This is Part 5 of the Turing Machines series. Here we walk through building Turing machines step by step, from simple examples to more complex problems. We'll use practical examples you can try on turingmachine.io.

Prerequisites

Before building your own machines, you should understand:

• Basic concepts: States, transitions, tape, head
• State diagrams: How to read and understand visual representations
• turingmachine.io: How to use the tool

If you need a refresher, review the earlier parts of this series.

Design Process

When building a Advanced Turing Machine Concepts, follow this process:

1. Understand the problem: What should the machine do?
2. Identify the approach: What algorithm will solve it?
3. Design states: What states are needed?
4. Define transitions: What happens in each state?
5. Test and debug: Try various inputs and fix issues
6. Refine: Optimize and improve

Example 1: Binary Complement

Let's start simple: create a machine that flips all bits in a binary number (0→1, 1→0).

Step 1: Understand the Problem

• Input: Binary string (e.g., 1011)
• Output: Complemented binary string (e.g., 0100)
• Operation: Flip each bit

Step 2: Design the Algorithm

1. Start at the leftmost symbol
2. For each symbol:
   • If 0, write 1
   • If 1, write 0
3. Move right
4. Repeat until blank
5. Halt

Step 3: Design States

We need:

• scan: State that scans and flips bits
• done: Halting state

Step 4: Define Transitions

scan:
  0: {write: 1, R}      # Flip 0 to 1, move right, stay in scan
  1: {write: 0, R}      # Flip 1 to 0, move right, stay in scan
  ' ': done             # Blank means done
done:                   # Halt

Step 5: Complete Machine

# Binary complement machine
input: '1011'
blank: ' '
start state: scan
table:
  scan:
    0: {write: 1, R}
    1: {write: 0, R}
    ' ': done
  done:

Try it: Input 1011 should become 0100

Example 2: Counting Symbols

Create a machine that counts how many 1s are in a binary string and writes the count in unary (as that many 1s).

Step 1: Understand the Problem

• Input: Binary string (e.g., 1011)
• Output: Unary count (e.g., 111 for three 1s)
• Operation: Count 1s, write count in unary

Step 2: Design the Algorithm

This is more complex. We need:

1. Scan through input
2. For each 1 found, mark it and add a 1 to a counter region
3. Continue until all input processed
4. Result is in the counter region

Step 3: Design States

• scan: Scanning for 1s
• mark: Marking a found 1
• count: Adding to counter
• return: Returning to scan position
• done: Halting

Step 4: Define Transitions

# Simplified approach: mark 1s as we find them, count them
scan:
  1: {write: X, R: count}  # Mark this 1, go count it
  0: R                      # Skip 0s
  X: R                      # Skip already marked
  ' ': {L: done}            # Done scanning
count:
  ' ': {write: 1, L: return}  # Add 1 to counter, return
  1: {write: 1, R}            # Extend counter if needed
return:
  X: {write: X, L}           # Return past marked symbols
  0: L
  1: L
  ' ': {R: scan}             # Back to start, continue scanning
done:
  # Clean up and halt

This is a simplified version. A complete implementation would need more states for proper counting.

Example 3: Palindrome Checker

Create a machine that checks if a string is a palindrome (reads same forwards and backwards).

Step 1: Understand the Problem

• Input: String (e.g., aba or abc)
• Output: Accept if palindrome, reject if not
• Operation: Compare first and last symbols, work inward

Step 2: Design the Algorithm

1. Mark the first symbol
2. Find and mark the last symbol
3. Compare them
4. If match, move inward and repeat
5. If all match, accept; if mismatch, reject

Step 3: Design States

• start: Beginning state
• markFirst: Mark first unmarked symbol
• findLast: Find last unmarked symbol
• compare: Compare marked symbols
• accept: Accepting state
• reject: Rejecting state

This is a complex machine requiring careful state design. A complete implementation would need many more states for edge cases (empty string, single symbol, etc.).

Example 4: Binary Addition (Simplified)

Create a machine that adds two binary numbers. This is more advanced but demonstrates important concepts.

Approach

1. Represent numbers separated by a delimiter (e.g., +)
2. Process from right to left (like manual addition)
3. Handle carries
4. Produce sum

Key States

• scanRight: Move to rightmost digits
• add: Add current digits
• carry: Handle carry
• done: Complete

This is a substantial machine. Start with simpler examples before attempting this.

Common Patterns

Pattern 1: Scanning

Many machines need to scan through input:

scan:
  symbol: R          # Move right, stay in scan
  ' ': nextState     # End of input, move to next phase

Pattern 2: Marking

Mark processed symbols:

process:
  X: {write: X, R}   # Skip already marked
  symbol: {write: X, R: nextState}  # Mark and continue

Pattern 3: Returning

Return to a previous position:

return:
  X: L               # Move left past markers
  symbol: L          # Move left
  ' ': {R: nextState}  # Found start, continue

Pattern 4: Conditional Logic

Different actions based on symbol:

check:
  0: {write: 0, R: stateA}   # If 0, go to stateA
  1: {write: 1, R: stateB}   # If 1, go to stateB
  ' ': done                  # If blank, halt

Debugging Tips

Common Issues

1. Missing transitions: Machine halts unexpectedly

   • Solution: Add transitions for all symbols in all states

2. Infinite loops: Machine never halts

   • Solution: Ensure progress toward halting condition

3. Wrong output: Machine produces incorrect result

   • Solution: Trace execution step by step, verify logic

4. Edge cases: Fails on empty input or single symbol

   • Solution: Test with various inputs, handle edge cases

Debugging Process

1. Use Step mode: Execute one step at a time
2. Check state: Verify current state is expected
3. Check tape: Verify tape contents are correct
4. Trace transitions: Follow which transitions fire
5. Test inputs: Try simple cases first, then complex

Practice Problems

Try building machines for these problems (in order of difficulty):

Beginner

1. Double a binary number: Multiply by 2 (shift left)
2. Check if string starts with specific symbol
3. Remove all 0s from binary string

Intermediate

4. Binary subtraction (simplified)
5. Find maximum of two numbers
6. Check if number is even (count 1s mod 2)

Advanced

7. Binary multiplication
8. Sort binary numbers
9. Recognize context-free languages

Learning Resources

Textbooks

1. Sipser's "Introduction to the Theory of Computation"

   • Chapter 3: Examples of Turing machines
   • Exercises with solutions
   • ISBN: 978-1133187790

2. Hopcroft, Motwani, Ullman - "Introduction to Automata Theory"

   • Detailed construction techniques
   • Many worked examples
   • ISBN: 978-0321455369

Online Resources

1. turingmachine.io Examples

   • Study pre-built machines
   • Modify and experiment
   • Learn from working code

2. JFLAP Examples

   • More complex machines
   • Formal verification
   • Available at: http://www.jflap.org/

Video Resources

1. Construction tutorials: Search YouTube for "Turing machine construction"
2. Problem-solving walkthroughs: See how experts approach problems
3. Course lectures: MIT OCW, Coursera courses

Next Steps

Now that you can build Turing machines, continue to:

• Part 6: Advanced Concepts and Applications - Explore computability, complexity, and related topics
• Practice: Build more machines, solve harder problems
• Share: Create machines and share via GitHub gists

Or return to the series index to see all articles.