Neuro-Symbolic AI Framework

DeepPSL

Differentiable Probabilistic Soft Logic bridging Perception and Reasoning.

Scenarios

First-Order Logic Rules

$\forall x, \text{Bird}(x) \rightarrow \text{Wings}(x)$

$\forall x, \text{Mammal}(x) \rightarrow \text{Fur}(x)$

$\forall x, \text{Fish}(x) \rightarrow \text{Aquatic}(x)$

1. Neural Perception

Soft truth values $\mathbf{p} \in [0, 1]^k$

$\mathbf{p} = \sigma(\text{NN}_\theta(\mathbf{x}))$

2. PSL Reasoning

Inferred atoms $\mathbf{y} = \arg\min E(\mathbf{y})$

$\phi(\mathbf{y}, \mathbf{p}) = \max(0, \ell(\mathbf{y}, \mathbf{p}))^2$

Mathematical Foundation

Mapping symbolic rules to Hinge-Loss Markov Random Fields (HL-MRFs).

1 Neural Grounding

The input features $\mathbf{x}$ are mapped to a set of observed atoms $\mathbf{p}$ via a parameterized neural network $f_\theta$.

\mathbf{p} = \sigma(\text{NN}_\theta(\mathbf{x}))

Where $\theta$ are learned weights, typically optimized end-to-end through the logic solver.

2 Lukasiewicz Relaxation

Logical rules are relaxed into continuous differentiable constraints. An implication $A \rightarrow B$ is represented as the linear inequality:

\ell_i(\mathbf{y}, \mathbf{p}) = y_{\text{head}} - p_{\text{body}} + 1 \ge 1

The violation $\phi_i$ is then the squared distance from the satisfied manifold.

Global MAP Inference

DeepPSL finds the Maximum A Posteriori (MAP) state of the joint distribution by solving a constrained Quadratic Program.

$$ \mathbf{y}^* = \arg\min_{\mathbf{y} \in [0,1]^n} \sum_{i=1}^m w_i \max(0, A_y^{(i)} \mathbf{y} + A_p^{(i)} \mathbf{p} + b^{(i)})^2 + \frac{\lambda}{2} \|\mathbf{y}\|^2 $$

Weighted Violations

Linear Grounding

Regularization