Day 2 — Expressions as Algebra: Tokens, Precedence & Postfix (RPN)

Teaching computers to understand expressions!

Note: This article uses technical terms and abbreviations. For definitions, check out the Key Terms & Glossary page.

Introduction

Humans read rules with ease. Computers need structure.

When we write something like score >= 0.85 and (stability > 0.9 or flag == 0), it looks natural to us but computers see a tangle of symbols. To evaluate this reliably, we teach machines three steps: tokenization, operator precedence, and postfix conversion.

TL;DR:

How to teach computers to read and evaluate expressions step by step — by tokenizing text, enforcing operator precedence, and converting rules to postfix (RPN) form for speed, clarity and consistency.

1. Tokenize – split text into meanings (words, numbers, operators).

2. Respect precedence – know which operators bind stronger.

3. Translate to postfix (RPN) – remove parentheses so evaluation is fast and unambiguous.

This gives us rules that are consistent, explainable, and lightning-fast to evaluate.

Where This Appears in Data Science

Rule-based labels & weak supervision – define labels from heuristics in a clear, reproducible way.

Cohort & segment definitions – select groups like "(active and high_quality) or (new_user and opted_in)".

Data-quality & feature checks – guard pipelines with rules like "not null" or value ranges.

Model monitoring & release criteria – e.g. "(precision ≥ X and recall ≥ Y) or (lift ≥ Z)".

Feature-engineering DSLs – describe derived features safely and consistently.

Governance & auditability – align rule text with its computation for traceable results.

Performance & scalability – postfix evaluation runs in O(n) with a tiny stack.

Step 1 — Tokenize the Rule

Let's start with:

(score >= 0.85 and stability > 0.9) or (flag == 0)

Break it into typed pieces:

IDs: score, stability, flag

Numbers: 0.85, 0.9, 0

Operators: >=, >, ==, and, or

Parentheses: (, )

Token stream:

( score >= 0.85 and stability > 0.9 ) or ( flag == 0 )

Step 2 — Operator Precedence

A consistent order keeps rules predictable (high → low):

1. * /
2. + -
3. Comparisons: >= <= > < == !=
4. not (unary)
5. and
6. or

Parentheses always override everything. Among equals, evaluate left to right.

Step 3 — Infix → Postfix (RPN)

Using the shunting-yard algorithm:

• Send values straight to output 🟩
• Push operators on a stack
• Pop higher/equal precedence ops before pushing a new one ↕
• Handle parentheses to group logic ()
• Pop everything left at the end

Our rule becomes:

score 0.85 >= stability 0.9 > and flag 0 == or

Same logic, zero ambiguity. Pure clarity.

How to Evaluate Postfix

Use a simple stack:

1. Read left → right.
2. Push values onto stack.
3. When you see an operator, pop the needed inputs, apply it, and push the result.
4. Return the final value (1 for True, 0 for False).

AND/OR use Boolean Logic on those 1s and 0s.

Worked Example 1 — Full Evaluation

Postfix:

score 0.85 >= stability 0.9 > and flag 0 == or

Worked Example 2 — Why Precedence Matters

Without parentheses:

score >= 0.85 and stability > 0.9 or flag == 0

Standard ladder → still evaluates as:

(score >= 0.85 and stability > 0.9) or (flag == 0)

Wrong precedence (e.g., "or" before "and") flips results entirely!

Always follow the ladder —or use explicit brackets.

Worked Example 3 — With Arithmetic

Infix:

(feature_x / feature_y > 2) and (z_score + bonus >= 3)

Postfix:

feature_x feature_y / 2 > z_score bonus + 3 >= and

Tip: guard against division by zero in feature_y.

Worked Example 4 — Adding Unary not

Infix:

(not drift) and (quality >= 0.95 or coverage >= 0.98)

Postfix:

drift not quality 0.95 >= coverage 0.98 >= or and

What You Gain

Consistency – same rule = same result everywhere.

Simplicity – easy to evaluate and debug.

Speed – O(n) evaluation with tiny memory footprint.

Clarity – no hidden precedence surprises.

Takeaway

Turning rule strings into tokens, honoring a clear precedence order, and evaluating postfix makes your logic solid, predictable, and explainable.

A small engineering habit that scales beautifully from data validation to full-blown rule engines

References

1. Aho, A. V., Lam, M. S., Sethi, R., & Ullman, J. D. (2006). Compilers: Principles, Techniques, and Tools (2nd ed.). Pearson Education.

2. Knuth, D. E. (1997). The Art of Computer Programming, Volume 1: Fundamental Algorithms (3rd ed.). Addison-Wesley.

3. Dijkstra, E. W. (1961). Algol 60 translation: An algol 60 translator for the x1. In Automatic Programming Information Centre (pp. 1-7).

4. Shunting Yard Algorithm. (n.d.). In Wikipedia. Retrieved from https://en.wikipedia.org/wiki/Shunting_yard_algorithm

5. Pratt, V. R. (1973). Top down operator precedence. In Proceedings of the 1st Annual ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages (pp. 41-51).

6. Aho, A. V., & Ullman, J. D. (1972). The Theory of Parsing, Translation, and Compiling (Vol. 1). Prentice-Hall.

7. Grune, D., & Jacobs, C. J. (2008). Parsing Techniques: A Practical Guide (2nd ed.). Springer.

8. Wirth, N. (1996). Compiler Construction. Addison-Wesley.

9. Louden, K. C. (1997). Compiler Construction: Principles and Practice. PWS Publishing.

10. Sedgewick, R., & Wayne, K. (2011). Algorithms (4th ed.). Addison-Wesley Professional.