Machine translation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Machine_translation

A mobile phone app translating Spanish text into English

Machine translation is use of either rule-based or probabilistic (i.e. statistical and, most recently, neural network-based) machine learning approaches to translation of text or speech from one language to another, including the contextual, idiomatic and pragmatic nuances of both languages.

History

Main article: History of machine translation

Origins

The origins of machine translation can be traced back to the work of Al-Kindi, a ninth-century Arabic cryptographer who developed techniques for systemic language translation, including cryptanalysis, frequency analysis, and probability and statistics, which are used in modern machine translation. The idea of machine translation later appeared in the 17th century. In 1629, René Descartes proposed a universal language, with equivalent ideas in different tongues sharing one symbol.

The idea of using digital computers for translation of natural languages was proposed as early as 1947 by England's A. D. Booth and Warren Weaver at Rockefeller Foundation in the same year. "The memorandum written by Warren Weaver in 1949 is perhaps the single most influential publication in the earliest days of machine translation." Others followed. A demonstration was made in 1954 on the APEXC machine at Birkbeck College (University of London) of a rudimentary translation of English into French. Several papers on the topic were published at the time, and even articles in popular journals (for example an article by Cleave and Zacharov in the September 1955 issue of Wireless World). A similar application, also pioneered at Birkbeck College at the time, was reading and composing Braille texts by computer.

1950s

The first researcher in the field, Yehoshua Bar-Hillel, began his research at MIT (1951). A Georgetown University MT research team, led by Professor Michael Zarechnak, followed (1951) with a public demonstration of its Georgetown-IBM experiment system in 1954. MT research programs popped up in Japan and Russia (1955), and the first MT conference was held in London (1956).

David G. Hays "wrote about computer-assisted language processing as early as 1957" and "was project leader on computational linguistics at Rand from 1955 to 1968."

1960–1975

Researchers continued to join the field as the Association for Machine Translation and Computational Linguistics was formed in the U.S. (1962) and the National Academy of Sciences formed the Automatic Language Processing Advisory Committee (ALPAC) to study MT (1964). Real progress was much slower, however, and after the ALPAC report (1966), which found that the ten-year-long research had failed to fulfill expectations, funding was greatly reduced. According to a 1972 report by the Director of Defense Research and Engineering (DDR&E), the feasibility of large-scale MT was reestablished by the success of the Logos MT system in translating military manuals into Vietnamese during that conflict.

The French Textile Institute also used MT to translate abstracts from and into French, English, German and Spanish (1970); Brigham Young University started a project to translate Mormon texts by automated translation (1971).

1975 and beyond

SYSTRAN, which "pioneered the field under contracts from the U.S. government" in the 1960s, was used by Xerox to translate technical manuals (1978). Beginning in the late 1980s, as computational power increased and became less expensive, more interest was shown in statistical models for machine translation. MT became more popular after the advent of computers. SYSTRAN's first implementation system was implemented in 1988 by the online service of the French Postal Service called Minitel. Various computer based translation companies were also launched, including Trados (1984), which was the first to develop and market Translation Memory technology (1989), though this is not the same as MT. The first commercial MT system for Russian / English / German-Ukrainian was developed at Kharkov State University (1991).

By 1998, "for as little as $29.95" one could "buy a program for translating in one direction between English and a major European language of your choice" to run on a PC.

MT on the web started with SYSTRAN offering free translation of small texts (1996) and then providing this via AltaVista Babelfish, which racked up 500,000 requests a day (1997). The second free translation service on the web was Lernout & Hauspie's GlobaLink. Atlantic Magazine wrote in 1998 that "Systran's Babelfish and GlobaLink's Comprende" handled "Don't bank on it" with a "competent performance."

Franz Josef Och (the future head of Translation Development AT Google) won DARPA's speed MT competition (2003). More innovations during this time included MOSES, the open-source statistical MT engine (2007), a text/SMS translation service for mobiles in Japan (2008), and a mobile phone with built-in speech-to-speech translation functionality for English, Japanese and Chinese (2009). In 2012, Google announced that Google Translate translates roughly enough text to fill 1 million books in one day.

Approaches

See also: Hybrid machine translation and Example-based machine translation

Before the advent of deep learning methods, statistical methods required a lot of rules accompanied by morphological, syntactic, and semantic annotations.

Rule-based

Main article: Rule-based machine translation

The rule-based machine translation approach was used mostly in the creation of dictionaries and grammar programs. Its biggest downfall was that everything had to be made explicit: orthographical variation and erroneous input must be made part of the source language analyser in order to cope with it, and lexical selection rules must be written for all instances of ambiguity.

Transfer-based machine translation

Main article: Transfer-based machine translation

Transfer-based machine translation was similar to interlingual machine translation in that it created a translation from an intermediate representation that simulated the meaning of the original sentence. Unlike interlingual MT, it depended partially on the language pair involved in the translation.

Interlingual

Main article: Interlingual machine translation

Interlingual machine translation was one instance of rule-based machine-translation approaches. In this approach, the source language, i.e. the text to be translated, was transformed into an interlingual language, i.e. a "language neutral" representation that is independent of any language. The target language was then generated out of the interlingua. The only interlingual machine translation system that was made operational at the commercial level was the KANT system (Nyberg and Mitamura, 1992), which was designed to translate Caterpillar Technical English (CTE) into other languages.

Dictionary-based

Main article: Dictionary-based machine translation

Machine translation used a method based on dictionary entries, which means that the words were translated as they are by a dictionary.

Statistical

Statistical machine translation tried to generate translations using statistical methods based on bilingual text corpora, such as the Canadian Hansard corpus, the English-French record of the Canadian parliament and EUROPARL, the record of the European Parliament. Where such corpora were available, good results were achieved translating similar texts, but such corpora were rare for many language pairs. The first statistical machine translation software was CANDIDE from IBM. In 2005, Google improved its internal translation capabilities by using approximately 200 billion words from United Nations materials to train their system; translation accuracy improved.

SMT's biggest downfall included it being dependent upon huge amounts of parallel texts, its problems with morphology-rich languages (especially with translating into such languages), and its inability to correct singleton errors.

Neural MT

Main article: Neural machine translation

A deep learning-based approach to MT, neural machine translation has made rapid progress in recent years. However, current consensus is that the so-called human parity achieved is not real, being based wholly on limited domains, language pairs, and certain test benchmarks i.e., it lacks statistical significance power.

Translations by neural MT tools like DeepL Translator, which is thought to usually deliver the best machine translation results as of 2022, typically still need post-editing by a human.

Prompt engineering is required in order to steer the GPT-3-generated translations.

Major issues

Machine translation could produce some non-understandable phrases, such as "鸡枞" (Macrolepiota albuminosa) being rendered as "Wikipedia".

Broken Chinese "沒有進入" from machine translation in Bali, Indonesia. The broken Chinese sentence sounds like "there does not exist an entry" or "have not entered yet".

Studies using human evaluation (e.g. by professional literary translators or human readers) have systematically identified various issues with the latest advanced MT outputs. Common issues include the translation of ambiguous parts whose correct translation requires common sense-like semantic language processing or context. There can also be errors in the source texts, missing high-quality training data and the severity of frequency of several types of problems may not get reduced with techniques used to date, requiring some level of human active participation.

Disambiguation

Main articles: Word-sense disambiguation and Syntactic disambiguation

Word-sense disambiguation concerns finding a suitable translation when a word can have more than one meaning. The problem was first raised in the 1950s by Yehoshua Bar-Hillel. He pointed out that without a "universal encyclopedia", a machine would never be able to distinguish between the two meanings of a word. Today there are numerous approaches designed to overcome this problem. They can be approximately divided into "shallow" approaches and "deep" approaches.

Shallow approaches assume no knowledge of the text. They simply apply statistical methods to the words surrounding the ambiguous word. Deep approaches presume a comprehensive knowledge of the word. So far, shallow approaches have been more successful.

Claude Piron, a long-time translator for the United Nations and the World Health Organization, wrote that machine translation, at its best, automates the easier part of a translator's job; the harder and more time-consuming part usually involves doing extensive research to resolve ambiguities in the source text, which the grammatical and lexical exigencies of the target language require to be resolved:

Why does a translator need a whole workday to translate five pages, and not an hour or two? ..... About 90% of an average text corresponds to these simple conditions. But unfortunately, there's the other 10%. It's that part that requires six [more] hours of work. There are ambiguities one has to resolve. For instance, the author of the source text, an Australian physician, cited the example of an epidemic which was declared during World War II in a "Japanese prisoners of war camp". Was he talking about an American camp with Japanese prisoners or a Japanese camp with American prisoners? The English has two senses. It's necessary therefore to do research, maybe to the extent of a phone call to Australia.

The ideal deep approach would require the translation software to do all the research necessary for this kind of disambiguation on its own; but this would require a higher degree of AI than has yet been attained. A shallow approach which simply guessed at the sense of the ambiguous English phrase that Piron mentions (based, perhaps, on which kind of prisoner-of-war camp is more often mentioned in a given corpus) would have a reasonable chance of guessing wrong fairly often. A shallow approach that involves "ask the user about each ambiguity" would, by Piron's estimate, only automate about 25% of a professional translator's job, leaving the harder 75% still to be done by a human.

Non-standard speech

One of the major pitfalls of MT is its inability to translate non-standard language with the same accuracy as standard language. Heuristic or statistical based MT takes input from various sources in standard form of a language. Rule-based translation, by nature, does not include common non-standard usages. This causes errors in translation from a vernacular source or into colloquial language. Limitations on translation from casual speech present issues in the use of machine translation in mobile devices.

Named entities

Main article: Named entity

In information extraction, named entities, in a narrow sense, refer to concrete or abstract entities in the real world such as people, organizations, companies, and places that have a proper name: George Washington, Chicago, Microsoft. It also refers to expressions of time, space and quantity such as 1 July 2011, $500.

In the sentence "Smith is the president of Fabrionix" both Smith and Fabrionix are named entities, and can be further qualified via first name or other information; "president" is not, since Smith could have earlier held another position at Fabrionix, e.g. Vice President. The term rigid designator is what defines these usages for analysis in statistical machine translation.

Named entities must first be identified in the text; if not, they may be erroneously translated as common nouns, which would most likely not affect the BLEU rating of the translation but would change the text's human readability. They may be omitted from the output translation, which would also have implications for the text's readability and message.

Transliteration includes finding the letters in the target language that most closely correspond to the name in the source language. This, however, has been cited as sometimes worsening the quality of translation. For "Southern California" the first word should be translated directly, while the second word should be transliterated. Machines often transliterate both because they treated them as one entity. Words like these are hard for machine translators, even those with a transliteration component, to process.

Use of a "do-not-translate" list, which has the same end goal – transliteration as opposed to translation. still relies on correct identification of named entities.

A third approach is a class-based model. Named entities are replaced with a token to represent their "class"; "Ted" and "Erica" would both be replaced with "person" class token. Then the statistical distribution and use of person names, in general, can be analyzed instead of looking at the distributions of "Ted" and "Erica" individually, so that the probability of a given name in a specific language will not affect the assigned probability of a translation. A study by Stanford on improving this area of translation gives the examples that different probabilities will be assigned to "David is going for a walk" and "Ankit is going for a walk" for English as a target language due to the different number of occurrences for each name in the training data. A frustrating outcome of the same study by Stanford (and other attempts to improve named recognition translation) is that many times, a decrease in the BLEU scores for translation will result from the inclusion of methods for named entity translation.

Somewhat related are the phrases "drinking tea with milk" vs. "drinking tea with Molly."

Translation from multiparallel sources

Some work has been done in the utilization of multiparallel corpora, that is a body of text that has been translated into 3 or more languages. Using these methods, a text that has been translated into 2 or more languages may be utilized in combination to provide a more accurate translation into a third language compared with if just one of those source languages were used alone.

Ontologies in MT

An ontology is a formal representation of knowledge that includes the concepts (such as objects, processes etc.) in a domain and some relations between them. If the stored information is of linguistic nature, one can speak of a lexicon. In NLP, ontologies can be used as a source of knowledge for machine translation systems. With access to a large knowledge base, systems can be enabled to resolve many (especially lexical) ambiguities on their own. In the following classic examples, as humans, we are able to interpret the prepositional phrase according to the context because we use our world knowledge, stored in our lexicons:

I saw a man/star/molecule with a microscope/telescope/binoculars.

A machine translation system initially would not be able to differentiate between the meanings because syntax does not change. With a large enough ontology as a source of knowledge however, the possible interpretations of ambiguous words in a specific context can be reduced. Other areas of usage for ontologies within NLP include information retrieval, information extraction and text summarization.

Building ontologies

The ontology generated for the PANGLOSS knowledge-based machine translation system in 1993 may serve as an example of how an ontology for NLP purposes can be compiled:

• A large-scale ontology is necessary to help parsing in the active modules of the machine translation system.
• In the PANGLOSS example, about 50,000 nodes were intended to be subsumed under the smaller, manually-built upper (abstract) region of the ontology. Because of its size, it had to be created automatically.
• The goal was to merge the two resources LDOCE online and WordNet to combine the benefits of both: concise definitions from Longman, and semantic relations allowing for semi-automatic taxonomization to the ontology from WordNet.
  • A definition match algorithm was created to automatically merge the correct meanings of ambiguous words between the two online resources, based on the words that the definitions of those meanings have in common in LDOCE and WordNet. Using a similarity matrix, the algorithm delivered matches between meanings including a confidence factor. This algorithm alone, however, did not match all meanings correctly on its own.
  • A second hierarchy match algorithm was therefore created which uses the taxonomic hierarchies found in WordNet (deep hierarchies) and partially in LDOCE (flat hierarchies). This works by first matching unambiguous meanings, then limiting the search space to only the respective ancestors and descendants of those matched meanings. Thus, the algorithm matched locally unambiguous meanings (for instance, while the word seal as such is ambiguous, there is only one meaning of seal in the animal subhierarchy).
• Both algorithms complemented each other and helped constructing a large-scale ontology for the machine translation system. The WordNet hierarchies, coupled with the matching definitions of LDOCE, were subordinated to the ontology's upper region. As a result, the PANGLOSS MT system was able to make use of this knowledge base, mainly in its generation element.

Applications

While no system provides the ideal of fully automatic high-quality machine translation of unrestricted text, many fully automated systems produce reasonable output. The quality of machine translation is substantially improved if the domain is restricted and controlled. This enables using machine translation as a tool to speed up and simplify translations, as well as producing flawed but useful low-cost or ad-hoc translations.

Travel

Machine translation applications have also been released for most mobile devices, including mobile telephones, pocket PCs, PDAs, etc. Due to their portability, such instruments have come to be designated as mobile translation tools enabling mobile business networking between partners speaking different languages, or facilitating both foreign language learning and unaccompanied traveling to foreign countries without the need of the intermediation of a human translator.

For example, the Google Translate app allows foreigners to quickly translate text in their surrounding via augmented reality using the smartphone camera that overlays the translated text onto the text. It can also recognize speech and then translate it.

Public administration

Despite their inherent limitations, MT programs are used around the world. Probably the largest institutional user is the European Commission. In the 2012, with an aim to replace a rule-based MT by newer, statistical-based MT@EC, The European Commission contributed 3.072 million euros (via its ISA programme).

Wikipedia

Machine translation has also been used for translating Wikipedia articles and could play a larger role in creating, updating, expanding, and generally improving articles in the future, especially as the MT capabilities may improve. There is a "content translation tool" which allows editors to more easily translate articles across several select languages. English-language articles are thought to usually be more comprehensive and less biased than their non-translated equivalents in other languages. As of 2022, English Wikipedia has over 6.5 million articles while the German and Swedish Wikipedias each only have over 2.5 million articles, each often far less comprehensive.

Surveillance and military

Following terrorist attacks in Western countries, including 9-11, the U.S. and its allies have been most interested in developing Arabic machine translation programs, but also in translating Pashto and Dari languages. Within these languages, the focus is on key phrases and quick communication between military members and civilians through the use of mobile phone apps. The Information Processing Technology Office in DARPA hosted programs like TIDES and Babylon translator. US Air Force has awarded a $1 million contract to develop a language translation technology.

Social media

The notable rise of social networking on the web in recent years has created yet another niche for the application of machine translation software – in utilities such as Facebook, or instant messaging clients such as Skype, GoogleTalk, MSN Messenger, etc. – allowing users speaking different languages to communicate with each other.

Online games

Lineage W gained popularity in Japan because of its machine translation features allowing players from different countries to communicate.

Medicine

Despite being labelled as an unworthy competitor to human translation in 1966 by the Automated Language Processing Advisory Committee put together by the United States government, the quality of machine translation has now been improved to such levels that its application in online collaboration and in the medical field are being investigated. The application of this technology in medical settings where human translators are absent is another topic of research, but difficulties arise due to the importance of accurate translations in medical diagnoses.

Ancient languages

The advancements in convolutional neural networks in recent years and in low resource machine translation (when only a very limited amout of data and examples are available for training) enabled machine translation for ancient languages, such as Akkadian and its dialects Babylonian and Assyrian.

Evaluation

Main article: Evaluation of machine translation

There are many factors that affect how machine translation systems are evaluated. These factors include the intended use of the translation, the nature of the machine translation software, and the nature of the translation process.

Different programs may work well for different purposes. For example, statistical machine translation (SMT) typically outperforms example-based machine translation (EBMT), but researchers found that when evaluating English to French translation, EBMT performs better. The same concept applies for technical documents, which can be more easily translated by SMT because of their formal language.

In certain applications, however, e.g., product descriptions written in a controlled language, a dictionary-based machine-translation system has produced satisfactory translations that require no human intervention save for quality inspection.

There are various means for evaluating the output quality of machine translation systems. The oldest is the use of human judges to assess a translation's quality. Even though human evaluation is time-consuming, it is still the most reliable method to compare different systems such as rule-based and statistical systems. Automated means of evaluation include BLEU, NIST, METEOR, and LEPOR.

Relying exclusively on unedited machine translation ignores the fact that communication in human language is context-embedded and that it takes a person to comprehend the context of the original text with a reasonable degree of probability. It is certainly true that even purely human-generated translations are prone to error. Therefore, to ensure that a machine-generated translation will be useful to a human being and that publishable-quality translation is achieved, such translations must be reviewed and edited by a human. The late Claude Piron wrote that machine translation, at its best, automates the easier part of a translator's job; the harder and more time-consuming part usually involves doing extensive research to resolve ambiguities in the source text, which the grammatical and lexical exigencies of the target language require to be resolved. Such research is a necessary prelude to the pre-editing necessary in order to provide input for machine-translation software such that the output will not be meaningless.

In addition to disambiguation problems, decreased accuracy can occur due to varying levels of training data for machine translating programs. Both example-based and statistical machine translation rely on a vast array of real example sentences as a base for translation, and when too many or too few sentences are analyzed accuracy is jeopardized. Researchers found that when a program is trained on 203,529 sentence pairings, accuracy actually decreases. The optimal level of training data seems to be just over 100,000 sentences, possibly because as training data increases, the number of possible sentences increases, making it harder to find an exact translation match.

Flaws in machine translation have been noted for their entertainment value. Two videos uploaded to YouTube in April 2017 involve two Japanese hiragana characters えぐ (e and gu) being repeatedly pasted into Google Translate, with the resulting translations quickly degrading into nonsensical phrases such as "DECEARING EGG" and "Deep-sea squeeze trees", which are then read in increasingly absurd voices; the full-length version of the video currently has 6.9 million views as of March 2022.

Machine translation and signed languages

Main article: Machine translation of sign languages

In the early 2000s, options for machine translation between spoken and signed languages were severely limited. It was a common belief that deaf individuals could use traditional translators. However, stress, intonation, pitch, and timing are conveyed much differently in spoken languages compared to signed languages. Therefore, a deaf individual may misinterpret or become confused about the meaning of written text that is based on a spoken language.

Researchers Zhao, et al. (2000), developed a prototype called TEAM (translation from English to ASL by machine) that completed English to American Sign Language (ASL) translations. The program would first analyze the syntactic, grammatical, and morphological aspects of the English text. Following this step, the program accessed a sign synthesizer, which acted as a dictionary for ASL. This synthesizer housed the process one must follow to complete ASL signs, as well as the meanings of these signs. Once the entire text is analyzed and the signs necessary to complete the translation are located in the synthesizer, a computer generated human appeared and would use ASL to sign the English text to the user.

Copyright

Only works that are original are subject to copyright protection, so some scholars claim that machine translation results are not entitled to copyright protection because MT does not involve creativity. The copyright at issue is for a derivative work; the author of the original work in the original language does not lose his rights when a work is translated: a translator must have permission to publish a translation.

AI-complete

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/AI-complete

In the field of artificial intelligence, the most difficult problems are informally known as AI-complete or AI-hard, implying that the difficulty of these computational problems, assuming intelligence is computational, is equivalent to that of solving the central artificial intelligence problem—making computers as intelligent as people, or strong AI. To call a problem AI-complete reflects an attitude that it would not be solved by a simple specific algorithm.

AI-complete problems are hypothesised to include computer vision, natural language understanding, and dealing with unexpected circumstances while solving any real-world problem.

Currently, AI-complete problems cannot be solved with modern computer technology alone, but would also require human computation. This property could be useful, for example, to test for the presence of humans as CAPTCHAs aim to do, and for computer security to circumvent brute-force attacks.

History

The term was coined by Fanya Montalvo by analogy with NP-complete and NP-hard in complexity theory, which formally describes the most famous class of difficult problems. Early uses of the term are in Erik Mueller's 1987 PhD dissertation and in Eric Raymond's 1991 Jargon File.

AI-complete problems

AI-complete problems are hypothesized to include:

• AI peer review (composite natural language understanding, automated reasoning, automated theorem proving, formalized logic expert system)
• Bongard problems
• Computer vision (and subproblems such as object recognition)
• Natural language understanding (and subproblems such as text mining, machine translation, and word-sense disambiguation)
• Autonomous driving
• Dealing with unexpected circumstances while solving any real world problem, whether it's navigation or planning or even the kind of reasoning done by expert systems.

Machine translation

Main article: Machine translation

To translate accurately, a machine must be able to understand the text. It must be able to follow the author's argument, so it must have some ability to reason. It must have extensive world knowledge so that it knows what is being discussed — it must at least be familiar with all the same commonsense facts that the average human translator knows. Some of this knowledge is in the form of facts that can be explicitly represented, but some knowledge is unconscious and closely tied to the human body: for example, the machine may need to understand how an ocean makes one feel to accurately translate a specific metaphor in the text. It must also model the authors' goals, intentions, and emotional states to accurately reproduce them in a new language. In short, the machine is required to have wide variety of human intellectual skills, including reason, commonsense knowledge and the intuitions that underlie motion and manipulation, perception, and social intelligence. Machine translation, therefore, is believed to be AI-complete: it may require strong AI to be done as well as humans can do it.

Software brittleness

Main article: Software brittleness

Current AI systems can solve very simple and/or restricted versions of AI-complete problems, but never in their full generality. When AI researchers attempt to "scale up" their systems to handle more complicated, real-world situations, the programs tend to become excessively brittle without commonsense knowledge or a rudimentary understanding of the situation: they fail as unexpected circumstances outside of its original problem context begin to appear. When human beings are dealing with new situations in the world, they are helped immensely by the fact that they know what to expect: they know what all things around them are, why they are there, what they are likely to do and so on. They can recognize unusual situations and adjust accordingly. A machine without strong AI has no other skills to fall back on.

DeepMind published a work in May 2022 in which they trained a single model to do several things at the same time. The model, named Gato, can "play Atari, caption images, chat, stack blocks with a real robot arm and much more, deciding based on its context whether to output text, joint torques, button presses, or other tokens."

Formalization

Computational complexity theory deals with the relative computational difficulty of computable functions. By definition, it does not cover problems whose solution is unknown or has not been characterised formally. Since many AI problems have no formalisation yet, conventional complexity theory does not allow the definition of AI-completeness.

To address this problem, a complexity theory for AI has been proposed. It is based on a model of computation that splits the computational burden between a computer and a human: one part is solved by computer and the other part solved by human. This is formalised by a human-assisted Turing machine. The formalisation defines algorithm complexity, problem complexity and reducibility which in turn allows equivalence classes to be defined.

The complexity of executing an algorithm with a human-assisted Turing machine is given by a pair ${\displaystyle ⟨{\mathrm{\Phi }}_H,{\mathrm{\Phi }}_M⟩}$, where the first element represents the complexity of the human's part and the second element is the complexity of the machine's part.

Results

The complexity of solving the following problems with a human-assisted Turing machine is:

• Optical character recognition for printed text: ${\displaystyle ⟨O(1),poly(n)⟩}$
• Turing test:
  • for an ${\displaystyle n}$-sentence conversation where the oracle remembers the conversation history (persistent oracle): ${\displaystyle ⟨O(n),O(n)⟩}$
  • for an ${\displaystyle n}$-sentence conversation where the conversation history must be retransmitted: ${\displaystyle ⟨O(n),O(n^2)⟩}$
  • for an ${\displaystyle n}$-sentence conversation where the conversation history must be retransmitted and the person takes linear time to read the query: ${\displaystyle ⟨O(n^2),O(n^2)⟩}$
• ESP game: ${\displaystyle ⟨O(n),O(n)⟩}$
• Image labelling (based on the Arthur–Merlin protocol): ${\displaystyle ⟨O(n),O(n)⟩}$
• Image classification: human only: ${\displaystyle ⟨O(n),O(n)⟩}$, and with less reliance on the human: ${\displaystyle ⟨O(\log n),O(n\log n)⟩}$.

Turing completeness

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Turing_completeness

In computability theory, a system of data-manipulation rules (such as a model of computation, a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any Turing machine (devised by English mathematician and computer scientist Alan Turing). This means that this system is able to recognize or decide other data-manipulation rule sets. Turing completeness is used as a way to express the power of such a data-manipulation rule set. Virtually all programming languages today are Turing-complete.

A related concept is that of Turing equivalence – two computers P and Q are called equivalent if P can simulate Q and Q can simulate P. The Church–Turing thesis conjectures that any function whose values can be computed by an algorithm can be computed by a Turing machine, and therefore that if any real-world computer can simulate a Turing machine, it is Turing equivalent to a Turing machine. A universal Turing machine can be used to simulate any Turing machine and by extension the computational aspects of any possible real-world computer.^{[NB 1]}

To show that something is Turing-complete, it is enough to show that it can be used to simulate some Turing-complete system. No physical system can have infinite memory, but if the limitation of finite memory is ignored, most programming languages are otherwise Turing-complete.

Non-mathematical usage

In colloquial usage, the terms "Turing-complete" and "Turing-equivalent" are used to mean that any real-world general-purpose computer or computer language can approximately simulate the computational aspects of any other real-world general-purpose computer or computer language. In real life, this leads to the practical concepts of computing virtualization and emulation.

Real computers constructed so far can be functionally analyzed like a single-tape Turing machine (the "tape" corresponding to their memory); thus the associated mathematics can apply by abstracting their operation far enough. However, real computers have limited physical resources, so they are only linear bounded automaton complete. In contrast, a universal computer is defined as a device with a Turing-complete instruction set, infinite memory, and infinite available time.

Formal definitions

In computability theory, several closely related terms are used to describe the computational power of a computational system (such as an abstract machine or programming language):

Turing completeness

A computational system that can compute every Turing-computable function is called Turing-complete (or Turing-powerful). Alternatively, such a system is one that can simulate a universal Turing machine.

Turing equivalence

A Turing-complete system is called Turing-equivalent if every function it can compute is also Turing-computable; i.e., it computes precisely the same class of functions as do Turing machines. Alternatively, a Turing-equivalent system is one that can simulate, and be simulated by, a universal Turing machine. (All known physically-implementable Turing-complete systems are Turing-equivalent, which adds support to the Church–Turing thesis.)

(Computational) universality

A system is called universal with respect to a class of systems if it can compute every function computable by systems in that class (or can simulate each of those systems). Typically, the term 'universality' is tacitly used with respect to a Turing-complete class of systems. The term "weakly universal" is sometimes used to distinguish a system (e.g. a cellular automaton) whose universality is achieved only by modifying the standard definition of Turing machine so as to include input streams with infinitely many 1s.

History

Turing completeness is significant in that every real-world design for a computing device can be simulated by a universal Turing machine. The Church–Turing thesis states that this is a law of mathematics – that a universal Turing machine can, in principle, perform any calculation that any other programmable computer can. This says nothing about the effort needed to write the program, or the time it may take for the machine to perform the calculation, or any abilities the machine may possess that have nothing to do with computation.

Charles Babbage's analytical engine (1830s) would have been the first Turing-complete machine if it had been built at the time it was designed. Babbage appreciated that the machine was capable of great feats of calculation, including primitive logical reasoning, but he did not appreciate that no other machine could do better. From the 1830s until the 1940s, mechanical calculating machines such as adders and multipliers were built and improved, but they could not perform a conditional branch and therefore were not Turing-complete.

In the late 19th century, Leopold Kronecker formulated notions of computability, defining primitive recursive functions. These functions can be calculated by rote computation, but they are not enough to make a universal computer, because the instructions that compute them do not allow for an infinite loop. In the early 20th century, David Hilbert led a program to axiomatize all of mathematics with precise axioms and precise logical rules of deduction that could be performed by a machine. Soon it became clear that a small set of deduction rules are enough to produce the consequences of any set of axioms. These rules were proved by Kurt Gödel in 1930 to be enough to produce every theorem.

The actual notion of computation was isolated soon after, starting with Gödel's incompleteness theorem. This theorem showed that axiom systems were limited when reasoning about the computation that deduces their theorems. Church and Turing independently demonstrated that Hilbert's Entscheidungsproblem (decision problem) was unsolvable, thus identifying the computational core of the incompleteness theorem. This work, along with Gödel's work on general recursive functions, established that there are sets of simple instructions, which, when put together, are able to produce any computation. The work of Gödel showed that the notion of computation is essentially unique.

In 1941 Konrad Zuse completed the Z3 computer. Zuse was not familiar with Turing's work on computability at the time. In particular, the Z3 lacked dedicated facilities for a conditional jump, thereby precluding it from being Turing complete. However, in 1998, it was shown by Rojas that the Z3 is capable of simulating conditional jumps, and therefore Turing complete in theory. To do this, its tape program would have to be long enough to execute every possible path through both sides of every branch.

The first computer capable of conditional branching in practice, and therefore Turing complete in practice, was the ENIAC in 1946. Zuse's Z4 computer was operational in 1945, but it did not support conditional branching until 1950.

Computability theory

Computability theory uses models of computation to analyze problems and determine whether they are computable and under what circumstances. The first result of computability theory is that there exist problems for which it is impossible to predict what a (Turing-complete) system will do over an arbitrarily long time.

The classic example is the halting problem: create an algorithm that takes as input a program in some Turing-complete language and some data to be fed to that program, and determines whether the program, operating on the input, will eventually stop or will continue forever. It is trivial to create an algorithm that can do this for some inputs, but impossible to do this in general. For any characteristic of the program's eventual output, it is impossible to determine whether this characteristic will hold.

This impossibility poses problems when analyzing real-world computer programs. For example, one cannot write a tool that entirely protects programmers from writing infinite loops or protects users from supplying input that would cause infinite loops.

One can instead limit a program to executing only for a fixed period of time (timeout) or limit the power of flow-control instructions (for example, providing only loops that iterate over the items of an existing array). However, another theorem shows that there are problems solvable by Turing-complete languages that cannot be solved by any language with only finite looping abilities (i.e., any language guaranteeing that every program will eventually finish to a halt). So any such language is not Turing-complete. For example, a language in which programs are guaranteed to complete and halt cannot compute the computable function produced by Cantor's diagonal argument on all computable functions in that language.

Turing oracles

Main article: Oracle machine

A computer with access to an infinite tape of data may be more powerful than a Turing machine: for instance, the tape might contain the solution to the halting problem or some other Turing-undecidable problem. Such an infinite tape of data is called a Turing oracle. Even a Turing oracle with random data is not computable (with probability 1), since there are only countably many computations but uncountably many oracles. So a computer with a random Turing oracle can compute things that a Turing machine cannot.

Digital physics

See also: Church–Turing thesis § Philosophical implications

All known laws of physics have consequences that are computable by a series of approximations on a digital computer. A hypothesis called digital physics states that this is no accident because the universe itself is computable on a universal Turing machine. This would imply that no computer more powerful than a universal Turing machine can be built physically.

Examples

The computational systems (algebras, calculi) that are discussed as Turing-complete systems are those intended for studying theoretical computer science. They are intended to be as simple as possible, so that it would be easier to understand the limits of computation. Here are a few:

• Automata theory
• Formal grammar (language generators)
• Formal language (language recognizers)
• Lambda calculus
• Post–Turing machines
• Process calculus

Most programming languages (their abstract models, maybe with some particular constructs that assume finite memory omitted), conventional and unconventional, are Turing-complete. This includes:

• All general-purpose languages in wide use.
  • Procedural programming languages such as C, Pascal.
  • Object-oriented languages such as Java, Smalltalk or C#.
  • Multi-paradigm languages such as Ada, C++, Common Lisp, Fortran, JavaScript, Object Pascal, Perl, Python, R.
• Most languages using less common paradigms:
  • Functional languages such as Lisp and Haskell.
  • Logic programming languages such as Prolog.
  • General-purpose macro processor such as m4.
  • Declarative languages such as XSLT.
  • VHDL and other hardware description languages.
  • TeX, a typesetting system.
  • Esoteric programming languages, a form of mathematical recreation in which programmers work out how to achieve basic programming constructs in an extremely difficult but mathematically Turing-equivalent language.

Some rewrite systems are Turing-complete.

Turing completeness is an abstract statement of ability, rather than a prescription of specific language features used to implement that ability. The features used to achieve Turing completeness can be quite different; Fortran systems would use loop constructs or possibly even goto statements to achieve repetition; Haskell and Prolog, lacking looping almost entirely, would use recursion. Most programming languages are describing computations on von Neumann architectures, which have memory (RAM and register) and a control unit. These two elements make this architecture Turing-complete. Even pure functional languages are Turing-complete.

Turing completeness in declarative SQL is implemented through recursive common table expressions. Unsurprisingly, procedural extensions to SQL (PLSQL, etc.) are also Turing-complete. This illustrates one reason why relatively powerful non-Turing-complete languages are rare: the more powerful the language is initially, the more complex are the tasks to which it is applied and the sooner its lack of completeness becomes perceived as a drawback, encouraging its extension until it is Turing-complete.

The untyped lambda calculus is Turing-complete, but many typed lambda calculi, including System F, are not. The value of typed systems is based in their ability to represent most typical computer programs while detecting more errors.

Rule 110 and Conway's Game of Life, both cellular automata, are Turing-complete.

Unintentional Turing completeness

Some games and other software are Turing-complete by accident, i.e. not by design.

Software:

• Microsoft Excel

Games:

• Dwarf Fortress
• Cities: Skylines
• Opus Magnum
• Minecraft
• Magic: The Gathering

Social media:

• Habbo Hotel

Computational languages:

• C++ templates
• printf format string
• TypeScript's type system
• x86 assembly's MOV instruction

Biology:

• Chemical reaction networks and enzyme-based DNA computers have been shown to be Turing-equivalent

Non-Turing-complete languages

Many computational languages exist that are not Turing-complete. One such example is the set of regular languages, which are generated by regular expressions and which are recognized by finite automata. A more powerful but still not Turing-complete extension of finite automata is the category of pushdown automata and context-free grammars, which are commonly used to generate parse trees in an initial stage of program compiling. Further examples include some of the early versions of the pixel shader languages embedded in Direct3D and OpenGL extensions.

In total functional programming languages, such as Charity and Epigram, all functions are total and must terminate. Charity uses a type system and control constructs based on category theory, whereas Epigram uses dependent types. The LOOP language is designed so that it computes only the functions that are primitive recursive. All of these compute proper subsets of the total computable functions, since the full set of total computable functions is not computably enumerable. Also, since all functions in these languages are total, algorithms for recursively enumerable sets cannot be written in these languages, in contrast with Turing machines.

Although (untyped) lambda calculus is Turing-complete, simply typed lambda calculus is not.