2nd/10_Wiki/Topic_Programming/AI_and_ML/Symbols.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-symbols	Symbols	10_Wiki/Topics	verified	self	Symbolic Representation Symbolic AI GOFAI Symbol Manipulation	none	A	0.9	applied	symbolic-ai neuro-symbolic knowledge-representation reasoning		2026-05-10	pending	language framework python scallop

Symbols

매 한 줄

"매 symbol 은 discrete, manipulable token — meaning 의 abstract carrier". 매 Newell·Simon 의 Physical Symbol System Hypothesis 의 origin. 매 2026 의 modern usage: pure symbolic AI 의 retire, neuro-symbolic hybrid (Scallop, DeepProbLog, LLM+Lean) 의 mainstream.

매 핵심

매 Physical Symbol System Hypothesis (Newell & Simon 1976)

• "A physical symbol system has the necessary and sufficient means for general intelligent action."
• Symbol: physical pattern referring to entity.
• Expression: composition of symbols.
• Process: creation, modification, reproduction, destruction.

매 symbol vs subsymbol

• Symbol (GOFAI): discrete, interpretable, composable. e.g., Prolog clauses, Knowledge Graph triples.
• Subsymbol (connectionist): distributed, continuous, learned. e.g., transformer hidden states.
• Bridge: tokenization, embedding-of-symbol, neuro-symbolic.

매 modern symbolic 사용 영역

• Theorem proving: Lean 4, Coq, Isabelle. LLM (DeepSeek-Prover) 의 partner.
• Knowledge Graph: Wikidata, schema.org — RDF triples.
• Constraint solving: Z3 SMT, OR-Tools.
• Program synthesis: Sketch, Rosette.
• Symbolic regression: SymbolicRegression.jl, PySR.

매 neuro-symbolic 2026

• Scallop: differentiable Datalog.
• DeepProbLog: probabilistic logic + NN.
• AlphaProof / AlphaGeometry: LLM proposer + symbolic verifier.
• Tool-using LLM: Wolfram, Lean, Z3 의 call.

매 응용

1. Math/Physics: AlphaProof IMO 2024 silver — LLM + Lean.
2. KG QA: text2cypher / text2sparql with verification.
3. Constraint planning: LLM proposes, Z3 verifies.

💻 패턴

1. SymPy symbolic math

from sympy import symbols, diff, integrate, solve, simplify

x, y = symbols('x y')
expr = x**3 + 2*x**2 - 5*x + 1

derivative = diff(expr, x)
antideriv = integrate(expr, x)
roots = solve(expr, x)
print(simplify(derivative * 2))

2. Z3 constraint solving

from z3 import Int, Solver, And, sat

a, b, c = Int('a'), Int('b'), Int('c')
s = Solver()
s.add(a + b + c == 30, a >= 0, b >= 0, c >= 0,
      And(a*b*c == 1000))
if s.check() == sat:
    m = s.model()
    print(m[a], m[b], m[c])

3. Lean 4 theorem (LLM-suggested)

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

4. RDF / SPARQL knowledge graph

from rdflib import Graph

g = Graph()
g.parse("dbpedia_subset.ttl")

q = """
SELECT ?actor ?film WHERE {
  ?film dbo:starring ?actor .
  ?film dbo:director dbr:Christopher_Nolan .
}
"""
for row in g.query(q):
    print(row.actor, row.film)

5. Scallop neuro-symbolic

import scallopy

ctx = scallopy.ScallopContext()
ctx.add_relation("digit", (int, float))  # (digit, prob from NN)
ctx.add_rule("sum(s) :- digit(a, _), digit(b, _), s == a + b")

# NN provides probabilistic facts; Scallop reasons differentiably
ctx.add_facts("digit", [(3, 0.9), (5, 0.85)])
result = ctx.run().relation("sum")

6. Tool-using LLM (Wolfram-as-tool)

tools = [{
  "name": "wolfram_alpha",
  "description": "Symbolic math via Wolfram Alpha",
  "input_schema": {"type": "object", "properties": {
    "query": {"type": "string"}}, "required": ["query"]}
}]

# Claude calls wolfram_alpha("integrate(x^2 sin(x), x)")
# returns symbolic answer; Claude composes natural language explanation.

7. Symbolic regression (PySR)

from pysr import PySRRegressor

model = PySRRegressor(
    niterations=40,
    binary_operators=["+", "*", "-", "/"],
    unary_operators=["cos", "exp", "sin"],
)
model.fit(X, y)
print(model.sympy())  # human-readable formula

매 결정 기준

상황	Approach
Exact math	SymPy / Mathematica
Logical constraints	Z3 / OR-Tools
Theorem proving	Lean 4 + LLM proposer
Structured KB QA	KG + SPARQL + LLM rephrase
Pattern from data	symbolic regression (PySR)

기본값: 매 symbolic-only 의 X. LLM proposer + symbolic verifier hybrid.

🔗 Graph

• 부모: Symbolic-AI vs Connectionism · Knowledge-Representation
• 변형: Neural-Symbolic-Integration · Theorem-Proving · Knowledge Graph
• Adjacent: GOFAI

🤖 LLM 활용

언제: symbolic system 의 natural-language interface, proof step proposal, KG query generation. 언제 X: symbolic verification 그 자체 (LLM 의 hallucinate — Lean/Z3 의 사용).

❌ 안티패턴

• Pure symbolic AI 의 modern attempt: 매 brittleness — perception 의 connectionist 의 필요.
• Hand-crafted ontology 의 over-invest: 매 maintenance hell. KG 의 LLM-bootstrap.
• LLM 의 symbolic answer 의 trust: 매 verify 의 fail. 매 Lean/Z3/SymPy 의 ground.
• Embedding-only retrieval: 매 logical relationship 의 lose — KG triples 의 hybrid.

🧪 검증 / 중복

• Verified (Newell & Simon Turing lecture 1976, Marcus 2020 critique, AlphaProof Nature 2024, Scallop ICLR 2023).
• 신뢰도 A.

🕓 Changelog

날짜	변경
2026-05-08	Phase 1
2026-05-10	Manual cleanup — symbolic AI + modern neuro-symbolic hybrid