2nd/10_Wiki/Topics/AI_and_ML/Probability and Logic Fusion.md

---
id: wiki-2026-0508-probability-and-logic-fusion
title: Probability and Logic Fusion
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [Probabilistic Logic, StaR-AI, Statistical Relational Learning, Neuro-Symbolic AI]
duplicate_of: none
source_trust_level: A
confidence_score: 0.85
verification_status: applied
tags: [neuro-symbolic, probabilistic-programming, knowledge-representation, reasoning]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: Python
  framework: Pyro / PyMC / DeepProbLog / Scallop
---

# Probability and Logic Fusion

## 매 한 줄
> **"매 unify symbolic logic (rules, KGs) with probability (uncertainty) — and now neural networks"**. 1990s-2010s 의 Statistical Relational Learning (SRL) 의 lineage: PRMs, MLNs, ProbLog, Bayesian networks + FOL. 2020s 에 neuro-symbolic 으로 reborn (DeepProbLog, Scallop, Logical Neural Networks, Differentiable Theorem Provers). 2026 currently driving verifiable LLM reasoning.

## 매 핵심

### 매 problem statement
- **Logic alone**: brittle to noise, uncertainty, exceptions.
- **Probability alone**: no compositional / relational structure.
- **Neural alone**: opaque, no symbolic guarantees.
- **Goal**: compositional + uncertain + learnable.

### 매 historical landmarks
- **Bayesian networks** (Pearl 1988) — DAG of conditional dists.
- **PRMs** (Friedman et al 1999) — BNs over relational schemas.
- **Markov Logic Networks** (Richardson & Domingos 2006) — FOL formulas with weights.
- **ProbLog** (De Raedt et al 2007) — probabilistic Prolog.
- **PSL** (Bach et al 2017) — soft logic with hinge-loss inference.
- **DeepProbLog** (Manhaeve et al 2018) — neural predicates inside ProbLog.
- **Scallop** (Li et al 2023) — differentiable Datalog for ML.
- **Logical Neural Networks** (Riegel et al 2020 IBM).

### 매 representations
- **MLN**: weighted FOL formulas → ground Markov network.
  - P(world) ∝ exp(Σ w_i × #true_groundings(F_i)).
- **ProbLog**: Prolog clauses with probabilities `0.7::burglary.`
- **PSL**: soft truth values in [0,1], conjunction = Lukasiewicz t-norm.
- **DeepProbLog**: `nn(mnist_net, [X], Y, [0..9]) :: digit(X, Y).`

### 매 modern (2024-2026) directions
- **LLM + verifier** (Lean, Coq, Z3): generate → check → repair. AlphaProof, AlphaGeometry style.
- **Differentiable logic**: gradients through soft-logic for end-to-end training.
- **Neuro-symbolic agents**: LLM generates programs, symbolic engine executes.

## 💻 패턴

### Markov Logic Network (PRACMLN-style)
```python
# Formulas with weights
formulas = [
    (1.5, "Smokes(x) => Cancer(x)"),
    (1.1, "Friends(x,y) ^ Smokes(x) => Smokes(y)"),
]
# P(world) ∝ exp(Σ w * count_true_groundings)
# Inference: MC-SAT or Gibbs sampling over ground atoms.
```

### ProbLog example
```prolog
0.1 :: burglary.
0.2 :: earthquake.
alarm :- burglary.
alarm :- earthquake.
0.7 :: john_calls :- alarm.

query(burglary).
evidence(john_calls, true).
```

### DeepProbLog (neural predicate)
```python
# Recognize MNIST digits and add them
network = MNIST_Net()
nn(mnist_net, [X], Y, [0,1,2,3,4,5,6,7,8,9]) :: digit(X, Y).
addition(X, Y, Z) :- digit(X, A), digit(Y, B), Z is A + B.

# Train: end-to-end gradient flows through neural digit predicate
# from supervision on (image1, image2, sum_label).
```

### Pyro probabilistic program (Bayesian + structure)
```python
import pyro, pyro.distributions as dist, torch

def model(data):
    # Latent disease probability
    p_disease = pyro.sample("p_disease", dist.Beta(1., 9.))
    for i, (test, outcome) in enumerate(data):
        d = pyro.sample(f"d_{i}", dist.Bernoulli(p_disease))
        # logical rule: P(test+ | disease) = 0.95, P(test+ | not disease) = 0.1
        p_test = 0.95 * d + 0.1 * (1 - d)
        pyro.sample(f"t_{i}", dist.Bernoulli(p_test), obs=test)
```

### LLM + Z3 verifier (2024-2026 pattern)
```python
from z3 import Solver, Int, And, sat
def llm_solve_with_check(problem):
    code = claude.complete(f"Translate to Z3 Python: {problem}")
    s = Solver()
    exec(code)  # populates s
    if s.check() == sat:
        return s.model()
    else:
        return claude.complete(f"Z3 returned UNSAT. Repair: {code}")
```

### PSL soft-logic rule
```python
# Lukasiewicz t-norm: A ^ B = max(0, A + B - 1); A => B = min(1, 1 - A + B).
# Rule: similar(p,q) ^ likes(p, x) => likes(q, x)  [weight 5]
# Inference: minimize Σ w_i * max(0, body - head) over continuous truth values.
```

### Scallop differentiable Datalog
```python
import scallopy
ctx = scallopy.ScallopContext(provenance="difftopkproofs")
ctx.add_relation("digit", (int, float), input_mapping=[(0,), (1,), ...])
ctx.add_rule("sum(a + b) = digit(_, a), digit(_, b)")
# Plug into PyTorch; gradients flow through proof structure.
```

## 매 결정 기준
| 상황 | Approach |
|---|---|
| Discrete random variables, known structure | Bayesian network (pgmpy) |
| First-order rules + data | MLN / ProbLog |
| Soft constraints, large scale | PSL |
| Neural perception + symbolic reasoning | DeepProbLog / Scallop |
| LLM reasoning correctness | LLM + Z3/Lean verifier |
| Complex generative model | Pyro / PyMC |

**기본값**: For neuro-symbolic 2026 — Scallop 또는 LLM+verifier; for pure SRL, ProbLog.

## 🔗 Graph
- 부모: [[Knowledge-Representation]] · [[Logic]]
- 변형: [[Bayesian-Network]]
- 응용: [[Neural-Symbolic-Integration|Neuro-Symbolic-AI]] · [[Knowledge-Graphs]]

## 🤖 LLM 활용
**언제**: domain with both structured rules and uncertainty, verifiable LLM reasoning, knowledge-graph completion w/ noise.
**언제 X**: pure pattern recognition (use NN), purely deterministic logic (use Prolog/Datalog).

## ❌ 안티패턴
- **MLN at scale**: grounding explodes; use lifted inference or PSL.
- **Probabilities as confidence scores**: must reflect actual frequencies / coherent priors.
- **Mixing neural and symbolic without gradient story**: end-to-end requires differentiable bridge.
- **Ignoring computational cost**: many SRL inferences are #P-hard.

## 🧪 검증 / 중복
- Verified (Pearl 1988, Richardson & Domingos 2006 ML, De Raedt et al 2007 IJCAI, DeepProbLog NeurIPS 2018, Scallop ICLR 2023).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — full SRL → neuro-symbolic timeline + 2026 LLM+verifier |