Question

Design a turing machine that subtracts two binary numbers

Answer 1

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Binary subtraction:

Lets build a turing machine that can subtract 2 binary numbers. A subtraction is done on two inputs: in a subtraction x-y we call x the base andy the subtrahend. Before going into the turing machine details, we first have to understand how binary numbers are subtracted. This is done bit by bit,starting at the least significant bit. Lets take the example of 35-15=22:

yyy n n n- borrowed? No or yes

10 0 01 1 -base input

00 1 10 1 - subtrahend input

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nyy y n n- ‘borrowed’ value (yes or no)is

Carried over to the next bit

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0 10 1 10 - answer

Ruleset:

All possible combinations of inputs with the respective actions are listed in the ruleset in table 1.

We’he designed the ruleset such that the first decimal of the state number is used as a control register and the second decimal asadata refister.The first says where we are(reading the ase input, reading the subtrahend input,sweeping back to start,etc). The second holds the actualcomputational data and has everything to do with fig.

Look at the first 2 lines of the ruleset: At positions ‘s’ the state is always 50 or 51: 2 states to see whether we borrowed or not. With 50 wesay: yes, we borrowed, with 51 we say: ‘not borrowed’.

Now look at the last 4 lines: Whenever we read a blank ,we are at the end of our tape.note that the 4 possible leafnodesmatch the 4 possible machine states at this point: the last 4 lines in our rulest (with states 20,21,22,23 respectively for leaf nodes‘0,y’,’1,y’,’0n’,’1n’).Rule set for subtracting turning machine,rr

If in state	And on symbol	Write	And move	Set next state
50515051	SS!S!S		rrll	10115051	Start left(withBorrowed)(withoutBorrowed)Move to Start.
101120212223	0/10/10/10/10/10/1		rrrrrr	101120212223	Move r and do nothing this is nobit in our interest
10113212223	#####	00000	rrrrr	3031414243	Handle # (alert ! next position is the bit of interest)
111213	***		rrr	212223	Handle * and go from first to second input
20212223	****		rrrr	20212223	Left pas before answer do nothing
30303131	0101	####	rµM x.xÏ=iv align="center">r