# Help Identifying a Gate In Nielsen and Chuang

I am seeking help to identify the oracle gates listed in this example. I understand that the right-most one is a toffoli gate, but what are the other ones? Specifically, I do not understand what a black versus white circle means for gate representation. This example comes from Nielsen and Chuang page 256

Here are the corresponding decompositions of each gate with use of Toffoli and $$X$$ gates:

The first one applies $$X$$ gate on the third qubit if control qubits are in the $$|00\rangle$$ state. The second one applies $$X$$ gate on the third qubit if control qubits are in the $$|01\rangle$$ state. The third one applies $$X$$ gate on the third qubit if control qubits are in the $$|10\rangle$$ state. The fourth one is the Toffoli gate that applies $$X$$ gate if control qubits are in the $$|11\rangle$$ state. In all other cases, they do nothing. The black (white) circle means that the corresponding qubit should be $$|1\rangle$$ $$(|0\rangle)$$ to satisfy the condition of the gate. Multiple circles are for multiple conditions on the control qubits.

For more info look at Figure 4.11 and 4.12 and corresponding descriptions in the same book.