FP Algebra-Driven Design

Algebraic thinking for API design. Discover the right API before implementing by specifying rules (equations) that operations must satisfy.

Cross-references: fp-principles | fp-domain-modeling | fp-usable-design

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1. Why Algebraic Thinking

[STARTER]

Code is the wrong abstraction level for design. Starting with data structures inherits unnecessary constraints.

• Specify rules first, implement second. Implementation is a solution to a system of equations.
• Rules generate tests automatically. Every rule is directly a property test generating thousands of cases.
• Rules reveal missing features. Analysis often exposes operations you need but haven't designed.
• Rules catch contradictions early. Contradiction during design costs minutes; in production, days.

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2. The Design Process

[STARTER]

1. Start with scope, not implementation. Don't decide data structures upfront.
2. Define observations first. How do users extract information? Observations define equality: two values equal if no observation distinguishes them. Gives enormous implementation freedom.
3. Add operations incrementally. For each new operation, immediately write rules connecting it to existing ones. This web of rules IS the design.
4. Let messy rules signal problems. Complex rules mean coarse building blocks. Decompose until each rule is nearly trivial.
5. Generalize aggressively. Remove unnecessary type constraints. If most operations don't inspect contained values, parameterize over them.

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3. Common Algebraic Structures

[STARTER] -> [ADVANCED]

Recurring patterns in software. Recognizing them unlocks known rules and capabilities.

[STARTER] Combinable Values (Semigroup)

What: Type with one merge operation where grouping doesn't matter. Rule: (a merge b) merge c = a merge (b merge c) (associativity) When: Combining things where parenthesization shouldn't matter. Examples: String concatenation | config merging | min/max.

[STARTER] Combinable Values with Default (Monoid)

What: Combinable Value with a default element inert under combination. Rules: Associativity + default merge x = x and x merge default = x When: Safe defaults | fold operations | "nothing happened yet" values. Examples: (+, 0) | (*, 1) | (concat, []) | (and, true). Design signal: If you find an associative operation, look for a default element. Finding one enables fold/reduce over collections.

[INTERMEDIATE] Merge-and-Forget Values (Semilattice)

What: Combinable Value where merging is also order-independent and idempotent. When: Conflict resolution | eventually-consistent systems | CRDTs. Example: Status tracker with seen < failed < completed uses max as merge.

[INTERMEDIATE] Structure-Preserving Transformations (Functor)

What: Container type where you can transform contents without changing structure. Preserves identity and composition. When: Operations that work on data shape rather than values inside. Design signal: If most operations are agnostic to contained type, you likely have this.

[ADVANCED] Combinable Containers (Applicative)

What: Container where you can combine contents element-wise and fill with uniform values. When: Combining containers holding different content types.

[ADVANCED] Reversible Operations (Group)

What: Combinable Value with Default where every element has an inverse that cancels it. When: Undo operations | spatial transformations. Example: Clockwise/counter-clockwise rotation are inverses; horizontal flip is its own inverse.

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4. Properties of a Well-Designed API

[INTERMEDIATE]

Three categories, eight properties:

Clarity (communicates well):

• Compositional: Complex values built from simpler ones
• Task-relevant: Operations provide high-level functionality useful to actual users
• Interrelated: Rules connect every operation to others

Economy (no waste):

• Parsimonious: Each operation does one thing well
• Orthogonal: No half-baked concept shared across multiple operations
• Generalized: Unnecessary type constraints removed

Safety (prevents mistakes):

• Closed: Every syntactically valid construction is semantically valid
• Complete: Maximum algebraic structure discovered for every concept

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5. Refactoring with Rules

[INTERMEDIATE]

Decompose Large Operations

Complex rules mean coarse building blocks. Split operations with many parameters into smaller, orthogonal ones. If a rule ignores some parameters, those should be separate operations.

Find Contradictions

Algebraic manipulation reveals contradictions before code is written. Requiring order-independence while returning an ordered list is a contradiction. Fix: use unordered collection.

Unify Observations

When multiple observations share traversal logic, find one observation from which all others derive. Simplifies entire rule set.

Use Symmetry to Discover Missing Features

When you have a "both" operation (parallel composition), look for its symmetric "either" counterpart. Symmetry discovers functionality not explicitly requested but inevitably needed.

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6. Three-Phase Testing Strategy

[INTERMEDIATE]

1. Build naive implementation: Implement each operation as direct data constructor. Obviously correct because it mirrors the spec exactly.
2. Discover all rules: Feed naive implementation to a tool that enumerates well-typed expressions and finds observational equalities. Discovers specified AND emergent rules.
3. Freeze as regression tests: Convert rules into executable property tests protecting any future optimized implementation.

Key insight: Two values are equal if no observation distinguishes them. Testing through observations allows completely different implementations to pass the same suite.

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7. Decision Tree: When Is Algebraic Thinking Worth It?

Is your domain about COMBINING things?
  YES --> Algebraic thinking helps significantly
    Do combinations have rules (order irrelevant, defaults, inverses)?
      YES --> Known algebraic structures; use their rules directly
      NO --> Rules still help, but structures are not standard
  NO --> Is your domain about TRANSFORMING things?
    YES --> Look for Structure-Preserving Transformation patterns
    NO --> Is your API surface small and well-understood?
      YES --> Algebraic thinking adds overhead; use conventional design
      NO --> Rules can still clarify, even without standard structures

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8. Combining with Other Patterns

Rules + Property-Based Testing: Rules ARE property tests. Algebraic constructors become PBT generators. Example: empty merge x = x becomes forAll(x -> assertEquals(empty.merge(x), x)).

Rules + Domain Modeling (see fp-domain-modeling): Domain wrappers with smart constructors are algebraic rules. State machine transitions are rules about valid sequences. Example: items(addItem(cart, item)) contains item -- directly a property test.

Rules + Usable Design (see fp-usable-design): Simple algebraic rules map to simple, searchable, nameable operations -- improving navigability and learnability. Example: decomposing processOrder into validate, price, confirm gives three nameable functions instead of one opaque one.

Decomposition + Feature Organization: When algebraic decomposition splits a monolithic operation into orthogonal pieces, organize by feature domain. Example: splitting configureWidget(size, color, border) into resize, recolor, restyle lets each live in its own feature module.

Full Cycle: Rules to Tests

-- 1. Define operations and observations
CartOps = { empty, addItem, removeItem, itemCount, totalPrice }

-- 2. Write rules
rule: itemCount(empty) == 0
rule: itemCount(addItem(cart, item)) == itemCount(cart) + 1
rule: removeItem(addItem(empty, x), x) == empty
rule: totalPrice(addItem(cart, item)) == totalPrice(cart) + item.price

-- 3. Rules become property tests
forAll(item -> assertEquals(itemCount(addItem(empty, item)), 1))
forAll(cart, item -> assertEquals(totalPrice(addItem(cart, item)), totalPrice(cart) + item.price))

-- 4. Implement naive version, run tests, then optimize

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9. Design Heuristics

1. Don't start with an implementation. Rules first, code second.
2. Simple rules indicate good decomposition. Complex rules mean coarse building blocks.
3. Every new operation needs rules connecting it to existing ones. Isolated operations indicate missing relationships.
4. Look for algebraic structures. Associative? Look for default. Default found? Check for commutativity or inverses. Each upgrade brings new rules.
5. Generalize by removing type constraints. If rules don't mention a specific type, parameterize.
6. Build naive first, optimize second. Generate tests from naive implementation, then build optimized one against those tests.
7. Invariants belong in types, not business logic. When rules require commutativity but data structure is ordered, change the data structure.