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| :''This article is a temporary experiment to see whether it is feasible and desirable to merge the articles [[Recursive set]], [[Recursive language]], [[Decidable language]], [[Decidable problem]] and [[Undecidable problem]]. Input on how best to do this is very much welcome on [[Talk:Recursive languages and sets|the article's talk page]]. This is a work in progress so the current version may seem awkward.''
| | If you present photography effectively, it helps you look much more properly at the globe around you. Offshore expert Word - Press developers high level of interactivity, accessibility, functionality and usability of our website can add custom online to using. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. They found out all the possible information about bringing up your baby and save money at the same time. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day. <br><br>Right starting from social media support to search engine optimization, such plugins are easily available within the Word - Press open source platform. The higher your blog ranks on search engines, the more likely people will find your online marketing site. We also help to integrate various plug-ins to expand the functionalities of the web application. This is identical to doing a research as in depth above, nevertheless you can see various statistical details like the number of downloads and when the template was not long ago updated. W3C compliant HTML and a good open source powered by Word - Press CMS site is regarded as the prime minister. <br><br>It is also popular because willing surrogates,as well as egg and sperm donors,are plentiful. Word - Press has different exciting features including a plug-in architecture with a templating system. If Gandhi was empowered with a blogging system, every event in his life would have been minutely documented so that it could be recounted to the future generations. For more information on [http://surf632.com/members/wendenewling/profile/ backup plugin] visit the web-site. The animation can be quite subtle these as snow falling gently or some twinkling start in the track record which are essentially not distracting but as an alternative gives some viewing enjoyment for the visitor of the internet site. If you have any questions on starting a Word - Press food blog or any blog for that matter, please post them below and I will try to answer them. <br><br>If all else fails, please leave a comment on this post with the issue(s) you're having and help will be on the way. The SEOPressor Word - Press SEO Plugin works by analysing each page and post against your chosen keyword (or keyword phrase) and giving a score, with instructions on how to improve it. Exacting subjects in reality must be accumulated in head ahead of planning on your high quality theme. The company gains commission from the customers' payment. See a product, take a picture, and it gives you an Amazon price for that product, or related products. <br><br>There is no denying that Magento is an ideal platform for building ecommerce websites, as it comes with an astounding number of options that can help your online business do extremely well. Mahatma Gandhi is known as one of the most prominent personalities and symbols of peace, non-violence and freedom. As a result, it is really crucial to just take aid of some experience when searching for superior quality totally free Word - Press themes, Word - Press Premium Themes for your web site. If this is not possible you still have the choice of the default theme that is Word - Press 3. Definitely when you wake up from the slumber, you can be sure that you will be lagging behind and getting on track would be a tall order. |
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| In [[computability theory]], a set is '''decidable''', '''computable''', or '''recursive''' if there is an [[algorithm]] that terminates after a finite amount of time and correctly decides whether a given object belongs to the set. Decidability of a set is of particular interest when the set is viewed as a [[decision problem]]; a decidable set is also a '''decidable problem''', '''computable problem''', and '''recursive problem'''. The remainder of this article uses the term ''decidable'', although ''recursive'' and ''computable'' are equivalent in this context.
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| A '''language''' is a set of [[String (computer science)|finite strings]] over a particular [[Alphabet (computer science)|alphabet]]. A language is decidable (also computable, recursive) if it is a decidable set.
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| A set, language, or decision problem that is not decidable is '''undecidable''', '''non-recursive''', '''non-computable''', or '''uncomputable'''. There are many known undecidable sets; one of the earliest, and most famous, examples is the [[halting problem]].
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| Decidable sets and languages are a strict subclass of the class of [[recursively enumerable set]]s. For those sets, it is only required that there is an algorithm that correctly decides when an input ''is'' in the set; the algorithm may fail to terminate for inputs not belonging to the set.
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| ==Formal definition==
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| A subset ''S'' of the [[natural numbers]] is called '''decidable''' if there exists a [[total function|total]] [[computable function]] <math>f</math> such that
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| <math>f(x) = 0\,</math> if <math>x \in S</math> and <math>f(x) \not = 0</math> if <math>x \notin S</math>. In other words, the set ''S'' is decidable [[if and only if]] the [[indicator function]] <math>1_{S}</math> is [[computable function|computable]].
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| A parallel definition applies to sets of strings over some finite alphabet; these sets are often called '''languages'''. A language is decidable if there is a computable function taking strings over the alphabet as input, which returns 0 when presented with a string not in the language, and returns 1 when presented with a string in the language.
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| The definition can be extended to arbitrary countable sets via [[Gödel numbering]]s. If each element of a set ''U'' has a unique associated natural number, a subset ''C'' of ''U'' is called computable if the set of natural numbers corresponding to the elements of ''C'' is decidable under the definition above. A similar definition can be made in which elements of ''U'' are identified with finite strings rather than natural numbers.
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| The definition can also be extended to sets of ordered pairs, ordered triples, and more generally finite sequences of objects. One way to do this is to use computable functions taking more than one argument – for example, a set ''A'' of ordered pairs of elements of a set ''X'' is decidable if there is a computable function ''g'' taking two arguments, such that for all ''x'' and ''y'' in ''X'', ''g''(''x'',''y'')= 0 if the pair (''x'',''y'') is not in ''A'', and ''g''(''x'',''y'')=1 if the pair is in ''A''. Another way of defining decidability for sets of sequences is to use a [[Cantor pairing function|pairing function]] to identify each sequence with a single object (natural number or string). Then the definitions of decidability above can be directly applied. This second method is particularly useful when the set in question contains sequences of varying lengths.
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| ==Examples==
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| There are many examples of decidable sets:
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| * The [[empty set]] is decidable, and the entire set of natural numbers is decidable.
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| * Every finite or [[cofinite]] subset of the natural numbers is decidable.
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| * The set of [[prime number]]s is decidable.
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| * The finite binary strings with an even number of 1s is decidable.
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| * If ''f'' is a [[computable function]] then the set of pairs (''x'',''y'') such that ''f''(''x'') = ''y'' is decidable.
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| It is possible for a set to be decidable even if the precise algorithm that decides it is not known. For example, consider the set ''A'' containing all natural numbers ''n'' such that there is a pair of [[twin prime]]s larger than ''n''. It is not presently known whether there are infinitely many twin primes, or whether (otherwise) there is a largest pair of twin primes. But in either case, the set ''A'' is decidable. If there are infinitely many twin primes, ''A'' contains every natural number, and is thus decidable. Otherwise, there is a largest pair of twin primes, which means ''A'' is finite, and thus decidable. This means that, regardless of whether there are infinitely many twin primes, the set ''A'' is decidable, despite the fact that the correct algorithm has not been identified.
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| ==Properties==
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| The class of decidable sets has numerous closure properties.
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| *If ''A'' is a decidable set then the [[complement (set theory)|complement]] of ''A'' is also a decidable set.
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| *If ''A'' and ''B'' are decidable sets then ''A'' ∩ ''B'', ''A'' ∪ ''B'', and <math>A \setminus B</math> are decidable.
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| * If ''A'' and ''B'' are decidable sets then ''A'' × ''B'' is decidable; this is the set of pairs (''x'',''y'') such that ''x'' is in ''A'' and ''y'' is in ''B''. Moreover, the image of ''A'' × ''B'' under the [[Cantor pairing function]] is decidable.
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| *The [[preimage]] of a decidable set under a [[total function|total]] [[computable function]] is a decidable set.
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| * The image of a decidable set under a total computable [[bijection]] is decidable.
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| Sets of strings have additional closure properties. If ''L'' and ''P'' are two decidable languages, then the following languages are also decidable:
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| * The [[Kleene star]] ''L''<sup>∗</sup>. A string is in this set if and only if it can be obtained by concatenating zero or more elements of ''L'', with repetition allowed.
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| * The concatenation ''L''∘ ''P''. A string is in this set if and only if it can be written as an element of ''L'' followed by an element of ''P''.
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| Several characterizations of decidable sets are known.
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| * A set ''A'' is decidable if and only if both ''A'' and the [[complement (set theory)|complement]] of ''A'' are [[recursively enumerable set]]s.
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| * A set of natural numbers is decidable if and only if it is at level <math>\Delta^0_1</math> of the [[arithmetical hierarchy]].
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| * A set of natural numbers is decidable if and only if it is either the range of a nondecreasing total computable function or is the empty set. Conversely, the image of a decidable set under a nondecreasing total computable function is decidable.
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| ==Decidable languages==
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| A '''decidable language''' in [[mathematics]], [[logic]] and [[computer science]], is a type of [[formal language]] which is also called '''recursive''' or '''Turing-decidable'''. Since computational problems can be formulated in terms of testing membership in a language, showing that a language is decidable is thus considered the same as showing that an equivalent computational problem is decidable.<ref>{{cite book|last=Sipser|first=Michael|title=Introduction to the Theory of Computation, Third Edition|year=2013|publisher=Cengage Learning|location=Boston, MA|isbn=978-1-133-18779-0|page=195}}</ref> In other words, we can say that a language L is decidable if there is a [[Turing Machine]] which decides L and halts on every input (meaning it either accepts or rejects, but never enters an infinite loop). A computational problem is thus considered decidable if it can be solved by a computer.
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| The class of all decidable languages is often called '''[[R (complexity)|R]]''', although this name is also used for the class [[RP (complexity)|RP]]. All decidable languages are [[recursively enumerable language|recursively enumerable]], and all [[regular language|regular]], [[context-free language|context-free]] and [[context-sensitive language|context-sensitive]] languages are decidable. In other words, all languages recognized by [[Deterministic finite automaton|DFA]]’s, [[Nondeterministic finite automaton|NFA]]’s, and [[Context-free grammar|CFG]]’s are decidable. However, not all languages recognized by a [[Turing machine|TM]] are decidable. Similarly, not all languages that are recognizable are decidable.<ref>{{cite book|last=Sipser|first=Michael|title=Introduction to the Theory of Computation, Third Edition|year=2013|publisher=Cengage Learning|location=Boston, MA|isbn=978-1-133-18779-0|page=201}}</ref> | |
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| This type of language was not defined in the [[Chomsky hierarchy]] of {{Harv|Chomsky|1959}}, and there is no simple class of [[formal grammar]]s that capture the decidable languages.{{Citation needed|date=October 2008}}
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| === List of Common Decidable Languages and Problems <ref>{{cite book|last=Sipser|first=Michael|title=Introduction to the Theory of Computation, Third Edition|year=2013|publisher=Cengage Learning|location=Boston, MA|isbn=978-1-133-18779-0|pages=194–200}}</ref> ===
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| <math>
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| A_{DFA} = \{\langle B,w \rangle \mid B \text{ is a DFA that accepts input string } w \}
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| </math>
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| <math>
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| A_{NFA} = \{\langle B,w \rangle \mid B \text{ is a NFA that accepts input string } w \}
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| </math>
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| <math>
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| A_{REX} = \{\langle R,w \rangle \mid R \text{ is a regular expression that generates string } w \}
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| </math>
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| <math>
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| A_{CFG} = \{\langle G,w \rangle \mid G \text{ is a CFG that generates string } w \}
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| </math> | |
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| <math> | |
| E_{DFA} = \{\langle A \rangle \mid A \text{ is a DFA and } L \left( A \right)= \empty \}
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| </math>
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| <math>
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| EQ_{DFA} = \{\langle A, B \rangle \mid A \text { and } B \text{ are DFA's and } L \left( A \right)= L\left( B \right) \}
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| </math>
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| <math>
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| E_{CFG} = \{\langle G \rangle \mid G \text{ is a CFG and } L \left( G \right)= \empty \}
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| </math>
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| == Undecidability ==
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| A '''[[decision problem]]''' is, informally, a problem whose solution is either "yes" or "not". Each such problem is characterized by the set of inputs whose solution is "yes". As a result, decision problems are formally defined as being sets, either of strings or of natural numbers: any such set defines the problem of deciding whether a given object belongs to the set.
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| A decision problem ''A'' is called '''decidable''' or '''effectively solvable''' if ''A'' is a [[recursive set]], that is, there exists an algorithm for establishing the presence of the element in the set. A problem is called '''partially decidable''', '''semidecidable''', '''solvable''', or '''provable''' if ''A'' is a [[recursively enumerable set]]. Partially decidable problems and any other problems that are not decidable are called '''undecidable'''.
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| ===The halting problem ===
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| {{main|Halting problem}}
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| In [[computability theory (computer science)|computability theory]], the '''halting problem''' is a [[decision problem]] which can be stated as follows:
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| :''Given a description of a [[computer program|program]] and a finite input, decide whether the program eventually halts when started with that input, or whether it runs forever..''
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| [[Alan Turing]] proved in 1936 that a general [[algorithm]] to solve the halting problem for ''all'' possible program-input pairs cannot exist; the set of pairs (''e'',''n'') such that the program with description ''e'' halts on input ''n'' is undecidable.
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| === Decidability of logical theories ===
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| {{main|Decidability (logic)}}
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| In [[mathematical logic]], a '''[[theory (mathematical logic)|theory]]''' is a set of formal sentences that is closed under [[logical consequence]] (essentially, any sentence that can be proved from sentences in the theory is itself in the theory). Important examples are the set of all [[Peano arithmetic|arithmetical sentences]] that are satisfied by the set of [[natural numbers]], and the set of arithmetical sentences provable from the axioms of [[Peano arithmetic]].
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| Many formal theories have been studied in the context of decidability. For example, the theory of the real numbers (in the [[signature (logic)|signature]] of [[field (mathematics)|fields]]) is decidable, while the [[first-order logic|first-order]] theory of the natural numbers is not. [[Gödel's incompleteness theorem]] implies that no first-order theory capable of interpreting a sufficient amount of the theory of the natural numbers can be decidable.
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| ==List of undecidable problems==
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| In [[computability theory]], an '''undecidable problem''' is a problem whose language is not a [[recursive set]]. More informally, such problems cannot be solved in general by computers; see [[Decidability (logic)|decidability]]. This is a list of undecidable problems. Note that there are [[uncountable set|uncountably]] many undecidable problems, so this list is necessarily incomplete. Though undecidable languages are not recursive languages, they may be a [[subset]] of [[Alan Turing|Turing]] recognizable languages.
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| === Problems related to abstract machines ===
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| * The [[halting problem]] (determining whether a specified machine halts or runs forever).
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| * The [[busy beaver]] problem (determining the length of the longest halting computation among machines of a specified size).
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| * [[Rice's theorem]] states that for all non-trivial properties of partial functions, it is undecidable whether a machine computes a partial function with that property.
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| === Other problems ===
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| * The [[Post correspondence problem]].
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| * The [[word problem for groups]].
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| * The [[word problem (computability)|word problem]] for certain [[formal languages]].
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| * The problem of determining if a given set of [[Wang tile]]s can tile the plane.
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| * The problem whether a [[Tag system]] halts.
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| * The problem of determining the [[Kolmogorov complexity]] of a string.
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| * Determination of the solvability of a Diophantine equation, known as [[Hilbert's tenth problem]]
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| * Determining whether two finite [[simplicial complex]]es are homeomorphic
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| * Determining whether the [[fundamental group]] of a finite simplicial complex is trivial
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| * Determining if a [[context-free grammar]] generates all possible strings, or if it is ambiguous.
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| * Given two context-free grammars, determining whether they generate the same set of strings, or whether one generates a subset of the strings generated by the other, or whether there is any string at all that both generate.
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| == References ==
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| {{Reflist}}
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| * {{Citation | last = Chomsky | first = Noam | year = 1959 | title = On certain formal properties of grammars | journal = Information and Control | volume = 2 | issue = 2 | pages = 137–167 | doi = 10.1016/S0019-9958(59)90362-6 | postscript = .}}
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| *{{ citation | author=Cutland, N. |title=Computability.|publisher=Cambridge University Press|year=1980|isbn=0-521-29465-7 }}
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| * {{Citation | last1=Rogers | first1=Hartley | title=The Theory of Recursive Functions and Effective Computability | origyear=1967 | publisher=First MIT press paperback edition | isbn=978-0-262-68052-3 | year=1987}}
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| * {{Citation|author = [[Michael Sipser]] | year = 1997 | title = Introduction to the Theory of Computation | publisher = PWS Publishing | chapter = Decidability | pages = 151–170 | isbn = 0-534-94728-X}}
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| * {{Citation | last1=Soare | first1=R. | title=Recursively Enumerable Sets and Degrees | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1987}}
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| {{Formal languages and grammars}}
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| [[Category:Computability theory]]
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| [[Category:Theory of computation]]
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| [[Category:Formal languages]]
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If you present photography effectively, it helps you look much more properly at the globe around you. Offshore expert Word - Press developers high level of interactivity, accessibility, functionality and usability of our website can add custom online to using. Your parishioners and certainly interested audience can come in to you for further information from the group and sometimes even approaching happenings and systems with the church. They found out all the possible information about bringing up your baby and save money at the same time. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day.
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