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Some textbook show that the grammar $G=(N,T,P,S)$ below belongs to context-senstive grammars:

  1. S->aSBC
  2. S->abc
  3. cB->Bc
  4. bG->bb

where N={S,B}, T={a,b,c}.

I am confused by the fact that the production rule cB->Bc cannot satisfy the definition of a CSG. So, why is the above grammar belongs to CSG?

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Your confusion is not unexpected. That grammar is not a context-sensitive grammar in its usual strict sense.

However, that grammar can be considered as a context-sensitive grammar in some situations. The Wikipedia entry on context-sensitive grmmar reads the following.

Some definitions of a context-sensitive grammar only require that for any production rule of the form $u \to v$, the length of $u$ shall be less than or equal to the length of $v$. This seemingly weaker requirement is in fact weakly equivalent, see Noncontracting grammar#Transforming into context-sensitive grammar.

You may want to double check the definition of CFG and exposition thereafter in the textbook you mentioned.

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