Neurosymbolic Constraint Solver 2D: Loss Landscape & Logic Graph

Neural Confidence (leaves)

A — is-fraud score 0.50

B — new-device flag 0.50

D — verified-identity 0.50

Learning rate η 0.30

Fuzzy logic family

Show gradient arrows

Symbolic Rules (soft)

R1–
A ∧ B → D (fraud + new device implies verify)

R2–
¬(B ∧ D) (never skip verify on a new device)

R3–
A ∨ D (flag fraud unless verified)

Live Readouts

Loss (unsat²)

0.000

Iterations

Mean satisfaction

Status

idle

How it works

Neurosymbolic AI blends a neural network's soft, continuous predictions with hard symbolic rules by relaxing logic into a differentiable t-norm. Each proposition holds a truth value in [0,1] instead of {0,1}. Two rule families are on offer here:

Product (Reichenbach):
AND(x,y) = x·y
OR(x,y)  = x + y − x·y
x → y    = 1 − x·(1 − y)

Łukasiewicz:
AND(x,y) = max(0, x+y−1)
OR(x,y)  = min(1, x+y)
x → y    = min(1, 1−x+y)

Both are legitimate t-norm/co-norm families used in real logic-tensor-network literature; product is smooth everywhere, Łukasiewicz is piecewise-linear (a genuine kink at the clamp, gradient defined as 0 there). The three rules compile to satisfaction scores sat₁, sat₂, sat₃, and the semantic loss is how far the system is from satisfying all of them:

L = Σᵢ (1 − satᵢ)²

Gradient descent nudges A, B, D directly against ∂L/∂(leaf), computed by the chain rule through the chosen t-norm — exactly the mechanism used in Logic Tensor Networks and semantic-loss training.

Sliders A / B / D — starting neural confidence for each proposition.
Drag on the heatmap — sets A (horizontal) and B (vertical) directly; the background shading is the actual loss landscape L(A,B) at the current D.
η (learning rate) — step size of each gradient-descent update.
Product / Łukasiewicz — swap the whole differentiable-logic family live; the landscape and every gradient recompute under the new formulas.
Gradient arrows — on the graph panel, an arrow at each leaf node shows the direction and (relative) magnitude of −∂L/∂(leaf), i.e. exactly where the next gradient step will push it.

Top panel: a node graph — A, B, D leaves feeding R1/R2/R3 rule nodes, colored red→green by truth value. Bottom panel: the loss landscape over (A,B) with a bright trajectory trail showing where gradient descent has walked.

Logic graph

Loss landscape L(A,B) — drag to set

Truth value → 0 (false) Truth value → 1 (true)