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Phrase-structure grammar

Phrase-structure grammar. A phrase-structure grammar is a quadruple G = (V, T, P, S) where V is a finite set of symbols called nonterminals, T is a set of terminals, P is the set of productions  → ,   (V  T)*V(V  T)*,   (V  T)*, S is a member of V called the start symbol.

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Phrase-structure grammar

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  1. Phrase-structure grammar A phrase-structure grammar is a quadruple G = (V, T, P, S) where V is a finite set of symbols called nonterminals, T is a set of terminals, P is the set of productions  → ,   (V  T)*V(V  T)*,   (V  T)*, S is a member of V called the start symbol.

  2. S → ABC AB → aAD AB → bAE DC → BaC EC → BbC Da → aD Db → bD Ea → aE Eb → bE AB →  C →  aB → Ba bB → Bb

  3. Context-sensitive Grammar Definition: A grammar G = (V, T, P, S) is context sensitive if ||  || for every production  in P. Definition: A “true” context sensitive grammar G = (V, T, P, S) is a grammar in which each production is of the form A  , where  and  are in (V T)*,  in (V  T)+, and A in V. The production A   is also written as A  / _

  4. Example: S  ABC S  ABCS AB  BA AC  CA BC  CB BA  AB CA  AC CB  BC A  a B  b C  c

  5. S  ASCB • S  ACB • CB  CR • CR  BR • BR  BC • AB  Ab • Ab  ab • Aa  aa • bB  bb • bC  bc • cC  cc

  6. Definition: A language L is context-sensitive language if it is generated by a context-sensitive grammar. Theorem: Every context-sensitive language can be generated by a true context-sensitive grammar.

  7. Step 1: convert all rules of the grammar G to the form    where  is a string of non terminals, let G1 be the new grammar replace a by Na and add a production Na  a, where Na is a new nonterminal Step 2: Let w(G) = max {||, where  is in P} Convert the grammar G1 to a grammar G2 such that   V + and w(G2)  2

  8. Let : A1 …Am  B1 .. Bn be a production If n  2, add it to G2 If 2  m < n, create two productions: A1 … Am  B1 … Bm-1X X  Bm … Bn If m = 1 and n  3 create n-1 productions A1  B1X1 X1  B2X2 … Xn-2  Bn-1Bn

  9. If m = n and n  3, create the n-1 productions A1A2  B1X1 X1A3  B2X2 … Xn-2An  Bn-1Bn Step 3: Convert G2 to a new grammar G3 Add productions of the form A   to G3 If AB  CD is a production and if A= C or B = D, add to G3 if AB  CD and A  C and B  D, then add the productions AB  XB, XB  XY, XY  CY CY  CD

  10. Definition: Let G = (V, T, P, S) be a context-sensitive grammar and let w  Tn for some n  1. Define a sequence of sets Wi  (V  T)* as follows: W0 = {S} for each i  0, Wi+1 = Wi  {  (VT)+ |    in G,  is in Wi, and ||  n}

  11. Proposition: Let Wi be as defined before. Then we have the following: 1) for each i  0, Wi  Wi+1 2) if Wk = Wk+1 for some k, Wk = Wk+m for all m > 0 3) for each i  0, Wi = {  (V  T)* | S m , ||  n, m  i} 4) there exists k < max(2*|VT|n, n+1) such that Wk = Wk+1 5) let k be the least integer such that Wk = Wk+1, then Wk = {  (VT)+ | S * , ||  n}

  12. Theorem: Let G = (V, T, P, S) be a context-sensitive grammar. Then there is an algorithm which, given any w  T*, decides whether or not w  L(G). (L(G) is recursive)

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