2-Sat Sub-Clauses and the Hypernodal Structure of the 3-Sat Problem
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6 T extual equivalence: cohesionCohesion:衔接A network of lexical, grammatical, and other relations which provide links between various parts of a text, and which organize and create a text; a surface relation which connects together the actual words and expressions that we can see or hear.(p180)[The connection between successive sentences in texts, conversations, etc., in so far as it can be described in terms of specific syntactic units.]Cohesive devicesHalliday and Hasan:five main cohesive devices in English:reference,substitution,ellipsis,conjunction,lexical cohesion.These devices are probably common to a large number of languages. However, different languages have different preferences for using specific devices more frequently than others or in specific combinations which may not correspond to English patterns of cohesion. For instance, pronominalization is very frequent in English but is rarely used in Japanese and Chinese. Lexical repetition is far more frequent in Hebrew than it is in English. p211Cohesion is also achieved by a variety of devices other than those mentioned by Halliday and Hasan and discussed above. These include, for instance, continuity of tense, consistency of style, and punctuation devices such as colons and semi-colons, which, like conjunctions, indicate how different parts of the text relate to each other. It is worth noting here that unmotivated shifts in style, a common pitfall in translation, can seriously disrupt the cohesion and coherence of a text. p2116.1 ReferenceDefinition: the relationship of identity which holds between two linguistic expressions. P181E.g.Mrs Thatcher has resigned. She announced her decision this morning.Mrs. Thatcher has resigned. This delighted her opponents.p181Coreferencea matter of realworld knowledge--Hoey (1988: 162)e.g.There‟s a boy climbing that tree.a. The boy‟s going to fall if he doesn‟t take care. (repetition)b. The lad‟s going to fall if he doesn‟t take care. (synonym)c. The child‟s going to fall if he doesn‟t take care. (superordinate)d. The idiot‟s going to fall if he doesn‟t take care. (general word)e. He‟s going to fall if he doesn‟t take care. (pronominal reference) p182-183Reference—differences between English and ChineseChinese: repetition, zero reference; fewer or zero pronouns;English: pronominal reference; obligatory pronounse.g.“But if a woman is partial to a man, and does not endeavor to conceal it, he must find it out.”“Perhaps he must, if he sees enough of her.”(Pride and Prejudice)?“如果一个女人爱上了一个男人,而又不故意瞒着它的话,他一定会发现它的。
语言学教程(第四版)练习-第4章Chapter Four From Word to TextI. Mark the choice that best completes the statement.1.Which of the following term does NOT mean the same as the relation of substitutability ?A. Associative relationB. Paradigmatic relationC. Vertical relationD. Horizontal relation2. Clauses can be used as subordinate constituents and the three basic types of subordinate clauses are complement clauses, adjuncts clauses and _______.A. relative clausesB. adverbial clausesC. coordinate clausesD. subordinate clauses3. Names of the syntactic functions are expressed in all the following terms EXCEPT ______.A. subjects and objectsB. objects and predicatorsC. modifiers and complementsD. endocentric and exocentric4. In English, case is a special form of the noun which frequently corresponds toa combination of perception and noun and it is realized in all the following channels EXCEPT _______.A. inflectionB. following a prepositionC. word orderD. vertical relation5. In English, theme and rheme are often expressed by _____ and ____.A. subject; objectB. subject; predicateC. predicate; objectD. object; predicate6. Phrase structure rules have _____ properties.A. recursiveB. grammaticalC. socialD. functional7. Which of the following is NOT among the three basic ways to classify languages in the world ?A. Word orderB. Genetic classificationC. Areal classificationD. Social classification8. The head of the phrase the city Rome is ______.A. the cityB. RomeC. cityD. the city Rome9. The phrase on the shelf belongs to ______ construction.A. endocentricB. exocentricC. subordinateD. coordinate10. The sentence They were wanted to remain quiet and not to expose themselves isa _____ sentence.A. simpleB. coordinateC. compoundD. complexII. Mark the following statements with “T” if they are true or “F” if they are false.1.The relation of co-occurrence partly belong to syntagmatic relations, partlyto paradigmatic relations.2.One property coordination reveals is that there is a limit on the number ofcoordinated categories that can appear prior to the conjunction.3.According to Standard Theory of Chomsky, deep structure contain all theinformation necessary for the semantic interpretation of sentences.4.In English, the object is recognized by tracing its relation to word order andby inflections of pronouns.5.Classes and functions determine each other, but not in any one-to-onerelation.ually noun phrases, verb phrases and adverbial phrases belong toendocentric types of constriction.7.In English the subject usually precedes the verb and the direct object usuallyfollows the verb.8.In the exocentric construction John kicked the ball, neither constituent standsfor the verb-object sequence.9. A noun phrase must contain a noun, but other elements are optional.10.In a coordinate sentence, two (or more) S constituents occur as daughters andco-heads of a higher S.III. Fill in each of the following blanks with an appropriate word. The first letter of the word is already given.1.The subordinate constituents are words which modify the Heads andconsequently, they can be called m____________.2.John believes (that the airplane was invented by an Irishman). The part in thebracket is a c_________ clause.3.In order to account for the case of the subject in passive voice, we haveanother two terms, p____________ and n__________.4.There is a tendency to make a distinction between phrase and w_______,which is an extension of word of a particular class by way of modification with its main features of the class unchanged.5.Recursiveness, together with o_______, is generally regarded as the core ofcreativity of language.6.Traditionally, p_________ is seen as part of a structural hierarchy, positionedbetween clause and word.7.The case category is used in the analysis of word classes to identity thes______ relationship between words in a sentence.8.Clause can be classifies into FINITE and NON-FINITE clauses, the latterincluding the traditional infinitive phrase, p__________, and gerundial phrase.9.Gender displays such contrasts as masculine: feminine: n_______.10.English gender contrast can only be observed in g__________ and a smallnumber of l__________ and they are mainly of the natural gender type.IV. Explain the following concepts or theories.1.Syntax2.IC analysis3.Relation of co-occurrence4.Category5.RecursivenessVI. Answer the following question.1.What are endocentric construction and exocentric construction?2.What are the basic functional terms in syntax?VII. Essay question.1.Explain an comment on the following sentence a and b.a.John is easy to please.b.John is eager to please.ment on the statement, “Linguistic structure is hiearchical”I. Mark the following statements with “T” if they are true or “F” if they are false.1.The syntactic rules of any language are finite in number, but they are capableof yielding an infinite number of sentences.2.Although, a single word can also be uttered as a sentence, normally asentence consists of at least a subject, its predicate and an object.\3.The sentences are linearly structured, so they are composed of sequence ofwords arranged in a simple linear order.4. a.John his upon an idea.b.An idea hit upon John.In the above sentences, the subject and object constituent by the sentences switch their position. Although sentence b is absurd, it is still grammatical, because John and an idea are of the same phrasal category.5.Though they are of a small number, the combinational rules are powerfulenough to yield all the possible sentences and rule out the impossible ones. 6.In a sentence like Mary likes flowers, both Mary and flowers are not onlyNouns, but also Noun Phrases.7.The recursive property can basically be discussed in a category-basedgrammar, but not in a word-based grammar.8.An XP must contain an X which is called the phrasal head.9.In the phrase this very tall girl, tall girl is an obligatory element and the headof the phrase.10.a. The man beat the child. b. The child was beaten by the man.In the above sentences, the movement of the child from its original place to a new place is a WH- movement.11.Tense and aspect, the two important categories of the verb, nowadays areviewed as separate notions in grammar.12.The structuralists regard linguistic units as isolated bits in a structure (orsystem).13.IC analysis can help us to see the internal structure of a sentence clearly andit can also distinguish the ambiguity of a sentence.14.Structural linguists hold that a sentence does only have a linear structure, butit has a hierarchical structure, made up of layers of word groups.15.In Saussure’s view, the linguist cannot attempt to explain individual signs in apiecemeal fashion. Instead he must try to find the value of a sign from its relation to others, or rather, its position in the system.16.The theme-rheme order is the usual one in unemotional narration, which is asubjective order.17.What is new in Halliday is that he has tried to relate the functions oflanguage to its structure.18.Sentence is a basic unit of structure in functional grammar.19.The interpersonal function of language refers to the idea held by Hallidaythat language serves ot establish and maintain social relations.20.Finite is a function in the clause as a representation, both the representationof outer experience and inner experience.21.The relations of co-occurrence partly belong to syntagmatic relations, partlyto paradigmatic relations.22.According to Chomsky, grammar is a mechanism that should be able togenerate all and only the grammatical sentences of a language.23.In English, the subject of a sentence is said to be the doer of an action, whilethe object is the person or thing acted upon by the doer. Therefore, the subject is always an agent and the patient is always the object.24.In English, the object is recognized by tracing its relation to word order andby inflections of pronouns.25.Classes and functions determine each other, but not in any one-to-onerelation.26.The syntactic rules of a language are finite in number, and there are a limitednumber of sentences which can be produced.27.Structuralism views language as both linearly and hierarchically structured.28.Phrase structure rules provide explanations on how syntactic categories areformed and sentences generated.29.UG is a system of linguistic knowledge and a human species-specific giftwhich exists in the mind of a normal human being.30.Tense and aspect are two important categories of the verb, and they wereseparated in traditional grammar.II. Fill in each of the following blanks with (an) appropriate word(s).1.As is required by the ______, a noun phrase must have case and case isassigned by verb, or preposition to the _________ position or by auxiliary to the ________ position.2.Adjacency condition states that a case _________ and a case _______ shouldstay adjacent to each other.3.The general movement rule accounting for the syntactic behavior of anyconstituent movement is called __________.4.The phrase structure rules, with the insertion of the lexicon, generatesentences at the level of _________.5.The application of syntactic movement rules transforms a sentence from thelevel of ________ to that of ______.6.In English there are two major types of movement, one involving the_____.A.(1)B.(2)C. both (1) and (2)D. neitherC.Imitation accounts for language acquisition.D.Phonological information must form part of syntactic movement.8.The symbol N indicates a/an ________.A.lexical categoryB.phrasal categoryC.intermediate categoryD. lexical insertion rule9.Of the following combination possibilities, ______ can NOT be generated from the following rule: NP→(Det)(Adj)N(PP)(S).A. NP→→→→Det Adj N PPS.10.An advantage of X-bar syntax over phrase structure syntax is that X-bar.A.avoid a ploliferation of redundant intermediate categories.B.allows us to identify indefinitely long embedded sentences.C. allows as to postulate categories other than lexical and phrasal.D. forces us to conclude that the ambiguity of phrases like the English Kingis lexical rather than structural.11. Which set of rules generates the following tree structures?A. S→NP VPB. NP→VPNP→N PP NP→NP NP PPVP→V NP VP→V NP PPPP →P NP PP →P NPNP→N NP →NC.S VP VP D, S NP VPNP→(NP/PP) NP →NP (NP /PP)VP →V NP VP →V NPPP →P NP PP →P NPNP→N NP →N12.a.It seems they are quite fit for the job.b. They seem quite fit for the job.Sentence b is a result of ______ movement.A.NPB.WHC.AUX.D. None13. The head of the phrase underneath the open window is _______.A.underneathB.theC.openD.window14.The following statements are in accordance with Hallliday’s opinion on language EXCEPT _______.he use of language involves a network of systems of choices.B. Language is never used as a mere mirror of reflected thought.nguage is a system of abstract forms and signs.nguage functions as a piece of human behavior.15.Chomsky is more concerned with ____ relations in his approach to syntax.16.______ is a type of control over the form of some words by other words in Certain syntactic constructions and in terms of certain category.A.ConcordernmentC. BindingD. Co-command17. Clauses can be used as subordinate constituents and the three basic types of subordinate clauses are complement clauses, adjunct clauses and _____.A.relative clausesB.adverbial clausesC.coordinate clausesD.subordinate clausess of the syntactic functions are expressed in all the following terms EXCEPT_____.A.subjects and objectsB.objects and predicatorsC.modifiers and complementsD. endocentric and exocentric19.In English, case is a special form of the noun which frequently corresponds toa combination of preposition and noun and it is realized in all the following channels EXCEPT ______.A.inflectionB.following a prepositionC.word orderD.vertical relation20. Clauses can be classified into finite and non-finite clauses, _____ including the traditional infinitive phrases, participial phrase and gerundial phrase.A. the formerB. the latterC.bothD. neither21.It is the _______ on case assignment that states that a case assignor and a case recipient should stay adjacent to each other.A. Case ConditionB.Adjacent ConditionC.Parameter ConditionD.Adjacent Parameter.22.Predication analysis is a way to analyze _______ meaning.A. phonemeB. wordC. phrase…d. sentence23.Which of the following italic parts is NOT an idiom?A. How to you do?B. How did you do ?C. He went to it hammer and tongs.D. They kept tabs on the Russian spy.24.When we say that we can change the second word in the sentence she is singing in the room with another word or phrase, we are talking about ______.A. governmentB. linear relationsC. syntactic relationsD. paradigmatic relations25.IN the phrase structure rule S→NP VP, the arrow can be read as ______.A. hasB. generatesC. consists ofD. is equal toIV. Answer the following questions as comprehensively as possible, giving examples if necessary.1.The following two sentences are ambiguous. Show the two readings of eachby drawing its respective tree diagrams.(1)The ball man and woman left(2) Visiting professor can be interestinge an example to show what a tree diagram is (as it is used inTransformational-Generative Grammar).e an example to show what IC analysis is.4.What are the three general functions of language according to Halliday?5.What distinguishes the structural approach to syntax from the traditionalone?6.Some grammar books say there are three basic tenses in English-the present,the past and the future; others say there are only two basic tenses –the present and the past. Explain what tense is and whether it is justifiable to say there is a future tense in English.。
Division OneI.有可能出的填空,选择题及判断题的内容:.1.European Culture is composed of _____________ and_____________________ as two major elements.2.Greek culture reached its high point of development/climax in __________ (century).3.All Greece was brought under the rule of Alexander, Kingof Macedon in _________(century).4.In _______, the Romans conquered Greece.5.Homer wrote __________ and _____________, twofamous epics.6.______________ is called “Father of History” in GreekCulture, who wrote the wars between ____________ and________________________.7.The famous writer who wrote comedy in Greek culture isnamed ______________.8.“The great historian that ever lived” is said to praise_____________________(name).9.The Dialectal Method is put forward by___________.10.Academy was established by ______________________,whose philosophy is called ____________________.11.Lyceum was established by _______________.12.Leader of the Cynics is ____________________, and“cynic” means _____________.13.The most important thing in life to the Stoics is_________________.14.Elements, a textbook of geometry, was written by___________.15.The famous temples in ancient Greece are____________________ and ______________.16.The Romans enjoyed two hundred years of peaceful time,known as ________________.17.In the 4th century, the emperor Constantine moved thecapital from Rome to Byzantium, renamed it ____________________(modern _____________).18.After 395, the Roman Empire was divided into_____________ and _________.19.The East Roman Empire fell to the ___________ in 1453.20.“I came, I saw, I conquered” was written by_________________.21.Two famous representatives of Roman architecture are________________ and _________________________.22.________ said, “Captive Greece took her rude conquerorcaptive”.23.Greek Culture has exerted great, positive influence on thelater-on literature, such as Shelly’s ______________ and Byron’s ___________________.24.The languages that were spoken by Greeks and Romansbelong to ___________________(哪个语系).25.The Roman Law, to some extent, protected the rights of______________.26.With the fall of the Western Roman Empire, there came______________(什么时代) lasting one thousand years.27._____________’s (whose) “Prometheus Bound”is aparody adapted by _______________ (he wrote “Prometheus Unbound”).28.Freud put forward _________________ based on thework, “Oedipus the King ” written by _____________. 29.The _________ was one of the similarities shared byGreeks and Romans as far as politics is concerned.30.__________’s (whose) political speeches and discoursesare the model of Latin diction.31.The largest amphitheatre in ancient Rome is known as______________.32.________ established the atomic theory.33.________ believed fire to be the primary element of theuniverse.34.________ established the dialectical method.35.________ was the founder of scientific mathematics.36.The humanistic ideas can be traced back to ___(the nameof a philosopher).37.“The master of those who know” was written by Dante topraise ___________ (the name of a philosopher).38.Zeno的介绍II.名词解释democracy of ancient Greece Pax Romana Doric StyleIonic Style the Sophists the Cynics the Scepticsthe Epicureans the Stoics Academy LyceumIII.简答与论述题:1.三种问法:What is the limitation of “Democracy” in ancientGreece?How do you understan d “Democracy” in ancientGreece?What is the difference between “Democracy” in ancientGreece and modern democracy?2.How did the Greek Culture originate and develop?3.How did the ancient Greek philosophy develop?4.How did the ancient Greek science develop?5.What is the difference between Plato and Aristotle interms of their philosophical ideas?Or What is the difference between Plato and Aristotle interms of their philosophical system?6.What is the great significance of Greek Culture on thelater-on cultural development?Or What positive influence did the Greek Culture exerton the world civilization?7.What are difference and similarity between Greekculture and Roman culture?8.How did Roman culture originate and develop?9.How was Roman Empire disintegrated?10.What styles can Greek architecture be classified into?Or How does Greek architecture take different styles?11.Explain Pax Romana.12.What contribution did the Roman make to the rule oflaw?Division TwoI.special Terms Explanation:the Hebrews the Old Testament the New TestamentPentateuch Genesis Exodus Prophets the Book ofDaniel the Historical BookII.有可能出填空,选择题及判断题的内容:1.Judaism and Christianity are closely related because__________________________________.2.The ancestors of the Jews are ________________, whichmeans ___________ in English.3.The history of the Hebrew people is recorded in__________________ of the Bible.4.The Hebrew people came to settle down in Palestine, knownas _____________ at the time.5.The Old Testament is about_____________________, and“Testament” means _______________________________.6.The New Testament is about _______________________.7.The oldest and most important part of the Old Testament iscalled _______________ or ____________.8.It was ______________ who brought Hebrew people safelyback to Palestine.9.It was ______________who united Hebrews.10.It was ______________who had Jerusalem established aspolitical and religious capital.11.The former church in 6th century B.C. is called_______________.12.Pentateuch is also called __________________.13._______________ made Christianity legal, as he issued_____________________ in 313.14.In 392, Emperor _______________made Christianity officialreligion.15.By 300, each local church is called ______________with itsfulltime leader, known as _____________________.16.Several parishes were grounded together into a large unitcalled _______________, which were headed by a_____________.17.The Last Supper is recorded in __________________, andwas put into painting by __________________.18.The Cruxifixion is recorded in _____________________.19.The oldest extant Greek translation of the Old Testament isknown as _________________.20.The most extant Latin version of the whole Bible is_______________________ which became the official Bibleof the Roman Catholic Church throughout the world.21.The first English version of the whole Bible was translatedby _________________.22.The most important and influential of English Bible is_______________ or _______________________________.23._____________________ and _______________________are called two great reservoirs of Modern English.24.The development of system of landed nobles is recorded in_________________ of the Old Testament.25._________________ deals with the history of the Hebrewpeople from their entry into Palestine around 1200B.C. tillthe fall of Palestine into hands of Assyrians and Chaldeans in586 B.C.26.The New Testament contains _______ (how many) accountswhich tell about the beginning of Christianity.III.论述题:1.三种问法:How did Christianity come into being and develop inEurope?How did Christianity become the official religion ofRoman Empire?How did the relations between Christians and theRoman government change?2.What are the different translation editions of the Bible?3.Why do we say the Bible has shaped western culture more decisively than anything else ever written?4.What are the great influences that the English Bible has on the American and British literature?Division ThreeI.Special Terms Explanation:Age of Faith Feudalism Manor/castle Code of ChivalryMonasticism The Crusades Carolingian RenaissanceScholasticism National Epics Vernacular LanguageGothic Architectural Style the Canterbury TalesII.有可能出选择题,填空与判断题的内容:1.The Middle Ages is also called______________________, __________________ and ____________________________.(考试中,会任选其中一种说法作为选择题来考) 2.The Middle Ages is the period in which three cultures were merged: _____________________, _________________and_____________________________, respectively referring to_____________________________, ____________________and _____________________________.3.Western Roman Empire was overthrown in __________(year).4.__________________(who) gave his soldiers estates known as fiefs as a reward for their services in 732.5.____________________ (who) can be vassals.6.____________________ (when) was a noble crowned as a knight in the Middle Ages in western Europe.7.People in the western Europe in the Middle Ages were divided into three classes: _____________, _______________ and_________________________.8.Between ______________ century and ________________ century, western Europe was the scene of frequent wars. 9.Feudalism comes from the Latin word, ________________, which means ______________________ in Latin. 10.Knight trained for war by fighting each other in mock battles called ____________________.11.Roger Bacon was one of the earliest advocates of _________________, calling for _________________ and_________________.12._____________(which tribe) swept into Europe from central Asia in the latter part of fourth century, robbing and killing alarge number of the half civilized Germanic tribes. 13.Those who devoted their lands to large-land owners in return for protection, but still had their own freedom were called________________.14.Those who came form cities or towns and devoted their freedom to large-land owners in return for protection werecalled ________________.15.Roman Catholic Church was divided into five classes: _____________, ________________, ______________,_________________ and __________________.16.Opus Maius was written by _____________________. 17.In Roman Catholic Church, “Catholic” means ____________. 18.The monk, St. Benedict founded _______________________ in about 529 A.D.19.In the Middle Ages, the Church even set up their own court called _____________________________.20.In the National Epics, vernacular languages meant _____________________________ with representative works:____________________in early English and _____________in early French.21.In the Divine Comedy, Dante expressed ________________ ideas and foreshadowed the spirit of _________________. 22.Chaucer was regarded as the first _________________ and the first _____________________ with his famous work_____________________, as the representative of______________(英语的哪个发展时期)23. In 1054, Christianity was divided into _____________and___________________________.24.The Middle Ages is a transitional period between ___________ times and ____________.25. The ceremony to grant the title “knight” is called ___________.26. ________________ translated the Old and New Testamentsfrom Hebrew and Greek originals into Latin. His translationwork, _______________, became the official Latin Bible usedby Roman Catholic Church of this day.27. “The Confession” and “the City of the God” were written by___________________.28. Since _____________ (哪个历史事件),Arabic numerals andalgebra were introduced into Europe.29. The mock battles for knight training are known as __________.30. ___________ paved the way for Modern English.31. The language used by Chaucer was _________________.III.间答与论述题:1.Why is the Middle Ages called Age of Faith?Or Why do we say the Middle Ages is a religious age?2.How did feudalism derive and develop in the Middle Ages?Or How did feudalism establish its firm ruling position in Europe?3.What is the great power and influence of the Roman Catholic church in the Middle Ages?4.What positive influence does the Crusades exert on the European Culture?Or What is the great significance of the Crusades?5. How did learning and science develop in the Middle Ages?6.How did literature develop in the Middle Ages?7.What is the importance of using vernacular languages in theMedieval Time?8.What is the difference between the vernacular languagesused in the National Epics during the Middle Ages and thevernacular language used by Mark Twain?9.What was the merit shared by Charlemagne and Alfred theGreat?10.How did the English Language develop and what is therepresentation of each stage?Or What stages did the English Language undergo?11.Why do we say using of vernacular languages in the MiddleAges signifies a transition?Division FourI.Special Terms Explanation:Renaissance Renaissance Art ReformationCounter-Reformation Calvinism the JesuitsProtestantismII.有可能出填空,选择与判断题的内容:1.During Renaissance, humanist thinkers and scholarsintroduced new ideas that expressed the interests of__________________.2.Renaissance stared in ___________ and ______________with the flowering of painting, sculpture and architecture.3.At the heart of Renaissance philosophy was the assertion of_____________________which is also the core ofhumanism.4.Many of Petrarch’s ________________ are written for Laura.5.The most representative painter of humanism was________________________ with his famous painting work____________________ and _______________.6.The statesman, ____________________ of Italy in theRenaissance period has greatly influenced the politicalscience in the west, called “Father of Western PoliticalScience”.7.Under the reign of ______________________, Englishbegan to embark on the road to colonization and foreign control that was to take it onto its heyday of capitalist development.8.________________ is the birth place of Renaissance.9.Madonna was _______________ (什么性质的油画),paintedby _________________.10.Man with the Glove was painted by __________________.11.Calvinism insisted on constructing a type of governmentknown as ______________________________.12.After Reformation, in England a new form of church wasestablished known as _______________ or _____________.13.During Counter-Reformation, __________________(教派名称) was devoted to defending the Roman Catholic Church with its head, _____________________(首领名字).14.______________ stressed hard work and thrifty way of life.15.Montaigne was a French humanist known for his___________.16.The representative author of Renaissance in Spain was______________________ with his famous literary work, _____________________ with which the European novel entered a new stage17.The most representative author and humanist duringRenaissance in England was ________________________, and his four great tragedies are respectively ____________________,________________________,______________________and______________________ 18.Columbus was the discoverer of ____________________in1492.19.______________ discovered the Cape of Good Hope in1487.20.______________ discovered the route to India round theCape of Good Hope between 1497 and 1498.21.America was named in ______________(whose) honor.22.__________________________put forward revolutionaryideas in astronomy during the Middle Ages, and his theorywas __________________________ proved later by_______________________and _______________________in 17th century.23.As you like it and Twelfth Night are two comedies written by________________.24.The idea that only those specially elected by God are savedwas advocated by ___________(school).25.It was during the reign of _________(whom) that the EnglishBible was adopted by the Church of England.26.The idea that the Bible was the supreme authority and manwas only bound to the law of the world of the God was putforward by _____________(whom).27.During Renaissance, __________(whom) preached in CzechLanguage.28.During Renaissance, With the reign of ___________(whom)England began to embark on the road to _____________ andforeign control.29.William Shakespeare, _________________, Thomas Moreand __________________ were humanist scholars duringRenaissance.30.David is a sculpture done by _____________.III.论述题:1.Why did Renaissance first begin in Italy?2.Why did Renaissance decline in Italy in the end?3.What is Renaissance Art ? What are the characteristicsand features of Renaissance Art?4.How did Reformation develop? (即:发展的四个阶段)5.How did capitalism come into being and develop?Or What are the propositions for Capitalism to rise?6.What is the great significance of Reformation?What are the great influences of Reformation?7.Why did England come later than other countries duringthe Renaissance but reach its climax in England? In whatway was English Renaissance different from that of othercountries?8.What type of culture did Renaissance create?9.What is the great significance of Reformation?10.What are the great geographical discoveries (or ofnavigation) in the Middle Ages?11.From what was the English essay derived?Division FiveI.Special Terms Explanation:Francis Bacon’s practical philosophyDeductive methodInductive methodThe Great InstaurationThomas Hobbes’ Social ContractJohn Lock’s Social Contractthe Natural State of Warthe Laws of NatureThe English Revolution (the English Bourgeoisie Revolution)Descartes’ Theory of KnowledgeDescartes’ DualismFrench ClassicismBaroque ArtII.有可能出填空,选择题与判断题的内容:1._______________ century was the transitional period fromthe Middle Ages to the Modern Times.2. The Modern World begins in __________(century).3. ________________ and _________________proved thatCopernicus’ heliocentric theory is t rue.4. There is the first breakthrough in __________and________________ in the 17th century.5. There was a profound change in the conception of men’splace in the universe in ___________(century).6. _________ looks at men’s position in the universe in a newway.7. Kepler was well known for his discovery of ___________.8. Newton discovered the theory of _________________, and he also invented the method called ___________________.9. ____________ distinguishes three levels of understanding: _______________, _____________________ and _______ _________________.6. ____________________ psychology and ______________ physics originated from Leibniz.7. In December, 1689, __________________________ was enacted by the English Parliament.8. After the English Revolution, _____________________(制度) was established in the Great Britain.9. The event that took place in 1688 in the Great Britain was called ________________.10.The men of the action and the leader in the English Revolution were _____________ and _____________, and the man of thought with his famous literary work _______.11. ____________________ was the representative author of French Classicism.12.The representative painter of Baroque Art was __________ with his famous painting work _______________________ and _____________________.13. ____________________ was the representative painter of Dutch Protestant Art.14. The law of falling bodies is established by __________.15.The importance of acceleration in dynamics is discoveredby ___________.16.__________ sated that there should be a common poweror a government backed by force able to punish.17. __________ believed that government was not createdby God, but by men themselves.18. _________ believed that sensation and reflection makeup experience.19. Paradise Lost was written by ____________, who wasthe man of ___________ in the English Revolution. 20. The Calling of St. Mathew and The Cardsharps werepainted by ______________, which belong to _________________(哪种艺术流派).21. “I doubt, therefore I think: I think, therefore I am.” wasput forward by ______________, which belongs to _______________________(什么理论).22. Both idealism and materialism are included in _______’_______________(谁的,什么理论).23. By the end of 16th century, _______________ had spreadto England. As a result, _______________was staredin England.III.论述题:1.Why do we say the 17th century was a transitional periodfrom the Middle Ages to the Modern World?2.How did science develop in the 17th century?3.What are the two merits shared by the Great Scientists of the17th century?What are the similarities among the 17th -century scientists?4.What is Baconian Materialism?5.What is the difference between Thomas Hobbes and JohnLock in terms of their social contract?6.How many stages did the English Revolution undergo?7.What is the great significance of the English Revolution?8.What is French Classicism? What are the characteristics?9.How did Lock justify rebellion against government?10. What are the causes of the English Revolution?。
小学下册英语第1单元期末试卷考试时间:80分钟(总分:140)A卷一、综合题(共计100题共100分)1. 选择题:Which animal is known as the "King of the Jungle"?A. ElephantB. LionC. TigerD. Bear2. 选择题:What is the color of snow?A. BlackB. WhiteC. BlueD. Green3. 听力题:The Hubble Space Telescope has provided many stunning images of _______.4. 填空题:My __________ (玩具名) is very __________ (形容词) and fun.5. 填空题:The puppy is _______.6. 听力题:The sun is _____ (shining/raining) today.7. 听力题:I like to visit the ___. (farm)8. 听力题:The _______ grows from a small seed.What is the name of the famous clock tower in London?A. Big BenB. Eiffel TowerC. Leaning Tower of PisaD. Statue of Liberty答案: A. Big Ben10. 选择题:What is 8 x 4?A. 24B. 32C. 40D. 4811. 选择题:What do we call a person who studies weather patterns?A. GeologistB. MeteorologistC. BiologistD. Climatologist答案: B. Meteorologist12. 听力题:A __________ is a way to symbolize a chemical reaction using letters and numbers.13. 填空题:We are having ________ (早餐) together.14. 听力题:What do you want for ________?15. 选择题:What do you call a person who investigates crimes?A. DetectiveB. OfficerC. InvestigatorD. Policeman答案: A16. 填空题:The rabbit hops around the _______ (花园).17. 填空题:The ________ was a significant event in the history of labor movements.What is the temperature at which water freezes?A. 0 degrees CelsiusB. 32 degrees CelsiusC. 100 degrees CelsiusD. 212 degrees Celsius19. 听力题:The chemical formula for sodium hydrogen carbonate is _____.20. 选择题:What is the opposite of 'bitter'?A. SweetB. SourC. SpicyD. Salty答案:A21. 填空题:The country known for its volcanoes is ________ (印尼).22. 填空题:We should learn about _____ (当地) plants and their uses.23. 填空题:There are many ______ in the park.24. 填空题:The kitten loves to chase its _______ (小猫喜欢追逐它的_______).25. 听力题:I saw a ________ in the sky.26. 填空题:My pet rabbit loves to dig in the ______ (土壤).27. 选择题:What do we call the act of building something?A. ConstructionB. CreationC. AssemblyD. Design答案:A28. 选择题:What is the name of the fairy tale character who lost her glass slipper?A. Snow WhiteB. CinderellaC. RapunzelD. Sleeping Beauty答案: B29. 选择题:What is the term for a person who studies insects?A. EntomologistB. ZoologistC. BotanistD. Biologist答案:A30. 听力题:The baby is _____ (crying/sleeping) in the crib.31. 选择题:What is the main ingredient in yogurt?a. Milkb. Creamc. Sugard. Water答案:A32. 选择题:What do frogs eat?A. MeatB. PlantsC. InsectsD. Fruits答案:C. Insects33. 听力题:We have ________ (homework) to do.34. 填空题:The bee buzzes around the ______.35. 填空题:The ______ (自然) world is full of wonders related to plants.36. 选择题:What is 5 x 6?A. 30B. 25C. 35D. 40答案: A37. 选择题:What do we call the place where we keep books?A. LibraryB. GymC. ParkD. Museum38. 填空题:_____ (植被覆盖) prevents soil erosion.39. 填空题:A _______ (鲨鱼) is a powerful predator.40. 选择题:Which instrument is used to measure temperature?A. BarometerB. ThermometerC. SpeedometerD. Ruler答案:B41. 填空题:I enjoy participating in ______ (学校活动) because it allows me to meet new people and make friends.42. 听力题:The process of making hydrogen gas from electrolysis involves _______.43. 选择题:What do you call a building where you can watch movies?A. MuseumB. TheaterC. LibraryD. School答案: B44. 选择题:What is the opposite of ‘high’?A. LowB. TallC. BigD. Short45. 听力题:We go _____ (cycling) on weekends.46. 填空题:A ________ (金鱼) swims gracefully in its bowl.47. 选择题:What is the name of the famous explorer who discovered America?A. Ferdinand MagellanB. Christopher ColumbusC. Marco PoloD. Vasco da Gama48. 听力题:The first modern Olympics were held in _______.49. 选择题:What is the name of the ancient Egyptian writing system?A. HieroglyphicsB. CuneiformC. LatinD. Sanskrit答案: A. Hieroglyphics50. 填空题:The _____ (乌鸦) is known for its intelligence and problem-solving skills. 乌鸦以其聪明才智和解决问题的能力而闻名。
Finding Minimal Unsatis¯able Subformulae inSatis¯ability instancesRenato Bruni and Antonio SassanoDipartimento di Informatica e Sistemistica,Universitµa di Roma\La Sapienza",via Buonarroti12-I-00185Roma,Italy.f bruni,sassano g@dis.uniroma1.itAbstract.A minimal unsatis¯able subformula(MUS)of a given CNF isa set of clauses which is unsatis¯able,but becomes satis¯able as soon as weremove any of its clauses.In practical scenarios it is often useful to know,in addition to the unsolvability of an instance,which parts of the instancecause the unsolvability.An approach is here proposed to the problem ofautomatic detection of such a subformula,with the double aim of¯ndingquickly a small-sized one.We make use of an adaptive technique in order torapidly select an unsatis¯able subformula which is a good approximation ofa MUS.Hard unsatis¯able instances can be reduced to remarkably smallerproblems,and hence e±ciently solved,through this approach.1IntroductionWe call minimal unsatis¯able subformula(MUS)of a logic CNF formula a set of clauses which is unsatis¯able,but becomes satis¯able removing any of its clauses(see sec.2).An approach is here proposed to the problem of automatic detection of a MUS,with the double aim of¯nding it small,and of proving unsatis¯ability faster.As for the¯rst point,in fact,in practical scenarios it is often useful to know,in addition to the unsolvability of an instance,which parts of the instance cause the unsolvability.As for the second point,the task of proving unsatis¯ability usually turns out to be computationally harder than proving satis¯ability.We can,roughly speaking,classify the solution methods for the Satis-¯ability problem in exact and heuristic ones.The proposed procedure is a complete one.Most of complete methods are branching procedures based on case splitting.On the contrary,we use a new branching scheme,whose branching rule consists in trying to satisfy at¯rst the hardest clauses while visiting a clause-based branching tree[1](see sec.3).Moreover,we use an Adaptive Core Search in order to rapidly select an unsatis¯able subformula, which is a good approximation of a MUS(see sec.4).Small but hard un-satis¯able instances can be e±ciently solved through this approach.More important,unsatis¯able subformulae are detected in all problems solved. Their size is often remarkably smaller than the original formulae.22Minimal Unsatis¯able SubformulaeLet A be the ground set of the literals a i.De¯ne:a l=a n+l and:a n+l=a l.A´f a i:a i=®i for i=1;:::;n;a i=:®i¡n for i=n+1;:::;2n g Every clause is a set C j=f a i:i2I jµI´f1;:::;2n gg.A CNF formula is collection F of sets C j over A:F=f C j:j=1;:::;m g.A truth assignment for the logical variable is a set S=f a i:a i2S) :a i2S g.It is partial if j S j<n,i.e.some variables are not assigned, and complete if j S j=n.Given a partial truth assignment S,the set of possible completion is C(S)=f a i:a i2S^:a i2S g.S satis¯es C if and only if S\C=Á.S falsi¯es C if and only if C(S)\C=Á.A minimal unsatis¯able subformula(MUS)is a collection GµF of clauses of the original instance having the following properties:1.8S;9C j2G such that S\C j=Á(unsatis¯able)2.8H½G;9S such that8C j2H;S\C j=Á(every subset is sat.)Of course,if any subformula is unsatis¯able,the whole problem is.On the other hand,an unsatis¯able formula always contains a MUS.There are procedures that,given a set of clauses,recognize whether it is a MUS or not in polynomial time[3].The key point is how to select a MUS.We propose a procedure to rapidly select a good approximation of a MUS,that means an unsatis¯able set of clauses having almost as few clauses as the smallest MUS.3The Branching schemeIn order to reduce backtracks,it's better to start assignment satisfying the more di±cult clauses[1],i.e.those which have the fewest satisfying truth assignments,or,in other words,represent the more constraining relations. The point is how to¯nd hardest clauses.An a priori parameter is the length: unit clauses are universally recognized to be hard,and the procedure of unit propagation,universally performed,satis¯es them at¯rst.Our evaluation of clause hardness is based on the history of the search, and keeps improving throughout the computation.We say that a clause C j is visited when we make a truth assignment aimed at satisfying C j,and that C j cause a failure either when an empty clause is generated due to truth assignment made on C j,or when C j itself becomes empty.Visiting C j many times shows that C j is di±cult,and failing on it shows even more clearly that C j is di±cult.Counting visits and failures requires a very little overhead.Therefore,we use the following branching rule:31.Perform all unit resolutions.2.When no unit clauses are present,make a truth assignment satisfyingthe clause C max which maximizes our clause hardness measure'(v j is the number of visits,f j the number of failures,p a parameter giving the penalty considered for failures,and l j the length of the clause).'(C j)=(v j+pf j)=l jTo satisfy C max,we add to the partial solution S a literal a l2C(S)such that a l2C max.If we need to backtrack,the next assignment is not just :a l,because it does not satisfy C max.Instead,we add another literal a k2Fig.1.Example of C max with the consistent branching possibilities.again to the truth assignments made to satisfy the previous clause,until we have another choice.This is a complete scheme,because,if a satisfying truth assignment exists,it will be reached,and,if the search tree is completely explored,the instance is unsatis¯able.4Adaptive Core SearchBy applying the above branching scheme,and'to evaluate clauses hard-ness,we develop the following procedure of Adaptive Core Search.0.(Preprocessing)Perform p branching iterations using just shortest clauserule(This gives initial values to').If instance is already solved,Stop.1.(Base)Select an initial collection of hardest clauses whose cardinalityis proportional to cardinality of F:C1=f C j:C j2F;'(C j)¸'(C k)8C k2F;jCj=c jFj;c<1gThis is our¯rst core,i.e.candidate to be a MUS.Remaining clauses form another collection O1=F n C1.4k.(Iteration)Perform h branching iteration on C k,ignoring O k.Obtain one of the following:{C k is unsatis¯able)F is unsatis¯able,then Stop.{no answer after h iteration)Re-start from(Base)selecting a new set of hardest clauses(allowed only a¯nite number t of times,in order to ensure termination:after t times,h is increased).{C k is satis¯ed by solution S k)Test S k on O k.One of the following:-8C j2O k;S k\C j=Á)F is satis¯ed,then Stop.-9C j2O k;j C(S k)\C j=Á.Call this collection of falsi¯edclauses N k.Add them to the core,obtaining C k+1=C k[N k,delete the partial solution S k,and apply again(Iteration).-9C j2O k;j S k\C j=Áand9C j2O k;j C(S k)\C j=ÁTryto extend the partial solution S k to F putting C k+1=F andapply again(Iteration).The main idea is that we select the clauses that resulted hard during the branching phase,and try to solve them as if they were our entire instance.If they really are an unsatis¯able instance,we have done.If,after h branching iterations we cannot solve them,this means that our instance is still too big,and it must be reduced more.Finally,if we¯nd a satisfying solution for the core,we try to extend it to the rest of the clauses.If some clauses are falsi¯ed,they are di±cult(together with the clauses of the core),and therefore they should be added to the core.The above algorithm is a complete one,and solves,in average case, smaller subproblems at the nodes of the search tree,hence operations per-formed,such like unit propagation,work only on current core C k.5Computational resultsColumns labeled'branch','ACS sel','ACS sol','ACS tot'respectively are the running times for the branching procedure only(without Adaptive core Search),core selection,core solving,and total times of Adaptive Core Search.n core and m core are the number of variables and clauses appearing in the unsatis¯able subformula selected.The test problems(all unsatis¯able)are from the DIMACS.Four of them were used in the test set of the Second DIMACS Implementation Challenge[2].Our running times on them are compared with those of the four faster complete algorithms of that challenge(C-sat,2cl,TabuS and BRR).Times are normalized according to the DIMACS benchmark dfmax, in order to compare them in a machine-independent way.5 6ConclusionsAn approach is here proposed to the problem of automatic detection of a MUS,with the double aim of¯nding a small-sized one,and of proving unsatis¯ability faster.In particular,we present a clause based tree search paradigm for Satis¯ability testing,a new heuristic to identify hard clauses, that are the clauses most likely to appear in a MUS,and the complete algorithm for approximating a Minimal Unsatis¯able Subformula.Smaller unsatis¯able subformulae are detected in all the solved unsatis-¯able instances.These can be remarkably smaller then the original formula, and give an approximation of a MUS in extremely short times.Table1.Results of C-SAT,2cl(limited resolution),DPL with Tabu Search,B-reduction and ACS on the aim series:3-SAT arti¯cially generated problems.Times are normalized according to dfmax results,as if they were obtained on our same machine. References1.J.N.Hooker and V.Vinay.Branching Rules for Satis¯ability.Journal of AutomatedReasoning,15:359{383,1995.2. D.S.Johnson and M.A.Trick,editors.Cliques,Coloring,and Satis¯ability,vol-ume26of DIMACS Series in Discrete Mathematics and Theoretical Computer Sci-ence.American Mathematical Society,1996.3.O.Kullmann.An application of matroid theory to the SAT Problem.ECCC TR00-018,Feb.2000.。
无动词分句(VERSLES S CLAUSE)是一种省略了主语和谓语动词的分句结构。
这种结构通常出现在句首,也可出现在其他位置。
它的被省略的主语通常也就是主句的主语;被省略的动词通常是连系动词be,因此无动词分句就其深层结构来说大多属于SVC句型。
例如:Close to the urban areas,this f arm mainlydoes marketgardening.由于靠近市区,这个农场主要生产商品蔬菜。
A science reporte r,Jim has a wide range of knowled ge.吉姆是一位科学新闻记者,所以知识面广。
在这里,Close to the urban areas=As it is close to the urban areas;A science report er=As he is a science reporte r,都是一种省略结构。
一、无动词分句的类型无动词分句,就其结构形式来说,分为以下三类:1.不带从属连词的无动词分句这类无动词分句通常只由形容词词组或名词词组构成。
例如:Speechless,Henry noddedand sat on a f olding seat.亨利一言不发,点点头,坐在折椅上。
Anxious f or a quick decisio n,the chairma n calledf or a v ote.主席急于迅速作出决定,便提交表决。
Ture or not,the reportprov ide d new ammunit ion f or his enemies.不管是真是假,这个报告都为他的敌人们提供了新的炮弹。
An excelle nt speaker,he was nev er at a loss for a word.作为一位卓越的演讲家,他从来不会无话可说。
-1-2-Sat Sub-Clauses and the Hypernodal Structure of the 3-Sat Problem Douglas PowellAbstractLike simpler graphs, nested (hypernodal) graphs consist of two components: a set of nodes and a set of edges,where each edge connects a pair of nodes. In the hypernodal graph model, however, a node may contain other graphs [1]; so that a node may be contained in a graph that it contains. The inherently recursive structure of the hypernodal graph model aptly characterizes both the structure and dynamic of the 3-sat problem, a broadly applicable, though intractable, computer science problem. In this paper I first discuss the structure of the 3-sat problem, analyzing the relation of 3-sat to 2-sat, a related, though tractable problem. I then discuss sub-clauses and sub-clause thresholds and the transformation of sub-clauses into implication graphs, demonstrating how combinations of implication graphs are equivalent to hypernodal graphs. I conclude with a brief discussion of the use of hypernodal graphs to model the 3-sat problem, illustrating how hypernodal graphs model both the conditions for satisfiability and the process by which particular 3-sat assignments either succeed or fail.1. Introduction and TerminologyIn its standard form (Conjunctive Normal Form, or CNF), the 3-sat problem consists of a 01m-1 formula F which is a conjunction of a set of m clauses {C , C , …, C }, where each clause is the disjunction of a set of three literals and each literal is the true or false instantiation of a member 01n-1of the set of n Boolean variables, {X , X , ..., X }. The set of n true literals is represented by 0, 1n-10, 1, n-1{x ,x , …, x } and the set of n false literals is represented by {-x ,-x ..., -x }. The symbol L refers to the set of true and false literals. The ‘-’ symbol represents negation, so that, forexample, -(-x0) = x0 and -(x0) = -x0. The ‘^’ symbol represents conjunction and the ‘v’ symbol 01Cm-1represents disjunction. Thus, a 3-sat formula has the form, (C ^ C ^ … ^ ),where each clause 123123takes the form (l v l v l ) and l , l , and l are true or false literals. Because contradictory pairs of literals are tautological and duplicate literals are redundant, the literals in a given clause are both distinct and non-contradictory. The 2-sat problem is defined similarly, though in 2-sat problems there are only two literals per clause.AssignmentsAn assignment for a particular 3-sat problem consists of a set of literals drawn from the sets of i i true and false literals for that problem. A consistent assignment contains either -x or x , but not i i both, for i 0 {0...n-1}, whereas an inconsistent assignment contains both -x and x for some i 0{0...n-1}. Formally, an assignment A is consistent if a 0A( -a óA ) but inconsistent if a 0A( -a 0A ). A partial assignment satisfies some subset of clauses in a 3-sat formula. Unless otherwise noted, in this paper the term assignment refers to a consistent (non-partial) assignment.-2-SatisfactionA literal satisfies the clauses in which it appears. For a given 3-sat formula, each literal appears in and satisfies some subset of the clauses of the formula, whereas its negation satisfies a distinct 0.n-10 subset of clauses. If c is a clause in F, L the complete set of literals such that L = {x ..x , -x ...-n-1x}, and a an arbitrary literal such that a 0 L , then the set of clauses satisfied by a is given by the formula satisfied(a) = c 0 F (a 0c). Likewise, satisfied(-a) = c 0 F (-a 0c).A clause is satisfied by an assignment (or partial assignment) when one or more of its threeconstituent literals is contained in that assignment. Thus, an assignment A satisfies a clause c iff (c v A) i . For example, an assignment containing -x0 satisfies the clause (-x0 v -x3 v -x4)but not the clause (x0 v -x3 v -x4).A 3-sat problem is satisfiable if there is a consistent assignment to some set of the n variables such that each clause is satisfied; conversely, the problem is unsatisfiable if no such assignment exists. Formally, an assignment A satisfies a formula F with clauses C if c 0C( (c v A) i ) v a 0A( -a óA ).3-sat Algorithms, Time Complexity and 2-satA complete algorithm for the 3-sat problem either finds an assignment for the problem instance or determines that the problem is unsatisfiable (see Gu [2] for a comprehensive study of existing sat algorithms). In the worst case, the performance of existing complete 3-sat algorithms degrades to exponential time, taking. 2 steps to solve the problem, unfeasible for evenn moderately large n [3]. The performance of existing 2-sat algorithms is far better: there are complete 2-sat algorithms that solve the 2-sat problem in linear time–trivial for even large n [4].In general, the difficulty of a set of 3-sat problems is defined by the ratio of clauses to variables r=m/n [5]. The problems used to investigate the concepts of sub-clauses and hypernodality were all generated with r=4.25, the region where roughly half the 3-sat problems are satisfiable and half are unsatisfiable [6].2. Satisfied Clause Sets and 2-Sat Sub-ClausesConsider the clause c = (-x1 v -x4 v x5). Assignment to the negation of any of the three literals in the clause reduces the clause to a two-literal sub-clause. For example, an assignment of x1reduces the clause c to (-x4 v x5), whereas an assignment of x4 reduces c to (-x1 v x5) and an assignment of -x5 reduces c to (-x1 v -x4). Since each 3-sat clause contains three literals, each clause contains three potential 2-sat sub-clauses, derived by negating, one at a time, each of the three literals in the clause (fig.1). Thus, if -a is used in an assignment, each three-literal clause in 2323satisfied(a) is reduced to a two-literal sub-clause , since -a ^ (a v l v l ) = (l v l ) .Clause^Negated Literal=>Sub-Clausec = (l v l v l )-l S={l v l}1231023-l S={l v l}2113-l S={l v l}3212Figure 1. Negated literals and sub-clauses.The Sub-Clause Space, SA 3-sat problem in n variables contains 2n literals. Since a literal may not appear in a sub-clause with either itself or its negation, each literal may be combined with 2(n-1) literals, yielding a total of 2n(n-1) potential sub-clauses. If r is the ratio of clauses to variables, then a given 3-sat problem F contains (r* n) clauses and 3r*n sub-clauses (since each clause contains 3 sub-clauses), assuming that there are no duplicate sub-clauses in F. Thus the ratio of actual to possible sub-clauses = , so the probability that any two literals produce the same sub-clause diminishes with increasing n. Given a 3-sat formula F, the sub-clause space S denotes the set of sub-clauses in F, equal to the union of the sub-clauses contained in the clauses of F.The Relation of Literals to Sub-ClausesLet S denote the set of sub-clauses in a given 3-sat formula F. Each sub-clause in S is activated by one or more literals, solved by either of its two member literals and converted into a unit clause (a clause containing one literal) by the negation of either of its two member literals. Similarly, each literal in a particular 3-sat problem activates a subset of the sub-clauses in S, solves a subset of those sub-clauses and converts a subset of those sub-clauses into unit clauses. Assuming that L = {x0...xn-1, -x0...-xn-1}, a is an arbitrary literal such that a0 L, satisfied(a) is the set of clauses satisfied by a, c is a clause in F, and Q is an arbitrary set of sub-clauses, the sets described in Fig. 2 show the relation of literals to their related sub-clauses:The term s ubclauses(a)denotes the set of two-literal clauses, interpreted as a partial CNF 2-sat formula, created if a is used in an assignment. Subclauses(a) is activated when a is assigned and is created by removing -a from each member clause in satisfied(a).Formally, subclauses(a) = s 0 satisfied(-a) (s v a) . Likewise, given a set of literals P, subclauses(P), and a literal a 0 P, subclauses(P) is the union of subclauses(a).The term subsat(a) denotes the set of sub-clauses satisfied by a. Note that a only satisfies sub-clauses that are activated. Formally, subsat(a) = s 0 S (a 0 s) .The term unitclauses(a) refers to the set of literals derived by removing a fromsubsat(-a). Formally, unitclauses(a) = s 0 subsat(-a) (s v a) .-3-The term creators(Q) refers to the set of literals that create (or activate) the set of sub-clauses in Q. Formally, creators(Q) = a 0 L ( (subclauses(a) v Q) i) .The term parent(s)refers to the set of clauses from which the sub-clause s derives.Likewise, if S is a set of sub-clauses, then parent(S) is the union of parent(s) for s0 S.Figure 2. Definitions relevant to sub-clauses.The satisfied(), subclauses(), subsat() and unitclauses() sets are interdependent in that an assignment to a literal creates a set of sub-clauses, which appear in the subsat() sets of other literals, a subset of which much be assigned to satisfy the sub-clauses in subclauses(), thereby creating unit clauses, which force the use of other literals, which, in turn, create more sub-clauses. As an algorithm uses literals in an assignment, this cycle continues, until either a satisfying assignment is found, and all activated sub-clauses are solved, or a contradiction is created, and some sub-clauses are left unsolved. (In dataflow terms, each literal produces a set of sub-clauses that must then be consumed, via satisfaction, by other literals.) The sub-clause interaction matrix (Fig. 2b) summarizes the connectivity of literals and sub-clauses, showing how literals create, solve and convert sub-clauses to unit variables. For example, the literal -x0 creates the sub-clause s0, which, when combined with -x2, creates the unit clause containing -x3.Figure 2b. Interaction Matrix for Literals and Sub-Clauses. Key: c = sub-clause created by literal; s = sub-clause solved by literal; a literal in a cell is a unit variable created by the combination of the corresponding row sub-clause and column literal(problem=k3n4Seq207p0).As an example of the interaction of clauses, literals and sub-clauses, suppose that F contains the 012clauses C = {-x1 v -x4 v x5}, C = {-x3 v -x4 v x8} and C = {-x1 v -x3 v -x4}, which, through 012assignment of x4, become the sub-clauses s = {-x1 v x5}, s = {-x3 v x8} and s = {-x1 v -x3}.Then, since012satisfied(-x4) = {C , C , C },012subclauses(x4) = { {C ^ x4}, {C ^ x4}, {C ^ x4}} = { {-x1 v x5}, {-x3 v x8},{-x1 v -x3} }012= {s , s , s },12subsat(-x3) = { s ,s }, and12unitclauses(x3)= { {s ^ x3}, {s ^ x3}} = { {x8}, {-x1}} .{x8, -x1}.012In summary, an assignment of x4 activates sub-clauses {s , s , s }, which, in combination with an assignment to x3 yield unit clauses containing -x1 and x8, which must thus be part of the assignment. 3. Sub-clause Thresholds and the use of Sub-clausesSub-Clause ThresholdsThe set of sub-clauses for a formula is given by a 0L ( subclauses(a) ). Given the set of 01n-1variables V = {X , X , ..., X }, an assignment contains either a true or false literal for eachvariable. Assuming that no two literals create the same sub-clause, the minimum number of sub-clauses in an assignment for the formula is given byminimum_threshold = | x 0V( min(|subclauses(x)|, |subclauses(-x)|) |,and the maximum number of sub-clauses in an assignment for the formula is given bymaximum_threshold = | x 0V( max(|subclauses(x)|, |subclauses(-x)|) |If the sub-clause count for an assignment A is given bysubclause_count = | a 0A( subclauses(a) ) |,then, for any satisfying assignment,minimum_threshold <= subclause_count <= maximum_threshold.Given a 3-sat formula F, any satisfying assignment A must satisfy at least as many sub-clauses as are in the minimum threshold, and at most as many sub-clauses as are in the maximum threshold. The rate at which a satisfying assignment must ultimately satisfy sub-clauses is given by the formula: | a 0A( subclauses(a) ) | ÷ | A |, which is equal to | a 0A( subclauses(a) ) | ÷ n. Finally, if an assignment A satisfies a 3-sat formula, then: ( a 0A( subclauses(a) )) = ( a 0A(subsat(a) )).Four Types of Sub-Clause-Derived AssignmentsBy comparing the number of sub-clauses created and solved by the true and false literals of a particular variable, it is trivial to derive the following four types of assignments: 1) minCreate, whose literals have the minimum number of created sub-clauses, 2) minCreateMaxSolve, whose literals are the lesser of ( | solved sub clauses | - | created sub clauses | ), 3) maxSolve, whose literals solve the maximum number of sub-clauses, and 4) maxCreate, whose literals create the maximum number of sub-clauses. In measures of assignment correctness (% clauses satisfied, % created but unsatisfiable sub-clauses, etc.), assignment types 1-3 are roughly equivalent to assignments generated using a simple greedy algorithm [7] which selects variables based on per-literal clause-satisfaction counts (fig. 2c). Thus, although the assignments generated by the various methods contain different literals, it is not clear whether or not algorithms using sub-clause-derived assignments will outperform those using assignments generated by greedy per-literal clause-satisfaction counts, though both methods appear to outperform pseudo-random assignment generation (fig. 2c).Assignment Type and % Satisfied Clauses(100 Problems, n=500, mixed satisfiable/unsatisfiable)Figure 2c. Percent of clauses satisfied by assignments generated using eitherMinCreateMaxSolve (mean=94.07%), Greedy (mean=94.12 %) or Random (mean=87.45 %).(100 problems, n=500, mixed satisfiable and unsatisfiable, m=4.25. Problem set:satk3n500_218.in.)Characteristic Inflection Point of the Unsolved Sub-Clause CurveSub-clauses created by a particular literal contain literals that are distributed throughout the complete set of literals. The inflection point (the point where the sub-clauses satisfied by the assigned literals begins to exceed the sub-clauses produced by the assigned literals) of the unsolved sub-clause curve occurs, for satisfying assignments, after approximately half (n/2) of the literals are assigned (fig. 2d).Mean Unsolved Sub-Clauses over 54 Satisfiable FormulaUnsatisfying Assignments and Excluded LiteralsBecause a satisfying assignment may not contain unsolved sub-clauses (see theorem),assignment literals that create unsolved sub-clauses must be excluded from an assignment. Thus,because the creators of each sub-clause are known, splitting an unsatisfying assignment into an allowed and an excluded set of literals is a fairly trivial matter:for (s in set of unsolved sub-clauses)excludedLiterals |= s.creatingLiteral.Figure 2d . Mean unsolved sub-clauses at each literal in satisfying assignments for 54 satisfiable problems (n=100,m=4.25, problem sets from the k3n100Seq211 series). The inflection point of the curve occurs at the literal where the assigned literals satisfy (consume) more sub-clauses than they create (produce), characteristically when around half the literals are assigned.4. The Relation of 3-sat to 2-satGiven an assignment A, each literal a0 A activates the set of sub-clauses in subclauses(a). Since subclauses(a) is a partial 2-sat problem, the conjunction of the subclauses(a) for all a0 A is a 2-sat problem. Clearly, if a literal is used in an assignment, then its activated sub-clauses must all be satisfied (see proof). This implies that each 3-sat problem contains a set of 2-sat problems, one of which is activated by a particular assignment. For example, consider the 3-sat formula in Fig. 3, with satisfying assignment A={-x0 -x1 x2}.Clauses for Satisfiable 3-sat Problem F(with assignment literals -x0, -x1 and x2 underlined):C0 ={-x0 -x1 -x2}, C1 ={-x0 -x1 x2},C2 ={-x0 x1 -x2}, C3 ={-x0 x1 x2},C4 ={x0 x1 x2}, C5 ={x0 -x1 -x2}C6 ={x0 -x1 x2}.Clauses Satisfied(a) for a0 L (numbers refer to satisfied clauses):Satisfied for True Literals Satisfied for False Literalssatisfied(-x0) = {0 12 3}satisfied( x0) = {4 5 6 }satisfied(-x1) = {0 1 5 6}satisfied( x1) = {2 3 4}satisfied(-x2) = {0 2 5}satisfied( x2) = {1 3 4 6}Sub-clauses for F:S0=(-x0 v -x1), S1=(-x0 v x1), S2=(-x0 v -x2), S3=(-x0 v x2),S4=(x0 v -x1), S5=(x0 v x1), S6=(x0 v -x2), S7={x0 v x2),S8=(-x1 v -x2), S9=(-x1 v x2), S10=(x1 v -x2), S11=(x1 v x2).Sub-clause thresholds for F:minimum_threshold = 9,maximum_threshold = 12,| subclauses(A) |= 11.Sub-clause sets, subclauses(a) for Literals a0 L (numbers refer to created sub-clauses):True Literals False Literalssubclauses(-x0)={8 9 11}subclauses( x0)={8 9 10 11}subclauses(-x1)={2 3 7}subclauses( x1)={2 3 6 7}subclauses(-x2)={0 1 5}subclauses( x2)={0 1 4}Figure 3. Satisfiable 3-sat Problem, with associated sub-clause sets.The assigned literals -x0, -x1 and x2 create the following sub-clause sets:subclauses(-x0)={8 9 11}{ S8=(-x1 v -x2), S9=(-x1 v x2), S11=(x1 v x2) }subclauses(-x1)={2 3 7}{ S2=(-x0 v -x2), S3=(-x0 v x2), S7={x0 v x2) } subclauses( x2)={0 1 4}{ S0=(-x0 v -x1), S1=(-x0 v x1), S4=(x0 v -x1) },which yield the set of subclauses={0 1 2 3 4 7 8 9 11}, corresponding to the following satisfiable two-sat problem (with satisfying literals underlined):(-x0 v -x1) ^ (-x0 v x1) ^ (-x0 v -x2) ^ (-x0 v x2) ^ (x0 v -x1) ^(x0 v x2) ^ (-x1 v -x2) ^ (-x1 v x2) ^ (x1 v x2).If x1 is substituted for -x1, the assigned literals resulting in the unsatisfying assignment A={-x0 x1 x2} create the following sub-clauses:subclauses(-x0)={8 9 10}{ S8=(-x1 v -x2), S9=(-x1 v x2), S10=(x1 v -x2) } subclauses( x1)={2 3 6 7}{ S2=(-x0 v -x2), S3=(-x0 v x2), S6=(x0 v -x2),S7={x0 v x2) }subclauses( x2)={0 1 4}{ S0=(-x0 v -x1), S1=(-x0 v x1), S4=(x0 v -x1) },which yield the set of subclauses={0 1 2 3 4 6 7 8 9 10}, corresponding to the following 2-sat problem (unsatisfied clauses in bold face):(-x0 v -x1) ^ (-x0 v x1) ^ (-x0 v -x2) ^ (-x0 v x2) ^ (x0 v -x1) ^(x0 v -x2) ^ (x0 v x2) ^ (-x1 v -x2) ^ (-x1 v x2) ^ (x1 v -x2).To more formally define the relation of 3-sat to 2-sat, let a be an arbitrary literal, and A an assignment for F. Let T be the 2-sat formula derived by taking the conjunction of subclauses(a) for all a0 A. If A satisfies F, then T is a satisfiable 2-sat formula (see proof). Conversely, if A does not satisfy F, then A does not satisfy T (see corollary). More generally, any satisfying assignment of a k-sat formula (where k is an integer), generates a satisfiable (k-1)-sat formula. (The proof for k-sa t follows the logic of the proof for 3-sat). It follows that a satisfying assignment for a 2-sat problem generates a satisfiable 1-sat problem which is satisfied by the satisfying assignment of the parent 2-sat problem. Thus, the 3-sat problem is inherently recursive: an assignment A for a 3-sat problem F generates a 2-sat problem T. The use of A as an assignment for T–a necessary condition for satisfiability of F–generates a 1-sat problem, which is satisfiable by A iff T is satisfiable by A. Likewise, T is satisfied by A iff F is satisfied by A.Theorem: Given a satisfying assignment A f or a 3-sat formula F, there is a satisfiable 2-sat formula T that is equal to the conjunction of the subclauses(a) for a 0A . T is satisfied by A. Proof: Let F be a satisfiable 3-sat formula. Let A be a satisfying assignment set for F. For any arbitrary literal a 0A there is a set of 2-sat subclauses, subclauses(a). Since the clauses of F are joined by conjunction, and subclauses(a) is merely the family of subclauses given by the conjunction {satisfied(-a) ^ a}, the subclauses(a) are also joined by conjunction. Thus the concatenation of subclauses(a) for a0 A is a 2-sat formula contained in F. Let T be the 2-sat formula in F given by v( subclauses(a) ) for a0 A. Let a be an arbitrary literal in A. Assume thatA satisfies F, but T is not satisfiable. If T is not satisfiable, then some literal a0 A created an unsatisfied sub-clause s0 T. Since s is created by a, s is equal to some clause c = {s v -a}. Since a 0 A, -a ó A. Since s is not satisfied, (s ^ A) = i, which means that (c ^ A) = i, which means that c is not satisfied. If c is not satisfied, then A does not satisfy F, which contradicts the premise that A satisfies F. So, c must be satisfied. Since -a ó A, at least one of the literals in s must be in A, or c is not satisfied. If one of the literals in s is in A, then s is satisfied, which means that T is satisfied, since s was the unsatisfying subclause of T. Thus, if A satisfies F, A also satisfies T, since otherwise there is an unsatisfied clause c0 F.Corollary 1: Given an unsatisfying assignment, a subset of the generated sub-clause sets are unsatisfied by the assignment.Proof: Let F be a satisfiable 3-sat formula. Let X be the set of variables for F. Let A be a consistent, unsatisfying assignment for F, containing either a true or false literal for all x0 X. Let SC=v( subclauses(a) ) for a0 A, and C be the set of clauses in F. Since A does not satisfy F, there is a clause c0 C that is unsatisfied, which means that (c ^ A) = i. By definition, the set of literals in c are true or false instantiations of three of the variables in X. Thus, for any literal q0c, -q 0 A. If -q 0 A, then subclauses(-q) contains a sub-clause s = (c ^ -q). Since (c ^ A) = i, and s f c, (s ^ A) = i. Since s 0 SC and s is activated by some a0 A, it follows that in an unsatisfying assignment, a subset of the generated sub-clause sets are unsatisfied by the assignment. Definition: Let literals(Q) denote the set of literals in a set of clauses Q.Corollary 2: If a partial, consistent assignment P, which contains less than n literals, satisfies all of its activated sub-clauses, but does not satisfy the corresponding 3-sat formula F, then F is divisible into at least two distinct constituent 3-sat sub-formulas, F1, which contains the set of clauses C1 that are satisfied by P, and F2, which contains the set of clauses C2, such that literals(C2) = {L - P}. Additionally, if L1 = literals(C1) c-(literals(C1) and L2 = literals(C2) c -(literals(C2), then L1 L2.Proof: Suppose that a 3-sat problem F contains the set of clauses C, and that P is a partial assignment for F. Suppose that P activates and satisfies a non-empty set of sub-clauses, but that P does not satisfy the corresponding 3-sat formula F. In other words:( a0 P ( subclauses(a) = a0P( subsat(a) ) ^ ( c 0 C(P ^ c = i)).If SC = subclauses(P), then P satisfies the set of clauses C1, where C1 = parent(SC). Since at least one clause in C is unsatisfied, C1f C and *C1* < *C*. Let (C2 = C - C1) denote the set of unsatisfied clauses of C. Suppose that c = (j v k v l) is an arbitrary unsatisfied clause in C. Clearly, c óC1, since all clauses in C1 are satisfied. Thus, c0C2. Additionally, P ^ c = i, since otherwise P satisfies c. Because có P, ( jóP ^ kóP ^ l óP). Similarly, (- jóP ^ - kóP ^ -l óP), since otherwise there would be an unsatisfied sub-clause in C1. Suppose, for example, that -j 0P. If -j 0P then P activates a sub-clause s = (k v l). However, s must be unsatisfied by P, since (kóP ^ l óP), which contradicts the premise that P satisfies all of its activated sub-clauses. So, (- jóP), and the same is true for both -k and -l. In summary, if F is a 3-sat problem, P a partial assignment that satisfies all of its activated sub-clauses, c an unsatisfied clause in F, and q the set of literals in c, then q ^ P = i and -q ^ P = i, from which it follows that F is divisible intoat least two distinct constituent 3-sat sub-formulas, with each sub-formula satisfied (or unsatisfied) by the literals in the corresponding sub-formula clauses.5. The Nested Structure of 3-SatIf an assignment A satisfies a 3-sat formula F, then, for an arbitrary literal a 0 A, the sets of sub-clauses in subclauses(a) must be satisfied. Thus, a is equivalent to ( a ^ subclauses(a) ). Thus,0, 1n-1for any 3-sat problem F , the set of literals {x ,x , …, x } is equivalent to the statements in Fig. 4:False LiteralsTrue Literals 0000-x ^ subclauses(-x )x ^ subclauses(x )1111-x ^ subclauses(-x )x ^ subclauses(x )...n-1n-1n-1n-1-x ^ subclauses(-x )x ^ subclauses(x )Figure 4. Literals and their associated subclause() sets.000001Consider the literal x , equivalent to x ^ subclauses(x ). Assume that subclauses(x ) = (s ^ s ^j 01i ... ^ s ). Let L be the set of true and false literals and assume that { a , a , ... ,a } 0 L . Assume 01245611122that s = (a v a ), and that subclauses(a1) = (s ^ s ^ s ). Since a =a ^ subclauses(a ), and a = a 20^ subclauses(a ), x is equivalent to any of the following equations:0001)x = x ^ subclauses(x )00102)x = x ^ (s ^ s ^ ... ^ sj)001123)x = x ^ ( (a v a ) ^ s ^ ... ^ sj).1Expanding a yields the following equation:0011124)x = x ^ ( ( (a ^ subclauses(a ) ) v a ) ^ s ^ ... ^ sj),1and expanding subclauses(a ) yields the following equation:001145625)x = x ^ ( ( (a ^ (s ^ s ^ s ) ) v a ) ^ s ^ ... ^ sj).4344If s = (a v a ), then expanding s yields the following equation:0011345626)x = x ^ ( ( (a ^ ( (a v a ) ^ s ^ s )) v a ) ^ s ^ ... ^ sj),333which, since a = a ^ subclauses(a ), expands to the following equation:Although a discussion of generative grammars is beyond the scope of this paper, the growth of equations 1-17 is an example of a Lindenmeyer system[9], a type of generative grammar that may be used to model 3-sat.3-sat sub-clauses may also be converted into finite state machines, where the antecedent of each sub-2clause-derived implication induces a transition between the creator of the sub-clause and the consequent of the implication. The transition function for this finite state machine is also recursive: it is a function of its states.00113345627)x = x ^ ( ( (a ^ ( ( (a ^ subclauses(a ) ) v a ) ^ s ^ s )) v a ) ^ s ^ ... ^ sj).0As equations 1-7 illustrate, x (or, by induction, any literal a 0 A ) is recursively defined in terms 0of itself, its set of induced sub-clauses, subclauses(x ), the constituent sub-clauses in0subclauses(x ), the constituent literals of those sub-clauses, their expansions, and so on. This nesting of literals within induced sub-clause sets is one of the key features of the structure of the 3-sat problem.16. Literals and their Sub-clause Sets are Hypernodal Implication GraphsAny 2-sat clause may be converted into an equivalent implication graph [8]. For example, the 2-12122sat clause c =(l v l ) is equivalent to the graph containing the two implications -l –> l and -l –>1l . Since each sub-clause is a 2-sat clause, sub-clauses may also be converted into implication a graphs (see Fig. 5). Thus, subclauses(a) is equivalent to I , the directed graph containing a and 2the connected implications of the sub-clauses in subclauses(a). Each (non-empty) implication graph contains instantiations of nodes that correspond to the literals in L. Since a nodecorresponds to a literal, and a literal is equivalent to an implication graph, each node reference is equivalent to an instantiation of the implication graph for the corresponding node.Hypernodal graphs are defined as H={N, G}, wher N is the set of nodes in the graph and G is a set of graphs containing the nodes. Unlike the nodes in a non-nested graph, which are terminal,the nodes in N of a hypernodal graph are defined as sets of nodes and edges. In short, nodes are themselves graphs. Thus, implication graphs, whose nodes are themselves other implication graphs, are hypernodal.Although implication graphs are hypernodal, the definition of an implication graph contains an additional component, the set of undirected cross-edges, which describes the connection of one a implication graph to another. Thus, the implication graph I of literal a consists of three a, a components: a set of nodes N where each node n 0 N corresponds to one of the literals in a, a;subclauses(a); a set of directed edges E such that each edge connects a pair of nodes in N and a a a set of undirected cross-edges CE , such that each cross-edge connects a node n 0N to a b a corresponding node n 0N , where b a . Thus, the implication graph for literal a, defined as I =a a a (N , E , Ce ), describes connections between its internal nodes and the nodes of other implication graphs.The properties of implication graphs are similar to the properties of literals and their associated sub-clause sets: the nodes in implication graphs are recursively defined (since nodes are。