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id: wiki-2026-0508-relational-algebra-in-databases
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title: Relational Algebra in Databases
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category: 10_Wiki/Topics
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status: verified
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canonical_id: self
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aliases: [Relational Algebra, RA, SQL Algebra]
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duplicate_of: none
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source_trust_level: A
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confidence_score: 0.95
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verification_status: applied
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tags: [database, theory, sql, query-optimization]
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raw_sources: []
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last_reinforced: 2026-05-10
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github_commit: pending
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tech_stack:
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language: sql
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framework: postgres
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---
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# Relational Algebra in Databases
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## 매 한 줄
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> **"매 SQL은 매 algebra 의 syntactic sugar"**. Codd(1970)의 relational algebra는 매 set-based operator(σ, π, ⋈, ∪, −, ×) 매 closed system. 매 modern query optimizer(Postgres, DuckDB, Snowflake)의 plan tree 매 그대로 RA expression.
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## 매 핵심
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### 매 6 primitive operators
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- **σ (Selection)**: row filter. `σ_{age>30}(R)` ≡ `WHERE age>30`.
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- **π (Projection)**: column subset. `π_{name,age}(R)` ≡ `SELECT name, age`.
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- **⋈ (Join)**: theta/equi/natural. `R ⋈_{R.id=S.rid} S`.
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- **∪ / − / ∩**: set ops on union-compatible relations.
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- **× (Cartesian product)**: `R × S` — 매 expensive.
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- **ρ (Rename)**: alias.
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### 매 Derived operators
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- **Outer joins** (⟕, ⟖, ⟗): null-padded.
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- **Division** (÷): "all-quantifier". `R ÷ S` = "tuples in R related to every S".
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- **Aggregation** (γ): `_{dept}γ_{avg(salary)}(Emp)`.
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### 매 응용
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1. Query optimizer 매 RA tree 의 rewrite (predicate pushdown, join reordering).
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2. View materialization 매 algebraic equivalence.
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3. Datalog / Differential dataflow의 incremental engine.
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## 💻 패턴
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### Selection pushdown
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```sql
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-- Logical: π_{name}(σ_{age>30}(Emp ⋈ Dept))
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-- Physical: σ pushed below ⋈ — 매 smaller intermediate
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SELECT name FROM Emp e JOIN Dept d ON e.dept_id=d.id WHERE e.age > 30;
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-- 매 optimizer 매 σ_{age>30} 의 Emp 매 push.
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```
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### Projection pushdown
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```sql
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-- π_{name,salary}(Emp ⋈ Dept) — Dept columns 매 unused
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EXPLAIN (FORMAT TEXT)
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SELECT e.name, e.salary FROM Emp e JOIN Dept d ON e.dept_id=d.id;
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-- Postgres: only e.name,e.salary,e.dept_id materialized.
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```
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### Join reordering (⋈ associative + commutative)
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```sql
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-- (A ⋈ B) ⋈ C ≡ A ⋈ (B ⋈ C) — but cost 매 다름
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SET join_collapse_limit = 12;
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EXPLAIN ANALYZE
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SELECT * FROM small s JOIN big b ON s.k=b.k JOIN huge h ON b.k=h.k;
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-- 매 small 매 build side 의 선택.
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```
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### Division via NOT EXISTS
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```sql
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-- "students who took every required course"
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-- Took ÷ Required
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SELECT s.id FROM Students s
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WHERE NOT EXISTS (
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SELECT 1 FROM Required r
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WHERE NOT EXISTS (
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SELECT 1 FROM Took t
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WHERE t.student_id=s.id AND t.course_id=r.course_id
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)
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);
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```
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### Aggregation (γ)
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```sql
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-- _{dept_id}γ_{count(*),avg(salary)}(Emp)
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SELECT dept_id, COUNT(*), AVG(salary)
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FROM Emp
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GROUP BY dept_id;
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```
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### Set operations
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```sql
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-- A − B (set difference)
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SELECT id FROM ActiveUsers
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EXCEPT
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SELECT id FROM BannedUsers;
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-- A ∩ B
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SELECT id FROM Premium INTERSECT SELECT id FROM Annual;
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```
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### Equivalence rewriting
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```sql
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-- σ_{p∧q}(R) ≡ σ_p(σ_q(R)) 매 split 의 가능
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-- σ_p(R ⋈ S) ≡ σ_p(R) ⋈ S if p references only R
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-- π_L(R ⋈ S) ≡ π_L(π_{L∪join}(R) ⋈ π_{L∪join}(S))
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```
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## 매 결정 기준
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| 상황 | Operator |
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|---|---|
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| Filter rows | σ |
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| Pick columns | π |
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| Combine relations on key | ⋈ |
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| Union-compatible merge | ∪ |
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| All-quantifier | ÷ |
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| Group + aggregate | γ |
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| Preserve unmatched | ⟕/⟖/⟗ |
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**기본값**: σ/π/⋈ 의 covers 매 95% of queries.
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## 🔗 Graph
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- 부모: [[SQL]]
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- Adjacent: [[Normalization]] · [[ACID]]
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## 🤖 LLM 활용
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**언제**: SQL → RA tree 변환 설명, query rewrite suggestion, 학습용 derivation.
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**언제 X**: production query plan — 매 EXPLAIN ANALYZE 의 사용.
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## ❌ 안티패턴
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- **Cartesian product 의 무심**: missing JOIN condition → N×M rows.
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- **σ above ⋈**: 매 optimizer 매 push 못 하는 case → manual rewrite.
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- **SELECT *** in subquery: π pushdown 매 방해.
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- **Bag vs set 의 혼동**: SQL은 bag(multiset). UNION ALL ≠ ∪.
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## 🧪 검증 / 중복
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- Verified (Codd 1970; Garcia-Molina *Database Systems* ch.2.4; Postgres planner docs).
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- 신뢰도 A.
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## 🕓 Changelog
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| 날짜 | 변경 |
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|---|---|
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| 2026-05-08 | Phase 1 |
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| 2026-05-10 | Manual cleanup — full content (operators + 7 patterns) |
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