Let i = 1, j = 1.

Notice:

L_{1,1} ^1 = {x ∈ I* | x is defined by regex (0|1)*} because there is no intermediate state between s_1 and s_1.
L_{1,1} ^2 = {x ∈ I* | x is defined by regex (0|1)*} because there is no intermediate state between s_1 and s_1.
L_{1,1} ^3 = {x ∈ I* | x is defined by regex (0|1)*} because there is no intermediate state between s_1 and s_1.
L_{1,1} ^4 = {x ∈ I* | x is defined by regex (0|1)*} because there is no intermediate state between s_1 and s_1.

Thus, L_{1,1} ^k = {x ∈ I* | x is defined by regex (0|1)*}.

Let i = 1, j = 2.

Notice:

L_{1,2} ^1 = {x ∈ I* | x is defined by regex (0|1)*0} because there is no intermediate state between s_1 and s_2.
L_{1,2} ^2 = {x ∈ I* | x is defined by regex (0|1)*0} because there is no intermediate state between s_1 and s_2.
L_{1,2} ^3 = {x ∈ I* | x is defined by regex (0|1)*0} because there is no intermediate state between s_1 and s_2.
L_{1,2} ^4 = {x ∈ I* | x is defined by regex (0|1)*0} because there is no intermediate state between s_1 and s_2.

Thus, L_{1,2} ^k = {x ∈ I* | x is defined by regex (0|1)*0}.

Let i = 1, j = 3.

Notice:

L_{1,3} ^1 = ∅ because A goes from state s_1 to state s_3 through an intermediate state s_2 such that 2 > 1.
L_{1,3} ^2 = {x ∈ I* | x is defined by regex (0|1)*01} because A goes from state s_1 to state s_3 through an intermediate state s_2 but 2 ≤ 2.
L_{1,3} ^3 = {x ∈ I* | x is defined by regex (0|1)*01} because A goes from state s_1 to state s_3 through an intermediate state s_2 but 2 ≤ 3.
L_{1,3} ^4 = {x ∈ I* | x is defined by regex (0|1)*01} because A goes from state s_1 to state s_3 through an intermediate state s_2 but 2 ≤ 4.

Thus, L_{1,3} ^k = {x ∈ I* | x is defined by regex (0|1)*01} .

Let i = 1, j = 4.

Notice:

L_{1,4} ^1 = ∅ because A goes from state s_1 to state s_4 through an intermediate states s_2 such that 2 > 1 and s_3 such that 3 > 1.
L_{1,4} ^2 = ∅ because A goes from state s_1 to state s_4 through an intermediate states s_2 such that 2 ≤ 2 and s_3 such that 3 > 2.
L_{1,4} ^3 = {x ∈ I* | x is defined by regex (0|1)*011} because A goes from state s_1 to state s_4 through intermediate states s_2 such that 2 ≤ 3 and s_3 such that 3 ≤ 3.
L_{1,4} ^4 = {x ∈ I* | x is defined by regex (0|1)*011} because A goes from state s_1 to state s_4 through intermediate states s_2 such that 2 ≤ 4 and s_3 such that 3 ≤ 4.

Thus, L_{1,4} ^k = {x ∈ I* | x is defined by regex (0|1)*011} .

Let i = 2, j = 2.

Notice:

L_{2,2} ^1 = {x ∈ I* | x is defined by regex (0|1)*0} because there is no intermediate state between s_2 and s_2.
L_{2,2} ^2 = {x ∈ I* | x is defined by regex (0|1)*0} because there is no intermediate state between s_2 and s_2.
L_{2,2} ^3 = {x ∈ I* | x is defined by regex (0|1)*0} because there is no intermediate state between s_2 and s_2.
L_{2,2} ^4 = {x ∈ I* | x is defined by regex (0|1)*0} because there is no intermediate state between s_2 and s_2.

Thus, L_{2,2} ^k = {x ∈ I* | x is defined by regex (0|1)*0} .

Let i = 2, j = 3.

Notice:

L_{2,3} ^1 = {x ∈ I* | x is defined by regex (0|1)*01} because there is no intermediate state between s_2 and s_3.
L_{2,3} ^2 = {x ∈ I* | x is defined by regex (0|1)*01} because there is no intermediate state between s_2 and s_3.
L_{2,3} ^3 = {x ∈ I* | x is defined by regex (0|1)*01} because there is no intermediate state between s_2 and s_3.
L_{2,3} ^4 = {x ∈ I* | x is defined by regex (0|1)*01} because there is no intermediate state between s_2 and s_3.

Thus, L_{2,3} ^k = {x ∈ I* | x is defined by regex (0|1)*01} .

Let i = 2, j = 4.

Notice:

L_{2,4} ^1 = ∅ because A goes from state s_2 to state s_4 through an intermediate state s_3 such that 3 > 1.
L_{2,4} ^2 = ∅ because A goes from state s_2 to state s_4 through an intermediate state s_3 such that 3 > 2.
L_{2,4} ^3 = {x ∈ I* | x is defined by regex (0|1)*011} because A goes from state s_2 to state s_4 through intermediate states s_3 such that 3 ≤ 3.
L_{2,4} ^4 = {x ∈ I* | x is defined by regex (0|1)*011} because A goes from state s_2 to state s_4 through intermediate states s_3 such that 3 ≤ 4.

Thus, L_{2,4} ^k = {x ∈ I* | x is defined by regex (0|1)*011}.

Let i = 3, j = 3.

Notice:

L_{3,3} ^1 = {x ∈ I* | x is defined by regex (0|1)*01} because there is no intermediate state between s_3 and s_3.
L_{3,3} ^2 = {x ∈ I* | x is defined by regex (0|1)*01} because there is no intermediate state between s_3 and s_3.
L_{3,3} ^3 = {x ∈ I* | x is defined by regex (0|1)*01} because there is no intermediate state between s_3 and s_3.
L_{3,3} ^4 = {x ∈ I* | x is defined by regex (0|1)*01} because there is no intermediate state between s_3 and s_3.

Thus, L_{3,3} ^k = {x ∈ I* | x is defined by regex (0|1)*01}.

Let i = 3, j = 4.

Notice:

L_{3,4} ^1 = {x ∈ I* | x is defined by regex (0|1)*011} because there is no intermediate state between s_3 and s_4.
L_{3,4} ^2 = {x ∈ I* | x is defined by regex (0|1)*011} because there is no intermediate state between s_3 and s_4.
L_{3,4} ^3 = {x ∈ I* | x is defined by regex (0|1)*011} because there is no intermediate state between s_3 and s_4.
L_{3,4} ^4 = {x ∈ I* | x is defined by regex (0|1)*011} because there is no intermediate state between s_3 and s_4.

Thus, L_{3,4} ^k = {x ∈ I* | x is defined by regex (0|1)*011}.

Let i = 4, j = 4.

Notice:

L_{4,4} ^1 = {x ∈ I* | x is defined by regex (0|1)*011} because there is no intermediate state between s_4 and s_4.
L_{4,4} ^2 = {x ∈ I* | x is defined by regex (0|1)*011} because there is no intermediate state between s_4 and s_4.
L_{4,4} ^3 = {x ∈ I* | x is defined by regex (0|1)*011} because there is no intermediate state between s_4 and s_4.
L_{4,4} ^4 = {x ∈ I* | x is defined by regex (0|1)*011} because there is no intermediate state between s_4 and s_4.

Thus, L_{4,4} ^k = {x ∈ I* | x is defined by regex (0|1)*011}.



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To recap:

L_{1,1} ^1 = {x ∈ I* | x is defined by regex (0|1)*}
L_{1,1} ^2 = {x ∈ I* | x is defined by regex (0|1)*}
L_{1,1} ^3 = {x ∈ I* | x is defined by regex (0|1)*}
L_{1,1} ^4 = {x ∈ I* | x is defined by regex (0|1)*}

L_{1,2} ^1 = {x ∈ I* | x is defined by regex (0|1)*0}
L_{1,2} ^2 = {x ∈ I* | x is defined by regex (0|1)*0}
L_{1,2} ^3 = {x ∈ I* | x is defined by regex (0|1)*0}
L_{1,2} ^4 = {x ∈ I* | x is defined by regex (0|1)*0}

L_{1,3} ^1 = ∅
L_{1,3} ^2 = {x ∈ I* | x is defined by regex (0|1)*01}
L_{1,3} ^3 = {x ∈ I* | x is defined by regex (0|1)*01}
L_{1,3} ^4 = {x ∈ I* | x is defined by regex (0|1)*01}

L_{1,4} ^1 = ∅
L_{1,4} ^2 = ∅
L_{1,4} ^3 = {x ∈ I* | x is defined by regex (0|1)*011}
L_{1,4} ^4 = {x ∈ I* | x is defined by regex (0|1)*011}

L_{2,2} ^1 = {x ∈ I* | x is defined by regex (0|1)*0}
L_{2,2} ^2 = {x ∈ I* | x is defined by regex (0|1)*0}
L_{2,2} ^3 = {x ∈ I* | x is defined by regex (0|1)*0}
L_{2,2} ^4 = {x ∈ I* | x is defined by regex (0|1)*0}

L_{2,3} ^1 = {x ∈ I* | x is defined by regex (0|1)*01}
L_{2,3} ^2 = {x ∈ I* | x is defined by regex (0|1)*01}
L_{2,3} ^3 = {x ∈ I* | x is defined by regex (0|1)*01}
L_{2,3} ^4 = {x ∈ I* | x is defined by regex (0|1)*01}

L_{2,4} ^1 = ∅
L_{2,4} ^2 = ∅
L_{2,4} ^3 = {x ∈ I* | x is defined by regex (0|1)*011}
L_{2,4} ^4 = {x ∈ I* | x is defined by regex (0|1)*011}

L_{3,3} ^1 = {x ∈ I* | x is defined by regex (0|1)*01}
L_{3,3} ^2 = {x ∈ I* | x is defined by regex (0|1)*01}
L_{3,3} ^3 = {x ∈ I* | x is defined by regex (0|1)*01}
L_{3,3} ^4 = {x ∈ I* | x is defined by regex (0|1)*01}

L_{3,4} ^1 = {x ∈ I* | x is defined by regex (0|1)*011}
L_{3,4} ^2 = {x ∈ I* | x is defined by regex (0|1)*011}
L_{3,4} ^3 = {x ∈ I* | x is defined by regex (0|1)*011}
L_{3,4} ^4 = {x ∈ I* | x is defined by regex (0|1)*011}

L_{4,4} ^1 = {x ∈ I* | x is defined by regex (0|1)*011}
L_{4,4} ^2 = {x ∈ I* | x is defined by regex (0|1)*011}
L_{4,4} ^3 = {x ∈ I* | x is defined by regex (0|1)*011}
L_{4,4} ^4 = {x ∈ I* | x is defined by regex (0|1)*011}