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Module 4 Music Section ⅡⅠ品句填词1．Can you ____________(解释) the meaning of what he remarked at the end of the meeting?答案：interpret2．We’ll get in touch with you as soon as a decision is made ____________(关于) your application.答案：regarding3．He was full of ____________(雄心) to become a great scientist and made up his mind to work harder.答案：ambition4．He began to ____________(创作) music at an early age, which showed his gift.答案：compose5．Only when theories are ____________(联合) with practice can we know how they work.答案：combined6．He felt greatly ____________(使沮丧) at the news that he had been turned down by the company.答案：depressed7．What ____________(特征) differ the Americans from the Canadians?答案：characteristics8．He says hundreds of thousands of homeless people are living with friends or ____________(亲戚) because of the terrible earthquake.答案：relativesⅡ单句改错1．By giving concert with other famous pianists, he collected enough money for saving the dying boy. ____________________答案：concert→concerts2．I don’t watch the news now because I find it too depressed. ____________________答案：depressed→depressing3．Some of them are dancing to the music, which gives lives to the party. ____________________答案：lives→life4．We have lost in contact with him since he went abroad in 2021. ____________________答案：去掉第一个in5．(2021·山东青岛一中月考)I had trouble overcoming my addiction to alcohol, and so was my friend Mike. ____________________答案：was→didⅢ课文语法填空Liu Fang，1.____________ famous international music star，had a great talent for music playing at a very early age. 2.____________ a child she was often taken to performances by her mother and learned to play the yueqin and the pipa. She 3.____________(give) concerts since she was eleven，4.____________(include) a performance for the Queen of England during her visit to China. At 15，she went to the Shanghai Conservatory of Music, 5.____________ she studied the pipa and the guzheng. After she graduated，she worked as a pipa soloist in her hometown—Kunming.As Liu Fang said，the biggest challenge in her performance is6.____________(respect) the traditions but to add her own style when7.____________(play) the pipa and the guzheng.Under the deep influences of 8.____________(tradition) Chinese singing，she can make listeners hear singing in her music. Liu Fang enjoys playing and performing in public. She likes to share feelings and 9.____________(idea) with friends and music lovers. And she 10.____________(wish) to compose her own music，using elements from different cultures.6．to respectⅠ单句语法填空1．He has not given up his ambition ____________(beat) the world record in high jump.答案：to beat2．Where have you been? I ____________(wait) for you in the rain for two hours.答案：have been waiting3．Nowadays, the adults have great pressure, and the same is true ____________ the children.答案：of/for4．The famous artist gave life ____________ the painting by adding some flying birds in the sky.答案：to5．He likes playing football，but he doesn’t like playing the piano，and it’s the same ____________Tom.答案：with6．A balanced diet can be better for our health when ____________(combine) with regular exercise.答案：combined7．The need to communicate is a key____________ (character) of human society.答案：characteristic8．We usually share the feelings and ideas ____________each other especially when we feel ____________(depress)．答案：with；depressed9．Don’t hesitate to make contact ____________me if you need any further information.答案：with10．It’s said that the amount of petrol a car uses is ____________(relate) to its speed.答案：relativeⅡ阅读理解A“Music is a moral law.It gives soul to the universe，wingsto the mind，flight to the imagination，and charm and joy to lifeand to everything，〞the ancient Greek philosopher Plato said.This is one of Don Spencer’s favorite mottos that he firmlybelieves.“We know that music brings joy and comfort，and makesus feel happy，〞Spencer says，“but research has also discovered music plays a powerful role in the mental development of children.Music inspires creativity，imagination and self－expression.It also builds self－respect and is good for memory skills.〞The power of musicMuch research supports both Spencer and Plato.A Stanford University study found that musical training improves the way the brain processes the spoken word.Research from Canada found that children aged four to six years old who had music lessons had better memories，as well as higher ability to read and write and Math levels.Not in the curriculumResearch from Australia shows it’s not that smart kids play music；it’s that music makes kids smarter.It supports Spencer’s call for music to be a standard part of the school curriculum，like English and Math.“Music is everywhere，but not at 75 percent of public schools around Australia who don’t have a devoted music teacher，〞Spencer says.“It’s said that many children don’t have access to formal musical education，particularly when it has such an effect on a child’s development.〞Ideally，Spencer would like every child to learn an instrument.“Kids can access cheap instruments like a ukulele，recorder or harmonica，〞Spencer says.He says the Internet has a lot of free videos which teach you how to play instruments.“I can’t stress enough how important music is，〞he says.“It builds relationships，unites people and，most importantly，it is fun.〞【解题导语】文章主要说明了音乐的重要性。
钢琴高一知识点英语Piano Knowledge in High School - EnglishAs high school students, it is important to expand our knowledge and explore various subjects. One subject that holds great significance is music, particularly the piano. The piano is not only a beautiful musical instrument but also a gateway to understanding music theory and enhancing our cognitive abilities. In this article, we will delve into some essential knowledge points about the piano in English.Introduction to the PianoThe piano, often referred to as the "king of musical instruments," is a keyboard instrument that produces sound by striking strings with hammers. It offers a wide range of notes and dynamics, making it versatile for playing a variety of genres, from classical to jazz and pop music. The piano has 88 keys, encompassing both white and black keys. The white keys represent the natural notes (A, B, C, D, E, F, G), while the black keys represent the sharps and flats.Pitch and OctavesIn piano terminology, pitch refers to the highness or lowness of a note. Each key on the piano corresponds to a specific pitch. The distance between two identical notes (e.g., from one C to the next C) is called an octave. Octaves play a crucial role in understanding scales, chords, and intervals on the piano.Reading Sheet MusicTo play the piano proficiently, it is essential to learn how to read sheet music. Sheet music consists of two main components: the musical staff and the notes. The staff is composed of five horizontal parallel lines, with each line or space representing a specific note. The placement of the note on the staff indicates the pitch, and the shape of the note determines its duration.Basic Music TheoryTo master piano playing, one must have a solid foundation in music theory. Key concepts to understand include scales, chords, and intervals. Scales are a series of notes played in a particular order, such as the major scale or the minor scale. Chords are formed by playing multiple notes simultaneously, while intervals refer to the distance between two notes.Technique and FingeringsProper technique and fingerings are vital for playing the piano efficiently. Learning the correct hand position, posture, and finger placement will help prevent strain and avoid injury. Additionally, understanding fingerings for specific musical passages will improve accuracy and overall performance.Famous Pianists and ComposersStudying the piano involves exploring the contributions of renowned pianists and composers. From the classical era to modern times, many individuals have shaped the development of piano music and left a lasting legacy. Exploring the works of composers such as Mozart, Beethoven, Chopin, and contemporary artists will provide inspiration and broaden our musical horizons.Benefits of Playing the PianoPlaying the piano offers numerous benefits beyond the musical aspect. It enhances cognitive skills, including fine motor skills, hand-eye coordination, and memory. Piano playing also serves as a stress reliever and promotes emotional expression and creativity. Moreover, it provides a sense of achievement and satisfaction through the mastery of a challenging skill.ConclusionThe piano is not just a musical instrument; it is a gateway to understanding music theory, expanding cognitive abilities, and expressing emotions creatively. By delving into the knowledge points mentioned above, high school students can embark on a fulfilling journey of piano-playing, exploring the works of famous composers, and reaping the numerous benefits that come with playing such a majestic instrument. So, let us embrace the enchanting world of the piano and fill our lives with its harmonious melodies.。
描写听觉的英语作文英文回答：The sense of hearing is an incredible gift that allows us to perceive and interpret sound. It is a complex process that involves the intricate interplay of our ears, brain, and nervous system. When sound waves reach our ears, they cause the eardrum to vibrate. These vibrations are then transmitted to the ossicles, a chain of three small bonesin the middle ear. The ossicles amplify the vibrations and send them to the inner ear, where they are converted into electrical signals by the cochlea. These signals are then sent to the auditory nerve, which carries them to the brain. The brain interprets these signals and allows us to hearand understand sound.Hearing is an essential sense that plays a crucial role in our daily lives. It allows us to communicate with others, navigate our environment, and enjoy music and otherauditory experiences. It also helps us to detect danger,such as the sound of an approaching car or a fire alarm.There are many different aspects to hearing, including pitch, loudness, and timbre. Pitch is the perceivedhighness or lowness of a sound, and is determined by the frequency of the sound waves. Loudness is the perceived strength of a sound, and is determined by the amplitude of the sound waves. Timbre is the perceived quality of a sound, and is determined by the shape of the sound waves.Hearing is a complex and fascinating sense that allows us to experience the world around us. It is a gift that we should cherish and protect.中文回答：听觉是一种不可思议的天赋，它让我们能够感知和理解声音。
superasegmental phonology举例-回复1. Introduction to Suprasegmental Phonology: Suprasegmental phonology refers to the study of features that extend beyond individual segments in speech, such as stress, intonation, pitch, and timing. These suprasegmental features play a crucial role in conveying meaning and expressing emotions in various languages. In this article, we will delve into the world of suprasegmental phonology, exploring different aspects and providing examples to illustrate their significance.2. Stress:One of the most prominent suprasegmental features is stress, which involves the emphasis placed on certain syllables or words in a sentence. English is a stress-timed language, meaning that stressed syllables are pronounced with greater length, loudness, and pitch prominence compared to unstressed syllables. For example, consider the sentence "He is going to the STORE." The capitalization indicates the stressed syllable in each word.3. Intonation:Intonation refers to the variation in pitch and tone patterns within speech. It plays a vital role in conveying different types ofinformation, such as indicating a question, expressing surprise, or conveying sarcasm. For instance, rising intonation at the end of a sentence typically denotes a question, as in "Are you coming?"4. Pitch:Pitch refers to the perceived highness or lowness of a sound. It can convey various meanings, such as differentiating between statements and questions, conveying emotions, or indicating sarcasm. For example, in Mandarin Chinese, different tones can completely change the meaning of a word. The word "ma" can mean "mother" (tone 1), "hemp" (tone 2), "horse" (tone 3), or "scold" (tone 4), depending on the pitch pattern.5. Timing:Timing refers to the duration and rhythm of speech. It plays a crucial role in determining the pace and flow of spoken language. For example, in Spanish, syllables within a word are typically evenly spaced with clear boundaries, whereas in English, timing can vary depending on stress and intonation patterns. Consider the difference in timing between the words "banana" and "bandana."6. Example 1 - Stress:Let's examine stress in English further. The word "record" can be pronounced with different stress patterns, leading to different meanings. With stress on the second syllable, it becomes a noun meaning an audio or visual recording. However, when stress is on the first syllable, it becomes a verb meaning to register or document something. The suprasegmental feature of stress significantly impacts the interpretation of words and sentences.7. Example 2 - Intonation:Intonation plays a crucial role in various languages to convey emotions and attitudes. Consider the question "You're leaving already?" with a rising intonation. This intonation pattern indicates surprise or disbelief, expressing the speaker's reaction to the situation. Changing the intonation pattern can alter the intended meaning and emotional content of the sentence.8. Example 3 - Pitch:As mentioned earlier, pitch variation can differentiate between words in tonal languages like Mandarin Chinese. Let's take the word "ma" as an example. When pronounced with a high level pitch (first tone), it means "mother." However, when pronounced with a rising pitch (second tone), it means "hemp." These differentpitch patterns take a basic syllable sound and completely change its meaning.9. Example 4 - Timing:Timing influences the rhythm and flow of spoken language. It helps determine the natural pauses and emphases within a sentence. Consider the sentence "I didn't say she stole my money." By placing emphasis on different words through timing, the sentence can convey seven different meanings, such as focusing on "I," "didn't," "she," "stole," "my," "money," or changing the overall emphasis on the sentence itself.10. Conclusion:Suprasegmental phonology encompasses stress, intonation, pitch, and timing, all crucial aspects that extend beyond individual segments in speech. These features play a fundamental role in conveying meaning, expressing emotions, and adding nuance to spoken language. Understanding and mastering suprasegmental phonology is essential for effective communication and language comprehension.。
用声音释放自己的情感英语Using Your Voice to Express Your Emotions in EnglishUnderstanding the Power of Your VoiceYour voice is a powerful tool for expressing your emotions.It can convey your feelings more effectively than words alone. When you speak English, remember that the tone, pitch, and volume of your voice can significantly impact how your emotions are perceived.1. Practice Emotional ExpressionStart by practicing expressing different emotions with your voice. Try to convey happiness, sadness, anger, and surprise using only your voice. Record yourself and listen back to identify areas for improvement.2. Use Tone to Convey EmotionThe tone of your voice can change the meaning of what you say. For instance, a high, excited tone can express joy, while a low, slow tone can communicate sadness.3. Vary Your PitchPitch is the highness or lowness of your voice. Varying your pitch can add emotion to your speech. A rising pitch can indicate a question or surprise, while a falling pitch can suggest finality or resignation.4. Control Your VolumeVolume is the loudness or softness of your voice. Speaking softly can show vulnerability or intimacy, while speaking loudly can show anger or excitement.5. Emphasize Key WordsWhen you want to emphasize a particular emotion, emphasizethe key words that carry the emotional weight of your message.6. Breathing TechniquesBreathing plays a crucial role in voice modulation. Deep, controlled breaths can help you maintain a steady voice,while quick, shallow breaths can convey urgency or panic.7. Body LanguageYour body language can also affect your voice. Stand or situp straight, and use gestures to support the emotions you're trying to express.8. Listen to Native SpeakersListening to native English speakers can help you understand how they use their voices to express emotions. Pay attentionto podcasts, movies, and songs.9. Role-PlayingEngage in role-playing exercises where you adopt different characters and express a range of emotions. This can help you become more comfortable with using your voice to convey feelings.10. Feedback and PracticeAsk for feedback from friends, teachers, or language partners. Practice regularly to improve your ability to use your voiceto express emotions effectively.By mastering these techniques, you can use your voice to release and express your emotions in English, making your communication more engaging and authentic.。
乐理英文专业术语English:Some basic music theory terms in English include pitch, rhythm, harmony, melody, key, scale, interval, chord, modulation, and form. Pitch refers to the highness or lowness of a sound, while rhythm isthe pattern of sounds and silences in music. Harmony is the combination of different musical notes played or sung simultaneously, while melody is a sequence of musical notes that is perceived as a single entity. The key of a piece of music is the groupof pitches, or scale, that forms the basis of the music, while scale is a series of notes in a specific pattern. An interval is the distance between two pitches, and a chord is a group of three or more notes played together. Modulation refers to the process of changing from one key to another within a piece of music, and form is the structure or organization of a piece of music.中文翻译:一些基本的音乐理论术语包括音高，节奏，和声，旋律，调式，音阶，音程，和弦，转调和形式。
描述唱歌常用英语English Answer:Terminology for Singing.Accompaniment: Instrumental music that accompanies a singer or choir.Aria: A solo vocal piece in an opera or oratorio.Bel canto: An Italian singing style characterized by beautiful tone production, flexibility, and agility.Chorus: A group of singers who perform together.Coloratura soprano: A soprano voice with exceptional agility and vocal range.Contralto: The lowest female voice type.Crescendo: A gradual increase in volume.Diminuendo: A gradual decrease in volume.Falsetto: A vocal register sung above the normal voice range, using the false vocal cords.Forte: A loud dynamic level.Fortissimo: The loudest dynamic level.Harmony: The combination of simultaneous musical notes that create pleasing sounds.Interval: The distance between two notes in terms of their pitch.Legato: A smooth and connected style of singing.Lyrics: The words to a song.Melody: A series of musical notes that form arecognizable tune.Opera: A dramatic musical work in which the characters sing the text.Oratorio: A large-scale choral work that tells a sacred or secular story.Piano: A soft dynamic level.Pianissimo: The softest dynamic level.Pitch: The highness or lowness of a musical note.Range: The range of notes a singer can produce.Rhythm: The pattern of beats and accents in music.Soprano: The highest female voice type.Tempo: The speed of a musical piece.Tenor: A mid-range male voice type.Tonality: The key or mode in which a piece of music is written.Vibrato: A slight oscillation in pitch that adds depth and richness to the voice.Chinese Answer:歌唱常用英语术语。
艺术类英文单词Art is a broad and diverse field that encompasses various genres, mediums, and expressions. As such, there are countless English words related to the arts, each holding its own significance and beauty. In this article, we will explore some essential vocabulary related to different forms of art, including painting, sculpture, literature, music, and cinema.1. PaintingPainting is a visual art form that utilizes pigments to create images on surfaces. Some key English words associated with painting are:- Canvas: A woven fabric used as a surface for painting.- Palette: A flat surface on which an artist mixes colors.- Brushstroke: A mark made by the brush on a canvas or painting surface.- Easel: A stand that holds the canvas or artwork while an artist is painting.2. SculptureSculpture involves creating three-dimensional artworks by carving, molding, or assembling materials. Here are some relevant vocabulary words:- Statue: A sculpture representing a person, animal, or object.- Marble: A type of stone often used in sculpting due to its smooth texture.- Chisel: A tool with a sharp cutting edge used for carving or shaping stone or wood.- Bronze: A metal alloy commonly used for casting sculptures.3. LiteratureLiterature encompasses written works, including novels, poems, plays, and essays. The following words are commonly used in discussions about literature:- Prose: Written or spoken language in its ordinary form, without metrical structure.- Metaphor: A figure of speech that draws comparison between two unrelated things.- Sonnet: A 14-line poem with a specific rhyme scheme and meter.- Plot: The sequence of events in a story.4. MusicMusic is an art form that uses sounds organized rhythmically, melodically, and harmonically. Some essential music-related vocabulary includes:- Melody: A sequence of musical notes arranged in a meaningful and memorable pattern.- Tempo: The speed at which a piece of music is played.- Pitch: The perceived frequency of a sound, determining highness or lowness.- Conductor: A person who leads an orchestra or choir during a musical performance.5. CinemaCinema refers to the art of creating and projecting motion pictures. The following words are frequently used in the context of cinema:- Director: The person responsible for overseeing the artistic aspects of a film and its production.- Cinematography: The art of capturing moving images on film or digital media.- Screenplay: The written script of a film, including dialogue, actions, and descriptions.- Editing: The process of selecting, arranging, and manipulating shots to create a coherent film.In conclusion, the world of art is rich with vocabulary that describes and defines various artistic endeavors. From painting to sculpture, literature to music, and cinema to photography, the English language offers a multitude of words to express the beauty and complexities of art. By familiarizing ourselves with these words, we can deepen our understanding and appreciation of the arts and their diverse forms of expression.。
乐理中英文对照English and Chinese Comparison" with a word count greater than 1000 words, written entirely in English without any additional punctuation marks in the body of the text.Music is a universal language that transcends cultural boundaries, allowing people from diverse backgrounds to connect and express themselves. At the core of this musical language lies the fundamental principles of music theory, which provide the framework for understanding and creating music. However, the way these principles are articulated and understood can vary significantly across different cultures and languages. In this essay, we will explore the nuances of music theory as it is expressed in both English and Chinese, examining the similarities, differences, and the insights that can be gained from this cross-cultural comparison.One of the most fundamental aspects of music theory is the concept of pitch, which refers to the highness or lowness of a sound. In the English language, pitches are typically named using the letters A through G, with additional modifiers such as sharp (#) or flat (b) to indicate the alteration of a pitch. This system is widely recognized and used in Western music traditions. In contrast, the Chinese musictheory employs a different approach to pitch naming, using a system of numbers and syllables. The seven main pitches are represented by the numbers 1 through 7, with additional modifiers such as "sharp" (shang) or "flat" (xia) to indicate the alterations. This numerical system, known as the Gong-Shang-Jue-Zhi-Yu (宫商角徵羽) system, is deeply rooted in the ancient Chinese philosophy of the five elements and the concept of the pentatonic scale.Another key aspect of music theory is the concept of scales, which are sequences of pitches arranged in a specific order. In the Western tradition, the most commonly used scale is the major scale, which consists of seven pitches and follows a specific pattern of whole and half steps. In the Chinese music theory, the concept of scales is closely tied to the Gong-Shang-Jue-Zhi-Yu system, with the pentatonic scale being the most fundamental and widely used. The pentatonic scale, which consists of five pitches, is believed to be closely connected to the natural harmonies of the universe and the emotional qualities of Chinese music.Another fascinating aspect of the comparison between English and Chinese music theory is the way in which chords, or the simultaneous sounding of multiple pitches, are conceptualized and understood. In the Western tradition, chords are typically classified based on their intervallic structure, with major, minor, diminished, and augmented chords being the most common. These chord types are oftenassociated with specific emotional qualities and are used to create harmonic progressions that drive the musical narrative.In the Chinese music theory, the concept of chords is less emphasized, and the focus is more on the melodic and modal aspects of music. Instead of relying on complex chord structures, Chinese music often explores the rich sonorities and emotional expressions that can be achieved through the interplay of different pitches within a modal framework. This modal approach to harmony is deeply rooted in the philosophical and cultural traditions of China, where music is often seen as a reflection of the natural world and the human experience.Despite these differences in the conceptual frameworks of music theory, there are also many areas of overlap and shared understanding between the English and Chinese traditions. For example, both systems recognize the importance of rhythm and the organization of musical time, with concepts such as meter, tempo, and rhythmic patterns being integral to the understanding and performance of music.Moreover, the universality of music as a means of human expression and communication is evident in the ways in which both English and Chinese music theory grapple with the emotional and expressive qualities of music. While the specific terminology and conceptualapproaches may differ, the underlying desire to capture and convey the emotional resonance of music is a common thread that binds these two musical traditions together.In conclusion, the comparison of music theory in English and Chinese reveals the rich diversity and complexity of the human experience of music. By understanding the nuances of these different approaches, we can gain a deeper appreciation for the cultural and philosophical underpinnings of musical traditions, and potentially uncover new avenues for cross-cultural collaboration and understanding. As we continue to explore the intersections and divergences between these two musical worlds, we may find that the true power of music lies in its ability to transcend linguistic and cultural boundaries, connecting us all through the universal language of sound.。
Generalized Lowness and Highness and ProbabilisticComplexity ClassesAndrew KlapperUniversity of ManitobaAbstractWe introduce generalized notions of low and high complexity classes and study their relation to structural questions concerning bounded probabilistic polynomial time com-plexity classes.We show,for example,that for a bounded probabilistic polynomial timecomplexity class C=BPΣP k,L C=H C implies that the polynomial hierarchy collapsesto C.This extends Sch¨o ning’s result for C=ΣP k(L C and H C are the low and highsets defined by C.)We also show,with one exception,that containment relations be-tween the bounded probabilistic classes and the polynomial hierarchy are preserved bytheir low and high counterparts.LBP P and LBP NP are characterized as NP∩BP Pand NP∩co-BP NP,respectively.These characterizations are then used to recover ofBoppana,Hastad,and Zachos’s result that if co-NP⊂BP NP,then the polynomialhierarchy collapses to BP NP,and Ko’s result that if NP⊂BP P,then the polynomialhierarchy collapses to BP P.Keywords:lowness,bounded probabilistic class,structural complexity theory,poly-nomial hierarchy1IntroductionRecently,increasing attention has been given to classes of problems for which fast solutions exist that involve probabilistic computation.Such problems lie in various probabilistic com-plexity classes,which can be defined in terms of sequences of existential,probabilistic,and universal quantifiers,in much the same way as the complexity classes in the polynomial hi-erarchy are defined by sequences of existential and universal quantifiers.Examples of such classes include BP P,originally defined by Gill[5],and further studied,e.g.,by Sipser[16], and Sch¨o ning[13],the various forms of Arthur-Merlin combinatorial game classes,defined by1Babai[2],the interactive proof systems,defined by Goldwasser,Micali,and Rackoff[7],and further studied,e.g.,in[8],[1],[6],[4],and others,BP NP,defined by Sch¨o ning[14],and, more generally,the probabilistic complexity classes defined by sensible pairs of sequences of quantifiers,defined by Hinman and Zachos[9],and studied in[17],[18],and[3].This list is not intended to be exhaustive,but rather to give an outline of some of the major papers in the growing body of work in thisfield.A great many identities have been proved among these classes.For example,BP NP= AM(see[14])=IP(see[8])=(∃+∃/∃+∀)(see[9]).Much of what is known about identities and inclusions between these classes is summarized in section2.It turns out that a large number of these classes are equal to classes in what can be called the bounded probabilisticpolynomial hierarchy,{BPΣPk ,BPΠPk,k=0,1,...}.It has been shown in[9]that theseclasses in fact lie in the polynomial hierarchy.In fact,for k≥1,BPΣPk ⊂ΠPk+1.A basicquestion is what containment relationship exists between BP P and NP.That is,which(if either)is stronger,randomness or nondeterminism?A major outstanding problem of complexity theory is whether the polynomial hierarchy collapses,and,if so,to what level.Much work has been done attempting to relate the collapse of the polynomial hierarchy to other structural phenomena.Sch¨o ning[12]definedtwo hierarchies of complexity classes-the low and high hierarchies{L Pk }and{H Pk}-lyinginside NP.He went on to show that the polynomial hierarchy collapses to the k th level ifand only if L Pk ∩H Pk=∅if and only if L Pk=H Pk.In[13]Sch¨o ning showed that(1)NP∩co-BP NP⊂L P2,and graph isomorphism is in co-BP NP.It follows that if graph isomorphismis NP-complete,then the polynomial hierarchy collapses toΣP2.This can also be proved usingthe result of Boppana,Hastad,and Zachos[3]that if co-NP is contained in BP NP,then the polynomial hierarchy collapses to BP NP.We note also that Ko[10]showed that if NP is contained in BP P,then the polynomial hierarchy collapses to BP P.The purpose of this paper is to extend Sch¨o ning’s lowness and highness techniques to the bounded probabilistic polynomial complexity classes.In particular,in section3we extend the definitions of lowness and highness to the bounded probabilistic polynomial classes(these definitions,in fact,apply to any complexity class for which we have a reasonable notion of extension by oracles).For each k≥0,this gives classes L C and H C for each class C in the bounded probabilistic polynomial hierarchy.We then show(with two exceptions)that the inclusions among the bounded probabilistic polynomial classes carry over to their low and high counterparts.Section4contains our main results.Wefirst show that for each such C, the polynomial hierarchy collapses to BP C if and only if LBP C∩HBP C=∅if and only if LBP C=HBP C.We then characterize LBP P as NP∩BP P,and characterize LBP NP as NP∩co-BP NP.These facts are then used to recover the hierarchy collapsing results mentioned above.It is a pleasure to thank Prof.S.Zachos for advice which helped lead to these results.I2am also grateful to Prof.U.Sch¨o ning and Prof.L.Longpr´e for their many suggestions,and to the participants in the joint Northeastern-BU complexity seminar for support and useful discussions.2Bounded Probabilistic ClassesIn this section the definition of bounded probabilistic complexity classes is recalled,as defined for example by Sch¨o ning[15],along with that of Hinman and Zachos’[9]more general classes defined by sensible pairs of quantifier sequences.The known structural relations among these classes are summarized.Definition1Let C be any complexity class.Then L∈BP C,if and only if there is a language K∈C and polynomials p(n)and q(n)>1such that x∈L implies P rob{y||y|=p(|x|)∧(x,y)∈K}>1−2−q(|x|),and x/∈L implies P rob{y||y|=p(|x|)∧(x,y)∈K}<2−q(|x|).More generally,we can consider complexity classes defined by sequences of quantifiers.We use a notation due to Hinman and Zachos[9].All quantifiers in this paper will be∃,∀,or ∃+,where∃+y|P(y)means P rob{y|P(y)}>2/3(or,more generally,∃+P(x,y)means there are polynomials p(n),q(n)such that,given input x,P rob{y,|y|=p(|x|)|P(x,y)}>2−q(|x|)). In general,quantifiers are assumed to have range bounded polynomially in the length of the input.Note that many of the definitions here also make sense for the quantifier ,meaning “at leastfifty percent”,but we will not discuss classes defined by here.We can now define quantifier classes as followsDefinition2(Zachos) 1.A pair(Q/Q )offinite sequences of quantifiers is sensible if they are of the same length and for any predicate P(x,y),(Q y:¬P(x,y)⇒¬Qy: P(x,y)).2.Let P(x,y)be a predicate and(Q/Q )be a sensible pair.If,for every x,either(Qy:P(x,y))or(Q y:¬P(x,y)),(these sets are disjoint by sensibility)then the language {x|Qy:P(x,y)}will be denoted by(Q/Q )P.The complement of L is{x|Q y:¬P(x,y)}.If K is a language,we will write(Q/Q )K for(Q/Q )((x,y)∈K),and,ifC is a complexity class,we will write(Q/Q )C for the class of all languages of the form(Q/Q )K,K∈C.We also write(Q/Q )for(Q/Q )P(where P is the class of polynomial time computable languages).Note that co-(Q/Q )K=(Q /Q)co-K.Many of the classes defined by sensible quantifier pairs are known by other names.For example,1.(∃/∀)=NP;32.(∃∀...Q k/∀∃...Qk )=ΣPk;(∀∃...Q k/∃∀...Qk)=ΠPk;3.(∃+/∀)=R;4.(∃+/∃+)=BP P;5.(∃+∃/∃+∀)=(∀∃/∃+∀)=BP NP=AM(the Arthur-Merlin games,defined originallyby Babai[2].This equality was proved by Hinman and Zachos[9]);6.(∃∃+/∀∃+)=(∃∀/∀∃+)=MA(the Merlin-Arthur games,also defined in[2],the equal-ity proved in[9]);7.(∃+∃∀...Q k/∃+∀∃...Qk )=(∀∃∀...Q k/∃+∀∃...Qk)=BPΣPk−1.It has been shown that all sensible quantifier pairs(either when considered as defining complexity classes,or as operators on classes which are closed downward under polynomial time many-one reducibility)reduce to the pairs in the above list,or their complements.We can think of sensible quantifier pairs as operators on complexity classes.The composition of two quantifier pairs is not,a priori,a quantifier pair.We do,however, have an inclusion in one direction:(Q/Q )(R/R )C⊂(QR/Q R )C.The reverse inclusion is only known to hold in special cases.To see the difference between these classes,note that L is in(QR/Q R )C if and only if there is a K∈C such that x∈L⇒(QyRz:(x,y,z)∈K) and x∈L⇒(Q yR z:(x,y,z)∈K).L is in(Q/Q )(R/R )C if and only if,there is a K such that,in addition,for every(x,y),either(Rz:(x,y,z)∈K)or(R z:(x,y,z)∈K)(that is, {(x,y)|Rz:(x,y,z)∈K}is in(R/R )C.)Definition3A pair(R/R )of quantifier sequences is complimentary if for every predicate P,(¬Ry:P(x,y))⇔(Ry:¬P(x,y)).Note that a complimentary quantifier pair is necessarily sensible.Of the pairs built from ∃,∀,and∃+,only(∃/∀),(∀/∃),and the various compositions of these two pairs are compli-mentary.Lemma1Let(R/R )be a complimentary quantifier pair,and let(Q/Q )be a sensible quan-tifier pair.Then for any class C,(Q/Q )(R/R )C=(QR/Q R )C.4Proof:The inclusion⊂holds in general.To see the reverse inclusion,let L∈(QR/Q R )C. Then there is a J∈C such thatx∈L⇒QyRz:(x,y,z)∈Jx∈L⇒Q yR z:(x,y,z)∈J.Let K be the set of pairs(x,y)such that Rz:(x,y,z)∈J.We will show that K∈(R/R )C, and that L=(Q/Q )K,proving the lemma.To see thefirst assertion,note that(x,y)∈K⇒Rz:(x,y,z)∈J(x,y)∈K⇒¬Rz:(x,y,z)∈J⇒R z:(x,y,z)∈JTo see the second assertion,note thatx∈L⇒QyRz:(x,y,z)∈J⇒Qy:(x,y)∈Kx∈L⇒Q yR z:(x,y,z)∈J⇒Q y¬Rz:(x,y,z)∈J⇒Q y:(x,y)∈K2 The following identities and inclusions are useful in proving the identifications above and in computing compositions of sensible pairs of quantifier sequences.They were proved by Hinman and Zachos[9].Theorem1(Hinman and Zachos) 1.(∃+/∃+)=(∃+∀/∀∃+)=(∀∃+/∃+∀);2.(∃∀/∃∃+)⊂(∀∃/∃+∀);3.(∃+∃/∃+∀)=(∀∃/∃+∀);4.(∃∃+/∀∃+)=(∃∀/∀∃+);5.For any quantifier Q,(∀/Q)⊂(∃+/Q)⊂(∃/Q)(where we treat pairs that are notsensible as defining empty classes);5In order to identify classes defined by oracles,it is desirable to be able to reduce such classes to classes defined by the composition of quantifier pairs.The following lemma allows us to do so in several important cases.First we need a definition.Definition4Let C be a complexity class.C will be said to be robust if for every language K in C the following two languages,K1and K2,are in C as well.1.Let#be an additional symbol.Define K1=K#∗.2.Let p(n)be a polynomial.Define K2={<x1,...,x m>|x1,...,x m∈K,|x1|=...=|x m|=n,m≤p(n)},where<,>is a pairing function.Note that in this definition K1is many-one equivalent to K,K2is d-reducible to K,and K is many-one reducible to K2.Thus any complexity class which is closed under d-reducibility is robust.In particular,all the classes in the polynomial hierarchy,as well as the bounded probabilistic classes derived from the classes in the polynomial hierarchy are robust.Lemma2Let C be a robust complexity class.Let K be any language in C,M a deterministic polynomial time oracle Turing machine,and let L=L(M,K).Then there is a language K ∈C and a polynomial time nondeterministic oracle Turing machine M which,after its initial guess,computes deterministically,generating a pair of query strings y1and y2,and accepts if and only if y1is accepted and y2is rejected by the oracle,such that L=L(M ,K ).Proof:By thefirst robustness condition,we may assume that all queries made by M are for strings of the same length.More precisely,we may assume that there is a polynomial p(n) such that,on input x,M produces only query strings of length p(|x|).We may also assume there is a polynomial q(n)such that on input x,M makes precisely q(|x|)queries,and that q(n)is of the form r(p(n))for some polynomial r(n).Let K be produced from K by the second robustness condition,using r(n)as the polynomial.Now we can define M as follows: on input x,guess a binary string y of length q(|x|).M maintains a pair of strings,u and v,initially null.M simulates the behavior of M,but when M queries K with the i th query string z,M examines the i th bit of y.If this bit is a1,then M concatenates z onto u and continues as if the query to K accepted.Otherwise,M concatenates z onto v and continues as if the query had rejected.If,at the end of the simulation,M rejects,then M rejects.If, however,M accepts,then M accepts if and only if u is in K and v is not in K .This machine has the desired properties.2 We refer to the two oracle queries y1and y2as positive and negative queries,respectively. We can apply this result to Zachos’quantifier classes.6Corollary1Let C be a robust complexity class.Let M be a deterministic polynomial time oracle Turing machine,let K∈C,and let(Q∃/Q ∀)be a sensible pair of quantifier sequences. Suppose L=(Q∃/Q ∀)L(M,K)is a well defined language,as in definition2.2.Then there is a deterministic polynomial time oracle Turing machine M and a language K ∈C,with L=(Q∃/Q ∀)L(M ,K ),such that M computes deterministically,then makes two oracle queries y1and y2,accepting if and only if y1is accepted and y2is rejected by the oracle.Proof:By lemma1,the guessing stage of the machine M in lemma2can be combined with the inner quantifier pair.2If C is closed under many-one reducibility,(i.e.,L≤Pm K∈C⇒L∈C)then so is(Q∃/Q ∀)C.Therefore,if C is closed under many-one reducibility,and the negative query can be eliminated,then,in fact,L(in the corollary)is in(Q∃/Q ∀)C.3Low and High ClassesIn this section we generalize the definitions of low and high complexity classes given by Sch¨o ning[12].The purpose of defining these classes is to provide a tool for studying the relationships among complexity classes in the polynomial hierarchy(PH).Sch¨o ning[12]has shown that the separation properties of certain classes in PH are reflected in separation properties of the low and high classes in NP.We will define notions of lowness and highness relative to complexity classes for which there is a reasonable notion of extendibility by adding oracles.This includes,for example,all complexity classes defined by Turing machines with some restrictions on computation paths (e.g.,certain numbers of paths accept for input in the given language,or paths below a certain point in the computation tree are polynomial time computable.)We do not have an axiomatic notion of extention of complexity classes by oracles.This is natural since oracles really ex-tend the machines models underlying complexity classes.Strictly speaking,when discussing extensions of complexity classes by oracles,we should define the extension of machine models by oracles.Different machine models for a given class may give different notions of oracle. For example,if P=NP,then deterministic and nondeterministic polynomial time bounded Turing machines are machine models for the same class.But it is well known that there is an oracle A such that P A=NP A,that is,the two machine models give different notions of oracle.However,at the very least,for a complexity class C containing P to be extendable by oracles we should have,for every language L,a complexity class C L such that1.If L∈P then C L=C.2.For every K,if L≤PTK,then C L⊂C K.73.C⊂C L.We will not explore further here the question of axioms for oracles.Definition5Let C be a complexity class with a notion of extension by oracle.Let L be a language in NP.Then L will be said to be low-C(respectively,high-C)if C L=C(resp., C L=C SAT).We denote by L C(respectively,HC)the set of languages which are low-C(resp., high-C.)For C=ΣPi orΠPi,i=0,1,...this definition gives the low and high classes L PiandH Pi as defined by Sch¨o ning[12].We will be particularly interested in the classes LBPΣPiand HBPΣPi ,i=0,1,...defined by the bounded probabilistic complexity classess BPΣPi.For any class C for which C L is defined for any language L,we define(BP C)L=BP(C L). More generally,if(Q/Q )is a sensible pair of quantifier sequences,we define((Q/Q )C)L= (Q/Q )(C L).In all cases in which these classes correspond to classical complexity classes, these definitions are consistent with the traditional definitions of oracles.Ourfirst goal is to show that the containment relationships that hold for the bounded probabilistic classes and PH hold for the low and high classes derived from them.Recall from[12]that for k≥0,L Pk ⊂L Pk+1,and H Pk⊂H Pk+1.We extend this result toTheorem2 1.For i≥0,L Pi ⊂LBPΣPi,and H Pi⊂HBPΣPi.2.For i≥1,LBPΣPi ⊂L Pi+1,and HBPΣPi⊂H Pi+1.3.LBP P⊂LMA⊂LBP NP,and HBP P⊂HMA⊂HBP NP. Proof:1.Let A∈L Pi .Then(ΣPi)A=ΣPi.By lemma1,(BPΣPi)A=(∃+/∃+)(ΣPi)A=(∃+/∃+)(ΣPi )=BPΣPi,so A∈LBPΣPi.Let A∈H Pi .Then(ΣPi)A=(ΣPi)SAT.Consequently,(BPΣPi)A=(∃+/∃+)(ΣPi)A=(∃+/∃+)(ΣPi )SAT=(BPΣPi)SAT,so A∈HBPΣPi.2.First note that for i≥1ΠPi+1=(∀/∃)ΣPi⊂(∀/∃)(∃+/∃+)ΣPi=(∀/∃)BPΣPi⊂(∀/∃)ΠPi+1by extending Lautemann[11]=ΠPi+1by lemma1.8Moreover,all these relations hold relative to an oracle.It follows that for any languageA,(ΠPi+1)A=((∀/∃)BPΣPi)A=(∀/∃)(BPΣPi)A.Now let A∈LBPΣPi .Then(BPΣPi)A=BPΣPi.We will show that(ΠPi+1)A=ΠPi+1,and it will follow that A∈L Pi+1.We have(ΠPi+1)A=(∀/∃)(BPΣPi)A=(∀/∃)BPΣPi=ΠPi+1The second inclusion is proved similarly.3.Let A∈LBP P,so BP P A=BP P.We haveMA A=(∃∃+/∀∃+)A⊂(∃∃+/∀∃+)(∃+/∃+)A=(∃∃+/∀∃+)(∃+/∃+)⊂(∃∃+/∀∃+)=MASo A∈LMA.Next let A∈LMA,so MA A=MA.We haveBP NP A=(∃+∃/∃+∀)A=(∃+/∃+)(∃/∀)A by lemma1⊂(∃+/∃+)(∃∃+/∀∃+)A=(∃+/∃+)MA A=(∃+/∃+)MA⊂(∃+∃∃+/∃+∀∃+)⊂(∃+∃+∃/∃+∃+∀)by theorem1=(∃+∃/∃+∀)=BP NPSo A∈LBP NP.The remaining inclusions are proved similarly.29r r r r r r r r r ¨¨¨¨¨¨¨¨¨d d d d d d dd d L P 0LBP P LMA LBP NP L ΣP 2=L P 2LBP ΣP 2LNP =L P 1Figure 1:Containment relations for low classes10Figure3describes the inclusions known among the low classes.Replacing“L”by“H”in thisfigure gives the known inclusions among the high classes.Note that while NP⊂MA, we have not been able to show that L1⊂LMA.On the other hand,we view the inability to prove a containment between LBP P and L1as a reflection of the inability to prove a containment between BP P and NP.More generally,given complexity classes C⊂D,both extendable by oracles,the question arises whether L C⊂L D and H C⊂H D.The dependence of the above proofs on characteristics of the specific complexity classes involved makes these general inclusions seem unlikely.The possibility remains that axioms can be found that make these inclusions true,while still holding for all generally accepted notions of oracle.4Collapsing the Polynomial HierarchySch¨o ning[12]has shown that,for every k≥0,if the polynomial hierarchy does not collapsetoΣPk ,then L Pkis disjoint from H Pk,and if the polynomial hierarchy collapses toΣPk,thenL P k =H Pk.We next extend these results to bounded probabilistic classes.Wefirst need toprove that if one level of the bounded probabilistic polynomial hierarchy collapses,then the entire hierarchy(and hence the polynomial hierarchy)collapses.Lemma3For all k≥0,if BPΣPk =BPΣPk+1,then the polynomial hierarchy collapses toBPΣPk.Proof:It suffices to show inductively that for all m≥k,BPΣPm =BPΣPm+1,the initialcase being true by hypothesis.If BPΣPm =BPΣPm+1,then BPΠPm=BPΠPm+1,soBPΣPm+2=(∃+∃/∃+∀)ΠPm+1⊂(∃+∃/∃+∀)BPΠPm+1=(∃+∃/∃+∀)BPΠPmby hypothesis⊂(∃+∃∃+/∃+∀∃+)ΠPm⊂(∃+∃+∃/∃+∃+∃)ΠPmby theorem1=(∃+∃/∃+∀)ΠPm=BPΣPm+1.2It can be shown similarly that ifΣPk+1⊂BPΣPk,then the polynomial hierarchy collapsesto BPΣPk ,and that ifΣPk=BPΣPk,then the polynomial hierarchy collapses toΣPk.Theorem3For each k≥0,111.If the polynomial hierarchy does not collapse to BPΣPk ,then LBPΣPk∩HBPΣPk=∅.2.If the polynomial hierarchy collapses to BPΣPk ,then LBPΣPk=HBPΣPk=NP.Proof:If A∈LBPΣPk ∩HBPΣPk,then BPΣPk+1⊂BP(ΣPk)A⊂BPΣPk,hence BPΣPk+1=BPΣPk .Thefirst assertion then follows from lemma3.Let A∈NP.Then BPΣPk ⊂BP(ΣPk)A⊂BPΣPk+1.If the polynomial hierarchy collapsesto BPΣPk ,then these three sets are equal,implying A is in LBPΣPkand in HBPΣPk.2It is well known that L0=P and L1=NP∩co-NP.The next result characterizes LBP NP and LBP P.Proposition1 1.BP NP BP NP∩co-BP NP=BP NP.2.LBP NP=NP∩co-BP NP=NP∩co-BP NP∩BP NP.3.(Zachos)BP P BP P=BP P.4.LBP P=NP∩BP P.Proof:To see the assertion,let L be a language in BP NP BP NP∩co-BP NP.By corollary1, we may assume that L is recognized by a BP NP oracle machine which,for each computation path,makes a single positive query to a language in BP NP,and a single negative query to a language in co-BP NP.A negative query to a language in co-BP NP is the same as a positive query to a language in BP NP,and,by robustness,two positive queries to BP NP can be replaced by one such query,so L is in(∃+∃/∃+∀)(∃+∃/∃+∀)=(∃+∃/∃+∀)=BP NP.This gives thefirst assertion.Next we prove the second assertion.By thefirst assertion,NP∩co-BP NP⊂LBP NP, since NP⊂BP NP.LBP NP⊂NP by definition,so next we show that LBP NP⊂co-BP NP∩BP NP.But L∈LBP NP⇒BP NP L=BP NP⇒L∈BP NP.Moreover,any machine that queries L can be replaced by one that recognizes the same language,but queries L.Thus L∈BP NP,i.e.,L∈co-BP NP.On the other hand,NP∩co-BP NP∩BP NP⊂NP∩co-BP NP,hence we get a series of inclusions NP∩co-BP NP⊂LBP NP⊂NP∩co-BP NP∩BP NP⊂NP∩co-BP NP. These inclusions collapse to give equality.The last assertion follows from the third by a similar argument.2 Finally,we can use proposition1to recover a result of Ko[10]and a result of Boppana, Hastad,and Zachos[3]on the collapse of the polynomial hierarchy.12Theorem4 1.(Ko)If NP is contained in BP P,then the polynomial hierarchy collapses to BP P.2.(Boppana,Hastad,Zachos)If co-NP is contained in BP NP then the polynomial hier-archy collapses to BP NP.Proof:If NP⊂BP P,thenLBP P=NP∩BP P,by proposition1=NPHence LBP P∩HBP P is nonempty.By theorem3,the polynomial hierarchy collapses to BP P.If co-NP⊂BP NP then NP⊂co-BP NP,soLBP NP=NP∩co-BP NP,by proposition1=NPHence LBP NP∩HBP NP is nonempty.By theorem3,the polynomial hierarchy collapses to BP NP.2 For a concrete example of the significance of this problem,recall the problem GRAPH ISO of determining whether two graphs are isomorphic.This problem is known to be in NP, but it is anopen problem whether it is NP-complete.It has been shown([13])that GRAPH ISO∈co−BP NP.It follows from proposition1and theorem3that if GRAPH ISO is in HBP NP(which holds,for example,if GRAPH ISO is NP-complete)then the polynomial hierarchy collapses to BP NP.5ConclusionsWe have extended Sch¨o ning’s lowness and highness techniques to bounded probabilistic poly-nomial time classes,and proved some basic properties.We have also showed how these techniques can be used to recover known results on the collapse of the polynomial hierarchy. Several questions remain unanswered.We have characterized LBP P and LBP NP,in termssimilar to the characterizations of L P1and L P-in all these cases,L C=co-C∩NP.Unlessthe polynomial hierarchy collapses,this pattern will not persist at or above C=ΣP2,forthen the intersection would give NP,which includes H C.It is not known,however,whether13LMA=co-MA∩NP.The reduction of oracle queries to a single positive and a single negative query is not adequate here,since the inner quantifier pair in MA is(∃+/∃+).More generally,we would like characterizations of the remaining low and high sets.High sets appear harder to characterize in general.The known results depend on the existence of complete sets of various types.It may be necessary to invent probabilistic reducibilities in order to characterize high bounded probabilistic sets.With or without such a characterization,we conjecture that if co-NP is a subset of MA, then the polynomial hierarchy collapses to MA.We would like to see a more universal result that implies all such collapsing conditions.Finally,we would like to apply these techniques to other classes in the poynomial hierarchy,such as D P,∆P2,and ZP P NP.By so doing,we would hope to gain insight into severalquestions raised by Zachos and F¨u rer[18]concerning containment between these classes and the bounded probabilistic classes.References[1]W.Aiello,S.Goldwasser,J.Hastad.“On the power of interaction”,Proceedings of27thAnnual IEEE Symposium on Foundations of Computer Science,1986,368-379.[2]L.Babai.“Trading group theory for randomness”,Proceedings of17th Symposium on theTheory of Computation,1985,421-429.[3]R.Boppana,J.Hastad,S.Zachos.Does co-NP have short interactive proofs?,Manuscript,1986.[4]A.Condon,dner.“Probabilistic game automata”,Proceedings of1st Annual Con-ference on Structure in Complexity Theory,1986,Lecture Notes in Computer Science, 223,Springer-Verlag,1986,144-162.[5]putational complexity of probabilistic Turing machines,SIAM Journal onComputing,6,1977,675-695.[6]O.Goldreich,S.Micali,A.Wigderson.“Proofs that yield nothing but the validity of theassertion and the methodology of cryptographic protocol design”,Proceedings of27th Annual IEEE Symposium on Foundations of Computer Science,1986,174-187.[7]S.Goldwasser,S.Micali,C.Rackoff.“The knowledge complexity of interactive proof-systems”,Proceedings of the17th Annual Symposium on the Theory of Computation, 1985,291-304.14[8]S.Goldwasser,M.Sipser.“Private coins versus public coins in interactive proof systems”,Proceedings of the18th Annual Symposium on the Theory of Computation,1986,59-68.[9]P.Hinman,S.Zachos.“Probabilistic machines,oracles,and quantifiers”,Proceedings ofthe Oberwolfach Recursion-theoretic Week,Lecture Notes in Mathematics,114,Springer-Verlag,1984,159-192.[10]K.Ko.Some observations on the probabilistic algorithms and NP-hard problems,Infor-mation Processing Letters,14,39-43.[11]utemann.BP P and the polynomial hierarchy,Information Processing Letters,17,1983,215-217.[12]U.Sch¨o ning.A low and high hierarchy within NP,Journal of Computer and SystemSciences,27,1983,14-28.[13]U.Sch¨o ning.“Graph isomorphism is in the low hierarchy”,Proceedings of the4th STACS,Lecture Notes in Computer Science,247,Springer-Verlag,1986,114-124.[14]U.Sch¨o ning.“Complexity and Structure”,Lecture Notes in Computer Science,211,Springer-Verlag,1986.[15]U.Sch¨o ning.“Probabilistic complexity classes and lowness”,Proceedings of Second An-nual Conference on Structure in Complexity Theory,1987,2-8.[16]M.Sipser.“A complexity theoretic approach to randomness”,Proceedings of the15thAnnual ACM Symposium on the Theory of Computation,1983,330-335.[17]S.Zachos.“Probabilistic quantifiers,adversaries,and complexity classes”,Proceedingsof the First Annual Conference on Structure in Complexity Theory,Lecture Notes in Computer Science,223,Springer-Verlag,1986,383-400.[18]S.Zachos,M.F¨u rer.Probabilistic quantifiers vs.distrustful adversaries,Manuscript,1985.15。
