The XOR, XNOR, Even Parity, and Odd Parity gates each compute the respective function of the inputs, and emit the result on the output. The two-input truth table for the gates is the following.
As you can see, the Odd Parity gate and the XOR gate behave identically with two inputs; similarly, the even parity gate and the XNOR gate behave identically. But if there are more than two specified inputs, the XOR gate will emit 1 only when there is exactly one 1 input, whereas the Odd Parity gate will emit 1 if there are an odd number of 1 inputs. The XNOR gate will emit 1 only when there is not exactly one 1 input, while the Even Parity gate will emit 1 if there are an even number of 1 inputs. The XOR and XNOR gates include an attribute titled Multiple-Input Behavior that allow them to be configured to use the Odd Parity and Even Parity behavior.
Any inputs that are unspecified (i.e., floating) are ignored. If all inputs are floating, then the output is floating, too. If any of the inputs are the error value (e.g., if conflicting values are coming into the same wire), then the output will be the error value, too. If, however, the "Gate Output When Undefined" option is "Error for undefined inputs," then the output will be the error value if one or more of the inputs are floating.
The multi-bit versions of each gate will perform its one-bit transformation bitwise on its inputs.
Note: Many authorities contend that the shaped XOR gate's behavior should correspond to the odd parity gate, but there is not agreement on this point. Logisim's behavior for XOR gates is based on the IEEE 91 standard. It is also consistent with the intuitive meaning underlying the term exclusive or: A waiter asking whether you want a side dish of mashed potatoes, carrots, peas, or cole slaw will only accept one choice, not three, whatever some authorities may tell you. (I must admit, though, that I have not subjected this statement to a rigorous test.)
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