Turing Machine Encoding of 3^{3^{2^0}}

This page outlines how to encode the nested exponentiation expression 3³²⁰ into a Turing Machine using unary representation and symbol rewriting.

🧮 Step 1: Understand the Expression

• 2^0 = 1
• 3^1 = 3
• 3^3 = 27

Final result: 27

🧵 Step 2: Tape Encoding in Unary

We represent numbers in unary using | strokes:

• 1 → |
• 3 → |||
• 27 → |||||||||||||||||||||||||||

Structured input tape:

^ (||| , ^ (||| , ^ (|| , 0)))

⚙️ Step 3: Turing Machine Components

Alphabet:

{ |, 0, ^, (, ), ,, _, # }

States:

{
  q_start, q_scan, q_mark_base, q_mark_exp,
  q_exp_loop, q_mult_loop, q_copy_base,
  q_cleanup, q_accept
}

Transition Function Overview:

1. Scan for innermost ^(a,b)
2. Evaluate exponentiation via repeated multiplication
3. Replace subexpression with result in unary
4. Repeat until no ^ remains

🧪 Step 4: Subroutine Sketch

Exponentiation:

• If b = 0, result is |
• Otherwise, repeat multiplication b times

Multiplication:

• Copy a, b times
• Append each copy to result zone

🧾 Step 5: Sample Transitions

Exponentiation for a^0 → 1:

State	Read	Write	Move	Next State
q_exp_loop	0	|	L	q_cleanup
q_exp_loop	|	|	R	q_copy_base

Multiplication Loop:

State	Read	Write	Move	Next State
q_mult_loop	|	#	R	q_copy_base
q_copy_base	|	|	R	q_mult_loop

🧠 Final Output

After all rewrites, the tape contains:

|||||||||||||||||||||||||||

That’s 27 strokes—our final result.

✨ Want to Go Deeper?

I can walk you through building the full transition table for one of the subroutines, simulate the tape step-by-step, or even visualize the state transitions. Just say the word!