Let
𝐿
be the language consisting of bit strings of the form
𝑤
𝑤
. More precisely,
𝐿
=
{
𝑤
𝑤
∣
𝑤
∈
Σ
∗
}
.

So,
𝐿
contains
𝜖
,
00
,
11
,
1010
,
0101
,
1111
,
101101
,
…

Select all options that are fooling sets of size at least k.

Question

Let 
𝐿
 be the language consisting of bit strings of the form 
𝑤
𝑤
. More precisely, 
𝐿
=
{
𝑤
𝑤
∣
𝑤
∈
Σ
∗
}
.

So, 
𝐿
 contains 
𝜖
,
00
,
11
,
1010
,
0101
,
1111
,
101101
,
…

Select all options that are fooling sets of size at least k.

BlackTom AI · Accepted Answer

We are given the language L = { w w | w ∈ Σ* }, i.e., strings that are exact concatenations of a string with itself. A fooling set F for L (of size at least k) typically means: there exist pairs (x_i, y_i) for i = 1..m with m ≥ k such that for every i, x_i y_i ∈ L, and for all i ≠ j, x_i y_j ∉ L. In other words, each pair (x_i, y_i) lands in L, but mixing the x_i with a different y_j breaks membership in L. The options provided list candidate sets of strings that could serve as the x_i parts of such fooling sets. We must analyze each option to see whether it can form a fooling set of size at least k, given appropriate choices of y_i, and while keeping in mind the structure of L (strings that are exactly twice some w). Below, I treat each option separately, noting the key structural observations and potential y_i choices, and I point out where the construction is plausible or likely to fail. I also show how the form of the strings in each option interacts with the requirement x_i y_i ∈ L and x_i y_j ∉ L for i ≠ j. I use diverse phrasing and clear, stepwise reasoning to illuminate the core ideas behind why an option would or would not support a fooling set of the required size.

Option 1: { 0, 00, 000, ..., 0^k }
- Observation: The x_i are prefixes consisting solely of 0’s of various lengths. To form a fooling set, we would need to pick corresponding y_i so that x_i y_i ∈ L, i.e., x_i y_i = w w for some w, and for i ≠ j, x_i y_j ∉ L.
- Plausible construction attempt: If we take y_i to be 0^{i}, then x_i y_i = 0^{i} 0^{i} = 0^{2i}, which is indeed of the form w w with w = 0^{i}. So for each i, x_i y_i ∈ L holds.
- Distinguishability check: For i ≠ j, x_i y_j would be 0^{i} 0^{j} = 0^{i + j}. This is again a double-copy of a string (w = 0^{min(i,j)} and the tail), but note: for a string to be in L, it must be exactly some w w, i.e., a first half equal to the second half. The string 0^{i+j} is of the form w w only when i = j (because the length must be even and the first half must equal the second half exactly). However, 0^{i+j} has a single division into halves only if i + j is even, and the halves must be equal as strings, which would require the first i+j over 2 symbols to equal the last i+j over 2 symbols. Since the entire string is all 0s, the halves are both all 0s of length (i+j)/2, so indeed 0^{i+j} is in L for every i, j with i+j even. That means x_i y_j ∈ L for some i ≠ j whenever i+j is even, which violates the required property x_i y_j ∉ L for i ≠ j. Therefore, this set cannot form a fooling set of the intended strength (size at least k) unless we restrict indices to force i+j always odd, which is not generally possible for all pairs in a natural enumeration. In short, the all-zeros strings option fails the strict fooling-set requirement in general.
- Conclusion for this option: The construction cannot guarantee x_i y_j ∉ L for all i ≠ j with a uniform, natural choice of y_i, so it is unlikely to be a valid fooling set of size ≥ k.

Option 2: { 1, 11, 111, ..., 1^{k−1} }
- Observation: The x_i are strings of 1’s of increasing length. Similar reasoning to Option 1: to get x_i y_i ∈ L, we could attempt y_i = 1^{i}, giving x_i y_i = 1^{i} 1^{i} = 1^{2i} ∈ L (with w = 1^{i}).
- Distinguishability check: For i ≠ j, x_i y_j = 1^{i} 1^{j} = 1^{i+j}. Again, 1^{i+j} is in L exactly when i+j is even (by the same half-structure argument). This means we do not have the necessary separation x_i y_j ∉ L for all i ≠ j. So this option also fails to establish a fooling set of the required form.
- Conclusion for this option: It does not reliably produce a fooling set of size at least k due to the same even-length divisibility issue as Option 1.

Option 3: { ε, εε, εεε, ..., ε^k }
- Observation: Each x_i is a string of ε concatenations, but ε is the empty string, and εε = ε, etc. All x_i are just ε, i.e., the same string repeated. A fooling set requires distinct x_i’s to work against distinct y_i’s in a way that preserves the separation property across i ≠ j.
- Consequences: Since all x_i are identical to ε, any choice of y_i yields x_i y_i = ε y_i = y_i, so the condition x_i y_i ∈ L reduces to choosing y_i ∈ L. Then for i ≠ j, x_i y_j = ε y_j = y_j, which is also in L if y_j ∈ L. This collapses the distinguishing condition: we cannot ensure x_i y_j ∉ L for i ≠ j unless we force some y_j not in L, which would violate the requirement that x_j y_j ∈ L. Therefore, this option cannot form a valid fooling set of size ≥ k.
- Conclusion for this option: Not viable as a fooling set for L of the required size.

Option 4: { 01, 0011, 000111, ..., 0^k 1^k }
- Observation: The x_i are strings of the form 0^i 1^i, i.e., i zeros followed by i ones. Such strings are not themselves of the form w w unless w is a string that, when doubled, yields 0^i 1^i. There is no w such that w w = 0^i 1^i, because the left half would have to be identical to the right half, which would force the boundary to align with a change from 0 to 1. In particular, 0^i 1^i cannot be split into two identical halves unless i = 0. Hence, for i > 0, x_i ∉ L.
- Fooling-set feasibility: We are allowed to choose y_i to try to satisfy x_i y_i ∈ L. A natural attempt would be to pick y_i to complete the pattern so that x_i y_i becomes some w w. But since x_i already changes alphabet from 0 to 1, the only way to get a perfect double would require a very specific y_i that mirrors the first half in a way that makes the entire string a perfect duplication. If we try y_i = 1^i, for example, we get x_i y_i = 0^i 1^i 1^i, which is not of the form w w unless the boundary falls exactly at a duplication point that matches the first half with the second half, which is not the case here. In general, because the prefix 0^i is followed by 1^i, constructing a y_i that yields a double (ww) becomes highly constrained or impossible for large i, making it hard to satisfy x_i y_i ∈ L for all i simultaneously. Moreover, ensuring x_i y_j ∉ L for i ≠ j across all pairs becomes even more delicate due to the mixed alphabet pattern.
- Conclusion for this option: It is unlikely to support a fooling set of size at least k given the lack of inherent self-doubling structure in the x_i strings and the difficulty of forcing all x_i y_i into L while keeping cross-pairs outside L.

Summary across options:
- Options 1 and 2 attempt to use uniform strings of all zeros or all ones; while we can force x_i y_i into L with y_i = 0^i or 1^i respectively, the cross-pair condition x_i y_j ∉ L fails for many i ≠ j because 0^{i+j} or 1^{i+j} can still be in L whenever i+j is even. This undermines the fooling-set requirement.
- Option 3 collapses all x_i to the same string (ε), which makes the crucial distinguishability condition impossible to satisfy for i ≠ j, so it cannot form a valid fooling set of size ≥ k.
- Option 4 lacks the self-doubling structure in the x_i themselves, making it impractical to guarantee x_i y_i ∈ L for all i while maintaining x_i y_j ∉ L for i ≠ j.

Overall assessment: Each option faces a structural obstacle with respect to constructing a fooling set of size at least k for L = { w w | w ∈ Σ* }. The key difficulty is ensuring that for every i, x_i y_i ∈ L while for every i ≠ j, x_i y_j ∉ L, given the way the x_i strings are formed. The common pitfall across the options is that even when one can make x_i y_i ∈ L for a chosen y_i, cross-combinations x_i y_j tend to land back in L for many pairs, violating the fooling-set condition. Because of this, none of the provided options can be confidently defended as a fooling set of size at least k without further constraints or refined constructions of the y_i’s, and at least one of these options is not viable for a fooling set of the required size.

COMP30026_2025_SM2 Worksheet 8

Let 𝐿 be the language consisting of bit strings of the form 𝑤 𝑤 . More precisely, 𝐿 = { 𝑤 𝑤 ∣ 𝑤 ∈ Σ ∗ } . So, 𝐿 contains 𝜖 , 00 , 11 , 1010 , 0101 , 1111 , 101101 , … Select all options that are fooling sets of size at least k.

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