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'''Head Grammar''' (HG) is a grammar formalism introduced in [[Carl Pollard]] (1984)<ref name="pollard1984">[[Carl Pollard|Pollard, C.]] 1984. ''Generalized Phrase Structure Grammars, Head Grammars, and Natural Language''. Ph.D. thesis, Stanford University, CA.</ref> as an extension of the [[Context-free grammar]] class of grammars. Head Grammar is therefore a type of [[phrase structure grammar]], as opposed to a [[dependency grammar]]. The class of head grammars is a subset of the [[Generalized_context-free_grammar#Linear_Context-free_Rewriting_Systems_.28LCFRSs.29|linear context-free rewriting systems]].
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One typical way of defining head grammars is to replace the terminal strings of [[Context-free grammar|CFGs]] with indexed terminal strings, where the index denotes the "head" word of the string. Thus, for example, a CF rule such as <math>A \to abc</math> might instead be <math>A \to (abc, 0)</math>, where the 0th terminal, the ''a'', is the head of the resulting terminal string. For convenience of notation, such a rule could be written as just the terminal string, with the head terminal denoted by some sort of mark, as in <math>A \to \widehat{a}bc</math>.
 
Two fundamental operations are then added to all rewrite rules: wrapping and concatenation.
 
==Operations on Headed Strings==
===Wrapping===
 
Wrapping is an operation on two headed strings defined as follows:
 
Let <math>\alpha \widehat{x} \beta</math> and <math>\gamma \widehat{y} \delta</math> be terminal strings headed by ''x'' and ''y'', respectively.
 
<math>w(\alpha \widehat{x} \beta, \gamma \widehat{y} \delta) = \alpha x \gamma \widehat{y} \delta \beta</math>
 
===Concatenation===
 
Concatenation is a family of operations on n > 0 headed strings, defined for n = 1, 2, 3 as follows:
 
Let <math>\alpha \widehat{x} \beta</math>, <math>\gamma \widehat{y} \delta</math>, and <math>\zeta \widehat{z} \eta</math> be terminal strings headed by ''x'', ''y'', and ''z'', respectively.
 
<math>c_{1,0}(\alpha \widehat{x} \beta) = \alpha \widehat{x} \beta</math>
 
<math>c_{2,0}(\alpha \widehat{x} \beta, \gamma \widehat{y} \delta) = \alpha \widehat{x} \beta \gamma y \delta</math>
 
<math>c_{2,1}(\alpha \widehat{x} \beta, \gamma \widehat{y} \delta) = \alpha x \beta \gamma \widehat{y} \delta</math>
 
<math>c_{3,0}(\alpha \widehat{x} \beta, \gamma \widehat{y} \delta, \zeta \widehat{z} \eta) = \alpha \widehat{x} \beta \gamma y \delta \zeta z \eta</math>
 
<math>c_{3,1}(\alpha \widehat{x} \beta, \gamma \widehat{y} \delta, \zeta \widehat{z} \eta) = \alpha x \beta \gamma \widehat{y} \delta \zeta z \eta</math>
 
<math>c_{3,2}(\alpha \widehat{x} \beta, \gamma \widehat{y} \delta, \zeta \widehat{z} \eta) = \alpha x \beta \gamma y \delta \zeta \widehat{z} \eta</math>
 
And so on for <math>c_{m,n} : 0 \leq n < m</math>. One can sum up the pattern here simply as "concatenate some number of terminal strings ''m'', with the head of string ''n'' designated as the head of the resulting string".
 
===Form of Rules===
 
Head Grammar rules are defined in terms of these two operations, with rules taking either of the forms
 
<math>X \to w(\alpha, \beta)</math>
 
<math>X \to c_{m,n}(\alpha, \beta, ...)</math>
 
where <math>\alpha</math>, <math>\beta</math>, ... are each either a terminal string or a non-terminal symbol.
 
==Example==
 
Head Grammars are capable of generating the language <math>\{ a^n b^n c^n d^n : n \geq 0 \}</math>. We can define the grammar as follows:
 
<math>S \to c_{1,0}(\widehat{\epsilon})</math>
 
<math>S \to c_{3,1}(\widehat{a}, T, \widehat{d})</math>
 
<math>T \to w(S, \widehat{b}c)</math>
 
The derivation for "abcd" is thus:
 
<math>S</math>
 
<math>c_{3,1}(\widehat{a}, T, \widehat{d})</math>
 
<math>c_{3,1}(\widehat{a}, w(S, \widehat{b}c), \widehat{d})</math>
 
<math>c_{3,1}(\widehat{a}, w(c_{1,0}(\widehat{\epsilon}), \widehat{b}c), \widehat{d})</math>
 
<math>c_{3,1}(\widehat{a}, w(\widehat{\epsilon}, \widehat{b}c), \widehat{d})</math>
 
<math>c_{3,1}(\widehat{a}, \widehat{b}c, \widehat{d})</math>
 
<math>a\widehat{b}cd</math>
 
And for "aabbccdd":
 
<math>S</math>
 
<math>c_{3,1}(\widehat{a}, T, \widehat{d})</math>
 
<math>c_{3,1}(\widehat{a}, w(S, \widehat{b}c), \widehat{d})</math>
 
<math>c_{3,1}(\widehat{a}, w(c_{3,1}(\widehat{a}, T, \widehat{d}), \widehat{b}c), \widehat{d})</math>
 
<math>c_{3,1}(\widehat{a}, w(c_{3,1}(\widehat{a}, w(S, \widehat{b}c), \widehat{d}), \widehat{b}c), \widehat{d})</math>
 
<math>c_{3,1}(\widehat{a}, w(c_{3,1}(\widehat{a}, w(c_{1,0}(\widehat{\epsilon}), \widehat{b}c), \widehat{d}), \widehat{b}c), \widehat{d})</math>
 
<math>c_{3,1}(\widehat{a}, w(c_{3,1}(\widehat{a}, w(\widehat{\epsilon}, \widehat{b}c), \widehat{d}), \widehat{b}c), \widehat{d})</math>
 
<math>c_{3,1}(\widehat{a}, w(c_{3,1}(\widehat{a}, \widehat{b}c, \widehat{d}), \widehat{b}c), \widehat{d})</math>
 
<math>c_{3,1}(\widehat{a}, w(a\widehat{b}cd, \widehat{b}c), \widehat{d})</math>
 
<math>c_{3,1}(\widehat{a}, ab\widehat{b}ccd, \widehat{d})</math>
 
<math>aab\widehat{b}ccdd</math>
 
==Formal Properties==
===Equivalencies===
 
Vijay-Shanker and Weir (1994)<ref name="vijayshankarAndWeir1994">Vijay-Shanker, K. and Weir, David J. 1994. ''The Equivalence of Four Extensions of Context-Free Grammars''. Mathematical Systems Theory 27(6): 511-546.</ref> demonstrates that [[Indexed grammar|Linear Indexed Grammars]], [[Combinatory categorial grammar|Combinatory Categorial Grammars]], [[Tree-adjoining grammar|Tree-adjoining Grammars]], and Head Grammars are [[weak equivalence (formal languages)|weakly equivalent]] formalisms, in that they all define the same string languages.
 
==References==
 
{{Reflist}}
 
{{DEFAULTSORT:Head Grammar}}
[[Category:Formal languages]]
[[Category:Grammar frameworks]]
[[Category:Syntax]]

Latest revision as of 19:18, 13 October 2014

Make sure you can hear your breathe (except in public places where you may want to be more discreet) and you can see the stomach moving as this will help you to build up a rhythm. Various means of transportation come second - from yachts or boats and cars to motorcycles. The contemporary bikes for mountains are provided with stronger and lighter frame types in addition to pioneering form and design. Mountain bike frames spend a lot of time on roads these days, too. Two hunters may choose to use trailers if their distance from the truck is over four or five miles.

The prices offered by these online stores are very competitive and realistic according to the market. Young children can hardly wait for their first Hot Wheels or tricycle. You can also purchase helmets and other safety equipment, lights, locks, saddles, racks, packs, trainers, bike maintenance gear, trailers, team wear, Trek souvenirs, and even computers for your bike. - Get a bike that is lightweight and made out of carbon fiber materials. The micro-brewery and pub sits right along the White Pine Trail, a paved rail-trail that runs from Grand Rapids to Cadillac.

Some also use the liner as a size aid, for example BMX helmets can come as one-size, with optional liner kits to change the helmets size. There are two main types of mountain and road bike stems. The blogs and news feeds can also be shared among the friends and family free of cost via the social media networks. Once you have answered the above questions and you see all these features present in a bike, you are on your way to choosing the road bike that is right for you. One of the obvious things to check on both a bike and car are brakes.

There are still other types of bikes that require different types of tires. If you want to learn more about Choosing the right ride for you mountain bike sizing. look into the website. There are also tires with detailed treads which are very prominent and these are very useful for the biker because of the grip they provide. Just remember your heart is a muscle and can be developed the more you practice. Now do your comparison-shopping based on five or six available bikes. Next to a good mountain bike helmet, mountain bike gloves to protect your hands are the best accessory you can invest in.

(many commute bikes come with front suspension and disc brakes these days). The bike should be examined to check it is working correctly. And the cyclist may certainly sense the distinction. The ratio is chosen based on the terrain in which the bike will be ridden, the size of the bike and the strength of the rides. Part of Park City's distinction as the International Mountain Bike Association's first and only gold-level Ride Center is its abundance of beginner trails.

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