Context Sensitive Grammar (Automata theory)

  1. Sorry if this is the wrong forum.

    I'm trying to get my head around Context Sensitive Grammar. In my coursebook, there's an example ('e' here is the empty string, epsilon, and 'E' is the "set member" symbol):

    The language {an bn cn | n E N} is described by the following rules:

    S -> e | abNSc
    bNa -> abN
    bNb -> bbN
    bNc -> bc

    And then it shows how to build aabbcc:

    S -> abNSc
    -> abNabNScc
    -> abNabNcc
    -> aabNbcc
    -> aabbNcc
    -> aabbcc

    The part I don't understand is why S becomes abNSc after the first substitution, while the second time it becomes an empty string. The rules state both substitutions are possible, but not when to use which one. Can this be choosen freely?

    Another language (with 4 non-deterministic symbols S, D, [, ]) describes {1, 1^2, 1^4, 1^8, ...}. Its rules:

    S -> [1]
    [ -> [D
    D1 -> 11D
    D] -> ]
    [ -> e
    ] -> e

    Example, building the string 11111111:

    S -> [1] -> [D1] ->* [DDD1]
    -> [DD1D]
    -> [D11D1D]
    -> [D1111DD]
    -> [11D111DD]
    -> [1111D11DD]
    -> [111111D1DD]
    -> [11111111DDD]
    -> [11111111DD]
    -> [11111111D]
    -> [11111111]
    -> [11111111
    -> 11111111

    Again, the number of times a substitution is carried out seems to be random, or at least arbitrary. Why do you not just keep doing the substitution from [ to [D indefinitely? Because you know how many times it needs to be done in order to get the resulting string you want? Can you choose to do the substitutions in any order you want (as long as it gets you the result you want), or is there a pattern to follow?

    Thanks for any help.
  2. jcsd
  3. Hurkyl

    Hurkyl 15,987
    Staff Emeritus
    Science Advisor
    Gold Member

    The language defined by a grammar consists of every string that you can produce by applying the production rules.
    So yeah.
  4. That's all I needed to hear. Thank you!
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