Suppose Alice's offer invites acceptance either by walking across the bridge, or by promising to do so. Imagine she says, "I offer you $500 to walk across the Michigan Avenue bridge. You may accept either by walking across the bridge or by promising to do so."
The rule is that, if the offer invites acceptance either by a performance or by a promise, then beginning performance is acceptance by performance of the offer, where the beginning of the performance is a promise to complete the performance. See Restatement (Second) Contracts ยง 62.
Thus, Bob