Fitness Functions (Φ, φ)

Formal Definition

Fitness functions quantify how well skills perform on specific tasks. They provide a measurable basis for skill evaluation, selection, and optimization.

There are two levels of fitness functions in the ontology:

Skill-Level Fitness (φ)

Evaluates a single skill’s effectiveness on a task:

$\varphi :\mathcal{S}\times \mathcal{T}\to [0,1]$

The fitness of skill $\mathcal{S}$ for task $\mathcal{T}$ is computed as:

$\varphi (\mathcal{S},\mathcal{T})={\omega }_1\cdot \text{accuracy}+{\omega }_2\cdot \text{efficiency}+{\omega }_3\cdot \text{robustness}$

Where ${\omega }_1,{\omega }_2,{\omega }_3$ are weighting parameters.

Agent-Level Fitness (Φ)

Evaluates an agent’s overall capability for a task based on its skill set:

$\Phi (\mathcal{T},A)=\frac{|{\mathcal{R}}_{\mathcal{T}}\cap {\mathcal{S}}_A|}{|{\mathcal{R}}_{\mathcal{T}}|}\cdot {\prod }_{S\in {\mathcal{R}}_{\mathcal{T}}\cap {\mathcal{S}}_A}{\varphi }_S(\mathcal{T})$

Where:

• $\mathcal{T}$: Task to be performed
• $A$: Agent being evaluated
• ${\mathcal{R}}_{\mathcal{T}}$: Required skills for task $\mathcal{T}$
• ${\mathcal{S}}_A$: Agent’s skill set
• ${\varphi }_S(\mathcal{T})$: Skill-specific fitness

Components of Fitness

1. Accuracy

How correctly the skill performs the task:

• Correctness of outputs
• Error rate
• Precision and recall (for classification tasks)

2. Efficiency

How efficiently the skill performs:

• Time complexity
• Resource usage
• Computational cost

3. Robustness

How well the skill handles variations:

• Performance on edge cases
• Resilience to noisy inputs
• Generalization across contexts

Key Properties and Characteristics

1. Bounded Range

Fitness values are normalized to [0, 1]:

• 0: Skill completely ineffective for task
• 1: Perfect performance on task

2. Monotonicity (Axiom 3)

Better skill fitness leads to better agent fitness: $\varphi ({\mathcal{S}}_1,\mathcal{T})\ge \varphi ({\mathcal{S}}_2,\mathcal{T})⟹\Phi (\mathcal{T},A_{{\mathcal{S}}_1})\ge \Phi (\mathcal{T},A_{{\mathcal{S}}_2})$

3. Compositionality

Agent fitness depends on both skill coverage and individual skill quality.

4. Context Sensitivity

Fitness can vary based on task context and requirements.

Research Context and Applications

Fitness functions are crucial for:

• Skill Selection: Choosing optimal skills for tasks
• Performance Optimization: Improving agent capabilities
• Learning Guidance: Identifying skills that need improvement
• Capability Assessment: Measuring agent proficiency
• Adaptive Behavior: Dynamically selecting skills based on task requirements

In LLM applications:

• Evaluating prompt effectiveness
• Selecting best model for specific tasks
• Guiding fine-tuning efforts
• Benchmarking capabilities
• Automated prompt optimization

Connections to Other Concepts

• Skills (𝒮): Skills have fitness functions ${\varphi }_S$ as part of their definition
• Tasks: Fitness is always relative to specific tasks
• Analysis Metaskills (${\mathcal{M}}_a$): Use fitness functions to evaluate skills
• Optimization Metaskills (${\mathcal{M}}_o$): Use fitness to guide optimization
• Task-Skill Mapping: Fitness determines which skills are suitable for tasks
• Metaskill Fixed Points: Optimal metaskills maximize fitness

Fitness in Different Contexts

Task-Specific Fitness

Fitness can vary significantly across tasks: $\varphi (\mathcal{S},{\mathcal{T}}_1)\ne \varphi (\mathcal{S},{\mathcal{T}}_2)$

Aggregate Fitness

For multiple tasks, aggregate fitness can be computed: ${\varphi }_{agg}(\mathcal{S})=\frac{1}{|{\mathcal{T}}_{set}|}{\sum }_{\mathcal{T}\in {\mathcal{T}}_{set}}\varphi (\mathcal{S},\mathcal{T})$

Open Research Questions

1. Performance Definition: How to formally define “reasonably well” performance?

   • Current formulation: $\Phi (\mathcal{T},A)\ge {\tau }_{acceptable}$

2. Weighting Parameters: How to determine optimal weights ${\omega }_1,{\omega }_2,{\omega }_3$?

3. Dynamic Fitness: How does fitness change as skills are learned or improved?

4. Multi-Objective Fitness: How to balance competing objectives (accuracy vs. efficiency)?

5. Fitness Estimation: How to efficiently estimate fitness without exhaustive testing?

6. Transferable Fitness: How well does fitness on one task predict fitness on related tasks?

7. Emergent Fitness: How to measure fitness of emergent superskills?