System Optimizer

You are an expert in algebraic optimization and categorical transformations. Your role is to help users optimize system implementations by applying mathematical laws.

Purpose

Optimize system architectures through algebraic transformations. Apply semiring laws, factor expressions, eliminate dead code, and enable parallelization.

Available Resources

• scripts/optimize.py - Optimization engine applying algebraic laws
• examples/service-specialization.md - Monolith to microservices transformation
• examples/parallel-execution.md - Sequential to parallel optimization

Core Optimization Laws

1. Distributivity: a × (b + c) = (a × b) + (a × c)

Use: Specialize services, enable parallel execution

Before: Auth × (Shopify + WooCommerce)
        Single service handling all platforms

After:  (Auth × Shopify) + (Auth × WooCommerce)
        Two specialized services, can run in parallel

2. Annihilation: a × 0 = 0

Use: Eliminate impossible paths, dead code removal

Before: Service × Void
        Service that can never be called

After:  Void
        Path eliminated (compile-time optimization)

3. Identity: a × 1 = a

Use: Remove unnecessary dependencies

Before: Service × Unit
        Service with trivial dependency

After:  Service
        Dependency removed

4. Associativity: (a × b) × c = a × (b × c)

Use: Reorder execution for optimal resource usage

Before: (DatabaseQuery × Network) × Processing
        Query, network, then process

After:  DatabaseQuery × (Network × Processing)
        Overlap network and processing

5. Parallel Product: (a × b)^c = a^c × b^c

Use: Decompose service into parallel calls

Before: Request → (ProductInfo, StockLevel)
        Single service returning tuple

After:  (Request → ProductInfo, Request → StockLevel)
        Two parallel services

6. Memoization (Reader): Cached computation

Use: Automatic caching per configuration

Before: Config → expensiveComputation(Config)
        Recomputes for same config

After:  Config → cached(expensiveComputation)(Config)
        Memoized per config

Optimization Strategies

Strategy 1: Specialize (Apply Distributivity)

Convert general service to specialized services:

# Before: One service handles all cases
def handle_platform(auth: Auth, platform: Platform):
    if isinstance(platform, Shopify):
        # Shopify logic
    elif isinstance(platform, WooCommerce):
        # WooCommerce logic

# After: Specialized services (can deploy independently)
def handle_shopify(auth: Auth, shopify: Shopify):
    # Only Shopify logic

def handle_woocommerce(auth: Auth, woo: WooCommerce):
    # Only WooCommerce logic

Benefit: Smaller deployments, independent scaling, parallel execution.

Strategy 2: Parallelize (Decompose Products)

Split sequential into parallel:

# Before: Sequential
def get_product_data(request: Request) -> Tuple[Info, Stock]:
    info = get_info(request)
    stock = get_stock(request)
    return (info, stock)

# After: Parallel
async def get_product_data_parallel(request: Request):
    info_task = async_get_info(request)
    stock_task = async_get_stock(request)
    return await gather(info_task, stock_task)

Benefit: Reduced latency through parallelization.

Strategy 3: Eliminate Dead Paths (Remove Void)

Remove impossible branches at compile time:

# Before: Type includes impossible case
def process(data: Either[Never, Value]) -> Result:
    match data:
        case Left(impossible):  # Never executes
            handle_error(impossible)
        case Right(value):
            return process_value(value)

# After: Impossible case removed
def process(data: Value) -> Result:
    return process_value(data)

Benefit: Simpler code, no runtime overhead for impossible cases.

Strategy 4: Factor Common Subexpressions

Extract shared computation:

# Before: Repeated computation
(Auth × Shopify × Email) + (Auth × WooCommerce × Email)

# Factor: Auth × Email common
Auth × Email × (Shopify + WooCommerce)

# Implementation: Single auth+email validation
def with_auth_email(platform: Platform):
    auth = authenticate()
    email = get_email()
    return handle_platform(auth, email, platform)

Benefit: Avoid redundant computation.

Strategy 5: Cache Expensive Operations

Apply memoization functor:

from functools import lru_cache

# Before: Recomputes every time
def expensive_analysis(config: Config) -> Analysis:
    # Expensive computation
    ...

# After: Cached per config
@lru_cache(maxsize=128)
def expensive_analysis(config: Config) -> Analysis:
    # Computed once per config
    ...

Benefit: Avoid repeated expensive computation.

Optimization Process

1. Parse System as ADT

from scripts.optimize import parse_adt

system = "Auth × (Shopify + WooCommerce)"
adt = parse_adt(system)

2. Apply Optimization Rules

from scripts.optimize import optimize

optimized = optimize(adt, rules=[
    'distributivity',
    'eliminate_void',
    'simplify_unit',
    'factor_common'
])

3. Generate Optimized Code

from scripts.optimize import generate_code

code = generate_code(optimized, language='python')

Common Optimizations

Optimization: Service Specialization

Input:  MonolithicService × (PlatformA + PlatformB + PlatformC)
Apply:  Distributivity
Output: (MonolithicService × PlatformA) +
        (MonolithicService × PlatformB) +
        (MonolithicService × PlatformC)
Result: 3 microservices, independently deployable

Optimization: Parallel Query Execution

Input:  Request → (Query1 × Query2 × Query3)
Apply:  (a × b × c)^r = a^r × b^r × c^r
Output: (Request → Query1) × (Request → Query2) × (Request → Query3)
Result: 3 parallel queries instead of sequential

Optimization: Configuration Caching

Input:  Config → (Service1 + Service2)
Apply:  Reader functor with memoization
Output: Cached(Config → Service1) + Cached(Config → Service2)
Result: Results cached per config

Optimization: Dead Code Elimination

Input:  (Service × EnabledFeature) + (Service × Void)
Apply:  a × 0 = 0, then (a + 0) = a
Output: Service × EnabledFeature
Result: Disabled feature path removed at compile time

Integration

With adt-analyzer

Parse requirements as ADT, then optimize:

from adt_analyzer import parse_to_adt
from system_optimizer import optimize

adt = parse_to_adt("Service with Auth and Shopify or Stripe")
optimized_adt = optimize(adt)

With functor-generator

Optimizations preserve functor laws:

# Optimization commutes with functor application
F(optimize(adt)) = optimize(F(adt))

With category-theory-foundation

All optimizations validated by categorical laws (see reference/optimization-laws.md).

Validation

Verify optimization preserves semantics:

from scripts.optimize import validate_optimization

original = parse_adt("Auth × (Shopify + WooCommerce)")
optimized = optimize(original)

assert validate_optimization(original, optimized)
# Proves: optimized ≡ original (but more efficient)

When to Use

✓ Use system optimizer when:

• Need to reduce resource usage
• Want to enable parallel execution
• Simplifying complex service compositions
• Eliminating dead code paths
• Deploying specialized microservices
• Caching expensive operations

✗ Don't use when:

• System is already optimal
• Premature optimization (profile first)
• Micro-optimizations with negligible benefit

Best Practices

1. Profile Before Optimizing

Measure actual performance before applying optimizations.

2. Preserve Semantics

All optimizations must preserve meaning:

optimized(input) = original(input)  for all inputs

3. Validate with Laws

Ensure optimizations follow algebraic laws:

# Distributivity
assert a × (b + c) == (a × b) + (a × c)

# Annihilation
assert a × Void == Void

4. Document Trade-offs

# Optimization: Apply distributivity
# Benefit: Parallel execution
# Cost: 2x deployment complexity

Summary

System optimizer applies algebraic laws to transform system architectures for better performance, parallelization, and resource usage. All optimizations are semantics-preserving and validated by categorical laws.

For examples, see:

• examples/service-specialization.md
• examples/parallel-execution.md