2nd/10_Wiki/Topic_Programming/Backend/Relational Algebra in Databases.md

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id	title	category	status	canonical_id	aliases	duplicate_of	source_trust_level	confidence_score	verification_status	tags	raw_sources	last_reinforced	github_commit	tech_stack
wiki-2026-0508-relational-algebra-in-databases	Relational Algebra in Databases	10_Wiki/Topics	verified	self	Relational Algebra RA SQL Algebra	none	A	0.95	applied	database theory sql query-optimization		2026-05-10	pending	language framework sql postgres

Relational Algebra in Databases

매 한 줄

"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.

매 핵심

매 6 primitive operators

• σ (Selection): row filter. σ_{age>30}(R) ≡ WHERE age>30.
• π (Projection): column subset. π_{name,age}(R) ≡ SELECT name, age.
• ⋈ (Join): theta/equi/natural. R ⋈_{R.id=S.rid} S.
• ∪ / − / ∩: set ops on union-compatible relations.
• × (Cartesian product): R × S — 매 expensive.
• ρ (Rename): alias.

매 Derived operators

• Outer joins (⟕, ⟖, ⟗): null-padded.
• Division (÷): "all-quantifier". R ÷ S = "tuples in R related to every S".
• Aggregation (γ): _{dept}γ_{avg(salary)}(Emp).

매 응용

1. Query optimizer 매 RA tree 의 rewrite (predicate pushdown, join reordering).
2. View materialization 매 algebraic equivalence.
3. Datalog / Differential dataflow의 incremental engine.

💻 패턴

Selection pushdown

-- Logical:  π_{name}(σ_{age>30}(Emp ⋈ Dept))
-- Physical: σ pushed below ⋈ — 매 smaller intermediate
SELECT name FROM Emp e JOIN Dept d ON e.dept_id=d.id WHERE e.age > 30;
-- 매 optimizer 매 σ_{age>30} 의 Emp 매 push.

Projection pushdown

-- π_{name,salary}(Emp ⋈ Dept) — Dept columns 매 unused
EXPLAIN (FORMAT TEXT)
SELECT e.name, e.salary FROM Emp e JOIN Dept d ON e.dept_id=d.id;
-- Postgres: only e.name,e.salary,e.dept_id materialized.

Join reordering (⋈ associative + commutative)

-- (A ⋈ B) ⋈ C  ≡  A ⋈ (B ⋈ C)  — but cost 매 다름
SET join_collapse_limit = 12;
EXPLAIN ANALYZE
SELECT * FROM small s JOIN big b ON s.k=b.k JOIN huge h ON b.k=h.k;
-- 매 small 매 build side 의 선택.

Division via NOT EXISTS

-- "students who took every required course"
-- Took ÷ Required
SELECT s.id FROM Students s
WHERE NOT EXISTS (
  SELECT 1 FROM Required r
  WHERE NOT EXISTS (
    SELECT 1 FROM Took t
    WHERE t.student_id=s.id AND t.course_id=r.course_id
  )
);

Aggregation (γ)

-- _{dept_id}γ_{count(*),avg(salary)}(Emp)
SELECT dept_id, COUNT(*), AVG(salary)
FROM Emp
GROUP BY dept_id;

Set operations

-- A − B  (set difference)
SELECT id FROM ActiveUsers
EXCEPT
SELECT id FROM BannedUsers;

-- A ∩ B
SELECT id FROM Premium INTERSECT SELECT id FROM Annual;

Equivalence rewriting

-- σ_{p∧q}(R) ≡ σ_p(σ_q(R))   매 split 의 가능
-- σ_p(R ⋈ S) ≡ σ_p(R) ⋈ S    if p references only R
-- π_L(R ⋈ S) ≡ π_L(π_{L∪join}(R) ⋈ π_{L∪join}(S))

매 결정 기준

상황	Operator
Filter rows	σ
Pick columns	π
Combine relations on key	⋈
Union-compatible merge	∪
All-quantifier	÷
Group + aggregate	γ
Preserve unmatched	⟕/⟖/⟗

기본값: σ/π/⋈ 의 covers 매 95% of queries.

🔗 Graph

• 부모: SQL
• Adjacent: Normalization · ACID

🤖 LLM 활용

언제: SQL → RA tree 변환 설명, query rewrite suggestion, 학습용 derivation. 언제 X: production query plan — 매 EXPLAIN ANALYZE 의 사용.

❌ 안티패턴

• Cartesian product 의 무심: missing JOIN condition → N×M rows.
• σ above ⋈: 매 optimizer 매 push 못 하는 case → manual rewrite.
• **SELECT *** in subquery: π pushdown 매 방해.
• Bag vs set 의 혼동: SQL은 bag(multiset). UNION ALL ≠ ∪.

🧪 검증 / 중복

• Verified (Codd 1970; Garcia-Molina Database Systems ch.2.4; Postgres planner docs).
• 신뢰도 A.

🕓 Changelog

날짜	변경
2026-05-08	Phase 1
2026-05-10	Manual cleanup — full content (operators + 7 patterns)