My codes: A = 00100, B = 00011, C = 00000, Parity property = Even 1

You are the central hub for a communication system. Messages come to you as sequences of As, Bs and Cs but coded in binary. Each such binary message is to be followed by a check digit. This is a final 0 or 1 so that the entire binary message satisfies your parity property.

The parity properties are:

Even 0	The entire message (including the check digit) has an even number of 0's.
Odd 0	The entire message (including the check digit) has an odd number of 0's.
Even 1	The entire message (including the check digit) has an even number of 1's.
Odd 1	The entire message (including the check digit) has an odd number of 1's.

For example if your codes are A = 101, B = 1101, C = 001 and your parity property is Odd0, the message ABAC would get encoded as 10111011010011. The final character is the check digit. It is a 1 because we want an odd number of 0s. So 10111011010011 is valid, but 10111011010010 and 10111011011100 are not. Make sure you correctly understand this example before you go further. ABAC is 1011101101001. It has 5 0s, so it already has an odd number of 0s. We have to add a check digit to keep this number odd, so the check digit in this case is 1. If the parity property had been Even0 the check digit would have been 0.

Your task is to design a binary finite state automaton (FSA) to accept all strings that represent valid messages (for your particular codes and parity property) and reject all others. This FSA must be DETERMINISTIC, REDUCED and must be in STANDARD FORM.

This project is machine marked. You can submit your attempts as many times as you like and your submission will be marked immediately. You will obtain one of 4 responses:

Your machine does not work. It does not process the string ... correctly. The string that your machine processes incorrectly may assist you in understanding why your machine does not work. (0 marks)

Your machine processes all strings correctly, but is not in reduced form. This means that your machine accepts precisely those messages that are valid, but has states which are equivalent. (5 marks)

Your machine processes all strings correctly. It is reduced but is not in standard form. This means that your machine accepts precisely those messages that are valid, has the right number of states, but they are not named in the correct order for standard form. (6 marks)

Your machine processes all strings correctly, and is in reduced standard form. Your machine is completely correct. (8 marks)

Let L denote the regular language (A + B + C)*(0 + 1), and let M denote the collection of all strings that satisfy the parity property. We then need to design a finite state automaton that accepts precisely all the strings of L ? M. However, we cannot handle the intersection directly.

The idea is to use De Morgan's law, that L ? M = L + M, where we have used + to denote union. We can then use the following simple idea:

Suppose that A is a deterministic finite state automaton that accepts precisely all the strings of a regular language L. Then to obtain a deterministic finite state automaton that accepts precisely all the strings of a regular language L, all we need to do is to change A in the following way make all accepting states non-accepting and all non-accepting states accepting.

Note, however, that this idea only works for deterministic finite state automata, and make sure not to leave the dumping (black hole) state out if it is needed.

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