Either Twenty or Twenty-Three

Alice and Bob are perfect logicians. Each has a positive integer written on their forehead; each sees the other's number but not their own. Both know that the sum of the two numbers is either 20 or 23, and this is common knowledge. A referee asks them alternately, starting with Alice: "Do you know the number on your own forehead?" Each answer is heard by both. The first seven answers (Alice, Bob, Alice, Bob, Alice, Bob, Alice) are all "no", and the eighth answer, from Bob, is "yes". What number is on Bob's forehead?

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Source: Variant of the classic foreheads-and-numbers common-knowledge puzzle in wide circulation; no traceable original source. Statement written for AxiomIQ.