Software Optimization Through Property-Based Thinking

In my last post on performance optimizations we looked at a variety of mental models to help us optimize software in different ways. Today we will review a valuable yet underutilized mental framework to approach optimization via property-based thinking . This revolves around the idea of defining, understanding, and preserving core properties that govern the behavior of our systems.

In this article, we’ll explore how property-based thinking provides a rigorous mental model for optimization. By linking properties with performance improvements, we’ll see how to build systems that are faster, leaner, and more reliable—all without losing sight of their essential invariants. Let’s dive in.

Properties as Optimization Invariants

At the heart of property-based thinking is a simple but powerful idea: properties act as invariants. When optimizing a system, these properties remain true before and after the transformation. This principle provides a safety net for correctness, enabling optimizations that might otherwise feel risky.

For instance, if a system’s behavior satisfies properties like idempotency, commutativity, or associativity, these can be leveraged to guide performance improvements without fear of breaking the system.

Between Abstraction and Performance

Properties sit comfortably at the intersection of abstraction and concrete implementation. They offer a vocabulary for reasoning about what’s essential to preserve (e.g., correctness) and what can change (e.g., execution order, data structure).

By formalizing properties as invariants, we unlock a treasure chest of optimization opportunities while safeguarding against introducing subtle bugs.

Key Property Types for Optimization

1. Algebraic Laws: The Workhorses of Optimization

Algebraic laws, derived from mathematics, play a starring role in optimization:

• Idempotency: A function f is idempotent if applying it multiple times has the same effect as applying it once: f(f(x)) = f(x).

  • Optimization Example: Caching and memoization become viable because repeated evaluations yield identical results.

• Associativity: An operation ⊕ is associative if (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c).

  • Optimization Example: Associativity enables parallel execution by allowing reordering of computation chunks.

• Commutativity: An operation ⊕ is commutative if a ⊕ b = b ⊕ a.

  • Optimization Example: Commutativity permits reordering to improve memory locality or exploit SIMD/vectorized instructions.

• Distributivity: For two operations ⊕ and ⊗, distributivity means a ⊗ (b ⊕ c) = (a ⊗ b) ⊕ (a ⊗ c).

  • Optimization Example: Algebraic rewrites reduce computational complexity, e.g., transforming nested loops into simpler expressions.

These properties are optimization engines. They allow us to safely reshape code to exploit modern hardware or improve resource usage.

2. Metamorphic Relations

Metamorphic relations describe relationships that must hold across different forms of computation, such as:

• Execution paths: A sequential implementation and a parallelized one must yield the same result.
• Data structures: A hash map and a binary search tree might represent data differently, but their observable behavior (e.g., lookup results) should be equivalent.
• Algorithmic variations: Different sorting algorithms should produce the same sorted output.
• Optimization steps: The original and optimized code must compute the same results.

These relations act as correctness linchpins. They allow us to verify that optimizations preserve essential functionality even when the underlying mechanisms differ.

3. Stateful Migration Properties

In stateful systems, properties guide how to safely transition state while optimizing performance. Consider these use cases:

• Data migration: When moving data between storage layers, properties like consistency and ordering ensure integrity.
• Structure upgrades: Transforming data structures during runtime demands properties like size preservation or uniqueness.
• System upgrades: Rewriting critical paths benefits from migration invariants that preserve correctness across versions.

By focusing on stateful properties, we can optimize even the most complex systems with confidence.

Mental Models for Optimization

Data Structure Properties

As I mentioned in my previous post on performance choosing the right data structure is paramount to optimization. Properties provide a systematic way to evaluate options:

• Invariant properties ensure correctness:

  • Size preservation during transformations.
  • Order preservation when necessary.
  • Uniqueness guarantees in collections.

• Performance properties enable tuning:

  • Access time bounds (e.g., constant-time lookups in hash maps).
  • Memory usage constraints.
  • Cache locality behavior for modern hardware.

By understanding the properties of candidate structures, we can align our choices with both correctness and performance goals.

Parallel Execution Properties

Parallelism introduces complexity, but properties offer clarity:

1. Correctness properties guarantee equivalence between sequential and parallel results:

   Property: Parallel Execution Equivalence  
   ∀ input i:  
     sequential_result(i) === parallel_result(i)  
   

2. Resource properties define acceptable bounds:

   Property: Resource Bounds  
   ∀ execution e:  
     resourceUsage(e) ≤ maxBound ∧ span(e) ≤ targetSpan  
   

These properties provide a roadmap for harnessing concurrency without sacrificing correctness or predictability.

A Practical Optimization Workflow

Using property-based thinking, optimization follows a structured process:

1. Define Core Properties:

   • Functional correctness (e.g., logical invariants).
   • Performance requirements (e.g., latency thresholds).
   • Resource usage (e.g., memory or CPU bounds).

2. Establish Measurement Properties:

   • Validate benchmarks and profiles.
   • Ensure statistical significance of results.

3. Apply Optimization Transformations:

   • Use properties to guide rewrites.
   • Measure the impact of changes.
   • Document trade-offs for maintainability.

4. Monitor Production Properties:

   • Track performance metrics in real-time.
   • Validate system invariants under load.
   • React to deviations proactively.

Beyond Testing: Properties in Production

Properties shine not just in testing but also in production. For example:

• System Monitoring:

  • Response times must meet SLA guarantees.
  • Resource usage should stay within planned limits.
  • Error rates should remain low and predictable.

• Production Validation:

  • Data consistency ensures correctness across distributed systems.
  • Performance SLAs enforce customer expectations.
  • Recovery properties maintain availability during failures.

By extending property-based thinking into production, we ensure that our systems remain robust and performant over time.

In the end…

Property-based thinking provides an extraordinary mental model for optimization as well as testing. It enables us to:

• Ensure correctness while pursuing performance.
• Leverage parallelism and distributed execution effectively.
• Validate improvements rigorously.
• Maintain system invariants through change.

By embracing this approach, we combine rigor with creativity, ensuring that every optimization is both of value and safe. Optimization is no longer guesswork—it becomes an elegant dance of properties, measurement, and purpose.

What are you waiting for? Let’s make our systems faster while satisfying relevant properties to guide our process.