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Language and Complexity. Read J & M Chapter 13. What Do We Mean by Complexity?. Informal: How hard is it to analyze and solve the problem? Formal: What is the average/best case/worst case running time of the best known algorithm to solve the problem?

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Language and complexity l.jpg

Language and Complexity

Read J & M Chapter 13.


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What Do We Mean by Complexity?

  • Informal:

  • How hard is it to analyze and solve the problem?

  • Formal:

  • What is the average/best case/worst case running time of the best known algorithm to solve the problem?

  • Is that the best we’ll ever be able to do?

  • What capabilities must a computing device to solve the problem have?


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The Chomsky Hierarchy

recursively enumerable

context sensitive

context free

regular


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The Chomsky Hierarchy - Modified

recursively enumerable

recursive

context sensitive

context free

regular


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Review of Grammars

Regular grammars: X  a, X  , X  aX

Context free grammars: X  

Context sensitive grammars: X  , |X|  | |

Unrestricted grammars (for RE languages):   ,  



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Where Does English Fall?

We need to refine the question:

The weak generative capacity of a grammar is the set of strings that the grammar generates.

The strong generative capacity of a grammar is the set of structures that the grammar generates.

Note that strong generative capacity mirrors linguistic and psychological reality much better than weak generative capacity does.

Example: The cat with the furry tail purred.


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Where Does English Fall – The Finiteness Question

  • Is the set of English sentences finite?

  • Issues:

    • Size of vocabulary

    • Length of sentences

      • I know that "1" isn't the largest number and I know that "2" isn't the largest number (...)

  • If the set of English sentences is finite, then a regular grammar has enough weak generative capacity.


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Is This English?

In the event that the Purchaser defaults in the payment of any instalment of purchase price, taxes, insurance, interest, or the annual charge described elsewhere herein, or shall default in the performance of any other obligations set forth in this Contract, the Seller may: at his option: (a) Declare immediately due and payable the entire unpaid balance of purchase price, with accrued interest, taxes, and annual charge, and demand full payment thereof, and enforce conveyance of the land by termination of the contract or according to the terms hereof, in which case the Purchaser shall also be liable to the Seller for reasonable attorney's fees for services rendered by any attorney on behalf of the Seller, or (b) sell said land and premises or any part thereof at public auction, in such manner, at such time and place, upon such terms and conditions, and upon such public notice as the Seller may deem best for the interest of all concerned, consisting of advertisement in a newspaper of general circulation in the county or city in which the security property is located at least once a week for Three (3) successive weeks or for such period as applicable law may require and, in case of default of any purchaser, to re-sell with such postponement of sale or resale and upon such public notice thereof as the Seller may determine, and upon compliance by the Purchaser with the terms of sale, and upon judicial approval as may be required by law, convey said land and premises in fee simple to and at the cost of the Purchaser, who shall not be liable to see to the application of the purchase money; and from the proceeds of the sale: First to pay all proper costs and charges, including but not limited to court costs, advertising expenses, auctioneer's allowance, the expenses, if any required to correct any irregularity in the title, premium for Seller's bond, auditor's fee, attorney's fee, and all other expenses of sale occurred in and about the protection and execution of this contract, and all moneys advanced for taxes, assessments, insurance, and with interest thereon as provided herein, and all taxes due upon said land and premises at time of sale, and to retain as compensation a commission of five percent (5%) on the amount of said sale or sales; SECOND, to pay the whole amount then remaining unpaid of the principal of said contract, and interest thereon to date of payment, whether the same shall be due or not, it being understood and agreed that upon such sale before maturity of the contract the balance thereof shall be immediately due and payable; THIRD, to pay liens of record against the security property according to their priority of lien and to the extent that funds remaining in the hands of the Seller are available; and LAST, to pay the remainder of said proceeds, if any, to the vendor, his heirs, personals representatives, successors or assigns upon the delivery and surrender to the vendee of possession of the land and premises, less costs and excess of obtaining possession.


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Where Does English Fall – The Grammar Size and Complexity Issue

  • Consider the rule:

    • S  NP VP

  • But how can we handle agreement:

    • S  SNP SVP

    • SNP  … vs.

    • PNP  …

    • SVP  …

    • PVP  …

    • PP  prep SNP

    • PP  prep PVP


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Assume Not Finite Issue

  • (Either because it really isn’t finite or because we care about strong generative capacity.)

  • English isn’t regular.

  • English can’t be characterized with a context-free grammar without sacrificing simplicity and elegance.

  • At least some natural languages aren’t context free at all.


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English isn’t Regular – An Example Issue

Examples: The boy she saw yesterday was crying.

The boy she saw coming down the road was crying.

Grammar:

S  NP VP (not allowed)

So we have to write something like:

S  the X

X  boy Y X  NP-det VP

Y  she Z Y  NP-head VP

Z  saw Q Z VP-gap VP


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English isn’t Regular – The Proof Issue

If S1 then S2

Either S3 or S4

The man who said S5 is arriving today.

Ifeither the man who said either quit or stay is arriving today or the man who said S5 is arriving tomorrow, then the man who said S5 is arriving the day after tomorrow.

Let: if   a then  a

either  b or  b

others  

Then this sentence is of the form abbbba, which is an instance of x xR.


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English isn’t Regular – Another Proof Issue

The cat the dog the rat the elephant admired bit chased likes tuna fish.

Form: (the noun)n (transitive verb)n-1 likes tuna fish.


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English isn’t CF – The Proof Issue

The nested structures that we’ve just seen can easily be described with a context-free grammar. But what about sentences of the form ww:

x1 x2 x3 x4 x5 …. y1 y2 y3 y4 y5 … (we call these cross serial

dependencies)

In Swiss German:

Jan säit das mer em Hans es huus hälfed aastriiche.

Jan says that we Hans/DAT the house/ACC helped paint


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Psychological Complexity Issue

See short discussion in J & M.