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  Context sensitive grammars and languages, Context Free Grammars & Languages

Context sensitive grammars and languages:

The rewriting rules B -> w of a CFG entail that the non-terminal B can be substituted by a word w ∈ (V ∪ A)* ‘in any context’. In disparity, the context sensitive grammar (or CSG) consists of rules of the form: u B v -> u w v, where u, v, w ∈ (V ∪ A)*, implying that B can be substituted by w just in the context ‘u on the left and v on the right’.

This turns out that this definition is equal (apart from null string ε) to needing that any CSG rule be of the form v -> w, where v, w ∈ (V ∪ A)*, and |v| ≤ |w|. This monotonicity property (in any derivation, the present string never gets shorter) entails that the word problem for CSLs: ‘given CSG G and given w, is w ∈ L(G)?’Is decidable. The exhaustive enumeration of all the derivations up to length |w| settles the issue.

Since an illustration of the greater power of CSGs over CFGs, remember that we employed the pumping lemma to verify that the language 0k 1k 2k is not CF. By way of disparity, we verify:

Theorem: L = {0k 1k 2k / k ≥ 1} is context sensitive.

The given CSG produces L. Function of non-terminals V = {S, B, C, Y, Z}: all Y and Z produces a 1 or a 0 at proper time; B initially marks the starting (left end) of the string, and later transforms the Zs to 0s; C is the counter which ensures an equivalent number of 0s, 1s, 2s are produced. Non-terminals play an identical role as markers in the Markov algorithms. While the latter encompass a deterministic control structure, grammars are non-deterministic.

S -> B K 2 at the final step in any derivation, B K produces 01, balancing this ‘2’.
K -> Z Y K 2 counter K produces (ZY)k 2k.
K -> C whenever k has been fixed, C might begin transforming Ys to 1s.
Y Z  -> Z Y Zs may move towards left, Ys towards right at any time.
B Z -> 0 B B might transform a Z to a 0 and shift it left at any time.
Y C -> C 1 C might transform a Y to a 1 and shift it right at any time.
B C -> 01 whenever B and C meet, all permutations, shifts and conversions have been completed.

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