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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 0

^{k}1^{k}2^{k}is not CF. By way of disparity, we verify:: L = {0Theorem^{k}1^{k}2^{k}/ 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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