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. 1)S  ASCB 2)S  ACB 3)CB  CR 4)CR  BR 5)BR  BC 6)AB  Ab 7)Ab  ab 8)Aa  aa 9)bB  bb 10)bC  bc 11)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 G 1 be the new grammar replace a by N a and add a production N a  a, where N a is a new nonterminal Step 2: Let w(G) = max {|  |, where  is in P} Convert the grammar G 1 to a grammar G 2 such that   V + and w(G 2 )  2

8. Let  : A 1 …A m  B 1.. B n be a production If n  2, add it to G 2 If 2  m < n, create two productions: A 1 … A m  B 1 … B m-1 X X  B m … B n If m = 1 and n  3 create n-1 productions A 1  B 1 X 1 X 1  B 2 X 2 … X n-2  B n-1 B n

9. If m = n and n  3, create the n-1 productions A 1 A 2  B 1 X 1 X 1 A 3  B 2 X 2 … X n-2 A n  B n-1 B n Step 3: Convert G 2 to a new grammar G 3 Add productions of the form A   to G 3 If AB  CD is a production and if A= C or B = D, add to G 3 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  T n for some n  1. Define a sequence of sets W i  (V  T) * as follows: W 0 = {S} for each i  0, W i+1 = W i  {   (V  T) + |    in G,  is in W i, and |  |  n}

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