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LOL
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This article is about the internet slang acronym. For other uses, see LOL (disambiguation).
"Laugh out loud" redirects here. For the CBC Radio One radio show, see Laugh Out Loud (radio). For the ABS-CBN television show, see Laugh Out Loud (TV series).
A lolcat using "LOL"
LOL, or lol, is an acronym for laugh(ing) out loud[1][2][3] or lots of laughs,[4][5][6] and a popular element of Internet slang. It was first used almost exclusively on Usenet, but has since become widespread in other forms of computer-mediated communication and even face-to-face communication. It is one of many initialisms for expressing bodily reactions, in particular laughter, as text, including initialisms for more emphatic expressions of laughter such as LMAO[7] ("laugh(ing) my ass off") and ROFL[8][9][10] (or its older form ROTFL;[11][12] "roll(ing) on the floor laughing"). Other unrelated expansions include the now mostly obsolete "lots of luck" or "lots of love" used in letter-writing.[13]
The list of acronyms "grows by the month",[9] and they are collected along with emoticons and smileys into folk dictionaries that are circulated informally amongst users of Usenet, IRC, and other forms of (textual) computer-mediated communication.[14] These initialisms are controversial, and several authors[15][16][17][18] recommend against their use, either in general or in specific contexts such as business communications.
LOL was first documented in the Oxford English Dictionary in March 2011.[19]
Contents
1 Analysis
2 Acceptance
3 Variations on the theme
3.1 Variants
3.2 Derivations
3.3 Related
3.4 Commonly used equivalents in other languages
3.5 The word "lol" in other languages
4 See also
5 References
6 Further reading
7 External links
Analysis
Laccetti (professor of humanities at Stevens Institute of Technology) and Molski, in their essay entitled The Lost Art of Writing, are critical of the terms, predicting reduced chances of employment for students who use such slang, stating that, "Unfortunately for these students, their bosses will not be 'lol' when they read a report that lacks proper punctuation and grammar, has numerous misspellings, various made-up words, and silly acronyms."[15][16] Fondiller and Nerone in their style manual assert that "professional or business communication should never be careless or poorly constructed" whether one is writing an electronic mail message or an article for publication, and warn against the use of smileys and abbreviations, stating that they are "no more than e-mail slang and have no place in business communication".[17]
Linguist John McWhorter stated, "Lol is being used in a particular way. It's a marker of empathy. It's a marker of accommodation. We linguists call things like that pragmatic particles…" Pragmatic particles are the words and phrases utilized to alleviate the awkward areas in casual conversation, such as oh in "Oh, I don’t know" and uh when someone is thinking of something to say. McWhorter stated that lol is utilized less as a reaction to something that is hilarious, but rather as a way to lighten the conversation.[20]
Yunker and Barry in a study of online courses and how they can be improved through podcasting have found that these slang terms, and emoticons as well, are "often misunderstood" by students and are "difficult to decipher" unless their meanings are explained in advance. They single out the example of "ROFL" as not obviously being the abbreviation of "rolling on the floor laughing" (emphasis added).[18] Haig singles out LOL as one of the three most popular initialisms in Internet slang, alongside BFN[dubious – discuss] ("bye for now") and IMHO ("in my honest/humble opinion"). He describes the various initialisms of Internet slang as convenient, but warns that "as ever more obscure acronyms emerge they can also be rather confusing".[1] Bidgoli likewise states that these initialisms "save keystrokes for the sender but [...] might make comprehension of the message more difficult for the receiver" and that "[s]lang may hold different meanings and lead to misunderstandings especially in international settings"; he advises that they be used "only when you are sure that the other person knows the meaning".[21]
Shortis observes that ROFL is a means of "annotating text with stage directions".[10] Hershock, in discussing these terms in the context of performative utterances, points out the difference between telling someone that one is laughing out loud and actually laughing out loud: "The latter response is a straightforward action. The former is a self-reflexive representation of an action: I not only do something but also show you that I am doing it. Or indeed, I may not actually laugh out loud but may use the locution 'LOL' to communicate my appreciation of your attempt at humor."[9]
David Crystal notes that use of LOL is not necessarily genuine, just as the use of smiley faces or grins is not necessarily genuine, posing the rhetorical question "How many people are actually 'laughing out loud' when they send LOL?".[22] Franzini concurs, stating that there is as yet no research that has determined the percentage of people who are actually laughing out loud when they write LOL.[2]
Victoria Clarke, in her analysis of telnet talkers, states that capitalization is important when people write LOL, and that "a user who types LOL may well be laughing louder than one who types lol", and opines that "these standard expressions of laughter are losing force through overuse".[23] Egan describes LOL, ROFL, and other initialisms as helpful so long as they are not overused. He recommends against their use in business correspondence because the recipient may not be aware of their meanings, and because in general neither they nor emoticons are in his view appropriate in such correspondence.[3] June Hines Moore shares that view.[24] So, too, does Lindsell-Roberts, who gives the same advice of not using them in business correspondence, "or you won't be LOL".[25]
Acceptance
On March 24, 2011, LOL, along with other acronyms, was formally recognized in an update of the Oxford English Dictionary.[19][26] In their research, it was determined that the earliest recorded use of LOL as an initialism was for "little old lady" in the 1960s.[27] They also discovered that the oldest written record of the use of LOL in the contemporary meaning of "Laughing Out Loud" was from a message typed by Wayne Pearson in the 1980s, from the archives of Usenet.[28]
Gabriella Coleman references "lulz" extensively in her anthropological studies of Anonymous.[29][30]
A 2003 study of college students by Naomi Baron found that the use of these initialisms in computer-mediated communication (CMC), specifically in instant messaging, was actually lower than she had expected. The students "used few abbreviations, acronyms, and emoticons". Out of 2,185 transmissions, there were 90 initialisms in total;[31] 76 were occurrences of LOL.[32]
LOL, ROFL, and other initialisms have crossed from computer-mediated communication to face-to-face communication. David Crystal—likening the introduction of LOL, ROFL, and others into spoken language in magnitude to the revolution of Johannes Gutenberg's invention of movable type in the 15th century—states that this is "a brand new variety of language evolving", invented by young people within five years, that "extend[s] the range of the language, the expressiveness [and] the richness of the language".[33][31] However Geoffrey K. Pullum argues that even if interjections such as LOL and ROFL were to become very common in spoken English, their "total effect on language" would be "utterly trivial".[34]
Variations on the theme
Variants
lul: phonetic spelling of LOL. "LUL" is also commonly used in the gaming community, due to it being an emote from BetterTTV, an extension of Twitch.tv.
lolz: Occasionally used in place of LOL.
lulz: Often used to denote laughter at someone who is the victim of a prank, or a reason for performing an action. Its use originated with Internet trolls. According to a New York Times article about Internet trolling, "lulz means the joy of disrupting another's emotional equilibrium."[35] Can be used as a noun – e.g. "do it for the lulz.", shortened into "ftlulz" (to distinguish it from "ftl" – "for the loss"). See also LulzSec.
LOLOLOLOLOLOLOLOLOLOL: For added emphasis, LOL can be appended with any number of additional iterations of "OL". In cases such as these, the abbreviation is not to be read literally (i.e., "Laughing out loud out loud out loud out loud"), but is meant to suggest several LOLs in a row.
trolololol or trollololol: A portmanteau of troll and LOL iterated. Indicates that the prank or joke was made by internet trolls, or the user thinks the prank or joke qualifies as internet trolling. See also Mr. Trololo.
Derivations
lolwut (sometimes "lulwut"): lol + wut, used to indicate bemused laughter, or confusion.
lawl, lawlz, or lal: Pseudo-pronunciation of LOL. Saying "lawl" is sometimes meant in mockery of those who use the term LOL, and is not meant to express laughter.
lel: A satirical form of lol, often used to denote bemusement or lack of amusement in response to a statement.[citation needed]
Related
An animated ASCII art image popularized in 2004 by memes using the word "roflcopter".
*G* or *g*: For "grins".[36] Like "lulz" it is used in the initialism "J4G" ("just for grins").[37]
kek: A term for laughter that originated in online games, possibly either World of Warcraft or Starcraft, the latter in which Korean players would type "kekeke" as onomatopoeia for laughter.[38] It later became associated with alt-right politics,[39] in the form of a parody religion surrounding the character Pepe the Frog by analogy with the frog-headed ancient Egyptian god Kek.[40]
LMAO: For "laughing my ass off".[7] Variant: LMFAO ("Laughing my fucking ass off").
lqtm: For "Laughing quietly to myself".[41]
ROFL: For "rolling on the floor laughing". It is often combined with LMAO for added emphasis as ROFLMAO ("Rolling on the floor laughing my ass off") or ROFLMFAO (Rolling on the floor laughing my fucking ass off).[42]
roflcopter: A portmanteau of ROFL and helicopter. A popular glitch in the Microsoft Sam text-to-speech engine enables the voice to make a sound akin to the rotation of rotor blades when 'SOI' is entered, and the phrase 'My ROFLcopter goes soi soi soi..." is often associated with the term as a result.
PMSL: For "pissing myself laughing".
Commonly used equivalents in other languages
In some languages with a non-Latin script, the abbreviation LOL itself is also often transliterated. See for example Arabic لول and Russian лол.[citation needed]
Pre-dating the Internet and phone texting by a century, the way to express laughter in morse code is "hi hi". The sound of this in morse ('di-di-di-dit di-dit, di-di-di-dit di-dit') is thought to represent chuckling.[43][44]
Most of these variants are usually found in lowercase.
555: the Thai variation of LOL. "5" in Thai is pronounced "ha", three of them being "hahaha".
asg: Swedish abbreviation of the term asgarv, meaning intense laughter.
g: Danish abbreviation of the word griner, which means "laughing" in Danish.[45]
ha3: Malaysian variation of LOL. ha3 means pronouncing ha three times, "hahaha".
jajajá: in Spanish, the letter "j" is pronounced /x/.[46]
jejeje: in the Philippines is used to represent "hehehe". "j" in Filipino languages is pronounced as /h/, derived from the Spanish /x/. Its origins can be traced to SMS language. It is widely used in a Filipino youth subculture known as Jejemons.[47][48]
kkkk: in Somali and Ethiopian languages spoken in the Horn of Africa, iterations of the letter "k", usually ranging between 2 and 8 k's, are used as a variation of LOL. These iterations are also used by Shona, Ndebele and other Zimbabwean languages speakers, with the longer variant being "kikiki" (emulating a laughing sound).
mdr: Esperanto version, from the initials of "multe da ridoj", which translates to "lot of laughs" in English.
mdr: French version, from the initials of "mort de rire" which roughly translated means "died of laughter", although many French people also use LOL instead as it is the most widely used on the internet.[49][50]
mkm: in Afghanistan "mkm" (being an abbreviation of the phrase "ma khanda mikonom"). This is a Dari phrase that means "I am laughing".
ptdr: French variant from pété de rire - literally meaning "broken with laughter"
rs: in Brazil "rs" (being an abbreviation of "risos", the plural of "laugh") is often used in text based communications in situations where in English LOL would be used, repeating it ("rsrsrsrsrs") is often done to express longer laughter or laughing harder. Also popular is "kkk" (which can also be repeated indefinitely), due to the pronunciation of the letter k in Portuguese sounding similar to the ca in card, and therefore representing the laugh "cacacacaca" (also similar to the Hebrew version above).[51]
wkwkwk: in Indonesian, "wkwkwk" is often used to express laughter.
חחח/ההה: Hebrew version of LOL. The letter ח is pronounced [/x/ /x/] and ה is pronounced [/h/ /h/]. Putting them together (usually three or more in a row) makes the word khakhakha or hahaha (since vowels in Hebrew are generally not written), which is in many languages regarded as the sound of laughter.
ههههه: Arabic script repetition of the "هـ" character meaning "Hahahaha" or "Hehehehe". "هـ" is equivalent to the letter "H". The equivalents of the "a" or "e" short vowels are Arabic diacritics and are optional to write.
ㅋㅋㅋ ("kkk" or "kekeke")[38] and ㅎㅎㅎ ("hhh") are usually used to indicate laughter in Korean. 'ㅋ', is a Korean Jamo consonant representing a "k" sound, and 'ㅎ' represents an "h" sound. Both "ㅋㅋㅋ" and "ㅎㅎㅎ" represent laughter which is not very loud. However, if a vowel symbol is written, louder laughter is implied: 하하 "haha" 호호, "hoho."[52]
哈哈哈/呵呵呵: in Chinese. Although 大笑 (da xiao; "big laugh") is used, a more widespread usage is "哈哈哈" (ha ha ha) or "呵呵呵" (he he) on internet forums. The phonetic rendering, "haha", is also common.
(笑): in Japanese, the kanji for laugh (笑) is used in the same way as lol. It can be read as kakko warai (literally "parentheses laugh") or just wara. w is also used as an abbreviation, and it is common for multiple w to be chained together.[53]. The resulting shape formed from multiple w leads to the usage of 草 (read as kusa), due to its resemblance to the shape of grass.
The word "lol" in other languages
In Dutch, lol is a word (not an acronym) which, coincidentally, means "fun" ("lollig" means "funny")
In Welsh, lol means "nonsense" or "ridiculous" – e.g., if a person wanted to say "utter nonsense" in Welsh, they would say "lol wirion" or "rwtsh lol"[54]
See also
Internet portal
Internet meme
Leet
References
^ Jump up to: a b Matt Haig (2001). E-Mail Essentials: How to Make the Most of E-Communications. Kogan Page. p. 89. ISBN 0-7494-3576-3.
^ Jump up to: a b Louis R. Franzini (2002). Kids Who Laugh: How to Develop Your Child's Sense of Humor. Square One Publishers. pp. 145–146. ISBN 0-7570-0008-8.
^ Jump up to: a b Michael Egan (2004). Email Etiquette. Cool Publications Ltd. pp. 32, 57–58. ISBN 1-84481-118-2.
Jump up ^ Ann Hewings, Martin Hewings, Grammar and Context, An advanced Resource Book, Routledge Applied Linguistics, 2005, ISBN 0-415-31080-6
Jump up ^ Jimmi Harrigan, Robert Rosenthal, Methods in Nonverbal Behavior Research, Oxford University Press, 2005, ISBN 978-0-19-852962-0
Jump up ^ John P. Sloan, Instant English 2, Gribaudo, 2011 ISBN 978-88-580-0448-7
^ Jump up to: a b LMAO – entry at Netlingo.com
Jump up ^ Ryan Goudelocke (August 2004). Credibility and Authority on Internet Message Boards (PDF) (M.M.C. thesis). Louisiana State University and Agricultural and Mechanical College. p. 22. Archived from the original (PDF) on 2007-03-16.
^ Jump up to: a b c Hershock, Peter (2003). Technology and cultural values : on the edge of the third millennium. Honolulu: University of Hawaii Press East-West Philosophers Conference. p. 561. ISBN 9780824826475.
^ Jump up to: a b Tim Shortis (2001). The Language of ICT. Routledge. p. 42. ISBN 978-0-415-22275-4.
Jump up ^ Eric S. Raymond and Guy L. Steele (1996). The New Hacker's Dictionary. MIT Press. p. 435. ISBN 0-262-68092-0.
Jump up ^ Robin Williams and Steve Cummings (1993). Jargon: An Informal Dictionary of Computer Terms. University of Michigan. p. 475. ISBN 978-0-938151-84-5.
Jump up ^ American Heritage Abbreviations Dictionary 3rd Edition. Houghton Mifflin. 2005. |access-date= requires |url= (help)
Jump up ^ Steven G. Jones (1998). Cybersociety 2.0: Revisiting Computer-Mediated Community and Technology. Sage Publications Inc. p. 52. ISBN 0-7619-1462-5.
^ Jump up to: a b Silvio Laccetti and Scott Molski (September 6, 2003). "Cost of poor writing no laughing matter". Atlanta Journal-Constitution.
^ Jump up to: a b "Article co-authored by Stevens professor and student garners nationwide attention from business, academia" (Press release). Stevens Institute of Technology. October 22, 2003. Archived from the original on March 15, 2007.
^ Jump up to: a b Shirley H. Fondiller and Barbara J. Nerone (2007). Health Professionals Style Manual. Springer Publishing Company. p. 98. ISBN 0-8261-0207-7.
^ Jump up to: a b Frank Yunker and Stephen Barry. "Threaded Podcasting: The Evolution of On-Line Learning". In Dan Remenyi. Proceedings of the International Conference on e-Learning, Université du Québec à Montréal, 22–23 June 2006. Academic Conferences Limited. p. 516. ISBN 1905305222.
^ Jump up to: a b Anna Stewart (March 25, 2011). "OMG! Oxford English Dictionary adds new words". CNN. Archived from the original on April 1, 2011. Retrieved March 28, 2011.
Jump up ^ McWhorter, John. "Txtng is killing language. JK!!!".
Jump up ^ Hossein Bidgoli (2004). The Internet Encyclopedia. John Wiley and Sons. p. 277. ISBN 0-471-22201-1.
Jump up ^ David Crystal (September 20, 2001). Language and the Internet. Cambridge University Press. p. 34. ISBN 0-521-80212-1.
Jump up ^ Victoria Clarke (January 30, 2002). "Internet English: an analysis of the variety of language used on Telnet talkers" (PDF).
Jump up ^ June Hines Moore (2007). Manners Made Easy for Teens. B&H Publishing Group. p. 54. ISBN 978-0-8054-4459-9.
Jump up ^ Sheryl Lindsell-Roberts (2004). Strategic Business Letters and E-Mail. Houghton Mifflin. p. 289. ISBN 0-618-44833-0.
Jump up ^ Marsia Mason (April 4, 2011). "OMG, K.I.D.S., IMHO, Needs to Go". Moorestown Patch. Retrieved April 9, 2011.
Jump up ^ Graeme Diamond (March 24, 2011). "New initialisms in the OED". Oxford English Dictionary. Retrieved March 28, 2011.
Jump up ^ James Morgan (April 8, 2011). "Why did LOL infiltrate the language?". BBC News. Retrieved April 9, 2011.
Jump up ^ Norton, Quinn. "Why Do Anonymous Geeks Hate Scientologists?". Gizmodo. Retrieved 17 February 2012.
Jump up ^ Coleman, Gabriella. "Our Weirdness Is Free: The logic of Anonymous — online army, agent of chaos, and seeker of justice". Triple Canopy. Retrieved 17 February 2012.
^ Jump up to: a b Kristen Philipkoski (February 22, 2005). "The Web Not the Death of Language". Wired News.
Jump up ^ Naomi Baron (February 18, 2005). "Instant Messaging by American College Students: A Case Study in Computer-Mediated Communication" (PDF). American Association for the Advancement of Science.
Jump up ^ Neda Ulaby (February 18, 2006). "OMG: IM Slang Is Invading Everyday English". Digital Culture. National Public Radio.
Jump up ^ Geoffrey K. Pullum (January 23, 2005). "English in Deep Trouble?". Language Log. Retrieved 2007-05-03.
Jump up ^ Schwartz, Mattathias (2008-08-03). "The Trolls Among Us". The New York Times. pp. MM24. Retrieved 2009-04-06.
Jump up ^ "What does *G* mean?". Internet Slang. Retrieved April 16, 2011.
Jump up ^ "What does J4G stand for?". Acronym finder. Retrieved April 16, 2011.
^ Jump up to: a b Sarkar, Samit (September 14, 2017). "Bungie explains how Destiny 2 armor resembling hate symbol made it into the game". Polygon.com. Retrieved August 4, 2018.
Jump up ^ Moomaw, Graham (February 16, 2017). "In Charlottesville, GOP candidate for governor Corey Stewart allies with alt-right-inspired blogger who wants to protect 'glorious Western civilization'". Richmond Times-Dispatch.
Jump up ^ "How an ancient Egyptian god spurred the rise of Trump". The Conversation. March 7, 2017. Retrieved July 18, 2017.
Jump up ^ "What does LQTM mean?". Internet Slang. Retrieved April 12, 2011.
Jump up ^ "LMAO". NetLingo. Retrieved April 12, 2011.
Jump up ^ Dinkins, Rodney R. (2010). "AMATEUR RADIO GLOSSARY: JARGON, ABBREVIATIONS AND TERMINOLOGY". Retrieved 21 September 2010.
Jump up ^ Dinkins, Rodney R. (2007). "Origin Of HI HI". ORIGIN OF HAM SPEAK – FACT, LEGENDS AND MYTHS. Retrieved 21 September 2010.
Jump up ^ Elkan, Mikael (2002). "Chat, chatsprog og smileys". Archived from the original on 2011-07-19. Retrieved 2009-08-22.
Jump up ^ "¡ja, ja, ja!". SpanishDict. Retrieved April 9, 2011.
Jump up ^ Marcoleta, Harvey (2010-04-24). "Jejemons: The new 'jologs'". Philippine Daily Inquirer. Archived from the original on 2010-04-27. Retrieved 2010-04-30.
Jump up ^ Nacino, Joseph (2010-04-26). "Jejemon in the Philippines". CNET Asia. Archived from the original on 2012-08-28. Retrieved 2010-04-30.
Jump up ^ "MDR". The Free Dictionary. Retrieved April 9, 2011.
Jump up ^ "French-English translation for "mdr (mort de rire)"". babLa. Retrieved April 9, 2011.
Jump up ^ "Learning to laugh and smile online... Brazilian Portuguese, by Semantica". Brazilian Portuguese, by Semantica. 2010-06-09. Retrieved 2018-01-28.
Jump up ^ "Slang 속어". We Study Korean. Retrieved April 9, 2011.
Jump up ^ "LOL=wwwwww". Tokyo-Insider. Retrieved April 9, 2011.
Jump up ^ "Welsh-English Lexicon". Cardiff School of Computer Science. Retrieved 2008-07-15.
Further reading
Connery, Brian A. (February 25, 1997). "IMHO: Authority and egalitarian rhetoric in the virtual coffeehouse". In Porter, D. Internet Culture. New York: Routledge. pp. 161–179. ISBN 0-415-91684-4.
Russ Armadillo Coffman (January 17, 1990). "smilies collection". Newsgroup: rec.humor. Retrieved 2006-12-22. —an early Usenet posting of a folk dictionary of abbreviations and emoticons, listing LOL and ROTFL
Ryan Goudelocke (August 2004). "Credibility and Authority on Internet Message Boards" (PDF). Louisiana State University and Agricultural and Mechanical College. Archived from the original (PDF) on 2007-03-16.
Claim to first use.
External links
Look up LOL or lol in Wiktionary, the free dictionary.
Wikimedia Commons has media related to LOL.
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Estelle Harris
From Wikipedia, the free encyclopedia
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This biography of a living person needs additional citations for verification. Please help by adding reliable sources. Contentious material about living persons that is unsourced or poorly sourced must be removed immediately, especially if potentially libelous or harmful. (August 2011) (Learn how and when to remove this template message)
Estelle Harris
Harris at the premiere of Ponyo in 2009.
Born
Estelle Nussbaum
(1928-04-04) April 4, 1928 (age 89)
Manhattan, New York, U.S.
Occupation
Actress, voice artist, comedian
Years active
1977–present
Spouse(s)
Sy Harris (m. 1953)[1]
Children
3[2]
Estelle Harris (née Nussbaum; April 4, 1928)[3] is an American actress, voice artist and comedian. Easily recognized by her distinctive, high-pitched voice, she is best known for her roles as Estelle Costanza on Seinfeld, the voice of Mrs. Potato Head in the Toy Story franchise, and Muriel on The Suite Life of Zack & Cody.
Contents
[hide]
1 Early life
2 Career
3 Personal life
4 Selected filmography
4.1 Film
4.2 Television
5 References
6 External links
Early life[edit]
Estelle Nussbaum was born in Manhattan, the younger of two daughters of Isaac and Anna Nussbaum, Polish Jewish immigrants who owned a candy store.[4]
Career[edit]
Harris is best known for her supporting role as Estelle Costanza, George's (Jason Alexander) mother and Frank's (Jerry Stiller) wife on Seinfeld, her voice role as Mrs. Potato Head in the Toy Story franchise, and her role as Muriel on The Suite Life of Zack & Cody. She appeared in Brad Paisley's music video for the song "Online".[5] In 2013, she played the mother of CBGB founder Hilly Kristal in the film CBGB.
Harris's voice work also includes Lula from Dave the Barbarian, Mama Lipsky on Kim Possible, Thelma on The Proud Family, Old Lady Bear in Brother Bear (2003), Audrey in Home on the Range (2004), and Death's nagging mother on Family Guy.
Personal life[edit]
Harris has been married to her husband Sy Harris since 1953; together they have three children.[6]
Selected filmography[edit]
Film[edit]
Year
Title
Role
Notes
1984 Once Upon a Time in America Peggy's Mother
1988 Stand and Deliver Secretary
1992 This Is My Life Aunt Harriet
1995 Perfect Alibi Aunt Dorothy
1995 The West Side Waltz
1997 Out to Sea Bridget
1998 Chairman of the Board Ms. Krubavitch
1998 The Odd Couple II Flirting Woman
1998 My Giant Aunt Pearl
1999 Lost & Found Mrs. Stubblefield
1999 Toy Story 2 Mrs. Potato Head Voice
2000 What's Cooking? Aunt Bea
2000 Dancing in September Sally
2000 Playing Mona Lisa Aunt Velva
2001 Good Advice Iris
2003 Brother Bear Old Lady Bear Voice
2004 Home on the Range Audrey Voice
2004 Teacher's Pet Mrs. Boogin Voice
2005 Tarzan II Mama Gunda Voice
2006 Queer Duck: The Movie Mrs. Duckstein Voice
2007 The Grand Ruth Melvin
2010 Toy Story 3 Mrs. Potato Head Voice
2010 Movin' In Arlene Taylor
2011 Hawaiian Vacation Mrs. Potato Head Voice
Short film
2011 Small Fry Mrs. Potato Head Voice
Short film
2012 Partysaurus Rex Mrs. Potato Head Voice
Short film
2013 CBGB Bertha Kristal
2015 Promoted Sylvia Silver
2019 Toy Story 4 Mrs. Potato Head Voice
Television[edit]
Night Court .... Easy Mary (1985–1986)
Married... with Children .... Delilah (1987)
Mathnet .... Postal Worker (1990)
Brooklyn Bridge .... Esther Shapiro (1991)
Law & Order .... Mrs. Rachlin (1992)
Mad About You .... Mother (1992)
Good Advice .... Ronnie Cohen (1993)
Chicago Hope .... Speaker at Funeral (Uncredited) (1995)
Aladdin .... Additional Voices (1995)
Timon & Pumbaa .... Timon's Mother (1995)
In the House .... Mrs. Claus (1995)
The Mask .... Lt. Kellaway's Mother (1996)
The Tick .... Arthur's Mother (1996)
Star Trek: Voyager .... Old Woman (1996)
Moesha .... Frances Howie (1997)
Living Single .... Esther Brooks (1997)
Cybill .... Woman (1997)
Seinfeld .... Estelle Costanza (1992–1998)
Addams Family Reunion .... Grandma Addams (1998)
Hercules .... Phil's Mother (1998)
The Wild Thornberrys .... Iguana/Turtle (1998)
Godzilla: The Series (1998)
The Rosie O'Donnell Show .... Herself (1998)
Queer Duck .... Mrs. Duckstein (1999)
The Brothers Flub .... Additional Voices (1999)
Sunset Beach .... Margaret Raynor (1999)
The Parkers .... Miss Agnes (1999)
Providence .... Darlene (2000)
Mickey Mouse Works .... Mrs. Turtle (1999–2000)
Sabrina, the Teenage Witch .... Dora (2001)
Family Guy .... Death's Mother (2001)
House of Mouse .... Mrs. Turtle (2002)
Weakest Link .... Herself (2002)
Whammy! The All-New Press Your Luck .... Herself (2002)
Half & Half .... Grandma Sophie (2003)
The Proud Family .... Helga (2003)
Test the Nation (2003)
Kim Possible .... Mama Lipsky (2004)
Kramer vs. Kramer: Kenny to Cosmo .... Herself (2004)
Dave the Barbarian .... Lula (2004–05)
Tripping the Rift .... Estelle (2005)
Phil of the Future .... Older Pim (2005)
Mind of Mencia (2006)
ER .... Mrs. Markovic (2006)
iCarly .... Mrs. Halberstadt (2007)
Can You Teach My Alligator Manners? .... Nana (2008)
The Suite Life of Zack & Cody .... Muriel (2005–2008)
American Dad! .... Marty's Wife (2009)
Curb Your Enthusiasm .... Estelle Costanza / Herself (2009)
Fanboy & Chum Chum .... Oz's Mom (2010)
Sonny with a Chance .... Grace Gallagher (2010)
The Bonnie Hunt Show .... Herself (2010)
KTLA Morning News .... Herself (2010)
Made in Hollywood .... Herself (2009–10)
Totally Tracked Down .... Herself (2010)
The Looney Tunes Show .... Sylvester's Mother (2011)
Futurama .... Velma Farnsworth (2012)
References[edit]
Jump up ^ Profile, latimes.com, January 29, 1995; accessed May 7, 2017.
Jump up ^ Profile, latimes.com, January 29, 1995; accessed May 7, 2017.
Jump up ^ Born in 1928 as per both the 1930 United States census, which lists Isaac Nussbaum (age 30) as "Head", Anna Nussbaum (age 30) as "Wife" and Minnie (age 9) and Estelle (age 2) as "Daughter[s]", New York City. Ward 6th Assembly District; Date: April 6, 1930.
Jump up ^ Per 1930 U.S. census data
Jump up ^ "Brad Paisley gets in tune with "Online" video". Country Standard Times. Retrieved 2012-12-09.
Jump up ^ Profile, latimes.com, January 29, 1995; accessed May 7, 2017.
External links[edit]
Estelle Harris on IMDb
Authority control
WorldCat Identities
VIAF: 36582940
LCCN: no2002039290
Retrieved from "https://en.wikipedia.org/w/index.php?title=Estelle_Harris&oldid=825856108"
Categories:
1928 births
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American people of Polish-Jewish descent
American film actresses
Jewish American comedians
American musical theatre actresses
American television actresses
American voice actresses
Jewish American actresses
Jewish comedians
People from Manhattan
American women comedians
Comedians from New York City
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Turing's proof
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Turing's proof is a proof by Alan Turing, first published in January 1937 with the title "On Computable Numbers, with an Application to the Entscheidungsproblem." It was the second proof of the assertion (Alonzo Church's proof was first) that some decision problems are "undecidable": there is no single algorithm that infallibly gives a correct "yes" or "no" answer to each instance of the problem. In his own words: "...what I shall prove is quite different from the well-known results of Gödel ... I shall now show that there is no general method which tells whether a given formula U is provable in K [Principia Mathematica]..." (Undecidable, p. 145).
Turing preceded this proof with two others. The second and third both rely on the first. All rely on his development of type-writer-like "computing machines" that obey a simple set of rules and his subsequent development of a "universal computing machine".
Contents
[hide]
1 Richard's paradox
2 Complications
3 Summary of the proofs
3.1 Summary of the first proof
3.2 Summary of the second proof
3.3 Details of second proof
3.4 Summary of the third proof
3.5 Details of the third proof
3.6 An example of what "complete configuration" means
4 Glossary of terms used by Turing
5 References
Richard's paradox[edit]
In 1905, Jules Richard presented this profound paradox. Alan Turing's first proof constructs this paradox with his so-called computing machine and proves that this machine cannot answer a simple question: will this machine be able to determine if any computing machine (including itself) will become trapped in an unproductive "infinite loop" (i.e. it fails to continue its computation of the diagonal number).
A succinct definition of Richard's paradox is found in Whitehead and Russell's Principia Mathematica:
Richard's paradox is as follows. Consider all decimals that can be defined by means of a finite number of words; let E be the class of such decimals. Then E has [aleph-sub-zero — an infinity of] terms; hence its members can be ordered as the 1st, 2nd, 3rd, ... Let N be a number defined as follows [Whitehead & Russell now employ the Cantor diagonal method]; if the nth figure in the nth decimal is p, let the nth figure in N be p+1 (or 0, if p = 9). Then N is different from all the members of E, since, whatever finite value n may have, the nth figure in N is different from the nth figure in the nth of the decimals composing E, and therefore N is different from the nth decimal. Nevertheless we have defined N in a finite number of words [i.e. this very word-definition just above!] and therefore N ought to be a member of E. Thus N both is and is not a member of E (Principia Mathematica, 2nd edition 1927, p. 61).
Complications[edit]
Turing's proof is complicated by a large number of definitions, and confounded with what Martin Davis called "petty technical details" and "...technical details [that] are incorrect as given" (Davis's commentary in Undecidable, p. 115). Turing himself published "A correction" in 1937: "The author is indebted to P. Bernays for pointing out these errors" (Undecidable, p. 152).
Specifically, in its original form the third proof is badly marred by technical errors. And even after Bernays' suggestions and Turing's corrections, errors remained in the description of the universal machine. And confusingly, since Turing was unable to correct his original paper, some text within the body harks to Turing's flawed first effort.
Bernays' corrections may be found in Undecidable, pp. 152–154; the original is to be found as:
"On Computable Numbers, with an Application to the Entscheidungsproblem. A Correction," Proceedings of the London Mathematical Society (2), 43 (1936–37), 544-546.
The on-line version of Turing's paper has these corrections in an addendum; however, corrections to the Universal Machine must be found in an analysis provided by Emil Post.
At first, the only mathematician to pay close attention to the details of the proof was Post (cf. Hodges p. 125) — mainly because he had arrived simultaneously at a similar reduction of "algorithm" to primitive machine-like actions, so he took a personal interest in the proof. Strangely (perhaps World War II intervened) it took Post some ten years to dissect it in the Appendix to his paper Recursive Unsolvability of a Problem of Thue, 1947 (reprinted in Undecidable, p. 293).
Before readers tackle "Proof #3" they are adivsed to place those corrections on their copy of the proof.
Other problems present themselves: In his Appendix Post commented indirectly on the paper's difficulty and directly on its "outline nature" (Post in Undecidable, p. 299) and "intuitive form" of the proofs (ibid.). Post had to infer various points:
"If our critique is correct, a machine is said to be circle-free if it is a Turing computing ... machine which prints an infinite number of 0s and 1s. And the two theorems of Turing's in question are really the following. There is no Turing ... machine which, when supplied with an arbitrary positive integer n, will determine whether n is the D.N of a Turing computing ... machine that is circle-free. [Secondly], There is no Turing convention-machine which, when supplied with an arbitrary positive integer n, will determine whether n is the D.N of a Turing computing ... machine that ever prints a given symbol (0 say)" (Post in Undecidable, p. 300)
Anyone who has ever tried to read the paper will understand Hodges' complaint:
"The paper started attractively, but soon plunged (in typical Turing manner) into a thicket of obscure German Gothic type in order to develop his instruction table for the universal machine. The last people to give it a glance would be the applied mathematicians who had to resort to practical computation..." (Hodges p. 124)
Summary of the proofs[edit]
In his proof that the Entscheidungsproblem can have no solution, Turing proceeded from two proofs that were to lead to his final proof. His first theorem is most relevant to the halting problem, the second is more relevant to Rice's theorem.
First proof: that no "computing machine" exists that can decide whether or not an arbitrary "computing machine" (as represented by an integer 1, 2, 3, . . .) is "circle-free" (i.e. goes on printing its number in binary ad infinitum): "...we have no general process for doing this in a finite number of steps" (p. 132, ibid.). Turing's proof, although it seems to use the "diagonal process", in fact shows that his machine (called H) cannot calculate its own number, let alone the entire diagonal number (Cantor's diagonal argument): "The fallacy in the argument lies in the assumption that B [the diagonal number] is computable" (Undecidable, p. 132). The proof does not require much mathematics.
Second proof: This one is perhaps more familiar to readers as Rice's theorem: "We can show further that there can be no machine E which, when supplied with the S.D ["program"] of an arbitrary machine M, will determine whether M ever prints a given symbol (0 say)" (his italics, Undecidable, p. 134).
Third proof: "Corresponding to each computing machine M we construct a formula Un(M) and we show that, if there is a general method for determining whether Un(M) is provable, then there is a general method for determining whether M ever prints 0" (Undecidable, p. 145)
[Readers who brave Proof #3 should come equipped with a solid background in (i) logic (ii) the paper of Kurt Gödel On Formally Undecidable Propositions of Principia Mathematica and Related Systems (reprinted in Undecidable, p. 5). For assistance with Gödel's paper they should consult e.g. Ernest Nagel and James R. Newman, Godel’s Proof, New York University Press, 1958.]
This proof requires the use of formal logic to prove a first lemma, followed by a brief word-proof of the second:
"Lemma 1: If S1 [symbol "0"] appears on the tape in some complete configuration of M, then Un(M) is provable" (Undecidable, p. 147)
"Lemma 2: [The converse] If Un(M) is provable then S1 [symbol "0"] appears on the tape in some complete configuration of M" (Undecidable, p. 148)
Finally, in only 64 words and symbols Turing proves by reductio ad absurdum that "the Hilbert Entscheidungsproblem can have no solution" (Undecidable, p. 145).
Summary of the first proof[edit]
Turing created a thicket of abbreviations; see the glossary at the end of this for help.
Some key clarifications:
Turing's machine H is attempting to print a diagonal number of 0s and 1s
This diagonal number is created when H actually "simulates" each "successful" machine under evaluation and prints the R-th "figure" (1 or 0) of the R-th "successful" machine
Turing spent much of his paper actually "constructing" his machines to convince us of their truth. This was required by his use of the reductio ad absurdum form of proof.
We must emphasize the "constructive" nature of this proof. Turing describes what could be a real machine, really buildable. The only questionable element is the existence of machine D, which this proof will eventually show to be impossible.
Turing begins the proof with the assertion of the existence of a “decision/determination” machine D. When fed any S.D (string of symbols A, C, D, L, R, N, semicolon “;”) it will determine if this S.D (symbol string) represents a "computing machine" that is either "circular" — and therefore "un-satisfactory u" — or "circle-free" — and therefore "satisfactory s".
Turing has previously demonstrated in his commentary that all "computing machines — machines that compute a number as 1s and 0s forever — can be written as an S.D on the tape of the “universal machine” U. Most of his work leading up to his first proof is spent demonstrating that a universal machine truly exists, i.e.
There truly exists a universal machine U
For each number N, there truly exists a unique S.D,
Every Turing machine has an S.D
Every S.D on U’s tape can be “run” by U and will produce the same “output” (figures 1, 0) as the original machine.
Turing makes no comment about how machine D goes about its work. For sake of argument, we suppose that D would first look to see if the string of symbols is "well-formed" (i.e. in the form of an algorithm and not just a scramble of symbols), and if not then discard it. Then it would go “circle-hunting”. To do this perhaps it would use “heuristics” (tricks: taught or learned). For purposes of the proof, these details are not important.
Turing then describes (rather loosely) the algorithm (method) to be followed by a machine he calls H. Machine H contains within it the decision-machine D (thus D is a “subroutine” of H). Machine H’s algorithm is expressed in H’s table of instructions, or perhaps in H’s Standard Description on tape and united with the universal machine U; Turing does not specify this.
In the course of describing universal machine U, Turing has demonstrated that a machine’s S.D (string of letters similar to a “program”) can be converted to an integer (base 8) and visa versa. Any number N (in base 8) can be converted to an S.D with the following replacements: 1 by A, 2 by C, 3 by D, 4 by L, 5 by R, 6 by N, 7 by semicolon ";".
As it turns out, machine H's unique number (D.N) is the number "K". We can infer that K is some hugely long number, maybe tens-of-thousands of digits long. But this is not important to what follows.
Machine H is responsible for converting any number N into an equivalent S.D symbol string for sub-machine D to test. (In programming parlance: H passes an arbitrary "S.D” to D, and D returns “satisfactory” or “unsatisfactory”.) Machine H is also responsible for keeping a tally R (“Record”?) of successful numbers (we suppose that the number of "successful S.D's i.e. R is much less than the number of S.D's tested, i.e. N). Finally, H prints on a section of its tape a diagonal number “beta-primed” B’. H creates this B’ by “simulating” (in the computer-sense) the “motions” of each “satisfactory” machine/number; eventually this machine/number under test will arrive at its Rth “figure” (1 or 0), and H will print it. H then is responsible for “cleaning up the mess” left by the simulation, incrementing N and proceeding onward with its tests, ad infinitum.
Note: All these machines that H is hunting for are what Turing called "computing machines". These compute binary-decimal-numbers in an endless stream of what Turing called "figures": only the symbols 1 and 0.
An example: Suppose machine H has tested 13472 numbers and produced 5 satisfactory numbers, i.e. H has converted the numbers 1 through 13472 into S.D’s (symbol strings) and passed them to D for test. As a consequence H has tallied 5 satisfactory numbers and run the first one to its 1st “figure”, the second to its 2nd figure, the third to its 3rd figure, the fourth to its 4th figure, and the fifth to its 5th figure. The count now stands at N = 13472, R = 5, and B’ = “.10011” (for example). H cleans up the mess on its tape, and proceeds:
H increments N = 13473 and converts "13473" to symbol string ADRLD. If sub-machine D deems ADLRD unsatisfactory, then H leaves the tally-record R at 5. H will increment the number N to 13474 and proceed onward. On the other hand, if D deems ADRLD satisfactory then H will increment R to 6. H will convert N (again) into ADLRD [this is just an example, ADLRD is probably useless] and “run” it using the universal machine U until this machine-under-test (U "running" ADRLD) prints its 6th “figure” i.e. 1 or 0. H will print this 6th number (e.g. “0”) in the “output” region of its tape (e.g. B’ = “.100110”).
H cleans up the mess, and then increments the number N to 13474.
The whole process unravels when H arrives at its own number K. We will proceed with our example. Suppose the successful-tally/record R stands at 12. H finally arrives at its own number minus 1, i.e. N = K-1 = 4335...3214, and this number is unsuccessful. Then H increments N to produce K = 4355...3215, i.e. its own number. H converts this to “LDDR...DCAR” and passes it to decision-machine D. Decision-machine D must return “satisfactory” (that is: H must by definition go on and on testing, ad infinitum, because it is "circle-free"). So H now increments tally R from 12 to 13 and then re-converts the number-under-test K into its S.D and uses U to simulate it. But this means that H will be simulating its own motions. What is the first thing the simulation will do? This simulation K-aka-H either creates a new N or “resets” the “old” N to 1. This "K-aka-H" either creates a new R or “resets” the “old” R to 0. Old-H “runs” new "K-aka-H" until it arrives at its 12th figure.
But it never makes it to the 13th figure; K-aka-H eventually arrives at 4355...3215, again, and K-aka-H must repeat the test. K-aka-H will never reach the 13th figure. The H-machine probably just prints copies of itself ad infinitum across blank tape. But this contradicts the premise that H is a satisfactory, non-circular computing machine that goes on printing the diagonal numbers's 1’s and 0’s forever. (We will see the same thing if N is reset to 1 and R is reset to 0.)
If the reader does not believe this, they can write a "stub" for decision-machine D (stub "D" will return "satisfactory") and then see for themselves what happens at the instant machine H encounters its own number.
Summary of the second proof[edit]
Less than one page long, the passage from premises to conclusion is obscure.
Turing proceeds by reductio ad absurdum. He asserts the existence of a machine E, which when given the S.D (Standard Description, i.e. "program") of an arbitrary machine M, will determine whether M ever prints a given symbol (0 say). He does not assert that this M is a "computing machine".
Given the existence of machine E, Turing proceeds as follows:
If machine E exists then a machine G exists that determines if M prints 0 infinitely often, AND
If E exists then another process exits [we can call the process/machine G' for reference] that determines if M prints 1 infinitely often, THEREFORE
When we combine G with G' we have a process that determines if M prints an infinity of figures, AND
IF the process "G with G'" determines M prints an infinity of figures, THEN "G with G'" has determined that M is circle-free, BUT
This process "G with G'" that determine if M is circle-free, by proof 1, cannot exist, THEREFORE
Machine E does not exist.
Details of second proof[edit]
The difficulty in the proof is step 1. The reader will helped by realizing that Turing is not explaining his subtle handiwork. (In a nutshell: he is using certain equivalencies between the “existential-“ and “universal-operators” together with their equivalent expressions written with logical operators.)
Here’s an example: Suppose we see before us a parking lot full of hundreds of cars. We decide to go around the entire lot looking for: “Cars with flat (bad) tires”. After an hour or so we have found two “cars with bad tires.” We can now say with certainty that “Some cars have bad tires”. Or we could say: “It’s not true that ‘All the cars have good tires’”. Or: “It is true that: ‘not all the cars have good tires”. Let us go to another lot. Here we discover that “All the cars have good tires.” We might say, “There’s not a single instance of a car having a bad tire.” Thus we see that, if we can say something about each car separately then we can say something about ALL of them collectively.
This is what Turing does: From M he creates a collection of machines {M1, M2, M3, M4, …, Mn} and about each he writes a sentence: “X prints at least one 0” and allows only two “truth values”, True = blank or False = :0:. One by one he determines the truth value of the sentence for each machine and makes a string of blanks or :0:, or some combination of these. We might get something like this: “M1 prints a 0” = True AND “M2 prints a 0” = True AND “M3 prints a 0” = True AND “M4 prints a 0” = False, ... AND “Mn prints a 0” = False. He gets the string
BBB:0::0::0: ... :0: ... ad infinitum
if there are an infinite number of machines Mn. If on the other hand if every machine had produced a "True" then the expression on the tape would be
BBBBB....BBBB ... ad infinitum
Thus Turing has converted statements about each machine considered separately into a single "statement" (string) about all of them. Given the machine (he calls it G) that created this expression, he can test it with his machine E and determine if it ever produces a 0. In our first example above we see that indeed it does, so we know that not all the M's in our sequence print 0s. But the second example shows that, since the string is blanks then every Mn in our sequence has produced a 0.
All that remains for Turing to do is create a process to create the sequence of Mn's from a single M.
Suppose M prints this pattern:
M => ...AB01AB0010AB…
Turing creates another machine F that takes M and crunches out a sequence of Mn’s that successively convert the first n 0’s to “0-bar” (0):
M1 = ...AB01AB0010AB...
M2 = ...AB01AB0010AB...
M3 = ...AB01AB0010AB...
M4 = ...AB01AB0010AB...
He claims, without showing details, that this machine F is truly build-able. We can see that one of a couple things could happen. F may run out of machines that have 0’s, or it may have to go on ad infinitum creating machines to “cancel the zeros”.
Turing now combines machines E and F into a composite machine G. G starts with the original M, then uses F to create all the successor-machines M1, M2,. . ., Mn. Then G uses E to test each machine starting with M. If E detects that a machine never prints a zero, G prints :0: for that machine. If E detects that a machine does print a 0 (we assume, Turing doesn’t say) then G prints :: or just skips this entry, leaving the squares blank. We can see that a couple things can happen.
G will print no 0’s, ever, if all the Mn’s print 0’s, OR,
G will print ad infinitum 0’s if all the M’s print no 0’s, OR,
G will print 0’s for a while and then stop.
Now, what happens when we apply E to G itself?
If E(G) determines that G never prints a 0 then we know that all the Mn’s have printed 0’s. And this means that, because all the Mn came from M, that M itself prints 0’s ad infinitum, OR
If E(G) determines that G does print a 0 then we know that not all the Mn’s print 0’s; therefore M does not print 0’s ad infinitum.
As we can apply the same process for determining if M prints 1 infinitely often. When we combine these processes, we can determine that M does, or does not, go on printing 1’s and 0’s ad infinitum. Thus we have a method for determining if M is circle-free. By Proof 1 this is impossible. So the first assertion that E exists, is wrong: E does not exist.
Summary of the third proof[edit]
Here Turing proves "that the Hilbert Entscheidungsproblem can have no solution" (Undecidable, p. 145). Here he
“…show(s) that there can be no general process for determining whether a given formula U of the functional calculus K is provable.” (ibid.)
Both Lemmas #1 and #2 are required to form the necessary "IF AND ONLY IF" (i.e. logical equivalence) required by the proof:
"A set E is computably decidable if and only if both E and its complement are computably enumerable" (Frankél, p. 67)
Summary of the proof
Turing demonstrates the existence of a formula Un(M) which says, in effect, that "in some complete configuration of M, 0 appears on the tape" (p. 146). This formula is TRUE, that is, it is "constructible", and he shows how to go about this.
Then Turing proves two Lemmas, the first requiring all the hard work. (The second is the converse of the first.) Then he uses reductio ad absurdum to prove his final result:
There exists a formula Un(M). This formula is TRUE, AND
If the Entscheidungsproblem can be solved THEN a mechanical process exists for determining whether Un(M) is provable (derivable), AND
By Lemmas 1 and 2: Un(M) is provable IF AND ONLY IF 0 appears in some "complete configuration" of M, AND
IF 0 appears in some "complete configuration" of M THEN a mechanical process exists that will determine whether arbitrary M ever prints 0, AND
By Proof 2 no mechanical process exists that will determine whether arbitrary M ever prints 0, THEREFORE
Un(M) is not provable (it is TRUE, but not provable) which means that the Entscheidungsproblem is unsolvable.
Discussion of the third proof
If readers intend to study the proof in detail they should correct their copies of the pages of the third proof with the corrections that Turing supplied.
To (even attempt to) follow the technical details, the reader will need to understand the definition of "provable" and be aware of important "clues".
"Provable" means, in the sense of Gödel, that (i) the axiom system itself is powerful enough to produce (express) the sentence "This sentence is provable", and (ii) that in any arbitrary "well-formed" proof the symbols lead by axioms, definitions, and substitution to the symbols of the conclusion.
First clue: "Let us put the description of M into the first standard form of §6". Section 6 describes the very specific "encoding" of machine M on the tape of a "universal machine" U. This requires the reader to know some idiosyncrasies of Turing's universal machine U and the encoding scheme.
(i) The universal machine is a set of "universal" instructions that reside in an "instruction table". Separate from this, on U's tape, a "computing machine" M will reside as "M-code". The universal table of instructions can print on the tape the symbols A, C, D, 0, 1, u, v, w, x, y, z, : . The various machines M can print these symbols only indirectly by commanding U to print them.
(ii) The "machine code" of M consists of only a few letters and the semicolon, i.e. D, C, A, R, L, N, ; . Nowhere within the "code" of M will the numerical "figures" (symbols) 1 and 0 ever appear. If M wants U to print a symbol from the collection blank, 0, 1 then it uses one of the following codes to tell U to print them. To make things more confusing, Turing calls these symbols S0, S1, and S2, i.e.
blank = S0 = D
0 = S1 = DC
1 = S2 = DCC
(iii) A "computing machine", whether it is built directly into a table (as his first examples show), or as machine-code M on universal-machine U's tape, prints its number on blank tape (to the right of M-code, if there is one) as 1s and 0s forever proceeding to the right.
(iv) If a "computing machine" is U+"M-code", then "M-code" appears first on the tape; the tape has a left end and the "M-code" starts there and proceeds to the right on alternate squares. When the M-code comes to an end (and it must, because of the assumption that these M-codes are finite algorithms), the "figures" will begin as 1s and 0s on alternate squares, proceeding to the right forever. Turing uses the (blank) alternate squares (called "E"- "eraseable"- squares) to help U+"M-code" keep track of where the calculations are, both in the M-code and in the "figures" that the machine is printing.
(v) A "complete configuration" is a printing of all symbols on the tape, including M-code [?] and "figures" up to that point, together with the figure currently being scanned (with a pointer-character printed to the left of the scanned symbol ?). If we have interpreted Turing's meaning correctly, this will be a hugely long set of symbols. But whether the entire M-code must be repeated is unclear; only a printing of the current M-code instruction is necessary plus the printing of all figures with a figure-marker).
(vi) Turing reduced the vast possible number of instructions in "M-code" (again: the code of M to appear on the tape) to a small canonical set, one of three similar to this: {qi Sj Sk R ql} e.g. If machine is executing instruction #qi and symbol Sj is on the square being scanned, then Print symbol Sk and go Right and then go to instruction ql: The other instructions are similar, encoding for "Left" L and "No motion" N. It is this set that is encoded by the string of symbols qi = DA...A, Sj = DC...C, Sk = DC...C, R, ql = DA....A. Each instruction is separated from another one by the semicolon. For example, {q5, S1 S0 L q3} means: Instruction #5: If scanned symbol is 0 then print blank, go Left, then go to instruction #3. It is encoded as follows
; D A A A A A D C D L D A A A
Second clue: Turing is using ideas introduced in Gödel's paper, that is, the "Gödelization" of (at least part of) the formula for Un(M). This clue appears only as a footnote on page 138 (Undecidable): "A sequence of r primes is denoted by ^(r)" (ibid.) [Here, r inside parentheses is "raised".] This "sequence of primes" appears in a formula called F^(n).
Third clue: This reinforces the second clue. Turing's original attempt at the proof uses the expression
(Eu)N(u) & (x)( ... etc. ...) (Undecidable, p. 146)
Earlier in the paper Turing had previously used this expression (p. 138) and defined N(u) to mean "u is a non-negative integer" (ibid.) (i.e. a Gödel number). But, with the Bernays corrections, Turing abandoned this approach (i.e. the use of N(u)) and the only place where "the Gödel number" appears explicitly is where he uses F^(n).
What does this mean for the proof? The first clue means that a simple examination of the M-code on the tape will not reveal if a symbol 0 is ever printed by U+"M-code". A testing-machine might look for the appearance of DC in one of the strings of symbols that represent an instruction. But will this instruction ever be "executed?" Something has to "run the code" to find out. This something can be a machine, or it can be lines in a formal proof, i.e. Lemma #1.
The second and third clues mean that, as its foundation is Gödel's paper, the proof is difficult.
Details of the third proof[edit]
In the example below we will actually construct a simple "theorem"—a little Post–Turing machine program "run it". We will see just how mechanical a properly designed theorem can be. A proof, we will see, is just that, a "test" of the theorem that we do by inserting a "proof example" into the beginning and see what pops out at the end.
Both Lemmas #1 and #2 are required to form the necessary "IF AND ONLY IF" (i.e. logical equivalence) required by the proof:
"A set E is computably decidable if and only if both E and its complement are computably enumerable" (Frankél, p. 67)
To quote Frankel:
"A sentence A is said to be decidable in a formal system S if either A or its negation is provable in S" (Frankél, p. 65)
Frankel has defined "provable" earlier in his book:
"A formal system is a system of axioms (expressed in some formally defined language) and rules of reasoning (also called inference rules), used to derive the theorems of the system. A theorem is any statement in the language of the system obtainable by a series of applications of the rules of reasoning, starting from the axioms. A proof is a finite sequence of such applications, leading to a theorem as its conclusion" (ibid p. 17).
Thus a "sentence" is a string of symbols, and a theorem is a string of strings of symbols.
Turing is confronted with the following tasks:
to convert a Universal Turing machine "program", and the numerical symbols on the tape (Turing's "figures", symbols "1" and "0"), into a "theorem"—that is, a (monstrously long) string of sentences that define the successive actions of the machine, (all) the figures of the tape, and the location of the "tape head".
Thus the "string of sentences" will be strings of strings of symbols. The only allowed individual symbols will come from Godel's symbols defined in his paper.(In the following example we use the "<" and ">" around a "figure" to indicate that the "figure" is the symbol being scanned by the machine).
An example of what "complete configuration" means[edit]
In the following, we have to remind ourselves that every one of Turing’s “computing machines” is a binary-number generator/creator that begins work on “blank tape”. Properly constructed, it always cranks away ad infinitum, but its instructions are always finite. In Turing’s proofs, Turing’s tape had a “left end” but extended right ad infinitum. For sake of example below we will assume that the “machine” is not a Universal machine, but rather the simpler “dedicated machine” with the instructions in the Table.
Our example is based on a modified Post–Turing machine model of a Turing Machine. This model prints only the symbols 0 and 1. The blank tape is considered to be all b’s. Our modified model requires us to add two more instructions to the 7 Post–Turing instructions. The abbreviations that we will use are:
R, RIGHT: look to the right and move tape to left, or move tape head right
L, LEFT : look to the left and to move tape right, or move tape head left
E, ERASE scanned square (e.g. make square blank)
P0,: PRINT 0 in scanned square
P1,: PRINT 1 in scanned square
Jb_n, JUMP-IF-blank-to-instruction_#n,
J0_n, JUMP-IF-0-to-instruction_#n,
J1_n, JUMP-IF-1-to-instrucntion_#n,
HALT.
In the cases of R, L, E, P0, and P1 after doing its task the machine continues on to the next instruction in numerical sequence; ditto for the jumps if their tests fail.
But, for brevity, our examples will only use three squares. And these will always start as there blanks with the scanned square on the left: i.e. bbb. With two symbols 1, 0 and blank we can have 27 distinct configurations:
bbb, bb0, bb1, b0b, b00, b01, b1b, b10, b11, 0bb, 0b0, 0b1, 00b, 000, 001, 01b, 010, 011, 1bb, 1b0, 1b1, 10b, 100, 101, 11b, 110, 111
We must be careful here, because it is quite possible that an algorithm will (temporarily) leave blanks in between figures, then come back and fill something in. More likely, an algorithm may do this intentionally. In fact, Turing’s machine does this—it prints on alternate squares, leaving blanks between figures so it can print locator symbols.
Turing always left alternate squares blank so his machine could place a symbol to the left of a figure (or a letter if the machine is the universal machine and the scanned square is actually in the “program”). In our little example we will forego this and just put symbols ( ) around the scanned symbol, as follows:
b(b)0 this means, “Tape is blanks-to-the-left of left blank but left blank is “in play”, the scanned-square-is-blank, “0”, blanks-to-right”
1(0)1 this means, “Tape is blanks-to-the-left, then 1, scanned square is “0”
Let us write a simple program:
start: P1, R, P1, R, P1, H
Remember that we always start with blank tape. The complete configuration prints the symbols on the tape followed by the next instruction:
start config: (b) P1,
config #1: (1) R,
config #2: 1(b) P1,
config #3: 1(1) R,
config #4: 11(b) P1,
config #5: 11(1) H
Let us add “jump” into the formula. When we do this we discover why the complete configuration must include the tape symbols. (Actually, we see this better, below.) This little program prints three “1”s to the right, reverses direction and moves left printing 0’s until it hits a blank. We will print all the symbols that our machine uses:
start: P1, R, P1, R, P1, P0, L, J1_7, H
(b)bb P1,
(1)bb R,
1(b)b P1,
1(1)b R,
11(b) P1,
11(1) P0,
11(0) L,
1(1)0 J1_7
1(1)0 L
(1)10 J0_7
(1)10 L
(b)110 J0_7
(b)110 H
Here at the end we find that a blank on the left has “come into play” so we leave it as part of the total configuration.
Given that we have done our job correctly, we add the starting conditions and see “where the theorem goes”. The resulting configuration—the number 110—is the PROOF.
Turing's first task had to write a generalized expression using logic symbols to express exactly what his Un(M) would do.
Turing’s second task is to "Godelize" this hugely long string-of-string-of-symbols using Godel's technique of assigning primes to the symbols and raising the primes to prime-powers, per Godel’s method.
Glossary of terms used by Turing[edit]
1 computable number — a number whose decimal is computable by a machine, i.e. by finite means (e.g. an algorithm)
2 M — a machine with a finite instruction table and a scanning/printing head. M moves an infinite tape divided into squares each “capable of bearing a symbol”. The machine-instructions are only the following: move one square left, move one square right, on the scanned square print symbol p, erase the scanned square, if the symbol is p then do instruction aaa, if the scanned symbol is not p then do instruction aaa, if the scanned symbol is none then do instruction aaa, if the scanned symbol is any do instruction aaa [where “aaa” is an instruction-identifier].
3 computing machine — an M that prints two kinds of symbols, symbols of the first type are called “figures” and are only binary symbols 1 and 0; symbols of the second type are any other symbols.
4 figures — symbols 1 and 0, a.k.a. “symbols of the first kind”
5 m-configuration — the instruction-identifier, either a symbol in the instruction table, or a string of symbols representing the instruction- number on the tape of the universal machine (e.g. "DAAAAA = #5")
6 symbols of the second kind — any symbols other than 1 and 0
7 circular — an unsuccessful computating machine. It fails to print, ad infinitum, the figures 0 or 1 that represent in binary the number it computes
8 circle-free — a successful computating machine. It prints, ad infinitum, the figures 0 or 1 that represent in binary the number it computes
9 sequence — as in “sequence computed by the machine”: symbols of the first kind a.k.a. figures a.k.a. symbols 0 and 1.
10 computable sequence — can be computed by a circle-free machine
11 S.D – Standard Description: a sequence of symbols A, C, D, L, R, N, “;” on a Turing machine tape
12 D.N — Description Number: an S.D converted to a number: 1=A, 2=C, 3 =D, 4=L, 5=R, 6=N, 7=;
13 M(n) — a machine whose D.N is number “n”
14 satisfactory — a S.D or D.N that represents a circle-free machine
15 U — a machine equipped with a “universal” table of instructions. If U is “supplied with a tape on the beginning of which is written the S.D of some computing machine M, U will compute the same sequence as M.”
16 β’—“beta-primed”: A so-called “diagonal number” made up of the n-th figure (i.e. 0 or 1) of the n-th computable sequence [also: the computable number of H, see below]
17 u — an unsatisfactory, i.e. circular, S.D
18 s — satisfactory, i.e. circle-free S.D
19 D — a machine contained in H (see below). When supplied with the S.D of any computing machine M, D will test M’s S.D and if circular mark it with “u” and if circle-free mark it with “s”
20 H — a computing machine. H computes B’, maintains R and N. H contains D and U and an unspecified machine (or process) that maintains N and R and provides D with the equivalent S.D of N. E also computes the figures of B’ and assembles the figures of B’.
21 R — a record, or tally, of the quantity of successful (circle-free) S.D tested by D
22 N — a number, starting with 1, to be converted into an S.D by machine E. E maintains N.
23 K — a number. The D.N of H.
Required for Proof #3
5 m-configuration — the instruction-identifier, either a symbol in the instruction table, or a string of symbols representing the instruction's number on the tape of the universal machine (e.g. "DAAAAA = instruction #5"). In Turing's S.D the m-configuration appears twice in each instruction, the left-most string is the "current instruction"; the right-most string is the next instruction.
24 complete configuration — the number (figure 1 or 0) of the scanned square, the complete sequence of all symbols on the tape, and the m-configuration (the instruction-identifier, either a symbol or a string of symbols representing a number, e.g. "instruction DAAAA = #5")
25 RSi(x, y) — "in the complete configuration x of M the symbol on square y is Si. "complete configuration" is definition #5,
26 I(x, y) — "in the complete configuration x of M the square y is scanned"
27 Kqm(x) — "in the complete configuration x of M the machine-configuration (instruction number) is qm"
28 F(x,y) — "y is the immediate successor of x" (follows Gödel's use of "f" as the successor-function).
29 G(x,y) — "x precedes y", not necessarily immediately
30 Inst{qi, Sj Sk L ql} is an abbreviation, as are Inst{qi, Sj Sk R ql}, and Inst{qi, Sj Sk N ql}. See below.
Turing reduces his instruction set to three “canonical forms” – one for Left, Right, and No-movement. Si and Sk are symbols on the tape.
tape Final
m-config Symbol Operations m-config
qi Si PSk, L qm
qi Si PSk, R qm
qi Si PSk, N qm
For example, the operations in the first line are PSk = PRINT symbol Sk from the collection A, C, D, 0, 1, u, v, w, x, y, z, :, then move tape LEFT.
These he further abbreviated as: (N1) qi Sj Sk L qm (N2) qi Sj Sk R qm (N3) qi Sj Sk N qm
In Proof #3 he calls the first of these “Inst{qi Sj Sk L ql}”, and he shows how to write the entire machine S.D as the logical conjunction (logical OR): this string is called “Des(M)”, as in “Description-of-M”. i.e. if the machine prints 0 then 1’s and 0’s on alternate squares to the right ad infinitum it might have the table (a similar example appears on page 119):
q1, blank, P0, R, q2
q2, blank, P-blank, R, q3
q3, blank, P1, R, q4
q4, blank, P-blank, R, q1
(This has been reduced to canonical form with the “p-blank” instructions so it differs a bit from Turing’s example.) If put them into the “ Inst( ) form” the instructions will be the following (remembering: S0 is blank, S1 = 0, S2 = 1):
Inst {q1 S0 S1 R q2}
Inst {q2 S0 S0 R q3}
Inst {q3 S0 S2 R q4}
Inst {q4 S0 S0 R q1}
The reduction to the Standard Description (S.D) will be:
; D A D D C R D A A ; D A A D D R D A A A ; D A A A D D C C R D A A A A ; D A A A A D D R D A ;
This agrees with his example in the book (there will be a blank between each letter and number). Universal machine U uses the alternate blank squares as places to put "pointers".
References[edit]
Turing, A.M. (1936), "On Computable Numbers, with an Application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, 2 (published 1937), 42 (1), pp. 230–65, doi:10.1112/plms/s2-42.1.230 (and Turing, A.M. (1938), "On Computable Numbers, with an Application to the Entscheidungsproblem: A correction", Proceedings of the London Mathematical Society, 2 (published 1937), 43 (6), pp. 544–6, doi:10.1112/plms/s2-43.6.544 ). (Online version.) This is the epochal paper where Turing defines Turing machines, shows that the Entscheidungsproblem is unsolvable.
Martin Davis, The Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, Raven Press, New York, 1965. The two papers of Post referenced above are included in this volume. Other papers include those by Godel, Church, Rosser, and Kleene.
Andrew Hodges, Alan Turing: The Enigma, Simon and Schuster, New York. Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.
Hans Reichenbach, Elements of Symbolic Logic, Dover Publications, Inc., New York, 1947.
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