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Antigravity Agent
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id: wiki-2026-0508-probability-and-logic-fusion
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title: Probability and Logic Fusion
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category: 10_Wiki/Topics
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status: verified
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canonical_id: self
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aliases: [Probabilistic Logic, StaR-AI, Statistical Relational Learning, Neuro-Symbolic AI]
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duplicate_of: none
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source_trust_level: A
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confidence_score: 0.85
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verification_status: applied
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tags: [neuro-symbolic, probabilistic-programming, knowledge-representation, reasoning]
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raw_sources: []
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last_reinforced: 2026-05-10
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github_commit: pending
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tech_stack:
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language: Python
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framework: Pyro / PyMC / DeepProbLog / Scallop
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---
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# Probability and Logic Fusion
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## 매 한 줄
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> **"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.
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## 매 핵심
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### 매 problem statement
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- **Logic alone**: brittle to noise, uncertainty, exceptions.
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- **Probability alone**: no compositional / relational structure.
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- **Neural alone**: opaque, no symbolic guarantees.
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- **Goal**: compositional + uncertain + learnable.
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### 매 historical landmarks
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- **Bayesian networks** (Pearl 1988) — DAG of conditional dists.
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- **PRMs** (Friedman et al 1999) — BNs over relational schemas.
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- **Markov Logic Networks** (Richardson & Domingos 2006) — FOL formulas with weights.
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- **ProbLog** (De Raedt et al 2007) — probabilistic Prolog.
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- **PSL** (Bach et al 2017) — soft logic with hinge-loss inference.
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- **DeepProbLog** (Manhaeve et al 2018) — neural predicates inside ProbLog.
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- **Scallop** (Li et al 2023) — differentiable Datalog for ML.
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- **Logical Neural Networks** (Riegel et al 2020 IBM).
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### 매 representations
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- **MLN**: weighted FOL formulas → ground Markov network.
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- P(world) ∝ exp(Σ w_i × #true_groundings(F_i)).
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- **ProbLog**: Prolog clauses with probabilities `0.7::burglary.`
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- **PSL**: soft truth values in [0,1], conjunction = Lukasiewicz t-norm.
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- **DeepProbLog**: `nn(mnist_net, [X], Y, [0..9]) :: digit(X, Y).`
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### 매 modern (2024-2026) directions
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- **LLM + verifier** (Lean, Coq, Z3): generate → check → repair. AlphaProof, AlphaGeometry style.
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- **Differentiable logic**: gradients through soft-logic for end-to-end training.
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- **Neuro-symbolic agents**: LLM generates programs, symbolic engine executes.
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## 💻 패턴
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### Markov Logic Network (PRACMLN-style)
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```python
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# Formulas with weights
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formulas = [
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(1.5, "Smokes(x) => Cancer(x)"),
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(1.1, "Friends(x,y) ^ Smokes(x) => Smokes(y)"),
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]
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# P(world) ∝ exp(Σ w * count_true_groundings)
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# Inference: MC-SAT or Gibbs sampling over ground atoms.
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```
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### ProbLog example
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```prolog
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0.1 :: burglary.
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0.2 :: earthquake.
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alarm :- burglary.
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alarm :- earthquake.
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0.7 :: john_calls :- alarm.
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query(burglary).
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evidence(john_calls, true).
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```
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### DeepProbLog (neural predicate)
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```python
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# Recognize MNIST digits and add them
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network = MNIST_Net()
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nn(mnist_net, [X], Y, [0,1,2,3,4,5,6,7,8,9]) :: digit(X, Y).
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addition(X, Y, Z) :- digit(X, A), digit(Y, B), Z is A + B.
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# Train: end-to-end gradient flows through neural digit predicate
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# from supervision on (image1, image2, sum_label).
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```
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### Pyro probabilistic program (Bayesian + structure)
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```python
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import pyro, pyro.distributions as dist, torch
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def model(data):
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# Latent disease probability
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p_disease = pyro.sample("p_disease", dist.Beta(1., 9.))
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for i, (test, outcome) in enumerate(data):
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d = pyro.sample(f"d_{i}", dist.Bernoulli(p_disease))
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# logical rule: P(test+ | disease) = 0.95, P(test+ | not disease) = 0.1
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p_test = 0.95 * d + 0.1 * (1 - d)
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pyro.sample(f"t_{i}", dist.Bernoulli(p_test), obs=test)
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```
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### LLM + Z3 verifier (2024-2026 pattern)
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```python
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from z3 import Solver, Int, And, sat
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def llm_solve_with_check(problem):
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code = claude.complete(f"Translate to Z3 Python: {problem}")
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s = Solver()
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exec(code) # populates s
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if s.check() == sat:
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return s.model()
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else:
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return claude.complete(f"Z3 returned UNSAT. Repair: {code}")
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```
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### PSL soft-logic rule
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```python
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# Lukasiewicz t-norm: A ^ B = max(0, A + B - 1); A => B = min(1, 1 - A + B).
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# Rule: similar(p,q) ^ likes(p, x) => likes(q, x) [weight 5]
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# Inference: minimize Σ w_i * max(0, body - head) over continuous truth values.
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```
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### Scallop differentiable Datalog
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```python
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import scallopy
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ctx = scallopy.ScallopContext(provenance="difftopkproofs")
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ctx.add_relation("digit", (int, float), input_mapping=[(0,), (1,), ...])
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ctx.add_rule("sum(a + b) = digit(_, a), digit(_, b)")
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# Plug into PyTorch; gradients flow through proof structure.
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```
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## 매 결정 기준
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| 상황 | Approach |
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|---|---|
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| Discrete random variables, known structure | Bayesian network (pgmpy) |
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| First-order rules + data | MLN / ProbLog |
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| Soft constraints, large scale | PSL |
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| Neural perception + symbolic reasoning | DeepProbLog / Scallop |
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| LLM reasoning correctness | LLM + Z3/Lean verifier |
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| Complex generative model | Pyro / PyMC |
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**기본값**: For neuro-symbolic 2026 — Scallop 또는 LLM+verifier; for pure SRL, ProbLog.
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## 🔗 Graph
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- 부모: [[Knowledge-Representation]] · [[Logic]]
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- 변형: [[Bayesian-Network]]
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- 응용: [[Neural-Symbolic-Integration|Neuro-Symbolic-AI]] · [[Knowledge-Graphs]]
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## 🤖 LLM 활용
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**언제**: domain with both structured rules and uncertainty, verifiable LLM reasoning, knowledge-graph completion w/ noise.
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**언제 X**: pure pattern recognition (use NN), purely deterministic logic (use Prolog/Datalog).
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## ❌ 안티패턴
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- **MLN at scale**: grounding explodes; use lifted inference or PSL.
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- **Probabilities as confidence scores**: must reflect actual frequencies / coherent priors.
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- **Mixing neural and symbolic without gradient story**: end-to-end requires differentiable bridge.
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- **Ignoring computational cost**: many SRL inferences are #P-hard.
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## 🧪 검증 / 중복
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- Verified (Pearl 1988, Richardson & Domingos 2006 ML, De Raedt et al 2007 IJCAI, DeepProbLog NeurIPS 2018, Scallop ICLR 2023).
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- 신뢰도 A.
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## 🕓 Changelog
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| 날짜 | 변경 |
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| 2026-05-08 | Phase 1 |
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| 2026-05-10 | Manual cleanup — full SRL → neuro-symbolic timeline + 2026 LLM+verifier |
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