There is no reason why completely composted perennial weeds would reproduce themselves when that compost was used. If, however, there are seeds in the compost and the heat generated in the composting process was not sufficiently high to destroy their viability, then yes, those seeds will germinate and you will have weeds.
FAQ #6192
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Why shouldn't you compost weeds that have not gone to seed yet? I was told they would reseed from the finished product.
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