PLT: Lecture 8

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PLT Lecture 8

• Generative Grammars

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Last Lecture

• Described Chomskys adaptations to Posts
  Production Systems
• Generated some simple grammars for covering some
  simple English instances (Discuss)
• Gave a formal definition of a generative grammar

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Phase Structure, Generative Grammars

• 4-tuple VT, VN, P, S
• VT
• VN
• VN n VT
• P binary relations called productions or re-write
  rules
• Domain V x VN X V
• Codomain V
• Eventually some rules of type
• Domain -gt VT
• S element of VN

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Conventions

• Individual terminal symbols a,b,ct
• Strings of terminal symbols u,v,w,x,y,z
• Individual non-terminals A,B,X,Y,Z //
  includes S
• Strings that may contain terminal and
  non-terminal symbols a, b, , w
• p e P (a, b)
• a -gt b, yat -gt ybt

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Trivial Generation Example

• G(0,1, S, S-gt0S1,S-gt01, S)
• a1S, b10S1
• a2S, b201
• Generation
• Start string is S
• t and y are
• Applying P1, new a, t, y
• Reapplying P1
• Eventual application of P2
• More generally,
• If G is a grammar, a, b e of V, p e G.P such
  that a-gtb, we can say a-gtb

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Sentential Forms and Language Instances

• Sentential forms
• Let S be the start symbol for Grammar G
• S e V gt S e V
• If a is derivable from S (Sgt a) then a is a
  sentential form of G
• Written SF(G)
• Language Instance
• Sentential form made up of only VT
• Written L(G)
• L(G) subset of SF(G)

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Prove L(G) ww0n1n n gt 0

• Let P(n) be equivalent to Sgt0nS1n
• P(0) holds (Sgt00S10 (P1)), SgtS
• Assume P(n) holds Sgt0nS1n
• Show for P(n1)
• Sgt00nS11n (P1)
• Sgt0n1S1n1
• Keeping Sgt01 as the final substitution

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A More Complex Grammar

• Grammar
• S-gt aSBC
• S-gt aBC
• CB-gtBC
• aB-gt ab
• bB-gt bb
• bC-gtbc
• cC -gt cc
• Observations
• Generate aaaabbbbcccc

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Classifying Grammars

• Unrestricted Grammars
• a -gt b
• Otherwise no restrictions
• Context Sensitive Grammars
• a -gt b
• a !gt b
• Context Free Grammars
• A-gt b
• Regular Grammars
• A-gtaB
• A-gta

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The Drunkards Lament

• On Thursday nights, a panic strikes the stables
  bar, a place to which students withdraw, in order
  to enhance their unhealthy pallor prior to
  visiting home for the weekend to look for
  sympathy, food, laundry services, and cash.
• The crowds mill around the bar. There is a
  communal terror that the bar may run dry. All
  able bodied men and women re-double their efforts
  to get what they are convinced is the very last
  pint to emerge from the taps.
• Some unscrupulous individuals know no difference
  between meum and tuum. These people keep their
  eyes open for pints which may be neglected
  because-
• Their owners are engaged in a heated discussion
  on the wave/particle duality of matter or the
  neo-realist school of painting(unlikely), or
• Their owners have gone to fetch another pint
• Their owners having had alcohol affect their
  anti-diuretic hormone now badly need to reduce
  their liquidity.
• Whatever the reason, it is truly heartrending to
  see their distress when the loss of the loved
  object is discovered. Many of them break into a
  lament, whose grammatical structure is now being
  studied keenly by linguists because of its simple
  yet highly effective poetic structure.

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The Drunkards Lament

• Instances
• Some dirty swine stole my drink
• A pig stole my drink
• A lousy rat took my drink
• A rotten lousy dirty rat swiped my drink
• The lousy dirty rotten swine took my drink
• Try to make a Context Free Grammar out of it
• Try to make a Regular Grammar out of it.

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Drunkards Lament CF Grammar

• ltLamentgt-gt ltArticlegtltAdjstuffgtltNoungtltVerbgt my
  drink
• ltArticlegt-gtSome A The
• ltAdjstuffgt-gtltAdjgtltAdjstuffgt
• ltAdjstuffgt-gtltAdjgt
• ltAdjstuffgt-gt l
• ltAdjgt-gt rotton lousy dirty
• ltNoungt-gt swine pig rat
• ltVerbgt-gt took swiped stole

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Drunkards Lament Regular Grammar

• ltLamentgt-gt Some ltRestOfStuffgt
• ltLamentgt-gt A ltRestOfStuffgt
• ltLamentgt-gt The ltRestOfStuffgt
• ltRestOfStuffgt-gtrotton ltRestOfStuffgt
• ltRestOfStuffgt-gtlousy ltRestOfStuffgt
• ltRestOfStuffgt-gtdirty ltRestOfStuffgt
• ltRestOfStuffgt-gtswine ltVerbObjectgt
• ltRestOfStuffgt-gtpig ltVerbObjectgt
• ltRestOfStuffgt-gtrat ltVerbObjectgt
• ltVerbObjectgt-gttook ltObjectgt
• ltVerbObjectgt-gtswiped ltObjectgt
• ltVerbObjectgt-gtstole ltObjectgt
• ltObjectgt-gt my ltDrinkWordgt
• ltDrinkwordgt-gt drink