# Tensor Logic: Why It Can Solve More Problems Than Tensors Alone

## Overview

**Tensor Logic** is a programming paradigm that integrates tensor operations with logical reasoning. While traditional tensor frameworks (PyTorch, TensorFlow, etc.) primarily focus on numerical computation and differentiable functions, Tensor Logic significantly expands the range of solvable problems by adding **logical structures**, **constraint satisfaction**, and **symbolic reasoning**.

---

## 1. Limitations of Traditional Tensor Operations

### 1.1 Problems with Numerical-Only Approaches

Traditional tensor operations face challenges such as:

```
# Typical deep learning pipeline
x = tensor([...])
y = model(x)  # forward pass
loss = criterion(y, target)
loss.backward()  # backpropagation
```

**Limitations**:
- **Discrete optimization**: Weak at integer solutions and combinatorial problems (TSP, graph coloring, etc.)
- **Constraint expression**: Cannot naturally express logical constraints like "at least one" or "exactly one"
- **Symbolic reasoning**: Cannot directly encode logical relationships like "if A then B"
- **Inverse problems**: Difficult to infer causes from results

### 1.2 Concrete Example: N-Queens Problem

Attempting to solve N-Queens with a pure tensor approach:

```python
# PyTorch-style approach
board = torch.softmax(logits, dim=-1)  # probability distribution per row
loss = -torch.log(board.max())  # ??? Which constraint to optimize?
```

This method struggles to naturally express and optimize constraints like "exactly one queen per row" and "no conflicts on diagonals."

---

## 2. Tensor Logic Extensions

Tensor Logic extends problem-solving capabilities by introducing the following concepts:

### 2.1 Tensorizing Logical Operations

**Representing propositions as tensors**:
```
// Each element represents "whether that proposition is true"
let queen = Tensor::zeros([N, N]); // queen[i,j] = 1.0 if queen at (i,j)
```

**Differentiable approximations of logical operators**:
| Logical Operation | Formula | Tensor Expression |
|------------------|---------|-------------------|
| NOT(A) | 1 - A | `1.0 - A` |
| AND(A,B) | A × B | `A * B` |
| OR(A,B) | A + B - A×B | `A + B - A * B` |
| A → B | 1 - A + A×B | `1.0 - A + A * B` |

This allows logical formulas to be converted into differentiable loss functions.

### 2.2 Converting Constraints to Loss Functions

**"Exactly one per row" constraint**:
```
// Row sum should equal 1
let row_sum = queen.sum(1);           // [N] sum of each row
let row_loss = (row_sum - 1.0).pow(2).sum();
```

**"No conflicts on diagonals" constraint**:
```
// Product of queens on diagonals should be 0
let diag_loss = compute_diagonal_conflicts(queen);
```

**Combined loss**:
```
let total_loss = row_loss + col_loss + diag_loss;
total_loss.backward();
```

### 2.3 Encoding Symbolic Structures

Tensor Logic can naturally encode graph structures and relations:

**Graph adjacency matrix**:
```
let adj = Tensor::zeros([N, N]);  // adj[i,j] = 1 if edge i→j
```

**Relational inference (transitive closure)**:
```
// Reasoning like "friends of friends are also friends"
let friends_of_friends = adj.matmul(adj);
let transitive = (adj + friends_of_friends).clamp(0.0, 1.0);
```

---

## 3. New Problem Classes Solvable with Tensor Logic

### 3.1 Combinatorial Optimization Problems

| Problem | Description | Tensor Logic Approach |
|---------|-------------|----------------------|
| **N-Queens** | Place queens on N×N board | Convert conflict constraints to loss |
| **TSP** | Find shortest tour | Approximate permutation matrix with Softmax |
| **Graph Coloring** | Assign different colors to adjacent vertices | Color matrix with adjacency constraints |
| **SAT** | Boolean satisfiability | Express clauses as products, literals as variables |

### 3.2 Inverse Problems

| Problem | Description | Tensor Logic Approach |
|---------|-------------|----------------------|
| **Inverse Life** | Infer initial state from result state | Make forward pass differentiable for inverse optimization |
| **Inverse Rendering** | Reconstruct scene from image | Propagate gradients through soft rendering |
| **Circuit Learning** | Learn circuit from truth table | Approximate gates with sigmoid |

### 3.3 Structured Reasoning

| Problem | Description | Tensor Logic Approach |
|---------|-------------|----------------------|
| **Knowledge Graph Completion** | Infer missing links | Encode relations as tensors |
| **Program Synthesis** | Synthesize functions from examples | Continuous relaxation of discrete operations |
| **Pattern Matching** | Detect structural patterns | Combine convolutions with logical operations |

---

## 4. The Power of Tensor Logic Through Examples

### 4.1 Example 1: Inverse Game of Life

**Problem**: Find an initial state that evolves to a specific state after 5 steps

**Why traditional approaches struggle**:
- Game of Life is discrete (0/1)
- Chaotic behavior—no analytical inverse function exists
- Combinatorial explosion (2^(N×N) possible initial states)

**Tensor Logic solution**:
```
// 1. Approximate rules differentiably
let neighbors = state.conv2d(kernel, 1, 1) - state;
let is_3 = ((neighbors - 3.0).pow(2) * -1.0).exp();  // Birth at 3 neighbors
let is_2 = ((neighbors - 2.0).pow(2) * -1.0).exp();  // Survival at 2 neighbors
let next_state = is_3 + state * is_2;

// 2. Optimize the inverse problem
let learnable_init = Tensor::randn([N, N], true);
let evolved = forward(learnable_init.sigmoid(), STEPS);
let loss = (evolved - target).pow(2).sum();
loss.backward();  // Gradient to initial state!
```

### 4.2 Example 2: Learning XOR Function

**Problem**: Learn a logic circuit that satisfies the XOR truth table

**Why it's interesting**:
- XOR is not linearly separable (cannot be solved with a single-layer perceptron)
- Digital logic is inherently discrete

**Tensor Logic solution**:
```
// Approximate logic gates with sigmoid
let hidden = (inputs.matmul(W1) + b1).sigmoid();
let output = (hidden.matmul(W2) + b2).sigmoid();

// Minimize error with truth table
let loss = (output - targets).pow(2).sum();
```

### 4.3 Example 3: Differentiable Ray Casting

**Problem**: Reconstruct a 2D scene from a 1D projection

**Why traditional approaches struggle**:
- Ray casting uses discrete intersection tests
- Index access is not differentiable

**Tensor Logic solution**:
```
// Soft sampling instead of hard indexing
let weights = ((grid - sample_pos).pow(2) * -sigma).exp();
let sample = (weights * scene).sum() / (weights.sum() + epsilon);
// → Gradients propagate to the entire scene!
```

---

## 5. Theoretical Foundations

### 5.1 Continuous Relaxation

By relaxing discrete variables x ∈ {0, 1} to continuous variables x̃ ∈ [0, 1]:
- Gradient methods become applicable
- Rounding to discrete solutions is done as post-processing

### 5.2 Structural Loss Functions

To find solutions that satisfy constraints:
```
Loss = Σ(penalty for constraint violations) + λ × objective function
```

### 5.3 Differentiable Simulation

Making physical simulations and dynamic systems differentiable enables:
- Optimization of initial conditions
- Parameter estimation
- Solving control problems

---

## 6. Summary of Tensor Logic Advantages

| Aspect | Traditional Tensors | Tensor Logic |
|--------|--------------------:|-------------:|
| **Problem Representation** | Purely numerical | Logic + numerical |
| **Constraints** | Added as regularization | Native support |
| **Discrete Problems** | Hard to approximate | Solved via continuous relaxation |
| **Inverse Problems** | Require special architectures | Natural via forward pass differentiation |
| **Symbolic Reasoning** | Not supported | Converted to tensor operations |
| **Explainability** | Black box | Readable logical structure |

---

## 7. Conclusion

Tensor Logic combines the power of **differentiable programming** with the expressiveness of **symbolic reasoning** to enable solving problem classes that were difficult for traditional tensor frameworks:

1. **Combinatorial Optimization**: Convert discrete constraints to continuous losses
2. **Inverse Problems**: Make forward simulations differentiable for inverse optimization
3. **Logical Reasoning**: Encode propositional logic as tensor operations
4. **Structure Learning**: Represent relations and patterns as tensors

This enables designing new algorithms that transcend the boundaries of machine learning, optimization, and reasoning.

---

## Reference Links

- [N-Queens Example](../examples/tasks/tensor_logic/n_queens/README.md)
- [TSP Solver Example](../examples/tasks/tensor_logic/tsp/README.md)
- [Inverse Game of Life Example](../examples/tasks/tensor_logic/inverse_life/README.md)
- [Digital Logic Example](../examples/tasks/tensor_logic/digital_logic/README.md)
- [Ray Caster Example](../examples/tasks/tensor_logic/raycast/README.md)
