Turing Machine algorithms

Question

I'm trying to determine how a Turing Machine (consisting of only 0's and 1's, no blanks) could recognize a sequence of 8 1's. Every algorithm I've found has a TM searching for an indeterminate number of 1's or 0's, not a specific number.

Essentially, if you have this tape:

1 1 1 1 1 1 1 1 0 0 0 1 0 0 0 1

How can you recognize that the 8 1's represent addition, and you want to add 0 0 0 1 and 0 0 0 1?

Answer 1

I take it that 11111111 is like an opcode and 0001 , 0001 are the operands for that opcode. At least, that's the only interpretation I am seeing.

A TM can look for a fixed, finite number of symbols by using a similar fixed, finite number of states, the sole purpose of each one being to recognize that another of the expected symbols has been seen. For instance, here's a four-tape TM that recognizes addition and does the binary addition:

|----|----|----|----|----||----|-----|-----|-----|-----|----|----|----|----|
| Q  | T1 | T2 | T3 | T4 || Q' | T1' | T2' | T3' | T4' | D1 | D2 | D3 | D4 |
|----|----|----|----|----||----|-----|-----|-----|-----|----|----|----|----|
// read the opcode /////////////////////////////////////////////////////////
| qA | 1  | x  | y  | z  || qB | 1   | x   | y   | z   | R  | S  | S  | S  |
| qB | 1  | x  | y  | z  || qC | 1   | x   | y   | z   | R  | S  | S  | S  |
| qC | 1  | x  | y  | z  || qD | 1   | x   | y   | z   | R  | S  | S  | S  |
| qD | 1  | x  | y  | z  || qE | 1   | x   | y   | z   | R  | S  | S  | S  |
| qE | 1  | x  | y  | z  || qF | 1   | x   | y   | z   | R  | S  | S  | S  |
| qF | 1  | x  | y  | z  || qG | 1   | x   | y   | z   | R  | S  | S  | S  |
| qG | 1  | x  | y  | z  || qH | 1   | x   | y   | z   | R  | S  | S  | S  |
| qH | 1  | x  | y  | z  || qI | 1   | x   | y   | z   | R  | S  | S  | S  |
// read the first addend ///////////////////////////////////////////////////
| qI | w  | x  | y  | z  || qJ | w   | w   | y   | z   | R  | R  | S  | S  |
| qJ | w  | x  | y  | z  || qK | w   | w   | y   | z   | R  | R  | S  | S  |
| qK | w  | x  | y  | z  || qL | w   | w   | y   | z   | R  | R  | S  | S  |
| qL | w  | x  | y  | z  || qM | w   | w   | y   | z   | R  | R  | S  | S  |
// read the second addend //////////////////////////////////////////////////
| qM | w  | x  | y  | z  || qN | w   | x   | w   | z   | R  | S  | R  | S  |
| qN | w  | x  | y  | z  || qO | w   | x   | w   | z   | R  | S  | R  | S  |
| qO | w  | x  | y  | z  || qP | w   | x   | w   | z   | R  | S  | R  | S  |
| qP | w  | x  | y  | z  || qQ | w   | x   | w   | z   | R  | S  | R  | S  |
// prepare the output tape /////////////////////////////////////////////////
| qQ | w  | x  | y  | z  || qR | w   | x   | y   | z   | S  | S  | S  | R  |
| qR | w  | x  | y  | z  || qS | w   | x   | y   | z   | S  | S  | S  | R  |
| qS | w  | x  | y  | z  || qT | w   | x   | y   | z   | S  | S  | S  | R  |
| qT | w  | x  | y  | z  || qU | w   | x   | y   | z   | S  | S  | S  | R  |
// handle addition when no carry ///////////////////////////////////////////
| qU | w  | 0  | 0  | z  || qU | w   | 0   | 0   | 0   | S  | L  | L  | L  |
| qU | w  | 0  | 1  | z  || qU | w   | 0   | 1   | 1   | S  | L  | L  | L  |
| qU | w  | 1  | 0  | z  || qU | w   | 1   | 0   | 1   | S  | L  | L  | L  |
| qU | w  | 1  | 1  | z  || qV | w   | 1   | 1   | 0   | S  | L  | L  | L  |
| qU | w  | B  | B  | B  || hA | w   | B   | B   | B   | S  | R  | R  | R  |
// handle addition when carry //////////////////////////////////////////////
| qV | w  | 0  | 0  | z  || qU | w   | 0   | 0   | 1   | S  | L  | L  | L  |
| qV | w  | 0  | 1  | z  || qV | w   | 0   | 1   | 0   | S  | L  | L  | L  |
| qV | w  | 1  | 0  | z  || qV | w   | 1   | 0   | 0   | S  | L  | L  | L  |
| qV | w  | 1  | 1  | z  || qV | w   | 1   | 1   | 1   | S  | L  | L  | L  |
| qV | w  | B  | B  | B  || hA | w   | B   | B   | B   | S  | R  | R  | R  |
|----|----|----|----|----||----|-----|-----|-----|-----|----|----|----|----|

Legend:

• Q: current state
• T1: current tape symbol, input tape
• T2: current tape symbol, scratch tape #1
• T3: current tape symbol, scratch tape #2
• T4: current tape symbol, output tape (not used)
• Q': state to transition into
• T1': symbol to write to input tape (not used)
• T2': symbol to write to scratch tape #1
• T3': symbol to write to scratch tape #2
• T4': symbol to write to output tape
• D1: direction to move input tape head
• D2: direction to move scratch tape #1 head
• D3: direction to move scratch tape #2 head
• D4: direction to move output tape head

Conventions:

• w, x, y and z are variables and represent either 0 or 1. A transition using all four of these can be thought of as a shorthand notation for writing sixteen (2^4) concrete transitions.
• directions are L=left, S=same, R=right.
• B is a blank symbol; it can be dispensed with if you add more states to assist U and V in the addition.