder of Boolean algebra,Boole[34].First applications of arithmetic relations for Boolean operations relate to the calculation of probability relations[35].Arithmetic expansions were used in calculation of logical network reliability by Malyugin[36].Recent applications of arithmetic transform in switching theory and logic designs originate in[2],[3],[30],[37].Malyugin developed the algebra of arithmetic polynomials for Boolean functions[38],[39], and their various characteristics that apply to some problems,in particular Petri networks modeling[40].In[38],he proposed representing an array of Boolean functions by an arithmetic polynomial and has proven uniqueness of such a representation. By assigning the operations of addition and multiplication on a set of arrays,it is possible to define algebra of arrays.The complexity of realization of an array is estimated by the number of terms of the polynomial.The classes of Boolean functions that can be realized by linear arithmetic polynomials are investigated in[39].The concept of a composition of arithmetic polynomials is also introduced.It is shown that every ordered system of Boolean functions,called a cortege,can be represented by a composition of linear arithmetic polynomials.The realization of a cortege is reduced to the sequential computation of the polynomials forming the composition.The concept of arithmetic polynomial was extended in[41]by introducing theaabsolute valueooperation. The condition for a system of Boolean functions to be represented by arithmetic polynomial is also defined there.Some new methods to construct arithmetic polynomials with absolute value operation are shown.

In[42],various arithmetic forms are synthesized.Also,different polynomial expansions of Boolean functions in Walsh,Haar,and arithmetic bases are classified and some of their mutual relations given.Detailed relations between arithmetic and Haar functions were given in[43].The same topic of polynomial expansions of Boolean functions in different bases was further elaborated in the book[44].The latter work also employed the methods of digital signal processing to synthesize the set of arithmetic polynomial forms of Boolean functions.Since the arithmetic transform matrix is defined recursively in form of Kronecker product,fast trans-forms based on Good's algorithm to form arithmetic polynomials have been considered[16],[44].The use of systolic arrays to synthesize Arithmetic polynomial was suggested in[44],[45].As shown in[38],[39],[40],[41],[46],parallelization of logical computations is possible and,under certain conditions,effective when we convert an arithmetic representation of a system of functions into a single form of an Arithmetic polynomial.This approach has been extensively developed in[41],[42],[46].The problem of obtaining the most complete use of the architectural capabilities of computers performing concurrent logical computa-tions was considered in[47],[48].This problem was converted to two problems:description of Boolean functions as arithmetic polynomials and the problem of selecting algorithms for intensive evaluation of such polynomials.In[49],a canonical structure is proposed that assures realization of Boolean functions in terms of linear arithmetic polynomials.Matrix relations for computation of the coefficients of the linear polynomials are derived.The linearity conditions of different orders are presented which need to be satisfied in order to obtain simplification of the final algorithmic structure.It is shown that the proposed approach is more efficient than the realization of Boolean functions by threshold devices[49].

Links between zero polarity Reed-Muller and arithmetic trans-form have been investigated in[3],[13],[41],[42],[44].Since there are P n different polarity Reed-Muller transforms of a Boolean function of n variables[12],the same concept may be applied to arithmetic transform.Arithmetic representation of logic functions with complements of variables was considered first in[3],[30].The basic properties of multipolarity generalized arithmetic transform were discussed in[20],[50],[51].By using generalized arithmetic transform in optimal polarity for a given Boolean function,the number of nonzero terms in arithmetic polynomial can be minimized,which results in faster calculation of the value of such a polynomial.A generalized adding transform is also useful when the incompletely specified Boolean functions are to be represented in the form of generalized Reed-Muller expansions.When such a case occurs,it is impossible to retrieve the information about the don't care minterms from the generalized Reed-Muller transform. On the other hand,using additional information about the incompletely specified Boolean functions in the form of general-ized adding transforms allows us to have all the information about the original Boolean function and the don't care minterms can be retrieved back when needed.The latter case can occur in the decomposition of systems of Boolean functions.In[50],the method of generation of forward and inverse transformation kernels for generalized arithmetic and adding transforms is presented. Different methods of generation of transformation matrices or spectra in arbitrary polarities from a known transformation matrix or spectrum in some polarity have been developed.Due to properties of dyadic convolution of the generalized arithmetic transform,greater computational efficiency may be achieved by the usage of subnumber operation.Based on the representation of generalized transform matrices in the form of Kronecker matrices, a unified approach to the fast algorithms in terms of strand matrices has been developed[51].In[52],[53],formulae to represent generalized composite arithmetic spectra of Boolean functions for basic logic connectives are derived.Other important operations used in classification and optimization of standard and tributary logical networks have been analyzed in the generalized arithmetic spectral domain[53].These operations include spectral decomposition,input and output negations,permutations of input variables,substitution of an input variable by a logical operation with some input variables or by the output of the function,and the variable itself.Based on the developed formulae,a new method to design tributary networks through operations on generalized arithmetic spectra was presented[53].

Heidtman[27]proposed an approach to derive the signatures for stuck-at faults in irredundant combinational networks by using arithmetic coefficients of the arithmetic polynomial expansion.A

suitable set of such arithmetic coefficients determines fault cover-age.Such spectral tests are not limited in the number of feasible inputs,as in the case of verification using Walsh spectral coefficients.They also detect numerous faults that are not covered by the presumed fault models,for example,bridging faults. Rahardja and Falkowski[54],[55],[56]have shown that the extensions of arithmetic polynomial logic to a mixed form where basis functions in the inverse mixed arithmetic transform matrices are arbitrary Boolean functions,provided that all such bases are linearly independent,is advantageous over the standard zero polarity Arithmetic transform used by Heidtman in testing.In particular,it was shown in[56]that the number of spectral tests is significantly reduced for both stuck-at and bridging faults for the majority of the classes of digital circuits.

Arithmetic polynomial forms for multiple-valued functions were considered in[3],[57]and their synthesis by the substitution method was considered in[58].The theories of logic algebra for multiple-valued functions in terms of the discrete Fourier trans-form were developed in the book[59].This book transfers almost all the results presented for binary Arithmetic polynomials in[44] into the multiple-valued domain.It allows a large number of analytical expressions to be derived that describe logic algebra of multiple-valued functions.The synthesized expressions take into account diverse required modes of operations such as arithmetic, Galois Fields,or modular.Hence,arithmetic polynomial expan-sions for multiple-valued functions are considered together with expansions in different algebraic structures known in the English literature under the name of multiple-valued Reed-Muller trans-forms.Similar to the work on Boolean functions,the authors use the names ofaarithmetical polynomialo[59],[60]andalogic polynomialo[59],[61]transforms and expansions when talking about arithmetic and Reed-Muller transforms and expansions for multiple-valued functions,respectively.In[62],a new generalized hybrid expansion in standard arithmetic with basis functions in GF(4)for quaternary logic is introduced.In contrast to arithmetic and logic polynomial forms[59],[60],[61]that used either only standard arithmetic or modulo algebra,the new expansion is based on both standard and GF(4)algebra.It is the first time than any hybrid expansion has been introduced.The relationships between the new expansion calledageneralized hybrid arithmetic expan-sionoand the known arithmetic expansions are explored in[62]. The concept of polarity applies to the new expansion and it generalizes the transformation.The way of obtaining the coeffi-cient vector of the expansion from some coefficient vector in known polarity is introduced.Generalized hybrid arithmetic expansions for known basis functions from quaternary Reed-Muller transforms are also considered.Finally,it should be noticed that,in the first paper in Russian on transforms for multiple-valued functions[63],the nameathe Fourier-Galois transformois used for the Reed-Muller transform for finite,but periodic binary and multiple-valued sequences.Under the same name,it is studied in detail in[64].In[65],a new algorithm for finding an Arithmetic representation for n-variable multiple-valued functions of m values(when m is a prime)was shown.The resulting arithmetic polynomial was of degree no greater than n? màI .The problem of realizing a cortege of functions in multiple-valued logic by evaluating an arithmetic expression in a manner similar to that presented for binary functions in[39]was also considered. Different ways of parallelization of the proposed algorithms and its implementation on an array processors were also discussed in [65].In[66],[67],another algorithm for representing a system of m-valued functions by linear arithmetic polynomials was shown.The approach used in this work was to represent the system of multiple-valued functions through the decomposition and a discrete orthogonal transformation by a system of linear arithmetic polynomials.Also,by using this representation,it is possible to restore completely the original system of multiple-valued func-tions.The same and related work presented the possibility of implementing considered algorithms on linear systolic arrays[67], [68].

Many problems of logic design can be reformulated in terms of solving logic equations and logic differential equations[2],[3],[12], [32].Discrete derivatives(differentials)and Taylor series of logic functions are studied in details in[12].The representation of the Reed-Muller expansion in the form of a Taylor series using discrete derivatives is given there.Implementation of Boolean differential calculus on systolic hardware was discussed in[44],[45]for the binary case and in[32],[59],[68],[69],[70]for the multiple-valued case,respectively.Tosic in[3],[30]introduced so-calledaarith-metic differenceoof a Boolean function with respect to its single variable.In contrast to this definition,an arithmetic analogue of discrete derivative for Taylor Arithmetic series for systems of Boolean functions was investigated in[32],[68],[69],[70].It follows from the definition in[32],[68],[69],[70]that it is a direct counterpart of Boolean discrete derivative.Also,this arithmetic equivalent of the Boolean derivative reflects not only the fact of the change in the function(what applies for Boolean derivatives)but also indicates the direction of the change.Its value is equal to0if the Boolean function has not changed with the change of the value of the variable from x i to x.Its value is equal to1if the Boolean function changed from0to1,and its value is equal to-1if the Boolean function changed from1to0.A class of arithmetic analogues of Boolean differential operators were extended for multiple-valued functions in[32],[68],[69],[70].In[70],two methods were presented for the synthesis of arithmetic canonical forms for multiple-valued functions in the form of matrix trans-forms.The main reason to use the matrix form was to map the presented algorithms into parallel architectures,in particular, linear systolic arrays.Arithmetic derivatives for multiple-valued functions were also considered as tools for fault detection in multiple-valued circuits[32].

4C ONCLUSION

This note describes in brief many developments in the theory of arithmetic transform and its extensions that were done mainly in Eastern Europe.Hence,the information presented here is a counterpart of the recent report presenting developments in the same area based on the work available in English and German only [1].It is believed that investigations of origins of some basic concepts and unification of the terminology will contribute to the further work in this area.As[1]shows,the following and other developments discussed in this note are unknown to Western researchers:arithmetic representation of systems of Boolean functions and conditions on existence of such a representation,aabsolute valueobased arithmetic expansions,multiple-valued arithmetic forms,the theory of arithmetic derivatives for Boolean and multiple-valued functions,as well as special systolic archi-tectures for their computation.With an ever increasing flow of technical information,it is difficult to keep up with new developments in one's own professional specialty and nearly impossible to survey totally such a research work,particularly if it has appeared in lesser known publications and originally in Russian.But,without guidance from past experience and current events,it is easy to expend resources on development paths already established as useful or abandoned elsewhere.The above review note is intended primarily to achieve greater visibility for the research work in the area of arithmetic transform published in Russian.

As always,this author apologizes to some Eastern European researchers who may have worked in this area and whose work is not mentioned in this review.If it has happened,it was caused

only by the limited access to the literature.Finally,it is hoped that this article,together with its earlier counterpart[1],will help in developing further the exciting theory of arithmetic transforms and expansions.

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Variables in Spectral Domain of Arithmetic Transform,oProc.IEEE Int'l Symp.Circuits and Systems,vol.4,pp.400-403,Atlanta,Ga.,May1996. [53] C.H.Chang and B.J.Falkowski,aLogical Manipulations and Design of

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on the contrary的解析

On the contrary Onthecontrary, I have not yet begun. 正好相反，我还没有开始。 https://www.wendangku.net/doc/f616378933.html, Onthecontrary, the instructions have been damaged. 反之，则说明已经损坏。 https://www.wendangku.net/doc/f616378933.html, Onthecontrary, I understand all too well. 恰恰相反，我很清楚 https://www.wendangku.net/doc/f616378933.html, Onthecontrary, I think this is good. ⑴我反而觉得这是好事。 https://www.wendangku.net/doc/f616378933.html, Onthecontrary, I have tons of things to do 正相反，我有一大堆事要做 Provided by jukuu Is likely onthecontrary I in works for you 反倒像是我在为你们工作 https://www.wendangku.net/doc/f616378933.html, Onthecontrary, or to buy the first good. 反之还是先买的好。 https://www.wendangku.net/doc/f616378933.html, Onthecontrary, it is typically american. 相反，这正是典型的美国风格。 222.35.143.196 Onthecontrary, very exciting.

恰恰相反，非常刺激。 https://www.wendangku.net/doc/f616378933.html, But onthecontrary, lazy. 却恰恰相反，懒洋洋的。 https://www.wendangku.net/doc/f616378933.html, Onthecontrary, I hate it! 恰恰相反，我不喜欢！ https://www.wendangku.net/doc/f616378933.html, Onthecontrary, the club gathers every month. 相反，俱乐部每个月都聚会。 https://www.wendangku.net/doc/f616378933.html, Onthecontrary, I'm going to work harder. 我反而将更努力工作。 https://www.wendangku.net/doc/f616378933.html, Onthecontrary, his demeanor is easy and nonchalant. 相反，他的举止轻松而无动于衷。 https://www.wendangku.net/doc/f616378933.html, Too much nutrition onthecontrary can not be absorbed through skin. 太过营养了反而皮肤吸收不了. https://www.wendangku.net/doc/f616378933.html, Onthecontrary, I would wish for it no other way. 正相反，我正希望这样 Provided by jukuu Onthecontrary most likely pathological. 反之很有可能是病理性的。 https://www.wendangku.net/doc/f616378933.html, Onthecontrary, it will appear clumsy. 反之，就会显得粗笨。 https://www.wendangku.net/doc/f616378933.html,

英语造句

一般过去式 时间状语：yesterday just now (刚刚) the day before three days ag0 a week ago in 1880 last month last year 1. I was in the classroom yesterday. I was not in the classroom yesterday. Were you in the classroom yesterday. 2. They went to see the film the day before. Did they go to see the film the day before. They did go to see the film the day before. 3. The man beat his wife yesterday. The man didn’t beat his wife yesterday. 4. I was a high student three years ago. 5. She became a teacher in 2009. 6. They began to study english a week ago 7. My mother brought a book from Canada last year. 8.My parents build a house to me four years ago . 9.He was husband ago. She was a cooker last mouth. My father was in the Xinjiang half a year ago. 10.My grandfather was a famer six years ago. 11.He burned in 1991

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●I wonder if it’s because I have been at school for so long that I’ve grown so crazy about going home. ●It is because she wasn’t well that she fell far behind her classmates this semester. ●I can well remember that there was a time when I took it for granted that friends should do everything for me. ●In order to make a difference to society, they spent almost all of their spare time in raising money for the charity. ●It’s no pleasure eating at school any longer because the food is not so tasty as that at home. ●He happened to be hit by a new idea when he was walking along the riverbank. ●I wonder if I can cope with stressful situations in life independently. ●It is because I take things for granted that I make so many mistakes. ●The treasure is so rare that a growing number of people are looking for it. ●He picks on the weak mn in order that we may pay attention to him. ●It’s no pleasure being disturbed whena I settle down to my work. ●I can well remember that when I was a child, I always made mistakes on purpose for fun. ●It’s no pleasure accompany her hanging out on the street on such a rainy day. ●I can well remember that there was a time when I threw my whole self into study in order to live up to my parents’ expectation and enter my dream university. ●I can well remember that she stuck with me all the time and helped me regain my confidence during my tough time five years ago. ●It is because he makes it a priority to study that he always gets good grades. ●I wonder if we should abandon this idea because there is no point in doing so. ●I wonder if it was because I ate ice-cream that I had an upset student this morning. ●It is because she refused to die that she became incredibly successful. ●She is so considerate that many of us turn to her for comfort. ●I can well remember that once I underestimated the power of words and hurt my friend. ●He works extremely hard in order to live up to his expectations. ●I happened to see a butterfly settle on the beautiful flower. ●It’s no pleasure making fun of others. ●It was the first time in the new semester that I had burned the midnight oil to study. ●It’s no pleasure taking everything into account when you long to have the relaxing life. ●I wonder if it was because he abandoned himself to despair that he was killed in a car accident when he was driving. ●Jack is always picking on younger children in order to show off his power. ●It is because he always burns the midnight oil that he oversleeps sometimes. ●I happened to find some pictures to do with my grandfather when I was going through the drawer. ●It was because I didn’t dare look at the failure face to face that I failed again. ●I tell my friend that failure is not scary in order that she can rebound from failure. ●I throw my whole self to study in order to pass the final exam. ●It was the first time that I had made a speech in public and enjoyed the thunder of applause. ●Alice happened to be on the street when a UFO landed right in front of her. ●It was the first time that I had kept myself open and talked sincerely with my parents. ●It was a beautiful sunny day. The weather was so comfortable that I settled myself into the

英语句子结构和造句

高中英语~词性~句子成分~语法构成 第一章节：英语句子中的词性 1.名词：n. 名词是指事物的名称，在句子中主要作主语.宾语.表语.同位语。 2.形容词;adj. 形容词是指对名词进行修饰~限定~描述~的成份，主要作定语.表语.。形容词在汉语中是（的）.其标志是: ous. Al .ful .ive。. 3.动词：vt. 动词是指主语发出的一个动作，一般用来作谓语。 4.副词：adv. 副词是指表示动作发生的地点. 时间. 条件. 方式. 原因. 目的. 结果.伴随让步. 一般用来修饰动词. 形容词。副词在汉语中是（地）.其标志是：ly。 5.代词：pron. 代词是指用来代替名词的词，名词所能担任的作用，代词也同样.代词主要用来作主语. 宾语. 表语. 同位语。 6.介词：prep.介词是指表示动词和名次关系的词，例如：in on at of about with for to。其特征：

介词后的动词要用—ing形式。介词加代词时，代词要用宾格。例如：give up her（him）这种形式是正确的，而give up she（he）这种形式是错误的。 7.冠词：冠词是指修饰名词，表名词泛指或特指。冠词有a an the 。 8.叹词：叹词表示一种语气。例如：OH. Ya 等 9.连词：连词是指连接两个并列的成分，这两个并列的成分可以是两个词也可以是两个句子。例如：and but or so 。 10.数词：数词是指表示数量关系词，一般分为基数词和序数词 第二章节：英语句子成分 主语：动作的发出者，一般放在动词前或句首。由名词. 代词. 数词. 不定时. 动名词. 或从句充当。 谓语：指主语发出来的动作，只能由动词充当，一般紧跟在主语后面。 宾语：指动作的承受着，一般由代词. 名词. 数词. 不定时. 动名词. 或从句充当. 介词后面的成分也叫介词宾语。 定语：只对名词起限定修饰的成分，一般由形容

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M A: Has the case been closed yet? B: No, the magistrate still needs to decide the outcome. magistrate n.地方行政官，地方法官，治安官 A: I am unable to read the small print in the book. B: It seems you need to magnify it. magnify vt.１．放大，扩大；２．夸大，夸张 A: That was a terrible storm. B: Indeed, but it is too early to determine the magnitude of the damage. magnitude n.１．重要性，重大；２．巨大，广大 A: A young fair maiden like you shouldn’t be single. B: That is because I am a young fair independent maiden. maiden n.少女，年轻姑娘，未婚女子 a.首次的，初次的 A: You look majestic sitting on that high chair. B: Yes, I am pretending to be the king! majestic a.雄伟的，壮丽的，庄严的，高贵的 A: Please cook me dinner now. B: Yes, your majesty, I’m at your service. majesty n.１．［Ｍ－］陛下（对帝王，王后的尊称）；２．雄伟，壮丽，庄严 A: Doctor, I traveled to Africa and I think I caught malaria. B: Did you take any medicine as a precaution? malaria n.疟疾 A: I hate you! B: Why are you so full of malice? malice n.恶意，怨恨 A: I’m afraid that the test results have come back and your lump is malignant. B: That means it’s serious, doesn’t it, doctor? malignant a.１．恶性的，致命的；２．恶意的，恶毒的 A: I’m going shopping in the mall this afternoon, want to join me? B: No, thanks, I have plans already. mall n.（由许多商店组成的）购物中心 A: That child looks very unhealthy. B: Yes, he does not have enough to eat. He is suffering from malnutrition.

base on的例句

意见应以事实为根据. 3 来自辞典例句 192. The bombers swooped ( down ) onthe air base. 轰炸机 突袭 空军基地. 来自辞典例句 193. He mounted their engines on a rubber base. 他把他们的发动机装在一个橡胶垫座上. 14 来自辞典例句 194. The column stands on a narrow base. 柱子竖立在狭窄的地基上. 14 来自辞典例句 195. When one stretched it, it looked like grey flakes on the carvas base. 你要是把它摊直, 看上去就象好一些灰色的粉片落在帆布底子上. 18 来自辞典例句 196. Economic growth and human well - being depend on the natural resource base that supports all living systems. 经济增长和人类的福利依赖于支持所有生命系统的自然资源. 12 1 来自辞典例句 197. The base was just a smudge onthe untouched hundred - mile coast of Manila Bay. 那基地只是马尼拉湾一百英里长安然无恙的海岸线上一个硝烟滚滚的污点. 6 来自辞典例句 198. You can't base an operation on the presumption that miracles are going to happen. 你不能把行动计划建筑在可能出现奇迹的假想基础上.

英语造句大全

英语造句大全English sentence 在句子中，更好的记忆单词！ 1、(1)、able adj. 能 句子：We are able to live under the sea in the future. (2)、ability n. 能力 句子：Most school care for children of different abilities. (3)、enable v. 使。。。能句子：This pass enables me to travel half-price on trains. 2、(1)、accurate adj. 精确的句子：We must have the accurate calculation. (2)、accurately adv. 精确地 句子：His calculation is accurately. 3、(1)、act v. 扮演 句子：He act the interesting character. （2）、actor n. 演员 句子：He was a famous actor. (3)、actress n. 女演员 句子：She was a famous actress. (4)、active adj. 积极的 句子：He is an active boy. 4、add v. 加 句子：He adds a little sugar in the milk. 5、advantage n. 优势 句子：His advantage is fight. 6、age 年龄n. 句子：His age is 15. 7、amusing 娱人的adj. 句子：This story is amusing. 8、angry 生气的adj. 句子：He is angry. 9、America 美国n.

(完整版)主谓造句

主语+谓语 1. 理解主谓结构 1) The students arrived. The students arrived at the park. 2) They are listening. They are listening to the music. 3) The disaster happened. 2．体会状语的位置 1) Tom always works hard. 2) Sometimes I go to the park at weekends.. 3) The girl cries very often. 4) We seldom come here. The disaster happened to the poor family. 3. 多个状语的排列次序 1) He works. 2) He works hard. 3) He always works hard. 4) He always works hard in the company. 5) He always works hard in the company recently. 6) He always works hard in the company recently because he wants to get promoted. 4. 写作常用不及物动词 1. ache My head aches. I’m aching all over. 2. agree agree with sb. about sth. agree to do sth. 3. apologize to sb. for sth. 4. appear (at the meeting, on the screen) 5. arrive at / in 6. belong to 7. chat with sb. about sth. 8. come (to …) 9. cry 10. dance 11. depend on /upon 12. die 13. fall 14. go to … 15. graduate from 16. … happen 17. laugh 18. listen to... 19. live 20. rise 21. sit 22. smile 23. swim 24. stay (at home / in a hotel) 25. work 26. wait for 汉译英： 1.昨天我去了电影院。 2.我能用英语跟外国人自由交谈。 3.晚上7点我们到达了机场。 4.暑假就要到了。 5.现在很多老人独自居住。 6.老师同意了。 7.刚才发生了一场车祸。 8.课上我们应该认真听讲。9. 我们的态度很重要。 10. 能否成功取决于你的态度。 11. 能取得多大进步取决于你付出多少努力。 12. 这个木桶能盛多少水取决于最短的一块板子的长度。

初中英语造句

【it's time to和it's time for】 ——————这其实是一个句型,只不过后面要跟不同的东西. ——————It's time to跟的是不定式（to do）.也就是说,要跟一个动词,意思是“到做某事的时候了”.如： It's time to go home. It's time to tell him the truth. ——————It's time for 跟的是名词.也就是说,不能跟动词.如： It's time for lunch.(没必要说It's time to have lunch) It's time for class.(没必要说It's time to begin the class.) They can't wait to see you Please ask liming to study tonight. Please ask liming not to play computer games tonight. Don’t make/let me to smoke I can hear/see you dance at the stage You had better go to bed early. You had better not watch tv It’s better to go to bed early It’s best to run in the morning I am enjoy running with music. With 表伴随听音乐 I already finish studying You should keep working. You should keep on studying English Keep calm and carry on 保持冷静继续前行二战开始前英国皇家政府制造的海报名字 I have to go on studying I feel like I am flying I have to stop playing computer games and stop to go home now I forget/remember to finish my homework. I forget/remember cleaning the classroom We keep/percent/stop him from eating more chips I prefer orange to apple I prefer to walk rather than run I used to sing when I was young What’s wrong with you There have nothing to do with you I am so busy studying You are too young to na?ve I am so tired that I have to go to bed early

The Kite Runner-美句摘抄及造句

《The Kite Runner》追风筝的人--------------------------------美句摘抄 1.I can still see Hassan up on that tree, sunlight flickering through the leaves on his almost perfectly round face, a face like a Chinese doll chiseled from hardwood: his flat, broad nose and slanting, narrow eyes like bamboo leaves, eyes that looked, depending on the light, gold, green even sapphire 翻译：我依然能记得哈桑坐在树上的样子，阳光穿过叶子，照着他那浑圆的脸庞。他的脸很像木头刻成的中国娃娃，鼻子大而扁平，双眼眯斜如同竹叶，在不同光线下会显现出金色、绿色，甚至是宝石蓝。 E.g.: A shadow of disquiet flickering over his face. 2.Never told that the mirror, like shooting walnuts at the neighbor's dog, was always my idea. 翻译：从来不提镜子、用胡桃射狗其实都是我的鬼主意。E.g.:His secret died with him, for he never told anyone. 3.We would sit across from each other on a pair of high

翻译加造句

一、翻译 1. The idea of consciously seeking out a special title was new to me., but not without appeal. 让我自己挑选自己最喜欢的书籍这个有意思的想法真的对我具有吸引力。 2.I was plunged into the aching tragedy of the Holocaust, the extraordinary clash of good, represented by the one decent man, and evil. 我陷入到大屠杀悲剧的痛苦之中，一个体面的人所代表的善与恶的猛烈冲击之中。 3.I was astonished by the the great power a novel could contain. I lacked the vocabulary to translate my feelings into words. 我被这部小说所包含的巨大能量感到震惊。我无法用语言来表达我的感情（心情）。 4，make sth. long to short长话短说 5.I learned that summer that reading was not the innocent(简单的) pastime(消遣) I have assumed it to be., not a breezy, instantly forgettable escape in the hammock(吊床)，( though I’ ve enjoyed many of those too ). I discovered that a book, if it arrives at the right moment, in the proper season, will change the course of all that follows. 那年夏天，我懂得了读书不是我认为的简单的娱乐消遣，也不只是躺在吊床上，一阵风吹过就忘记的消遣。我发现如果在适宜的时间、合适的季节读一本书的话，他将能改变一个人以后的人生道路。 二、词组造句 1. on purpose 特意，故意 This is especially true here, and it was ~. (这一点在这里尤其准确，并且他是故意的) 2.think up 虚构，编造，想出 She has thought up a good idea. 她想出了一个好的主意。 His story was thought up. 他的故事是编出来的。 3. in the meantime 与此同时 助记：in advance 事前in the meantime 与此同时in place 适当地... In the meantime, what can you do? 在这期间您能做什么呢？ In the meantime, we may not know how it works, but we know that it works. 在此期间，我们不知道它是如何工作的，但我们知道，它的确在发挥作用。 4.as though 好像，仿佛 It sounds as though you enjoyed Great wall. 这听起来好像你喜欢长城。 5. plunge into 使陷入 He plunged the room into darkness by switching off the light. 他把灯一关，房

改写句子练习2标准答案

The effective sentences:(improve the sentences!) 1.She hopes to spend this holiday either in Shanghai or in Suzhou. 2.Showing/to show sincerity and to keep/keeping promises are the basic requirements of a real friend. 3.I want to know the space of this house and when it was built. I want to know how big this house is and when it was built. I want to know the space of this house and the building time of the house. 4.In the past ten years,Mr.Smith has been a waiter,a tour guide,and taught English. In the past ten years,Mr.Smith has been a waiter,a tour guide,and an English teacher. 5.They are sweeping the floor wearing masks. They are sweeping the floor by wearing masks. wearing masks,They are sweeping the floor. 6.the drivers are told to drive carefully on the radio. the drivers are told on the radio to drive carefully 7.I almost spent two hours on this exercises. I spent almost two hours on this exercises. 8.Checking carefully,a serious mistake was found in the design. Checking carefully,I found a serious mistake in the design.

用以下短语造句

M1 U1 一. 把下列短语填入每个句子的空白处（注意所填短语的形式变化）： add up (to) be concerned about go through set down a series of on purpose in order to according to get along with fall in love (with) join in have got to hide away face to face 1 We’ve chatted online for some time but we have never met ___________. 2 It is nearly 11 o’clock yet he is not back. His mother ____________ him. 3 The Lius ___________ hard times before liberation. 4 ____________ get a good mark I worked very hard before the exam. 5 I think the window was broken ___________ by someone. 6 You should ___________ the language points on the blackboard. They are useful. 7 They met at Tom’s party and later on ____________ with each other. 8 You can find ____________ English reading materials in the school library. 9 I am easy to be with and _____________my classmates pretty well. 10 They __________ in a small village so that they might not be found. 11 Which of the following statements is not right ____________ the above passage? 12 It’s getting dark. I ___________ be off now. 13 More than 1,000 workers ___________ the general strike last week. 14 All her earnings _____________ about 3,000 yuan per month. 二.用以下短语造句： 1.go through 2. no longer/ not… any longer 3. on purpose 4. calm… down 5. happen to 6. set down 7. wonder if 三. 翻译： 1.曾经有段时间，我对学习丧失了兴趣。（there was a time when…） 2. 这是我第一次和她交流。（It is/was the first time that …注意时态） 3.他昨天公园里遇到的是他的一个老朋友。（强调句） 4. 他是在知道真相之后才意识到错怪女儿了。（强调句） M 1 U 2 一. 把下列短语填入每个句子的空白处（注意所填短语的形式变化）： play a …role (in) because of come up such as even if play a …part (in) 1 Dujiangyan(都江堰) is still ___________in irrigation(灌溉) today. 2 That question ___________ at yesterday’s meeting. 3 Karl Marx could speak a few foreign languages, _________Russian and English. 4 You must ask for leave first __________ you have something very important. 5 The media _________ major ________ in influencing people’s opinion s. 6 _________ years of hard work she looked like a woman in her fifties. 二.用以下短语造句： 1.make (good/full) use of 2. play a(n) important role in 3. even if 4. believe it or not 5. such as 6. because of

英语造句

English sentence 1、(1)、able adj. 能 句子：We are able to live under the sea in the future. (2)、ability n. 能力 句子：Most school care for children of different abilities. (3)、enable v. 使。。。能 句子：This pass enables me to travel half-price on trains. 2、(1)、accurate adj. 精确的 句子：We must have the accurate calculation. (2)、accurately adv. 精确地 句子：His calculation is accurately. 3、(1)、act v. 扮演 句子：He act the interesting character.（2）、actor n. 演员 句子：He was a famous actor. (3)、actress n. 女演员 句子：She was a famous actress. (4)、active adj. 积极的 句子：He is an active boy. 4、add v. 加 句子：He adds a little sugar in the milk. 5、advantage n. 优势 句子：His advantage is fight. 6、age 年龄n. 句子：His age is 15. 7、amusing 娱人的adj. 句子：This story is amusing. 8、angry 生气的adj. 句子：He is angry. 9、America 美国n. 句子：He is in America. 10、appear 出现v. He appears in this place. 11. artist 艺术家n. He is an artist. 12. attract 吸引 He attracts the dog. 13. Australia 澳大利亚 He is in Australia. 14.base 基地 She is in the base now. 15.basket 篮子 His basket is nice. 16.beautiful 美丽的 She is very beautiful. 17.begin 开始 He begins writing. 18.black 黑色的 He is black. 19.bright 明亮的 His eyes are bright. 20.good 好的 He is good at basketball. 21.British 英国人 He is British. 22.building 建造物 The building is highest in this city 23.busy 忙的 He is busy now. 24.calculate 计算 He calculates this test well. 25.Canada 加拿大 He borns in Canada. 26.care 照顾 He cared she yesterday. 27.certain 无疑的 They are certain to succeed. 28.change 改变 He changes the system. 29.chemical 化学药品