A technician suspects that one or more of the processors in a distributed system

Question

A technician suspects that one or more of the processors in a distributed system is not working properly. The processors, A, B, and C, are all capable of reporting information about the status (working or not working) of the processors in the system. The technician is unsure whether a processor is really not working, or whether the problem is in the status reporting routines in one or more of the processors. After polling each processor, the technician receives the following status reports. Bullet Processor A reports that processor B is not working and processor C is working.

Explanation / Answer

We translate each statement into the symbolic logic.

Here AA is the statement meaning "processor A is working", and similarly for the others.

The first part asks us to find which of AA, BB, and CC is true, assuming that all of the above statements are true. Your answer is correct here.

The second part asks us to find which of the above statements are false, assuming that AA, BB, and CCare true. We can just substitute 11 (meaning truth) into the above statements and find their values:

The third part asks us to find which of AA, BB, and CC are true, assuming the following:

We can think of an argument using valid argument forms, but I don't see it. A brute-force approach to find the status of processor A is to compute the truth value of

[(A(¬BC))(B(AB))(C(¬A¬B))]A[(A(¬BC))(B(AB))(C(¬A¬B))]A

with the help of a truth-table. Since the above implication is a tautology, then our assumptions can deduce AA to be true. To find the status of other processors, replace the right-hand-side of the above implication by BB and CC, and determine whether the implication is a tautology.

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