Here are minimal NFA diagrams for each language, described in text and ASCII art. --- ## a. Strings over {a, b} containing at least **3** a's ### States - \( q_0 \): start state, 0 a's seen - \( q_1 \): 1 a seen - \( q_2 \): 2 a's seen - \( q_3 \): 3 or more a's seen (accepting) ### Diagram ``` (q0) --a--> (q1) --a--> (q2) --a--> [q3] | | | / \ b b b a,b b | | | \ / v v v [q3] (q0) (q1) (q2) ``` - From \( q_3 \), any input (a or b) loops on \( q_3 \). --- ## b. Strings over {a, b} that **start with at least 3 consecutive a's** ### States - \( q_0 \): start - \( q_1 \): seen 1 a - \( q_2 \): seen 2 consecutive a's - \( q_3 \): seen 3 consecutive a's (accepting) - \( q_{dead} \): any non-a in the first three positions ### Diagram ``` (q0) --a--> (q1) --a--> (q2) --a--> [q3] --a,b--> [q3] | | | b b b | | | v v v [dead] [dead] [dead] ``` --- ## c. Strings over {a, b} that **end with at least 3 consecutive a's** ### States - \( q_0 \): start - \( q_1 \): last symbol was a - \( q_2 \): last two symbols were a's - \( q_3 \): last three symbols were a's (accepting) ### Diagram ``` (q0) --a--> (q1) --a--> (q2) --a--> [q3] | | | | | | b b b b b b | | | | | | v v v v v v (q0) (q0) (q0) (q0) (q0) (q0) ``` - Only \( q_3 \) is accepting. --- ## d. Strings over {a, b} containing at least **3 consecutive a's** (anywhere) ### States - \( q_0 \): start - \( q_1 \): 1 consecutive a - \( q_2 \): 2 consecutive a's - \( q_3 \): 3 consecutive a's (accepting) ### Diagram ``` (q0) --a--> (q1) --a--> (q2) --a--> [q3] | | | | b b b a,b | | | | v v v v (q0) (q0) (q0) [q3] ``` --- ## e. Strings over {a, b} with **no two consecutive a's** ### States - \( q_0 \): start or last was b - \( q_1 \): last was a ### Diagram ``` (q0) --a--> (q1) --b--> (q0) | | b a | | v v (q0) [dead] ``` - \( q_1 \) --a--> dead (trap state). --- ## f. Strings over {a, b} with **exactly one occurrence of "a", and that "a" is followed by "b"** (Assuming you mean: strings that contain "ab" as a substring and nowhere else does "a" occur.) Let’s suppose you mean all strings of the form \( xab \), where \( x \) is any string over \( \{a,b\} \). ### States - \( q_0 \): start - \( q_1 \): seen 'a' - \( q_2 \): seen 'ab' (accepting) ### Diagram ``` (q0) --a--> (q1) --b--> [q2] | | | / \ b b a a,b b | | | \ / v v v [q2] (q0) (q1) (dead) ``` - After reaching \( q_2 \), any further input stays in \( q_2 \). --- ## g. Strings over {0,1} that **end with 0001 or 1000** ### States - \( q_0 \): start - \( q_1 \): last symbol 1 - \( q_2 \): seen 10 - \( q_3 \): seen 100 - \( q_4 \): accepting (seen 1000) - \( q_5 \): seen 0 - \( q_6 \): seen 00 - \( q_7 \): seen 000 - \( q_8 \): accepting (seen 0001) ### Diagram ``` (q0) --1--> (q1) --0--> (q2) --0--> (q3) --0--> [q4] | | | 0 1 0 | v v (v5)<--------------------------------(q8)[accepting] (q0) --0--> (q5) --0--> (q6) --0--> (q7) --1--> [q8] ``` - Only \( q_4 \) and \( q_8 \) are accepting. --- ## h. Strings over {0,1} that **start with 000, or end with 000, or both** ### States - \( q_0 \): start - \( q_1 \): seen 0 - \( q_2 \): seen 00 - \( q_3 \): seen 000 (accepting for "start with 000") - \( q_4 \): seen last symbol 0 - \( q_5 \): seen last two symbols 00 - \( q_6 \): seen last three symbols 000 (accepting for "end with 000") ### Diagram ``` (q0) --0--> (q1) --0--> (q2) --0--> [q3] | / \ 1 0,1 0,1 | v v (q4)<-------------------------(q4) (q4) (q0) --(0,1)--> (q4) (q4) --0--> (q5) (q5) --0--> (q6)[accepting] (q4), (q5), (q6): on 1, go to q4 or q5 as appropriate. ``` - Accept if you reach \( q_3 \) (start with 000), or \( q_6 \) (end with 000). --- **Note:** If you need diagrams as TikZ or LaTeX or in another format, let me know.