2nd/10_Wiki/Topic_Programming/AI_and_ML/Formal Methods.md

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

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-formal-methods	Formal Methods	10_Wiki/Topics	verified	self	formal verification TLA+ model checking theorem proving Coq Lean Alloy refinement type	none	A	0.95	applied	formal-methods verification tla-plus model-checking coq lean software-correctness		2026-05-10	pending	language TLA+ / Lean / Coq / Alloy

Formal Methods

매 한 줄

"매 mathematical 의 의 의 software 의 correctness 의 prove". 매 model checking (TLA+), 매 theorem proving (Coq, Lean), 매 SMT (Z3), 매 refinement type (F*). 매 modern: 매 AWS TLA+ usage, 매 Lean 4 in math, 매 LLM-aided proof.

매 핵심

매 spectrum

• Lightweight: 매 design-by-contract, type system.
• Model checking: TLA+, SPIN, Alloy.
• Theorem proving: Coq, Lean, Isabelle.
• SMT solvers: Z3, CVC5.
• Verified compiler: CompCert, CakeML.

매 modern usage

• AWS S3, DynamoDB use TLA+ for distributed protocols.
• CompCert: 매 verified C compiler.
• seL4: 매 verified microkernel.
• CakeML: 매 verified ML.
• Cardano: 매 Plutus formal.
• Lean 4: 매 mathlib, Anthropic / OpenAI 의 LLM usage.

매 modern AI integration

• LLM-aided proof (PALM, GPT-f).
• AlphaProof / AlphaGeometry (DeepMind).
• Tactic prediction (Coq, Lean).

매 응용

1. Distributed protocol.
2. Cryptography.
3. Compiler / OS.
4. Hardware design.
5. Smart contract.
6. Mathematics.

💻 패턴

TLA+ specification

EXTENDS Naturals, TLC
VARIABLE counter

Init == counter = 0
Next == counter' = counter + 1
Spec == Init /\ [][Next]_counter
Invariant == counter >= 0

TLA+ distributed mutex (Lamport)

EXTENDS Sequences, Naturals
VARIABLES requests, granted

Request(c) == requests' = Append(requests, c) /\ UNCHANGED granted
Grant ==
    /\ Len(requests) > 0
    /\ granted' = Head(requests)
    /\ requests' = Tail(requests)

MutualExclusion == Cardinality({c \in granted}) <= 1

Lean 4 (theorem)

theorem add_comm (a b : Nat) : a + b = b + a := by
  induction a with
  | zero => simp
  | succ n ih => rw [Nat.succ_add, ih, Nat.add_succ]

Coq proof

Theorem plus_comm : forall n m : nat, n + m = m + n.
Proof.
  intros n m.
  induction n as [|n' IHn'].
  - simpl. rewrite <- plus_n_O. reflexivity.
  - simpl. rewrite IHn'. rewrite <- plus_n_Sm. reflexivity.
Qed.

Z3 SMT (Python)

from z3 import *

x, y = Ints('x y')
s = Solver()
s.add(x + y > 5)
s.add(x < 10)
s.add(y < 10)
if s.check() == sat:
    print(s.model())  # 매 e.g., x=6, y=0

Alloy (relational)

sig Person {
    spouse: lone Person
}
fact spouse_symmetric { all p, q : Person | p in q.spouse iff q in p.spouse }
fact no_self_marriage { all p : Person | p not in p.spouse }
run {} for 5

F* (refinement type)

val abs : x:int -> y:int{y >= 0 /\ (y = x \/ y = -x)}
let abs x = if x < 0 then -x else x

Refinement (Liquid Haskell)

{-@ type Pos = {v:Int | v > 0} @-}
{-@ divBy :: Int -> Pos -> Int @-}
divBy :: Int -> Int -> Int
divBy x y = x `div` y

Property-based test (QuickCheck)

import Test.QuickCheck

prop_reverse :: [Int] -> Bool
prop_reverse xs = reverse (reverse xs) == xs

main = quickCheck prop_reverse

LLM-aided proof (Lean Tactic Prediction)

def predict_tactic(state, llm):
    prompt = f"""You are a Lean 4 expert. Given this proof state:
{state}

Suggest the next tactic. Output Lean syntax only."""
    return llm.generate(prompt)

Verified C compiler (CompCert use)

ccomp -O hello.c -o hello
# 매 produces same semantic 의 unverified gcc -O0

Spec → impl (Dafny)

method Max(a: int, b: int) returns (m: int)
  ensures m >= a && m >= b
  ensures m == a || m == b
{
  if a >= b { m := a; } else { m := b; }
}

TLC model checker

tlc -workers auto -fp 64 Spec.tla
# 매 invariant violation 의 trace 의 print

Hoare logic (manual)

{P}  S  {Q}
{x = 0}  x := x + 1  {x = 1}

LLM proof verifier (sketch)

def llm_proof_then_verify(claim, llm):
    proof_attempt = llm.generate(f'Prove in Lean 4: {claim}')
    # 매 Lean check
    result = run_lean(proof_attempt)
    if result.error:
        return llm.refine_proof(proof_attempt, result.error)
    return proof_attempt

Proof assistant frontend (VSCode)

# 매 lean-tactic
extensions:
  - leanprover.lean4
  - jroesch.lean
shortcuts:
  - "Ctrl+Shift+Enter": run tactic

매 결정 기준

상황	Tool
Distributed protocol	TLA+
Crypto algorithm	Coq / EasyCrypt
Compiler	CompCert / CakeML
Math research	Lean 4 / Isabelle
Quick property check	QuickCheck / Hypothesis
Specific bug	Z3 / model check
Smart contract	Plutus / Vyper formal
Critical embedded	Frama-C / SPARK

기본값: 매 distributed = TLA+ + 매 lightweight = property-based + 매 critical = full formal proof + 매 LLM-aided 의 modern.

🔗 Graph

• 변형: Model-Checking · Theorem-Proving · Refinement-Type
• 응용: Coq · Lean
• Adjacent: CompCert · Property-Based-Testing · TypeScript 타입 시스템 (TypeScript Type System)

🤖 LLM 활용

언제: 매 critical / safety. 매 distributed protocol. 매 math research. 언제 X: 매 prototype / app code.

❌ 안티패턴

• Formal everything: 매 cost > benefit.
• Spec ≠ impl: 매 verify wrong.
• Trust model checker for liveness: 매 fairness need.
• Skip property test: 매 cheap form 의 lose.
• LLM proof 의 trust: 매 verify with assistant.

🧪 검증 / 중복

• Verified (Lamport TLA+, Pierce TAPL, AWS papers, Lean docs).
• 신뢰도 A.

🕓 Changelog

날짜	변경
2026-04-20	Auto-reinforced
2026-05-08	Phase 1
2026-05-10	Manual cleanup — methods + 매 TLA / Lean / Coq / Z3 / Dafny / Alloy code