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| A '''context-sensitive grammar''' ('''CSG''') is a [[formal grammar]] in which the left-hand sides and right-hand sides of any [[Production (computer science)|production rules]] may be surrounded by a context of [[terminal symbol|terminal]] and [[nonterminal symbol]]s. Context-sensitive grammars are more general than [[context-free grammar]]s but still orderly enough to be [[parsing|parsed]] by a [[linear bounded automaton]].
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| [[Noam Chomsky]] introduced context-sensitive grammars in the 1950s as a way to describe the syntax of [[natural language]] where it is indeed often the case that a word may or may not be appropriate in a certain place depending upon the context. A [[formal language]] that can be described by a context-sensitive grammar is called a [[context-sensitive language]].
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| ==Formal definition==
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| {{Disputed|date=August 2013}}
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| A [[formal grammar]] ''G'' = (''N'', Σ, ''P'', ''S''), where ''N'' is a set of nonterminal symbols, Σ is a set of terminal symbols, ''P'' is a set of production rules, and ''S'' is the start symbol, is context-sensitive if all rules in ''P'' are of the form
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| : α''A''β → αγβ
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| where ''A'' ∈ ''N'' (i.e., ''A'' is a single [[nonterminal]]), α,β ∈ (''N'' U Σ)* (i.e., α and β are strings of nonterminals and [[Terminal symbol|terminal]]s) and γ ∈ (''N'' U Σ)<sup>+</sup> (i.e., γ is a nonempty string of nonterminals and terminals).
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| Some definitions of a context-sensitive grammar only require that for any production rule of the form u → v, the length of u shall be less than or equal to the length of v. This seemingly weaker requirement is claimed to be in fact equivalent.<ref>{{cite book|authors=John E. Hopcroft, Jeffrey D. Ullman|title=Introduction to Automata Theory, Languages, and Computation|publisher=Addison-Wesley|year=1979}}; p.223-224; Exercise 9, p.230. In the 2003 edition, the chapter on CSG has been omitted.</ref>
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| In addition, a rule of the form
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| : S → λ
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| where λ represents the [[empty string]] and S does not appear on the right-hand side of any rule is permitted. The addition of the empty string allows the statement that the context sensitive languages are a proper superset of the context free languages, rather than having to make the weaker statement that all context free grammars with no →λ productions are also context sensitive grammars.
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| The name ''context-sensitive'' is explained by the α and β that form the context of ''A'' and determine whether ''A'' can be replaced with γ or not. This is different from a [[context-free grammar]] where the context of a nonterminal is not taken into consideration. (Indeed, every production of a context free grammar is of the form V → w where V is a ''single'' nonterminal symbol, and w is a string of terminals and/or nonterminals (w can be empty)).
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| If the possibility of adding the empty string to a language is added to the strings recognized by the noncontracting grammars (which can never include the empty string) then the languages in these two definitions are identical.
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| A formal language can be described by a context-sensitive grammar if and only if it is accepted by some linear bounded automaton.<ref>(Hopcroft, Ullman, 1979); Theorem 9.5, 9.6, p.225-226</ref>
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| == Examples ==
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| * This grammar generates the canonical non-[[context-free language]] { ''a''<sup>''n''</sup>''b''<sup>''n''</sup>''c''<sup>''n''</sup> : ''n'' ≥ 1 } :
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| # ''S'' → ''aSBC''
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| # ''S'' → ''aBC''
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| # ''CB'' → ''HB''
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| # ''HB'' → ''HC''
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| # ''HC'' → ''BC''
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| # ''aB'' → ''ab''
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| # ''bB'' → ''bb''
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| # ''bC'' → ''bc''
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| # ''cC'' → ''cc''
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| Rules 1 and 2 allow for blowing-up ''S'' to ''a''<sup>''n''</sup>(''BC'')<sup>''n''</sup>; rules 3 to 5 allow for successively exchanging each ''CB'' to ''BC'' (three rules are needed for that since a rule ''CB'' → ''BC'' wouldn't fit into the scheme α''A''β → αγβ); rules 6 to 9 allow for replacing a non-terminal ''B'' or ''C'' with its corresponding terminal ''b'' or ''c'', provided it is in the right place.
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| The generation chain for aaabbbccc is:
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| : ''S''
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| : →<sub>1</sub> ''aSBC''
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| : →<sub>1</sub> ''a''{{boldmono|aSBC}}''BC'' | |
| : →<sub>2</sub> ''aa''{{boldmono|aBC}}BCBC''
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| : →<sub>3</sub> ''aaaB''{{boldmono|HB}}CBC''
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| : →<sub>4</sub> ''aaaB''{{boldmono|HC}}CBC''
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| : →<sub>5</sub> ''aaaB''{{boldmono|BC}}CBC''
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| : →<sub>3</sub> ''aaaBBC''{{boldmono|HB}}C''
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| : →<sub>4</sub> ''aaaBBC''{{boldmono|HC}}C''
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| : →<sub>5</sub> ''aaaBBC''{{boldmono|BC}}C''
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| : →<sub>3</sub> ''aaaBB''{{boldmono|HB}}CC''
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| : →<sub>4</sub> ''aaaBB''{{boldmono|HC}}CC''
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| : →<sub>5</sub> ''aaaBB''{{boldmono|BC}}CC''
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| : →<sub>6</sub> ''aa''{{boldmono|ab}}BBCCC''
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| : →<sub>7</sub> ''aaa''{{boldmono|bb}}BCCC''
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| : →<sub>7</sub> ''aaab''{{boldmono|bb}}CCC''
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| : →<sub>8</sub> ''aaabb''{{boldmono|bc}}CC''
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| : →<sub>9</sub> ''aaabbb''{{boldmono|cc}}C''
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| : →<sub>9</sub> ''aaabbbc''{{boldmono|cc}}''
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| More complicated grammars can be used to parse { ''a''<sup>''n''</sup>''b''<sup>''n''</sup>''c''<sup>''n''</sup>''d''<sup>''n''</sup>: ''n'' ≥ 1 }, and other languages with even more letters.
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| A context-sensitive grammar for the language <math> \{ a^{2^i} : i \geq 1 \}</math> is constructed in Example 9.5 (p.224) of (Hopcroft, Ullman, 1979).
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| == Normal forms ==
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| Every context-sensitive grammar which does not generate the empty string can be transformed into an equivalent one in [[Kuroda normal form]]. "Equivalent" here means that the two grammars generate the same language. The normal form will not in general be context-sensitive, but will be a [[noncontracting grammar]].
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| == Computational properties and uses ==
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| The [[decision problem]] that asks whether a certain string ''s'' belongs to the language of a certain context-sensitive grammar ''G'', is [[PSPACE-complete]]. There are even some context-sensitive grammars whose fixed grammar recognition problem{{clarify|reason=What is a fixed grammar recognition problem? I couldn't find an article about that.|date=November 2013}} is PSPACE-complete.
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| The emptiness problem for context-sensitive grammars (given a context-sensitive grammar ''G'', is ''L''(''G'')=∅ ?) is [[Undecidable language|undecidable]].
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| It has been shown that nearly all [[natural language]]s may in general be characterized by context-sensitive grammars, but the whole class of CSG's seems to be much bigger than natural languages{{citation needed|date=November 2011}}. Worse yet, since the aforementioned decision problem for CSG's is PSPACE-complete, that makes them totally unworkable for practical use, as a polynomial-time algorithm for a PSPACE-complete problem would imply [[P=NP problem|P=NP]]. Ongoing research on [[computational linguistics]] has focused on formulating other classes of languages that are "[[mildly context-sensitive language|mildly context-sensitive]]" whose decision problems are feasible, such as [[tree-adjoining grammar]]s, [[combinatory categorial grammar]]s, [[coupled context-free language]]s, and [[Generalized context-free grammar#Linear Context-free Rewriting Systems (LCFRSs)|linear context-free rewriting system]]s. The languages generated by these formalisms properly lie between the context-free and context-sensitive languages.
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| ==See also==
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| * [[Chomsky hierarchy]]
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| * [[Growing context-sensitive grammar]]
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| ==References==
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| *{{cite book|last=Martin|first=John C.|title=Introduction to Languages and the Theory of Computation|year=2010|publisher=McGraw-Hill|location=New York, NY|isbn=9780073191461|url=http://www.mhprofessional.com/product.php?cat=39&isbn=0073191469|edition=4th|accessdate=10 December 2012}}
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| {{reflist}}
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| ==External links==
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| * [http://danielmattosroberts.com/earley/context-sensitive-earley.pdf Earley Parsing for Context-Sensitive Grammars]
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| {{Formal languages and grammars}}
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| {{DEFAULTSORT:Context-Sensitive Grammar}}
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| [[Category:Formal languages]]
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| [[Category:Grammar frameworks]]
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Have we ever heard that somebody said "My computer is getting slower, Assist me?" Are you seeking methods on how do I speed up my computer? Are you tired of wasting too much time considering the loading process is really slow? If you wish To recognize how to boost and speed up computer performance, then this short article may aid show we certain concepts plus tricks "What is the cause?" plus How to prevent your computer getting slower?
Windows Defender - this does come standard with various Windows OS Machines, yet otherwise is download from Microsoft for free. It can aid safeguard against spyware.
When you compare registry cleaners we require a fast acting registry cleaning. It's no wise spending hours and your PC waiting for a registry cleaning to complete its task. We desire the cleaner to complete its task inside minutes.
The issue with many of the individuals is that they never like to spend funds. In the cracked variation one does not have to pay anything and may download it from internet especially easily. It is easy to install as well. However, the problem comes whenever it is unable to detect all possible viruses, spyware and malware inside the system. This is because it is actually obsolete in nature plus does not receive any usual updates within the website downloaded. Thus, a system is accessible to problems like hacking.
The zookaware should come as standard with a back up and restore facility. This ought to be an easy to implement procedure.That signifies which should you encounter a issue with the PC after utilizing a registry cleaning you are able to merely restore a settings.
Turn It Off: Chances are if you are like me; then you spend a lot of time on a computer on a daily basis. Try offering your computer certain time to do absolutely nothing; this could sound funny yet should you have an elder computer you are asking it to do too much.
Reboot PC - Just reboot the PC to see if the error is gone. Frequently, rebooting the PC readjusts the internal settings plus software plus therefore fixes the problem. If it doesn't then move on to follow the instructions below.
Before we purchase a complete modern system; it happens to be time to get the older 1 cleaned up thus you can start getting more done online today! Visit our website below and access the many reputable registry cleaner software accessible.