2nd/10_Wiki/Topics/AI_and_ML/Symbols.md

---
id: wiki-2026-0508-symbols
title: Symbols
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [Symbolic Representation, Symbolic AI, GOFAI, Symbol Manipulation]
duplicate_of: none
source_trust_level: A
confidence_score: 0.9
verification_status: applied
tags: [symbolic-ai, neuro-symbolic, knowledge-representation, reasoning]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: 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
```python
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
```python
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)
```lean
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
```python
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
```python
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)
```python
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)
```python
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|Neuro-Symbolic-AI]] · [[Theorem-Proving]] · [[Knowledge Graph|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 |