CS 275 Automata and Formal Language Theory. Proof of Lemma II Lemma (II )

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1 CS 275 Automata and Formal Language Theory Course Notes Part II: The Recognition Problem (II) Additional Material (This material is no longer taught and not exam relevant) Sect II.2.: Basics of Regular Languages and Expressions Anton Setzer (Based on a book draft by J. V. Tucker and K. Stephenson) Dept. of Computer Science, Swansea University csetzer/lectures/ automataformallanguage/current/index.html April 19, 2018 CS 275 Sect. II.2. (Additional Material) 1/ 24 CS 275 Sect. II.2. (Additional Material) 2/ 24 Proof of Lemma II Lemma (II ) 1. Assume a grammar G which has only productions of the form A Bw or A w for some w T, A, B N. Then L(G) = L(G ) for some left-linear grammar G, which can be computed from G. 2. Assume a grammar G which has only productions of the form A wb or A w for some w T, A, B N. Then L(G) = L(G ) for some right-linear grammar G, which can be computed from G. CS 275 Sect. II / 24 CS 275 Sect. II / 24

2 Proof Step 1 End of Proof of II Let G be the result of applying Step 1 to the grammar as described in the lecture. Then one can easily see that for w T S G w iff S G w We have now obtained a grammar which doesn t contain productions of the form A B for nonterminals A, B. silent The following lemma shows that such languages are definable by left-linear or right-linear grammars. CS 275 Sect. II / 24 Proof of Lemma II CS 275 Sect. II / 24 Proof of Lemma II Lemma (II ) 1. Assume a grammar G which has only productions of the form A Bw or A w for some w T +, w T, A, B N. Then L(G) = L(G ) for some left-linear grammar G, and G can effectivly computed from G. 2. Assume a grammar G which has only productions of the form In (2) replace Productions A a1 a 2 a n B with n 2 by A a 1 A 1, A 1 a 2 A 2,..., A n 1 a n B for some new nonterminals A i. Productions A a1 a 2 a n with n 2 by A a 1 A 1, A 1 a 2 A 2,..., A n 1 a n for some new nonterminals A i. (1) is proved similarly. A wb or A w for some w T +, w T, A, B N. Then L(G) = L(G ) for some right-linear grammar G, and G can effectivly computed from G. CS 275 Sect. II / 24 CS 275 Sect. II / 24

3 Derivations in Regular Grammars Theorem (a) Let G = (N, T, S, P) be a left-linear grammar, A N, w (N T ), A w. Then the derivation of A w is Derivations in Regular Grammars Theorem (b) Let G = (N, T, S, P) be a right-linear grammar, A N, w (N T ), A w. Then the derivation of A w is A A 1 a 1 A 2 a 2 a 1 A n a n a 2 a 1 = w (1) or A A 1 a 1 A 2 a 2 a 1 A n a n a 2 a 1 (2) a n+1 a n a 2 a 1 = w or A A 1 a 1 A 2 a 2 a 1 A n a n a 2 a 1 a n a 2 a 1 = w (3) or or A a 1 A 1 a 1 a 2 A 2 a 1 a 2 a n A n = w (1) A a 1 A 1 a 1 a 2 A 2 a 1 a 2 a n A n (2) a 1 a 2 a n a n+1 = w A a 1 A 1 a 1 a 2 A 2 a 1 a 2 a n A n (3) a 1 a 2 a n = w for productions A i A i+1 a i+1 (in (1) (3)), A n a n+1 (in (2)) A n ɛ (in (3)) CS 275 Sect. II / 24 for productions A i a i+1 A i+1 (in (1) - (3)) A n a n+1 (in (2)) A n ɛ (in (3)). CS 275 Sect. II / 24 Proof The above are the only derivations possible. CS 275 Sect. II / 24 CS 275 Sect. II / 24

4 Proof of Lemma II Proof of Lemma II Lemma (II ) Let G, G be both left-linear grammars or both right-linear grammars. Then we can define a left-linear or right-linear grammars G i s.t. 1. L(G 1 ) = L(G) L(G ), 2. L(G 2 ) = L(G).L(G ), 3. L(G 3 ) = L(G). These grammars can be computed from G and G. Assume in 1./2./3. G = (T, N, S, P), G = (T, N, S, P ). After renaming of nonterminals we can assume N N =. Let S be a new symbol not in N N T T. We define multi-step left/right-linear grammars with those properties, from which one can construct ordinary (one-step) left/right-linear grammars with those properties. We only carry out the proof for right-linear grammars. Proof of 1. CS 275 Sect. II / 24 We define G 1 as follows: grammar G 1 terminals T T nonterminals N N {S } start symbol S productions S S S S P P CS 275 Sect. II / 24 Proof of 1. CS 275 Sect. II / 24 So G 1 has the productions from G and G plus Derivations in G 1 have the form and for derivations and S S and S S. S S w S S w S G w S G w So for w (T T ) we have S G 1 w iff S G w or S G w, so L(G ) = L(G) L(G ). CS 275 Sect. II / 24

5 Proof of 2. Proof of 2. We define G 2 as follows: grammar G 2 terminals T T nonterminals N N start symbol S productions A aa for A aa P (A, A N, a T ) A as for A a P (A N, a T ) P So G 2 has the productions from G, the productions of the form A aa from G and productions A as, if A a is a production from G. A derivation in G 2 starts with a derivation S a 1 A 1 a 1 a 2 A 2 a 1 a 2 a 3 A 3 a 1 a 2 a n 1 A n 1 a 1 a 2 a n S for derivations in G of the form S a 1 A 1 a 1 a 2 A 2 a 1 a 2 a 3 A 3 a 1 a 2 a n 1 A n 1 a 1 a 2 a n. Proof of 2. CS 275 Sect. II / 24 Proof of 3. CS 275 Sect. II / 24 Then this is followed by a derivation a 1 a 2 a n S a 1 a 2 a n b 1 B 1 a 1 a 2 a n b 1 b 2 B 2 a 1 a 2 a n b 1 b 2 b m 1 B m 1 a 1 a 2 a n b 1 b 2 b m, We define G 3 as follows: grammar G 3 terminals T for a derivation in G of the form S b 1 B 1 b 1 b 2 B 2 b 1 b 2 b m 1 B m 1 b 1 b 2 b m nonterminals start symbol N S Therefore S G 2 w for some w (T T ) if and only if S G 1 w and S G 2 w for some w, w s.t. w = ww. So L(G 2 ) = L(G).L(G ). productions S ɛ, A aa for A aa P (A, A N, a T ) A as for A a P (A N, a T ) CS 275 Sect. II / 24 CS 275 Sect. II / 24

6 Proof of 3. Derivations in G 3 are S ɛ or they start similarly as for concatenation with S ws for a derivation in G S w and w N +. In the latter case it can continue either (using S ɛ) with ws w or with ws ww S for a derivation in G S w Again in the latter case we can continue (using S ɛ) with ww S ww or with ww S ww w S for a derivation in G etc. S w CS 275 Sect. II / 24 Proof of Lemma II Proof of 3. We obtain that in G 3 we have if there exist derivations in G of S w 1 S w 2 S w n s.t. w = w 1 w 2 w n. So we get S w L(G 3 ) = {w 1 w 2 w n n 0, w 1,..., w n L(G)} = L(G) CS 275 Sect. II / 24 Proof of Lemma II Lemma (II ) Let E be a regular Expression. Then there exist both left-linear and right-linear grammars G, G s.t. G and G can be computed from L. L(E) = L(G) = L(G ) Proof: By Lemma II.2.2.1, and the fact that the finite languages, {ɛ} and {a} are regular. Induction on the definition of regular expressions. Case 1: L =, ɛ, a (where a T ). Then L is finite, therefore definable by a left/right-linear grammar. Case 2: L = (L 1 ) (L 2 ) or L = (L 1 )(L 2 ) or L = (L 1 ). By IH L i are defined by left/right-linear grammars G i. By Lemma II it follows that L can be defioned by a left/right-linear grammar. CS 275 Sect. II / 24 CS 275 Sect. II / 24