# DNA Computing Guide

A comprehensive guide to performing computation directly on DNA-encoded data using VibeDNA.

## Table of Contents

1. [Introduction to DNA Computing](#1-introduction-to-dna-computing)
2. [Nucleotide Values](#2-nucleotide-values)
3. [Logic Gates](#3-logic-gates)
4. [Arithmetic Operations](#4-arithmetic-operations)
5. [Comparison Operations](#5-comparison-operations)
6. [Shift Operations](#6-shift-operations)
7. [Expression Evaluation](#7-expression-evaluation)
8. [Practical Applications](#8-practical-applications)

---

## 1. Introduction to DNA Computing

### What is DNA Computing?

DNA computing is a form of computing that uses DNA molecules and biochemical reactions to perform computational operations. Unlike traditional silicon-based computers that use electrical signals, DNA computing leverages the molecular properties of nucleotides to store and process information.

```
Traditional Computing              DNA Computing
        |                                |
   Bits (0, 1)                    Nucleotides (A, T, C, G)
        |                                |
   Transistors                    DNA Molecules
        |                                |
   Electrical Signals             Molecular Reactions
```

### Why Compute Directly on DNA?

Computing directly on DNA-encoded data offers several compelling advantages:

```
+------------------------------------------------------------------+
|                    ADVANTAGES OF DNA COMPUTING                    |
+------------------------------------------------------------------+
|                                                                   |
|  1. MASSIVE PARALLELISM                                          |
|     - Trillions of DNA strands process simultaneously            |
|     - Natural parallel architecture                               |
|                                                                   |
|  2. STORAGE DENSITY                                              |
|     - 1 gram of DNA = ~215 petabytes of data                     |
|     - Far exceeds magnetic/optical storage                        |
|                                                                   |
|  3. ENERGY EFFICIENCY                                            |
|     - DNA operations consume minimal energy                       |
|     - Self-assembly reduces active computation                    |
|                                                                   |
|  4. LONGEVITY                                                    |
|     - DNA can be preserved for thousands of years                 |
|     - No bit rot or magnetic degradation                          |
|                                                                   |
|  5. ELIMINATE ENCODE/DECODE CYCLES                               |
|     - Compute without converting to binary                        |
|     - Reduced processing overhead                                 |
|                                                                   |
+------------------------------------------------------------------+
```

### VibeDNA's Approach

VibeDNA implements DNA computing through a software abstraction layer that maps computational operations to DNA sequence manipulations:

```
                         VibeDNA Architecture
+-----------------------------------------------------------------------+
|                                                                       |
|   +-----------------+    +-----------------+    +-----------------+   |
|   |   User Code     |    |  DNA Sequences  |    |    Results      |   |
|   |   (Python)      |--->|   (Computed)    |--->|  (DNA/Binary)   |   |
|   +-----------------+    +-----------------+    +-----------------+   |
|                               |                                       |
|                               v                                       |
|   +-----------------------------------------------------------+      |
|   |                    DNAComputeEngine                        |      |
|   |  +------------+  +------------+  +------------+           |      |
|   |  |   Logic    |  | Arithmetic |  | Comparison |           |      |
|   |  |   Gates    |  | Operations |  | Operations |           |      |
|   |  +------------+  +------------+  +------------+           |      |
|   |  +------------+  +------------+  +------------+           |      |
|   |  |   Shift    |  | Expression |  |  Parallel  |           |      |
|   |  | Operations |  | Evaluator  |  |  Strands   |           |      |
|   |  +------------+  +------------+  +------------+           |      |
|   +-----------------------------------------------------------+      |
|                                                                       |
+-----------------------------------------------------------------------+
```

---

## 2. Nucleotide Values

### A=0, T=1, C=2, G=3 Mapping

VibeDNA assigns numeric values to each nucleotide base, creating a natural quaternary (base-4) number system:

```
+-------------+----------------+----------------+----------------+
| Nucleotide  | Decimal Value  | Binary Value   | Description    |
+-------------+----------------+----------------+----------------+
|      A      |       0        |      00        | Adenine        |
|      T      |       1        |      01        | Thymine        |
|      C      |       2        |      10        | Cytosine       |
|      G      |       3        |      11        | Guanine        |
+-------------+----------------+----------------+----------------+
```

This mapping is defined in VibeDNA constants:

```python
from vibedna.utils.constants import NUCLEOTIDE_VALUE, VALUE_NUCLEOTIDE

# Nucleotide to numeric value
NUCLEOTIDE_VALUE = {
    "A": 0,  # Adenine  = 0
    "T": 1,  # Thymine  = 1
    "C": 2,  # Cytosine = 2
    "G": 3,  # Guanine  = 3
}

# Numeric value to nucleotide
VALUE_NUCLEOTIDE = {
    0: "A",
    1: "T",
    2: "C",
    3: "G",
}
```

### Base-4 (Quaternary) Number System

DNA naturally encodes numbers in base-4 (quaternary). Each nucleotide position represents a power of 4:

```
Position Values (right to left):
+-------+-------+-------+-------+-------+-------+
| Pos 5 | Pos 4 | Pos 3 | Pos 2 | Pos 1 | Pos 0 |
+-------+-------+-------+-------+-------+-------+
| 4^5   | 4^4   | 4^3   | 4^2   | 4^1   | 4^0   |
| =1024 | =256  | =64   | =16   | =4    | =1    |
+-------+-------+-------+-------+-------+-------+
```

**Example: Converting DNA to Decimal**

```
DNA Sequence: ATCG

Position:     A        T        C        G
Value:        0        1        2        3
Position:     3        2        1        0
Weight:      4^3      4^2      4^1      4^0
           = 64     = 16      = 4      = 1

Calculation: (0 x 64) + (1 x 16) + (2 x 4) + (3 x 1)
           =    0    +    16    +    8    +    3
           =   27 (decimal)
```

```
More Examples:
+------------------+------------------------------+-----------+
| DNA Sequence     | Calculation                  | Decimal   |
+------------------+------------------------------+-----------+
| AAAA             | 0+0+0+0                      |     0     |
| AAAT             | 0+0+0+1                      |     1     |
| AAAC             | 0+0+0+2                      |     2     |
| AAAG             | 0+0+0+3                      |     3     |
| AATA             | 0+0+4+0                      |     4     |
| AATT             | 0+0+4+1                      |     5     |
| GGGG             | 192+48+12+3                  |   255     |
| TAAT             | 64+0+0+1                     |    65     |
+------------------+------------------------------+-----------+
```

---

## 3. Logic Gates

VibeDNA implements a complete set of logic gates that operate directly on DNA sequences.

### DNA AND Gate (Minimum Operation)

The DNA AND gate returns the nucleotide with the **lower** value at each position:

```
AND Operation: result = min(a, b)

  +---+   +---+
  | A |---|   |
  +---+   |AND|----> min(A, B)
  | B |---|   |
  +---+   +---+
```

**Truth Table:**

```
+-------+-------+--------+
|   A   |   B   |A AND B |
+-------+-------+--------+
| A (0) | A (0) | A (0)  |
| A (0) | T (1) | A (0)  |
| A (0) | C (2) | A (0)  |
| A (0) | G (3) | A (0)  |
| T (1) | A (0) | A (0)  |
| T (1) | T (1) | T (1)  |
| T (1) | C (2) | T (1)  |
| T (1) | G (3) | T (1)  |
| C (2) | A (0) | A (0)  |
| C (2) | T (1) | T (1)  |
| C (2) | C (2) | C (2)  |
| C (2) | G (3) | C (2)  |
| G (3) | A (0) | A (0)  |
| G (3) | T (1) | T (1)  |
| G (3) | C (2) | C (2)  |
| G (3) | G (3) | G (3)  |
+-------+-------+--------+
```

**Code Example:**

```python
from vibedna.compute import DNAComputeEngine, DNALogicGate

engine = DNAComputeEngine()

# DNA AND operation
seq_a = "ATCG"
seq_b = "GCTA"

result = engine.apply_gate(DNALogicGate.AND, seq_a, seq_b)
print(f"{seq_a} AND {seq_b} = {result}")
# Output: ATCG AND GCTA = ACTA

# Step-by-step breakdown:
# Position 0: A(0) AND G(3) = A (min of 0,3 = 0)
# Position 1: T(1) AND C(2) = T (min of 1,2 = 1)
# Position 2: C(2) AND T(1) = T (min of 2,1 = 1)
# Position 3: G(3) AND A(0) = A (min of 3,0 = 0)
# Result: ATTA
```

### DNA OR Gate (Maximum Operation)

The DNA OR gate returns the nucleotide with the **higher** value at each position:

```
OR Operation: result = max(a, b)

  +---+   +---+
  | A |---|   |
  +---+   | OR|----> max(A, B)
  | B |---|   |
  +---+   +---+
```

**Truth Table:**

```
+-------+-------+--------+
|   A   |   B   | A OR B |
+-------+-------+--------+
| A (0) | A (0) | A (0)  |
| A (0) | T (1) | T (1)  |
| A (0) | C (2) | C (2)  |
| A (0) | G (3) | G (3)  |
| T (1) | A (0) | T (1)  |
| T (1) | T (1) | T (1)  |
| T (1) | C (2) | C (2)  |
| T (1) | G (3) | G (3)  |
| C (2) | A (0) | C (2)  |
| C (2) | T (1) | C (2)  |
| C (2) | C (2) | C (2)  |
| C (2) | G (3) | G (3)  |
| G (3) | A (0) | G (3)  |
| G (3) | T (1) | G (3)  |
| G (3) | C (2) | G (3)  |
| G (3) | G (3) | G (3)  |
+-------+-------+--------+
```

**Code Example:**

```python
from vibedna.compute import DNAComputeEngine, DNALogicGate

engine = DNAComputeEngine()

# DNA OR operation
seq_a = "ATCG"
seq_b = "GCTA"

result = engine.apply_gate(DNALogicGate.OR, seq_a, seq_b)
print(f"{seq_a} OR {seq_b} = {result}")
# Output: ATCG OR GCTA = GCCG

# Step-by-step breakdown:
# Position 0: A(0) OR G(3) = G (max of 0,3 = 3)
# Position 1: T(1) OR C(2) = C (max of 1,2 = 2)
# Position 2: C(2) OR T(1) = C (max of 2,1 = 2)
# Position 3: G(3) OR A(0) = G (max of 3,0 = 3)
# Result: GCCG
```

### DNA XOR Gate (Addition Mod 4)

The DNA XOR gate performs addition modulo 4 at each position:

```
XOR Operation: result = (a + b) mod 4

  +---+   +---+
  | A |---|   |
  +---+   |XOR|----> (A + B) mod 4
  | B |---|   |
  +---+   +---+
```

**Truth Table:**

```
+-------+-------+---------+------------------+
|   A   |   B   | A XOR B | Calculation      |
+-------+-------+---------+------------------+
| A (0) | A (0) |  A (0)  | (0+0) mod 4 = 0  |
| A (0) | T (1) |  T (1)  | (0+1) mod 4 = 1  |
| A (0) | C (2) |  C (2)  | (0+2) mod 4 = 2  |
| A (0) | G (3) |  G (3)  | (0+3) mod 4 = 3  |
| T (1) | A (0) |  T (1)  | (1+0) mod 4 = 1  |
| T (1) | T (1) |  C (2)  | (1+1) mod 4 = 2  |
| T (1) | C (2) |  G (3)  | (1+2) mod 4 = 3  |
| T (1) | G (3) |  A (0)  | (1+3) mod 4 = 0  |
| C (2) | A (0) |  C (2)  | (2+0) mod 4 = 2  |
| C (2) | T (1) |  G (3)  | (2+1) mod 4 = 3  |
| C (2) | C (2) |  A (0)  | (2+2) mod 4 = 0  |
| C (2) | G (3) |  T (1)  | (2+3) mod 4 = 1  |
| G (3) | A (0) |  G (3)  | (3+0) mod 4 = 3  |
| G (3) | T (1) |  A (0)  | (3+1) mod 4 = 0  |
| G (3) | C (2) |  T (1)  | (3+2) mod 4 = 1  |
| G (3) | G (3) |  C (2)  | (3+3) mod 4 = 2  |
+-------+-------+---------+------------------+
```

**Code Example:**

```python
from vibedna.compute import DNAComputeEngine, DNALogicGate

engine = DNAComputeEngine()

# DNA XOR operation
seq_a = "ATCG"
seq_b = "GCTA"

result = engine.apply_gate(DNALogicGate.XOR, seq_a, seq_b)
print(f"{seq_a} XOR {seq_b} = {result}")
# Output: ATCG XOR GCTA = GAGT

# Step-by-step breakdown:
# Position 0: A(0) XOR G(3) = G ((0+3) mod 4 = 3)
# Position 1: T(1) XOR C(2) = G ((1+2) mod 4 = 3)
# Position 2: C(2) XOR T(1) = G ((2+1) mod 4 = 3)
# Position 3: G(3) XOR A(0) = G ((3+0) mod 4 = 3)
# Result: GGGG (but actual positions...)

# Let's trace correctly:
# Position 0: A(0) XOR G(3) = (0+3) mod 4 = 3 = G
# Position 1: T(1) XOR C(2) = (1+2) mod 4 = 3 = G
# Position 2: C(2) XOR T(1) = (2+1) mod 4 = 3 = G
# Position 3: G(3) XOR A(0) = (3+0) mod 4 = 3 = G
```

### DNA NOT Gate (Complement)

The DNA NOT gate inverts each nucleotide using a complement mapping:

```
NOT Operation: A <-> G, T <-> C (bit inversion)

  +---+   +---+
  | A |---|NOT|----> complement(A)
  +---+   +---+

Complement Mapping:
  A (00) <---> G (11)    Bit flip: 00 -> 11
  T (01) <---> C (10)    Bit flip: 01 -> 10
```

**Truth Table:**

```
+-------+---------+------------------------+
|   A   |  NOT A  | Binary Transformation  |
+-------+---------+------------------------+
| A (0) |  G (3)  |   00 -> 11             |
| T (1) |  C (2)  |   01 -> 10             |
| C (2) |  T (1)  |   10 -> 01             |
| G (3) |  A (0)  |   11 -> 00             |
+-------+---------+------------------------+
```

**Code Example:**

```python
from vibedna.compute import DNAComputeEngine, DNALogicGate

engine = DNAComputeEngine()

# DNA NOT operation
seq_a = "ATCG"

result = engine.apply_gate(DNALogicGate.NOT, seq_a)
print(f"NOT {seq_a} = {result}")
# Output: NOT ATCG = GCTA

# Step-by-step breakdown:
# Position 0: NOT A = G (complement of 0 is 3)
# Position 1: NOT T = C (complement of 1 is 2)
# Position 2: NOT C = T (complement of 2 is 1)
# Position 3: NOT G = A (complement of 3 is 0)
# Result: GCTA
```

### NAND, NOR, XNOR Gates

These compound gates apply NOT to the result of AND, OR, and XOR respectively:

```
NAND = NOT(AND)     NOR = NOT(OR)      XNOR = NOT(XOR)

  +---+   +---+       +---+   +---+       +---+   +---+
  | A |---|   |       | A |---|   |       | A |---|   |
  +---+   |AND|--+    +---+   | OR|--+    +---+   |XOR|--+
  | B |---|   |  |    | B |---|   |  |    | B |---|   |  |
  +---+   +---+  |    +---+   +---+  |    +---+   +---+  |
                 |                   |                   |
              +--v--+             +--v--+             +--v--+
              | NOT |             | NOT |             | NOT |
              +--+--+             +--+--+             +--+--+
                 |                   |                   |
                 v                   v                   v
              Result              Result              Result
```

**NAND Truth Table:**

```
+-------+-------+---------+---------+
|   A   |   B   |A AND B  |A NAND B |
+-------+-------+---------+---------+
| A (0) | A (0) |  A (0)  |  G (3)  |
| A (0) | T (1) |  A (0)  |  G (3)  |
| T (1) | T (1) |  T (1)  |  C (2)  |
| C (2) | G (3) |  C (2)  |  T (1)  |
| G (3) | G (3) |  G (3)  |  A (0)  |
+-------+-------+---------+---------+
```

**NOR Truth Table:**

```
+-------+-------+---------+--------+
|   A   |   B   | A OR B  |A NOR B |
+-------+-------+---------+--------+
| A (0) | A (0) |  A (0)  |  G (3) |
| A (0) | T (1) |  T (1)  |  C (2) |
| T (1) | T (1) |  T (1)  |  C (2) |
| C (2) | G (3) |  G (3)  |  A (0) |
| G (3) | G (3) |  G (3)  |  A (0) |
+-------+-------+---------+--------+
```

**XNOR Truth Table:**

```
+-------+-------+---------+---------+
|   A   |   B   |A XOR B  |A XNOR B |
+-------+-------+---------+---------+
| A (0) | A (0) |  A (0)  |  G (3)  |
| A (0) | T (1) |  T (1)  |  C (2)  |
| T (1) | T (1) |  C (2)  |  T (1)  |
| C (2) | C (2) |  A (0)  |  G (3)  |
| G (3) | G (3) |  C (2)  |  T (1)  |
+-------+-------+---------+---------+
```

**Code Example:**

```python
from vibedna.compute import DNAComputeEngine, DNALogicGate

engine = DNAComputeEngine()

seq_a = "ATCG"
seq_b = "TCGA"

# NAND operation
nand_result = engine.apply_gate(DNALogicGate.NAND, seq_a, seq_b)
print(f"{seq_a} NAND {seq_b} = {nand_result}")

# NOR operation
nor_result = engine.apply_gate(DNALogicGate.NOR, seq_a, seq_b)
print(f"{seq_a} NOR {seq_b} = {nor_result}")

# XNOR operation
xnor_result = engine.apply_gate(DNALogicGate.XNOR, seq_a, seq_b)
print(f"{seq_a} XNOR {seq_b} = {xnor_result}")
```

---

## 4. Arithmetic Operations

### Addition with Carry

DNA addition works like traditional addition but in base-4 (quaternary):

```
Addition Algorithm:
  1. Start from rightmost position
  2. Add nucleotide values + carry
  3. If sum >= 4, carry = 1, result = sum mod 4
  4. Move left and repeat

     Carry:    0    1    0    0
             +------------------+
    Seq A:  |  A    T    C    G |  (0, 1, 2, 3)
    Seq B:  |  T    C    G    A |  (1, 2, 3, 0)
             +------------------+
    Result: |  T    A    C    G |  with carry = 0
             +------------------+

Step-by-step:
  Position 3 (rightmost): G(3) + A(0) + 0 = 3 -> G, carry=0
  Position 2: C(2) + G(3) + 0 = 5 -> 5 mod 4 = 1 -> T, carry=1
  Position 1: T(1) + C(2) + 1 = 4 -> 4 mod 4 = 0 -> A, carry=1
  Position 0: A(0) + T(1) + 1 = 2 -> C, carry=0
  Final: CATG with overflow=False
```

**Worked Example:**

```
Add TTGG + CCAA

Converting to decimal for verification:
  TTGG = 1*64 + 1*16 + 3*4 + 3*1 = 64 + 16 + 12 + 3 = 95
  CCAA = 2*64 + 2*16 + 0*4 + 0*1 = 128 + 32 + 0 + 0 = 160
  Sum = 95 + 160 = 255

DNA Addition:
  Position 3: G(3) + A(0) = 3 -> G, carry=0
  Position 2: G(3) + A(0) + 0 = 3 -> G, carry=0
  Position 1: T(1) + C(2) + 0 = 3 -> G, carry=0
  Position 0: T(1) + C(2) + 0 = 3 -> G, carry=0

Result: GGGG = 3*64 + 3*16 + 3*4 + 3*1 = 192 + 48 + 12 + 3 = 255
```

**Code Example:**

```python
from vibedna.compute import DNAComputeEngine

engine = DNAComputeEngine()

# DNA Addition
seq_a = "TTGG"
seq_b = "CCAA"

result, overflow = engine.add(seq_a, seq_b)
print(f"{seq_a} + {seq_b} = {result}")
print(f"Overflow: {overflow}")
# Output: TTGG + CCAA = GGGG
# Overflow: False

# Addition with overflow
seq_c = "GGGG"
seq_d = "AAAT"

result2, overflow2 = engine.add(seq_c, seq_d)
print(f"{seq_c} + {seq_d} = {result2}")
print(f"Overflow: {overflow2}")
# Output: GGGG + AAAT = AAAA
# Overflow: True (255 + 1 = 256 overflows 4 positions)
```

### Subtraction with Borrow

DNA subtraction borrows from higher positions when needed:

```
Subtraction Algorithm:
  1. Start from rightmost position
  2. Subtract (a - b - borrow)
  3. If result < 0, add 4 and set borrow = 1
  4. Move left and repeat

     Borrow:   0    0    1    0
             +------------------+
    Seq A:  |  G    C    T    A |  (3, 2, 1, 0)
    Seq B:  |  T    C    G    A |  (1, 2, 3, 0)
             +------------------+
    Result: |  C    A    T    A |  with underflow = 0
             +------------------+

Step-by-step:
  Position 3: A(0) - A(0) - 0 = 0 -> A, borrow=0
  Position 2: T(1) - G(3) - 0 = -2 -> -2+4 = 2 -> C, borrow=1
  Position 1: C(2) - C(2) - 1 = -1 -> -1+4 = 3 -> G, borrow=1
  Position 0: G(3) - T(1) - 1 = 1 -> T, borrow=0
  Final: TGCA with underflow=False
```

**Worked Example:**

```
Subtract GGAA - TTCC

Converting to decimal:
  GGAA = 3*64 + 3*16 + 0*4 + 0*1 = 192 + 48 = 240
  TTCC = 1*64 + 1*16 + 2*4 + 2*1 = 64 + 16 + 8 + 2 = 90
  Difference = 240 - 90 = 150

DNA Subtraction:
  Position 3: A(0) - C(2) - 0 = -2 -> -2+4 = 2 -> C, borrow=1
  Position 2: A(0) - C(2) - 1 = -3 -> -3+4 = 1 -> T, borrow=1
  Position 1: G(3) - T(1) - 1 = 1 -> T, borrow=0
  Position 0: G(3) - T(1) - 0 = 2 -> C, borrow=0

Result: CTTC = 2*64 + 1*16 + 1*4 + 2*1 = 128 + 16 + 4 + 2 = 150
```

**Code Example:**

```python
from vibedna.compute import DNAComputeEngine

engine = DNAComputeEngine()

# DNA Subtraction
seq_a = "GGAA"
seq_b = "TTCC"

result, underflow = engine.subtract(seq_a, seq_b)
print(f"{seq_a} - {seq_b} = {result}")
print(f"Underflow: {underflow}")
# Output: GGAA - TTCC = CTTC
# Underflow: False

# Subtraction with underflow
seq_c = "AATT"
seq_d = "TTCC"

result2, underflow2 = engine.subtract(seq_c, seq_d)
print(f"{seq_c} - {seq_d} = {result2}")
print(f"Underflow: {underflow2}")
# Output: Underflow occurred (negative result)
```

### Multiplication

DNA multiplication uses shift-and-add algorithm in base-4:

```
Multiplication: ATCG x TC

              ATCG        (27 in decimal)
           x    TC        (6 in decimal)
         ----------
      Partial products:
        ATCG x C(2) = ATCG shifted 0, multiplied by 2
        ATCG x T(1) = ATCG shifted 1, multiplied by 1
         ----------
              Result      (162 in decimal)

Verification: 27 x 6 = 162
```

**Worked Example:**

```
Multiply TC x CG

Converting to decimal:
  TC = 1*4 + 2*1 = 6
  CG = 2*4 + 3*1 = 11
  Product = 6 x 11 = 66

Converting 66 to DNA (base-4):
  66 / 4 = 16 remainder 2 (C)
  16 / 4 = 4  remainder 0 (A)
  4  / 4 = 1  remainder 0 (A)
  1  / 4 = 0  remainder 1 (T)

Result: TAAC (reading remainders from bottom to top)
Verification: 1*64 + 0*16 + 0*4 + 2*1 = 64 + 2 = 66
```

**Code Example:**

```python
from vibedna.compute import DNAComputeEngine

engine = DNAComputeEngine()

# DNA Multiplication
seq_a = "TC"
seq_b = "CG"

result = engine.multiply(seq_a, seq_b)
print(f"{seq_a} x {seq_b} = {result}")
# Output: TC x CG = TAAC

# Larger multiplication
seq_c = "ATCG"  # 27
seq_d = "AATC"  # 6

result2 = engine.multiply(seq_c, seq_d)
print(f"{seq_c} x {seq_d} = {result2}")
# Output: ATCG x AATC = AATGCAAC (27 x 6 = 162)
```

### Division

DNA division produces a quotient and remainder:

```
Division: TGCA / TC

              TGCA = 114 in decimal
           /    TC = 6 in decimal
         ----------
         Quotient  = 19
         Remainder = 0

Converting 19 to DNA:
  19 / 4 = 4 remainder 3 (G)
  4  / 4 = 1 remainder 0 (A)
  1  / 4 = 0 remainder 1 (T)

Quotient: TAG = 1*16 + 0*4 + 3*1 = 19
Remainder: AA = 0
```

**Worked Example:**

```
Divide GGCC / TC

Converting to decimal:
  GGCC = 3*64 + 3*16 + 2*4 + 2*1 = 192 + 48 + 8 + 2 = 250
  TC = 1*4 + 2*1 = 6

  250 / 6 = 41 remainder 4

Converting 41 to DNA:
  41 / 4 = 10 remainder 1 (T)
  10 / 4 = 2  remainder 2 (C)
  2  / 4 = 0  remainder 2 (C)

Quotient: CCT = 2*16 + 2*4 + 1*1 = 32 + 8 + 1 = 41
Remainder: TA = 1*4 + 0*1 = 4

Verification: 41 x 6 + 4 = 246 + 4 = 250
```

**Code Example:**

```python
from vibedna.compute import DNAComputeEngine

engine = DNAComputeEngine()

# DNA Division
seq_a = "GGCC"
seq_b = "TC"

quotient, remainder = engine.divide(seq_a, seq_b)
print(f"{seq_a} / {seq_b} = {quotient} remainder {remainder}")
# Output: GGCC / TC = AACT remainder AT

# Division by zero detection
try:
    engine.divide("ATCG", "AAAA")
except ZeroDivisionError as e:
    print(f"Error: {e}")
# Output: Error: Division by zero (all A's)
```

---

## 5. Comparison Operations

### Equal, Not Equal

Compare DNA sequences by their numeric values:

```
Comparison: ATCG vs ATCG

  ATCG = 27
  ATCG = 27
  Result: Equal (returns 0)

Comparison: ATCG vs GCTA

  ATCG = 27
  GCTA = 3*64 + 2*16 + 1*4 + 0*1 = 228
  Result: Not Equal (27 != 228)
```

**Code Example:**

```python
from vibedna.compute import DNAComputeEngine

engine = DNAComputeEngine()

# Equality comparison
seq_a = "ATCG"
seq_b = "ATCG"
seq_c = "GCTA"

print(f"{seq_a} == {seq_b}: {engine.equals(seq_a, seq_b)}")
# Output: ATCG == ATCG: True

print(f"{seq_a} == {seq_c}: {engine.equals(seq_a, seq_c)}")
# Output: ATCG == GCTA: False
```

### Greater Than, Less Than

Compare sequences numerically:

```
Comparison Diagram:

  AAAA  AAAT  AAAC  AAAG  AATA  ...  GGGG
    |     |     |     |     |         |
    0     1     2     3     4   ...  255
    <----- Less Than ------ Greater Than ----->
```

**Code Example:**

```python
from vibedna.compute import DNAComputeEngine

engine = DNAComputeEngine()

seq_a = "ATCG"  # 27
seq_b = "GCTA"  # 228
seq_c = "AATT"  # 5

# Less than comparison
print(f"{seq_a} < {seq_b}: {engine.less_than(seq_a, seq_b)}")
# Output: ATCG < GCTA: True

# Greater than comparison
print(f"{seq_a} > {seq_c}: {engine.greater_than(seq_a, seq_c)}")
# Output: ATCG > AATT: True

# Compare function returns -1, 0, or 1
print(f"compare({seq_a}, {seq_b}): {engine.compare(seq_a, seq_b)}")
# Output: compare(ATCG, GCTA): -1 (a < b)

print(f"compare({seq_b}, {seq_a}): {engine.compare(seq_b, seq_a)}")
# Output: compare(GCTA, ATCG): 1 (a > b)

print(f"compare({seq_a}, {seq_a}): {engine.compare(seq_a, seq_a)}")
# Output: compare(ATCG, ATCG): 0 (equal)
```

---

## 6. Shift Operations

### Left Shift, Right Shift

Shift operations move nucleotides and fill with A (zero):

```
Left Shift (multiply by 4^n):
+------------------+    +------------------+
|  A  T  C  G      | -> |  T  C  G  A      |  shift left by 1
+------------------+    +------------------+
      ATCG (27)              TCGA + "A" = TCGAA (108)
                              27 x 4 = 108

Right Shift (divide by 4^n):
+------------------+    +------------------+
|  A  T  C  G      | -> |  A  T  C         |  shift right by 1
+------------------+    +------------------+
      ATCG (27)              ATC (6)
                              27 / 4 = 6 (integer division)
```

**Code Example:**

```python
from vibedna.compute import DNAComputeEngine

engine = DNAComputeEngine()

seq = "ATCG"  # 27

# Left shift (multiply by powers of 4)
shifted_left_1 = engine.shift_left(seq, 1)
print(f"{seq} << 1 = {shifted_left_1}")
# Output: ATCG << 1 = ATCGA (27 x 4 = 108)

shifted_left_2 = engine.shift_left(seq, 2)
print(f"{seq} << 2 = {shifted_left_2}")
# Output: ATCG << 2 = ATCGAA (27 x 16 = 432)

# Right shift (divide by powers of 4)
shifted_right_1 = engine.shift_right(seq, 1)
print(f"{seq} >> 1 = {shifted_right_1}")
# Output: ATCG >> 1 = ATC (27 / 4 = 6)

shifted_right_2 = engine.shift_right(seq, 2)
print(f"{seq} >> 2 = {shifted_right_2}")
# Output: ATCG >> 2 = AT (27 / 16 = 1)
```

### Rotate Operations

Rotate operations wrap nucleotides around:

```
Rotate Left by 1:
+------------------+    +------------------+
|  A  T  C  G      | -> |  T  C  G  A      |
+------------------+    +------------------+
   First moves to end

Rotate Right by 1:
+------------------+    +------------------+
|  A  T  C  G      | -> |  G  A  T  C      |
+------------------+    +------------------+
   Last moves to front
```

**Code Example:**

```python
from vibedna.compute import DNAComputeEngine

engine = DNAComputeEngine()

seq = "ATCG"

# Rotate left
rotated_left_1 = engine.rotate_left(seq, 1)
print(f"rotate_left({seq}, 1) = {rotated_left_1}")
# Output: rotate_left(ATCG, 1) = TCGA

rotated_left_2 = engine.rotate_left(seq, 2)
print(f"rotate_left({seq}, 2) = {rotated_left_2}")
# Output: rotate_left(ATCG, 2) = CGAT

# Rotate right
rotated_right_1 = engine.rotate_right(seq, 1)
print(f"rotate_right({seq}, 1) = {rotated_right_1}")
# Output: rotate_right(ATCG, 1) = GATC

rotated_right_2 = engine.rotate_right(seq, 2)
print(f"rotate_right({seq}, 2) = {rotated_right_2}")
# Output: rotate_right(ATCG, 2) = CGAT

# Full rotation returns original
rotated_full = engine.rotate_left(seq, 4)
print(f"rotate_left({seq}, 4) = {rotated_full}")
# Output: rotate_left(ATCG, 4) = ATCG
```

---

## 7. Expression Evaluation

### Combining Operations

VibeDNA supports compound expressions with multiple operations:

```
Expression Evaluation Flow:

  Input: "(A AND B) XOR C"

  Variables: A = "ATCG", B = "GCTA", C = "AATT"

  Step 1: Parse expression tree

              XOR
             /   \
           AND    C
          /   \
         A     B

  Step 2: Evaluate bottom-up

         A AND B = ATTA
         (ATTA) XOR C = ATTT (result)
```

**Code Example:**

```python
from vibedna.compute import DNAComputeEngine

engine = DNAComputeEngine()

# Define variables
variables = {
    "A": "ATCG",
    "B": "GCTA",
    "C": "AATT",
}

# Simple expression
result1 = engine.execute_expression("A AND B", variables)
print(f"A AND B = {result1}")
# Output: A AND B = ATTA

# Compound expression with parentheses
result2 = engine.execute_expression("(A AND B) XOR C", variables)
print(f"(A AND B) XOR C = {result2}")

# NOT expression
result3 = engine.execute_expression("NOT A", variables)
print(f"NOT A = {result3}")
# Output: NOT A = GCTA

# Complex nested expression
result4 = engine.execute_expression("(A OR B) AND (NOT C)", variables)
print(f"(A OR B) AND (NOT C) = {result4}")
```

### Order of Operations

VibeDNA expression evaluation follows standard precedence rules:

```
Operator Precedence (highest to lowest):

  1. Parentheses ()     - evaluated first
  2. NOT                - unary operator
  3. AND                - binary operators
  4. OR, XOR, NAND, NOR, XNOR  - binary operators (left to right)

Example: A AND B OR C XOR NOT D

  Evaluated as: ((A AND B) OR C) XOR (NOT D)

  Step 1: NOT D
  Step 2: A AND B
  Step 3: (result of step 2) OR C
  Step 4: (result of step 3) XOR (result of step 1)
```

### Complex Expression Examples

**Example 1: Bit Masking**

```python
from vibedna.compute import DNAComputeEngine

engine = DNAComputeEngine()

# Using AND to mask specific positions
data = "ATCGATCG"
mask = "GGGGAAAA"  # Keep first 4, zero last 4

variables = {"DATA": data, "MASK": mask}

result = engine.execute_expression("DATA AND MASK", variables)
print(f"Masked result: {result}")
# Only positions with G in mask retain original values
```

**Example 2: Toggle Operation (XOR)**

```python
from vibedna.compute import DNAComputeEngine

engine = DNAComputeEngine()

# XOR for toggle/flip operation
original = "ATCGATCG"
toggle_pattern = "GGGGGGGG"  # XOR with max values

variables = {"ORIG": original, "TOGGLE": toggle_pattern}

# Toggle
toggled = engine.execute_expression("ORIG XOR TOGGLE", variables)
print(f"Toggled: {toggled}")

# Toggle again to restore
variables["TOGGLED"] = toggled
restored = engine.execute_expression("TOGGLED XOR TOGGLE", variables)
print(f"Restored: {restored}")
# restored == original
```

**Example 3: Conditional Selection**

```python
from vibedna.compute import DNAComputeEngine

engine = DNAComputeEngine()

# Mux-like selection: (A AND SEL) OR (B AND NOT SEL)
a_input = "GGGG"
b_input = "TTTT"
selector = "GGAA"  # G=max, A=min

variables = {
    "A": a_input,
    "B": b_input,
    "SEL": selector,
}

# When SEL is high (G), select A; when low (A), select B
result = engine.execute_expression(
    "(A AND SEL) OR (B AND (NOT SEL))",
    variables
)
print(f"Selected: {result}")
```

---

## 8. Practical Applications

### Use Cases for DNA Computation

```
+------------------------------------------------------------------+
|                    DNA COMPUTING APPLICATIONS                     |
+------------------------------------------------------------------+

1. DATA ENCRYPTION & SECURITY
   +----------------------------------------------------------+
   |  - XOR-based encryption directly on DNA sequences        |
   |  - DNA-native key generation and management              |
   |  - Tamper-evident checksums computed in DNA              |
   +----------------------------------------------------------+

2. PARALLEL DATA PROCESSING
   +----------------------------------------------------------+
   |  - Process millions of records simultaneously            |
   |  - DNA database queries without conversion               |
   |  - Pattern matching across massive datasets              |
   +----------------------------------------------------------+

3. LONG-TERM ARCHIVAL COMPUTATION
   +----------------------------------------------------------+
   |  - Verify data integrity without full decode             |
   |  - Update checksums in archived DNA                      |
   |  - Incremental modification of stored data               |
   +----------------------------------------------------------+

4. BIOLOGICAL DATA ANALYSIS
   +----------------------------------------------------------+
   |  - Native sequence comparison operations                 |
   |  - Mutation detection through XOR operations             |
   |  - Consensus building via parallel AND/OR                |
   +----------------------------------------------------------+

5. ERROR DETECTION & CORRECTION
   +----------------------------------------------------------+
   |  - Parity computation in DNA domain                      |
   |  - Syndrome calculation for error location               |
   |  - In-place error correction                             |
   +----------------------------------------------------------+
```

**Application Example: DNA Checksum**

```python
from vibedna.compute import DNAComputeEngine, ParallelStrandProcessor

engine = DNAComputeEngine()
processor = ParallelStrandProcessor(workers=4)

# Calculate XOR checksum of multiple DNA blocks
data_blocks = [
    "ATCGATCG",
    "GCTAGCTA",
    "AATTGGCC",
    "CCGGAATT",
]

# Parallel XOR reduction for checksum
checksum = processor.xor_reduce(data_blocks)
print(f"DNA Checksum: {checksum}")

# Verify data integrity
def verify_integrity(blocks, expected_checksum):
    computed = processor.xor_reduce(blocks)
    return engine.equals(computed, expected_checksum)

is_valid = verify_integrity(data_blocks, checksum)
print(f"Data integrity verified: {is_valid}")
```

### Performance Considerations

```
+------------------------------------------------------------------+
|                   PERFORMANCE CHARACTERISTICS                     |
+------------------------------------------------------------------+

Operation Type          Time Complexity    Space Complexity
------------------------------------------------------------------
Logic Gates (AND/OR/XOR)    O(n)              O(n)
NOT                         O(n)              O(n)
Addition/Subtraction        O(n)              O(n)
Multiplication              O(n*m)            O(n+m)
Division                    O(n*m)            O(n)
Comparison                  O(n)              O(1)
Shift                       O(1)              O(n)
Rotate                      O(n)              O(n)
Expression Evaluation       O(k*n)            O(n)

Where:
  n = length of DNA sequence
  m = length of second operand
  k = number of operations in expression

+------------------------------------------------------------------+
|                      OPTIMIZATION TIPS                            |
+------------------------------------------------------------------+

1. BATCH OPERATIONS
   - Use ParallelStrandProcessor for multiple sequences
   - Reduces per-operation overhead
   - Leverages multi-threading

2. MINIMIZE CONVERSIONS
   - Keep data in DNA format throughout pipeline
   - Convert to/from integers only when necessary
   - Use DNA-native comparisons

3. EXPRESSION CACHING
   - Reuse computed intermediate results
   - Store frequently used expressions
   - Pre-compile expression trees

4. CHOOSE APPROPRIATE PRECISION
   - Use shortest sequences that meet requirements
   - Longer sequences increase computation time
   - Balance precision vs. performance
```

**Performance Example:**

```python
from vibedna.compute import ParallelStrandProcessor, DNALogicGate
import time

# Parallel processing demonstration
processor = ParallelStrandProcessor(workers=8)

# Generate test data
strands_a = ["ATCGATCG" for _ in range(1000)]
strands_b = ["GCTAGCTA" for _ in range(1000)]

# Parallel XOR across 1000 strand pairs
start = time.time()
results = processor.map_gate(strands_a, strands_b, DNALogicGate.XOR)
elapsed = time.time() - start

print(f"Processed 1000 XOR operations in {elapsed:.4f} seconds")
print(f"Throughput: {1000/elapsed:.0f} operations/second")
```

---

## Summary

VibeDNA's DNA computing capabilities enable powerful data processing directly on DNA-encoded information:

```
+------------------------------------------------------------------+
|                          QUICK REFERENCE                          |
+------------------------------------------------------------------+

NUCLEOTIDE VALUES:
  A = 0    T = 1    C = 2    G = 3

LOGIC GATES:
  AND  = min(a, b)           NAND = NOT(AND)
  OR   = max(a, b)           NOR  = NOT(OR)
  XOR  = (a + b) mod 4       XNOR = NOT(XOR)
  NOT  = complement (A<->G, T<->C)

ARITHMETIC:
  add(a, b)       -> (result, overflow)
  subtract(a, b)  -> (result, underflow)
  multiply(a, b)  -> result
  divide(a, b)    -> (quotient, remainder)

COMPARISON:
  compare(a, b)   -> -1 (less), 0 (equal), 1 (greater)
  equals(a, b)    -> bool
  less_than(a, b) -> bool
  greater_than(a, b) -> bool

SHIFT:
  shift_left(seq, n)   -> seq * 4^n
  shift_right(seq, n)  -> seq / 4^n
  rotate_left(seq, n)  -> circular left
  rotate_right(seq, n) -> circular right

EXPRESSIONS:
  execute_expression("(A AND B) XOR C", variables)

+------------------------------------------------------------------+
```

For more examples and advanced usage, see the [API Documentation](../API.md) and [Examples Guide](../EXAMPLES.md).

---

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