Faulty Circuitlogician series

This problem illustrates the possibility of using logic as a tool for diagnosis. The solver has to deduce from the specification given together with some test data which component or components in a boolean circuit may be incorrect. Of course, the assumption that the number of incorrect components is minimal is not logically implied by the rest of the problem setup: in diagnosis it is common to make an assumption like this to the effect that the system is as near correct as is compatible with the data.

Faulty Circuit image

We consider logic circuits built up by connecting gates of the following sorts:

NOT	one input;	output the opposite value.
F	one input;	output (always) false.
T	one input;	output (always) true.
AND	two inputs;	output true if both inputs true, otherwise false.
OR	two inputs;	output true if at least one input true, otherwise false.
NAND	two inputs;	output true if at least one input false, otherwise false.
NOR	two inputs;	output true if both inputs false, otherwise false.
XNOR	two inputs;	output true if both inputs the same, false if they are different.
XOR	two inputs;	output true if the inputs are different, false if both the same.

The image above is the specification for a logic circuit for computing an output E in terms of four inputs A, B, C and D

Unfortunately, the actual device does not exactly match the above specification: at least one of the components has been incorrectly installed, so that where there should be one sort of gate there is actually another. Some test data are as follows:

	INPUTS				OUTPUT
	A	B	C	D	E
Trial 1	True	True	True	True	True
Trial 2	False	False	True	False	True
Trial 3	True	True	False	True	True
Trial 4	True	True	True	False	True
Trial 5	True	True	False	False	False

Assuming the set of faults is as small as possible given the data, identify the incorrect component(s).

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