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Structural Linguistics

Amoxicillin Over The Counter, Way back in the late 1800's and early 1900's, early linguists, probably better called philologists, would spend a great deal of time cataloging and describing the worlds languages, which ofcourse entailed that they analyzed the grammar of the languages they were studying. Grammars back then were essentially language specific, atleast more so than they are today, in that essentially everything in the grammar was presumed to be applicable only to the language it was intended to describe. The grammars were as purely descriptive as you could get — they brought no theory into it at all. The most that structural linguistics ever really got in the way of abstractions away from particular languages were the linguistic universals, which were statistically significant patterns in how word order phenomena were distributed.

The shape of these grammars was relatively simple, what we would call today a context free grammar (more on that later), plus some generic operations like agreement. Essentially this just means that the grammar of a language was described in terms of the kinds of phrases that you could say, Amoxicillin from mexico, and what was in those phrases — the order things came in, what was or wasn't necessary, and how many of them you could have. Maybe a noun phrase had to have a noun and as many optional adjective phrases after the noun as you like, and so forth, Amoxicillin Over The Counter. Or maybe a verb phrase can have any number of optional nouns preceding the verb but at most one following it. But that was essentially it — just descriptions of what phrases consist of. The standard way of describing these kinds of rules looks like this:

  1. Sentence -> NounPhrase VerbPhrase

  2. S -> NP VP

  3. NP -> D N

  4. D -> the

  5. N -> dog

  6. VP -> V

  7. V -> barked

(1) can be read "An Sentence can contain a NounPhrase followed by a VerbPhrase". (2) is more common for conciseness, but there's no difference between them in principle. Amoxicillin Over The Counter, A simple way to understand how these sorts of grammars work is to view them as a way to take a string of symbols and rewrite part of the string, to produce a new set of symbols (this is what's called a "rewrite system"). So for this toy language, we can think of this rule as saying "Given the string containing S, I can replace the S with NP VP and get further towards producing some sentence in this language", order Amoxicillin from mexican pharmacy. To get all the way to a valid sentence, a string of symbols has to be rewritten until no more phrasal symbols remain. For example, starting with S, we can rewrite it in stages to produce the sentence "the dog barked" like so:

  1. S (start)

  2. NP VP (rewrite S as NP VP by rule 2)

  3. D N VP (rewrite NP as D N by rule 3)

  4. the N VP (rewrite D as the by rule 4)

  5. the dog VP (rewrite N as dog by rule 5)

  6. the dog V (rewrite VP as V by rule 6)

  7. the dog barked (rewrite V as barked by rule 7)

Formal Languages

In about the mid 1950s, Noam Chomsky, who was then at Harvard, began working on a mathematical understanding of what a language was, and what a grammar was. Amoxicillin reviews, In Chomsky's system, which has gone on to become formal language theory, the primary way languages and grammars are understood in any formal sense, a language can be viewed as the hypothetical set of all sentences in that language, and a grammar is a set of rules that you can perform to produce sentences in the language. This perspective can be said to generate or enumerate the sentences in a language, and so languages and grammars describable this way are often described as being generative-enumerative, Amoxicillin Over The Counter. Chomsky formulated a hierarchy of languages based on the properties of the grammars needed to generate them.

The simplest languages in the Chomsky Hierarchy, sometimes called Type-4 languages or finite languages, are languages with a fixed number of sentences. For instance, a language with sentences that contain exactly one letter of the English alphabet will have only 26 valid sentences. Grammars for this kind of language, online buying Amoxicillin, called Type-4 grammars or finite grammars, generally look like so:

  1. S -> a
    S -> b
    S -> z

  2. S -> a | b | ... | z

Amoxicillin Over The Counter, Here the pipes (|'s) denote alternatives. These two notations are equivalent and usually the latter is preferred. Finite grammars have exactly one "non-terminal" (phrase-naming) symbol (which will henceforth denoted in general by uppercase letters) on the left of the arrow, and exactly one "terminal" (word-naming) symbol (henceforth denoted in general by lowercase letters) on the right. In general, then, finite languages have grammars with rules of the form:

A -> a

Finite languages are so useless in describing natural languages that they're often left off the Chomsky Hierarchy entirely, About Amoxicillin, and Chomsky himself didn't even consider them when formulating the hierarchy. If we wanted to say that animal calls form a kind of language, then it would be of this sort — a fixed repertoire of things that can be said.

One step above finite languages are what are called Type-3, or regular, languages. Regular languages can be infinitely large in size, but don't have to be (meaning all finite languages are regular, but not all regular languages are finite), Amoxicillin Over The Counter. The corresponding grammars for finite languages have rules of the forms:

  1. A -> a

  2. A -> aB

Regular languages are also useless in describing natural languages, but we're getting closer to something like what we need to describe them. For convenience, we can define some shorthands for rules of this sort:

  • a* denotes 0-or-more a's and can be replaced by A if the following rules are added in the grammar:

    1. A -> ε (where ε means 'no symbol', or 'the empty string')

    2. A -> aA

  • (a) denotes 0-or-1 a's and can be replaced by A if the following rule is added to the grammar:

    1. A -> ε | a

  • {...}, where .., kjøpe Amoxicillin på nett, köpa Amoxicillin online. is any collection of symbols separated by |'s, denotes any of the items between the |'s, and can be replaced by A if the following rule is added to the grammar:

    1. A -> ... (where ... is the same collection of symbols as in {...})

Amoxicillin Over The Counter, Above regular languages in the Chomsky Hierarchy are the Type-2, or context free (CF), languages (CFLs). Just as regular languages form a superset of finite languages, so too do context free languages for a superset of regular languages. Context free grammars (CFGs) in general have the form:

A -> v

Where v denotes any string of symbols, terminal or non-terminal. These grammars are called context free because no matter what is around a non-terminal symbol, Generic Amoxicillin, it can always be rewritten in the same way. Context free grammars can generate languages that regular grammars can't. A classic example of this being the set of sentences containing some number of a's followed by the same number of b's, as in { ε, ab, aabb, aaabbb, .., Amoxicillin Over The Counter. }. The grammar for such a language could be:

  1. S -> ε | aSb

Structuralist grammars are context free, if you ignore agreement phenomena, case assignment, and the like. A typical grammar of this sort for English might be something like:

  1. S -> NP VP

  2. NP -> (D) AdjP* N {PP | CP}*

  3. AdjP -> AdvP* Adj

  4. AdvP -> AdvP* Adv

  5. PP -> P (NP)

  6. CP -> Conj S

  7. VP -> AdvP* AuxV* V (NP) {NP | PP | CP}* AdvP*

  8. XP -> XP Conj XP (for XP equal to any non-terminal)

The last rule here is actually not technically a context free rule, but is instead a sort of meta-rule that tells us how to get the rules specific for each category. For instance, if XP = NP, Amoxicillin without prescription, then we can get the rule NP -> NP conj NP. Amoxicillin Over The Counter, Rules like this become especially prominent in more complicated kinds of grammars, but they're very useful for keeping things simple and easy to understand.

As it turns out, context free grammars are not enough to describe the phenomena of natural language. They either under-generate and don't produce all the correct sentences, or they over-generate and produce ungrammatical sentences. A more powerful kind of grammar forms a superset to context free grammars, called Type-1, or context sensitive, grammars (CSGs), describing context sensitive languages (CSLs). This extra power makes it possible to generate ever more finely tuned languages. Amoxicillin recreational, Context sensitive grammars have rules of the form:

vAw -> vqw

Where v, w, and q denote strings of symbols, and v and w are each the same on both sides of the arrow. These languages are called context sensitive because what you replace A with depends on what's around it, hence rewriting non-terminal symbols is sensitive to context, Amoxicillin Over The Counter. Many people believe that natural languages are at most context sensitive, or somewhere between context free and context sensitive. Just as context free grammars can generate languages that regular grammars can't, so too can context sensitive grammars generate languages that context free grammars can't. For example, the language consisting of sentences with some number of a's followed by the same number of b's then the same number of c's, as in { ε, Amoxicillin dose, abc, aabbcc, aaabbbccc, ... }. The grammar for such a language could be:

  1. S -> a(S)BC

  2. CB -> HB

  3. HB -> HC

  4. HC -> BC

  5. aB -> ab

  6. bB -> bb

  7. bC -> bc

  8. cC -> cc

Amoxicillin Over The Counter, The last kind of language is called a recursively enumerable language, and these languages an be produced by Type-0, or unrestricted, grammars. The rules in an unrestricted grammar have the form:

v -> w

Where v and w are any string of symbols, thus there is no restriction on what the rules can look like. In unrestricted grammars, there's no real distinction between terminal and non-terminal symbols. My Amoxicillin experience, Like the rest of the hierarchy, recursively enumerable languages are a superset of the less powerful types, so all regular, CF, and CS grammars are unrestricted grammars, but not the reverse. Some models of grammar are Type-0, but the rules for them are often not written like this. Type-0 grammars are the most powerful kinds of grammars, and can describe all describable languages.

No serious grammar formalism that I know of uses context-sensitive or unrestricted grammars without a custom notation (phonological rules, however, are context sensitive, though instead of writing them like vAw -> vqw they're written A -> q / v_w), Amoxicillin Over The Counter. The reason is that any grammar above a CFG has unwieldy notation for anything but the simplest of sentences. A very common kind of grammar formalism used for grammars more powerful than CFGs is what's called an Attribute Grammar (AG). Attribute grammars look very similar to CFGs, purchase Amoxicillin, however symbols can have attributes associated with them, and attributes can be derived, mutated, inherited, etc. during the course of a sentence production. For example, the language { ε, abc, Cheap Amoxicillin no rx, aabbcc, aaabbbccc, ... } form earlier might be described with the following attribute grammar:

  1. S[m] -> ε, where m = 0 | A[n] B[p] C[q], where n, p, q = m and m > 0

  2. A[m] -> a, where m = 1 | a A[n], where n = m-1

  3. B[m] -> b, where m = 1 | b B[n], Amoxicillin wiki, where n = m-1

  4. C[m] -> c, where m = 1 | c C[n], where n = m-1

The second rule, for example, can be read as meaning "An A[m] can be rewritten as a, wherever m = 0, otherwise it can be rewritten as an A[n], where n = m-1". A production for aaabbbccc would look like:

  • S[3]

  • A[3] B[3] C[3]

  • a A[2] B[3] C[3]

  • aa A[1] B[3] C[3]

  • aaa B[3] C[3]

  • aaab B[2] C[3]

  • aaabb B[1] C[3]

  • aaabbb C[3]

  • aaabbbc C[2]

  • aaabbbcc C[1]

  • aaabbbccc

Another kind of attribute grammar might not derive values in a top down fashion, What is Amoxicillin, but might instead propogate values around like so:

  1. S -> A[m] B[n], where m = n

  2. A[m] -> a, where m = 1 | aa, where m = 2

  3. B[m] -> b, where m = 1 | bb, where m = 2

Giving these two productions:

  • S

  • A[m] B[n] (requiring m = n)

  • a B[n] (where m = 1, making n = 1)

  • ab

  • S

  • A[m] B[n] (requiring m = n)

  • aa B[n] (where m = 2, making n = 2)

  • aabb

In this sort of attribute, producing A[m] for some value of m propagates the value over to n, which forces B[n] to producing only certain alternatives.

As we'll see later, many modern grammars can be viewed as very similar to attribute grammars, if not kinds of attribute grammars. Tree Adjoining Grammars, LFG, GPSG, and HPSG all have properties that are very reminiscent of attribute grammars.


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Comments (6)

  1. gerhard wrote::

    “Way back in the late 1800’s and early 1900’s, early linguists, probably better called philologis”
    do you really need to say way back? the late 1800s and early 1900 isn’t really that far back. in reality that time period represents 3 generations of people. 3 lifetimes really isn’t that far back in history to signify a significant passage of time.

    Great article tho. keep it up!

    Sunday, May 17, 2009 at 6:01 pm #
  2. augur wrote::

    It was meant to be mildly sarcastic. I guess it would’ve been more obvious it I’d said “Way back in the early 2000s” :P

    Sunday, May 17, 2009 at 7:26 pm #
  3. cxger wrote::

    Great article! One question: In the list of rules for a context free grammar, asterisks are placed in certain rules (i.e., AdjP -> AdvP* Adj). What does the asterisk denote?

    Wednesday, May 27, 2009 at 12:06 am #
  4. augur wrote::

    As noted above, certain regular rules have abbreviations given the utility of the rules. If you look above where it says “For convenience, we can define some shorthands for rules of this sort:”, you’ll find the fuller definitions, but in short:

    a* means 0-or-more a’s,
    (a) means 0-or-1 a (that is, an optional a),
    and { a | b | c } means any of a, b, c.

    So for instance, the rule

    S -> a*

    produces the language { ε, a, aa, aaa, aaaa, … }, summarizable as { an | n ≥ 0 }.

    S -> (a)

    gives you { ε, a } = { an | 1 ≥ n ≥ 0 }

    and finally:

    S -> { a | b | c }

    gives you the language { a, b, c }

    They’re all just shorthands for common rule structures, namely:

    a* is equivalent to A where:

    A -> ε | aA

    (a) is equivalent to A where:

    A -> ε | a

    and { a | b | c } is equivalent to A where:

    A -> a | b | c

    Wednesday, May 27, 2009 at 12:40 am #
  5. wrote::

    some more points on difference between type1 and type 3 languages

    Tuesday, September 1, 2009 at 1:37 pm #
  6. augur wrote::

    abhay, by that I presume you mean I should explain some more? what specifcally did you find needed clarification or expansion?

    Tuesday, September 1, 2009 at 7:11 pm #