Answer :
Final answer:
The question is about computational complexity theory. Specifically, it is theoretically proven that if two languages L1 and L2 belong to ZPP or BPP classes of complexity, the concatenation of these languages also belongs to the same class.
Explanation:
This question pertains to complexity theory, specifically zero-error probabilistic polynomial time (ZPP) and bounded-error probabilistic polynomial time (BPP) classes. These classes define sets of problems that can be solved by probabilistic Turing machines within respective constraints.
(a) If L1 and L2 are languages belonging to ZPP, then their concatenation L1L2 (defined as {xy: x∈L1, y∈L2}) also belongs to ZPP. This is because ZPP, as a complexity class, is closed under concatenation. Meaning that, if you can decide membership for x in L1 and y in L2 separately in probabilistic polynomial time with no errors, you can do the same for the concatenated string xy.
(b) The same reasoning applies to BPP. If L1 and L2 belong to BPP, their concatenation, L1L2, also belongs to BPP. BPP is also closed under concatenation, meaning that membership for x in L1 and y in L2 can be decided separately in probabilistic polynomial time with a bounded error probability. Hence, the same can be done for the concatenated string xy.
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