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the sentence about total Turing machines, since "algorithm" is very common usage for this concept in theoretical computer science. I do realize that this well-intended addition could also cause some confusion similar to what seems to have happened around the use of "recursive" here, but I did it for a good (at least in my opinion) reason. There is a debate about the term "algorithm" going on right now in the Talk page for another article in
Knowledge, and I intend to try to help resolve the debate and (more importantly) to improve that article. To do that, I needed this article to mention "algorithm" as it is commonly used in teaching theoretical computer science.
1976:. Notice that even NP-complete decision problems, such as 3SAT, are languages that can be accepted by some LBA. The little we know about separation of complexity classes makes it even more difficult to find a witness. For example, it is conjectured, but we cannot prove, that EXPTIME-complete languages are not context-sensitive. In order to have a chance of proving that a language is not in CSL, we have to resort to decision problems that are much more difficult. Another example of a decidable set, which is provably not context-sensitive, is the set of first-order formulas in
2258:. Obviously the article is written in a dense style but I think that's okay as long as the introduction clarifies the context. I believe that improving readability of such articles is much more work (though desirable) than having discussions like this one. Currently, I have no suggestion how to slightly modify the introduction without introducing negative statements ("Recursive languages are unrelated to ..."). What is your expectation of the first sentences, now that you probably know a bit more about the subject? Would it help if the article on
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2310:"Formal language" is defined in parentheses in the very first sentence of the article. In computer science terms: It's an arbitrary set of strings. Nothing more. No meaning attached whatsoever. This is a perfectly standard term in computer science, in mathematical logic and in linguistics, and anyone wo doesn't believe what he or she reads (I can understand why one wouldn't on first hearing that definition) can simply follow the link to
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being a shorthand for ww. For example, we have (abba)²= abbaabba, ((abcab)²)²= abbaabbaabbaabba, and so on. Thus, there are regular expressions that can be described very succinctly using repeated squaring. We can transform a regular expression with squaring into an ordinary regular expression, but the length of the output can be exponential in the input length. Now consider the set
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2109:, this connection is so remote that you had better think of them as unconnected. I don't think we have an article on SQL recursive queries, and I don't know if we should have one. But it's clear that recursive languages in the mathematics / pure computer science sense are a much more important topic and that this article should not be hijacked for the SQL topic.
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Thank you for your thoughtful reply. I suppose you have restated my point: I read the article and I could not make heads nor tails of it. My options are to a) assume that I am an idiot or b) assume that the article is opaque enough that a normally educated and aware reader cannot contextualize it. I
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a topic of theory and not a classification of programming languages (and could therefore be perceived less important for programmers). However, to help reduce the complexity of the article, you can try to explain what makes you think that SQL is a recursive language and what you would expect to read
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I agree that the treatment of formal language in the first paragraph is OK. Alas, computer science is rife with terms that have different meanings in different contexts. Along with "recursive", "algorithm" is another one. I took the liberty of slightly modifying this paragraph to add "algorithm" to
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If we are looking for a language that is decidable but provably not context-sensitive, we need to look for a more difficult language. Assume now we extend the syntax of the usual regular expressions with the operations union, concatenation and Kleene star by adding the squaring operator ², with w²
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This is too ambiguous. Obviously this set can not be enumerated, some of its members are actually infinite. I believe what the author intended was something along the lines of "the set of all Turing machines that accept recursive languages is not recursively enumerable", or "the set of all Turing
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For anyone keeping this article on their watchlist: there is currently an attempt at merging a number of articles on (non-)recursive/(un)decidable/(un)computable sets/languages/decision problems into a single, thorough article. The experiment currently sits at
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is a recursive language, and it is a language that virtually every programmer uses. These programmers should be able to read this article and understand the important aspects of recursive languages and how they are distinguished from
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to be a programming language with
Spanish keywords, and I haven't found a link from any programming language article to this one that could make you assume a relation. A recursively defined programming language uses
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recursive (not: recursively defined). The article should not be hijacked/bloated with all kinds of disclaimers what the article is not about (in particular it's not about natural languages, recursive jokes, ...).
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The concept of "recursive language" is an important one for all programmers and they need to be able to understand it. The article as it currently reads sheds little or no light on this issue. For example
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Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential space. In: Symposium on
Switching and Automata Theory (SWAT 1972), pp. 125–129. IEEE (1972)
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And by the way, I did note that the way I cited Prof. Sipser's textbook is not the way the moderators want us to do it, but I have not learned how to create proper citations yet. FWIW, it is on my to do
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There are even notoriously difficult decision problems, whose associated language is context-sensitive. One such example is the inequivalence problem of regular expressions. More precisely, the set
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implies the existence of recursive, but non-context-sensitive languages; I think this should be mentioned in the article. Apparently, a concrete language can't be given in a form as simple as, say,
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In order to illustrate the place of recursive languages in the (extended) Chomsky hierarchy, could somebody please give an example for a recursive language that isn't context-sensitive? -
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is decidable. Namely, a Turing machine with output tape can transform a regular expression with squaring of length 'n' into an equivalent ordinary regular expression of length at most
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I just reverted an anonymous edit that added the following sentence: "Note: The set of all recursive languages can not be enumerated effectively. The proof is by diagonalization."
1676:{\displaystyle S_{2}=\{\,r\mid r{\mbox{ is a regular expression with squaring over the alphabet }}\{a,b\}{\mbox{ and }}r{\mbox{ does not describe all words in }}\{a,b\}^{*}\,\}}
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I agree. There is a perfectly servicable introductory paragraph. It says: class of formal languages. What the heck is a formal language? That makes absolutely no sense.
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on
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1558:{\displaystyle S_{1}=\{\,r\mid r{\mbox{ is a regular expression over the alphabet }}\{a,b\}{\mbox{ and }}r{\mbox{ does not describe all words in }}\{a,b\}^{*}\,\}}
1297:{\displaystyle \{\,r\mid r{\mbox{ is a regular expression with negation over the alphabet }}\{a,b\}{\mbox{ and the language described by }}r{\mbox{ is empty}}\,\}}
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1423:. Your regular expression example is closest to that form; do you have a reference for it (which should also establish that the language is recursive)? -
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is about and then end up here, you might read the article with false expectations. But doesn't the first sentence clarify that the term is derived from
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Markus Holzer and Martin Kutrib: "The
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I looked again into this, and found another language, that is a bit easier to describe. In what follows, I refer to the original research paper
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went with b) and all I am suggesting is that you review this article for slight modification with a lay reader in mind. Thanks for your time.
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Anyway, such claims do not belong into the "closure properties" section, they should be stated more formally, and cite a suitable reference.
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2061:. This article might be too mathematical. I am not qualified to judge that aspect of it, but I suspect that this article is just wrong.--
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I am a programmer not a theorist. My understanding is that many languages were purposely designed to be recursive, such as
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On the other hand, using techniques from computational complexity theory, one can prove that there is a constant 'c: -->
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Most languages that can be given in a simple form as you mentioned can be computed in nondeterministic time
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can be decided by a LBA, and is therefore context-sensitive (cf. Theorem 15 in Holzer/Kutrib).
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I wonder why you actually thought the terms were related. You wouldn't expect the
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I don't think the article is wrong in general. The concept of recursive languages
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You can take any language that provably requires more than linear space, see here
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I think that what's missing from this page is an example of a recursive language.
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Find pictures for the biographies of computer scientists (see
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can be recursively defined ). Of course, if you stumble upon
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The term “recursive language” refers to a specific class of
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machines that are deciders is not recursively enumerable".
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is a regular expression with squaring over the alphabet
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was more readable in order to grasp the context? --
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486:Linguistics
281:Mathematics
272:mathematics
228:Mathematics
199:Start-class
148:free images
31:not a forum
2360:Categories
1947:and space
394:Philosophy
383:philosophy
333:Philosophy
2205:recursion
2197:recursion
2167:recursion
2129:Smalltalk
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1198:jonsafari
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874:Infobox
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154:WP refs
142:scholar
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