Extending into Computation: Algorithms and Experiments

From Symbolic System to Computation

Step 2 – Specify Your Data Structures

Step 3 – Formalize Your System (Paper Exercise)

Step 3 – Formalize Your System (Paper Exercise)

Use this guided exercise to turn your system into something a computer could store.

Task A – Letters as a list

On paper or in a text editor, write:

1. Your full letter set in order. Example format:

• `LETTERS = [A, B, G, D, ...]`

1. For each letter, list at least one attribute. Example:

• `A: value=1, element=Fire`
• `B: value=2, element=Water`

Task B – Sefirot as nodes

1. List your sefirot:

• `NODES = [Keter, Chokhmah, Binah, ...]`

1. If you drew coordinates in earlier modules, attach them:

• `Keter: (0, 3)`
• `Chokhmah: (-1, 2)`

Task C – Gates and Names

1. Write a sentence defining what counts as a gate in your system. For example:

• "A gate is an unordered pair of distinct letters. AB = BA."
• or "A gate is an ordered pair; AB is different from BA."

1. Write a sentence defining a Name in your system. For example:

• "A Name is a sequence of 3 letters drawn from LETTERS, repetitions allowed."

Reflection prompt

• Is there any part of your system that is still vague or ambiguous when you try to write it this way?
• If yes, note it. Ambiguity is a hint that you may need to refine your definitions before coding.

Step 4 – Algorithm for Generating 231 Gates

Step 4 – Algorithm for Generating 231 Gates

Now translate the classic 231 Gates idea (or your variant) into an algorithm. We will use simple Python-like pseudocode. You do not need to run it; focus on the logic.

Unordered gates (AB = BA)

If you treat AB and BA as the same gate, you want all unordered pairs of distinct letters.

```python

LETTERS = ["A", "B", "G", "D", "H", "V", "Z", "Ch", "T", "Y", "K", "L", "M", "N", "S", "O", "P", "Tz", "Q", "R", "Sh", "Th"]

GATES = []

for i in range(len(LETTERS)):

for j in range(i + 1, len(LETTERS)):

gate = (LETTERS[i], LETTERS[j])

GATES.append(gate)

print(len(GATES)) # For 22 letters, this is 231

```

Key ideas:

• The inner loop starts at `i + 1`, so you never repeat a pair or pair a letter with itself.
• With 22 letters, this gives `22 * 21 / 2 = 231` gates.

Ordered gates (AB != BA)

If you treat AB and BA as different, you want all ordered pairs of distinct letters.

```python

LETTERS = ["A", "B", "G", "D", ...] # your letters

GATES = []

for i in range(len(LETTERS)):

for j in range(len(LETTERS)):

if i == j:

continue # skip identical letters

gate = (LETTERS[i], LETTERS[j])

GATES.append(gate)

print(len(GATES)) # For 22 letters, this is 22 * 21 = 462

```

Your task

• Decide whether your system uses unordered or ordered gates.
• Adapt the pseudocode above to match your definition (e.g., change LETTERS, allow or forbid repeats, etc.).

Step 5 – Tree Traversal and Walks

Step 6 – Design a Search or Enumeration Experiment

Step 6 – Design a Search or Enumeration Experiment

Now you will plan one computational experiment on your system. Choose one of the scenarios below and outline it in your notes.

Option A – Exhaustive search over Names

Goal: Find all Names in your system that satisfy a pattern.

1. Define the pattern. Examples:

• Names whose total numeric value is prime.
• Names that start with a letter from one pillar and end on another.

1. Describe the algorithm in words:

• "Loop over all possible Names of length 3, compute their value, keep only those that are prime."

1. Predict what you might discover:

• Clusters, gaps, or symmetries in where such Names appear.

Option B – Random walk on the Tree

Goal: Explore how a random path behaves.

1. Define the rules:

• Start at Tiferet.
• At each step, randomly choose one connected sefirah.
• Record the sequence of visited nodes.

1. Decide what to measure:

• How often do you visit each sefirah in 1000 steps?
• How long does it take, on average, to reach Malkhut from Keter?

Option C – Gate statistics

Goal: Analyze the distribution of gate attributes.

1. Assign each letter a numeric value or category.
2. For each gate, compute something (sum, difference, category pair).
3. Look for patterns:

• Are some totals much more common than others?
• Do certain letter categories pair more often?

Write down your chosen option, rules, and what you expect to learn. This is your first computational research question about your own system.

Step 7 – Quick Check on Algorithms

Step 7 – Quick Check on Algorithms

Answer this question to check your understanding of gate generation and search.

Step 8 – Review Key Terms

Step 8 – Review Key Terms

Use these flashcards to reinforce the main computational ideas from this module.