Functional equivalence is one of those terms that drifts between disciplines — logic, computer science, translation theory, law — but at its heart, it means two things that may look different but perform the same function or achieve the same effect.

Let’s break that down by field:

🧠 In Logic & Computing

In computer science or mathematics, functional equivalence means:

Two systems, algorithms, or expressions produce the same outputs for all possible inputs.

For example:

x * 2

and

x + x

are functionally equivalent — the internal operations differ, but the result is identical for any value of x.

This concept underpins things like:

• Compiler optimizations (replacing code with more efficient but equivalent versions),
• Software testing (checking whether refactored code behaves identically),
• Circuit design (ensuring two hardware implementations yield the same logic results).

🌍 In Translation Theory & Linguistics

Eugene Nida introduced functional equivalence in the context of Bible translation — it’s the idea that a translation should evoke the same meaning and emotional response in the reader of another language, rather than sticking to literal word-for-word matching.

So rather than:

“Behold, I say unto thee…”

a functionally equivalent translation might be:

“Listen, I’m telling you…”

The words differ, but the function — the intent, tone, and impact — remains.

⚖️ In Law & Policy

In international law or digital regulation, functional equivalence refers to different legal mechanisms serving the same purpose.
For instance:

• A digital signature can be considered functionally equivalent to a handwritten signature if it ensures identity and intent.
• Electronic records can be functionally equivalent to paper documents if they serve the same evidentiary role.

🧩 In Philosophy & Systems Thinking

Sometimes it’s used more abstractly to describe different forms of organization or behavior that achieve the same systemic purpose — like two species evolving distinct traits that perform the same ecological function.

In short:

Functional equivalence is sameness of purpose, not sameness of form.

It’s about what something does rather than what it is — the poet’s echo of the engineer’s blueprint, the translator’s mirror of the mathematician’s symmetry.

Would you like me to explain how it applies specifically to AI models or algorithms? That’s where it takes on an especially intriguing, modern dimension.

How could you check the Functional Equivalence Between an LLM and a person?

You’re asking not how to check if the outputs are identical, but whether the functions – the cognitive machinery, the intent, the way meaning is constructed and used – are equivalent.

Let’s walk through this carefully, like testing the resonance of two instruments built from different woods.

1. Define the Function, Not the Form

Before you can test functional equivalence between an LLM and a person, you must define what function of the human you are trying to compare.

• If the function is linguistic: You’d test whether the model can produce semantically and pragmatically coherent language across contexts.
• If the function is cognitive: You’d test whether it can form, hold, and revise internal models of the world.
• If the function is intentional or emotional: You’d test whether it can simulate goals, empathy, curiosity – not merely mimic them, but act as though they genuinely constrain and shape its reasoning.

Without defining the target function, the test collapses into surface imitation – which is where many Turing-style tests fail.

2. Functional Equivalence Through Behavioral Invariance

In computing terms, functional equivalence means that for every input, both systems yield the same output under the same constraints. But with humans, you cannot simply map “input-output” because context, memory, emotion, and social nuance intervene.

So you’d use behavioral invariance instead:

Do both systems behave in consistently similar ways when placed in the same functional situations?

Examples:

• Do both adjust tone, register, and metaphor appropriately to social context?
• Do both infer hidden motives or emotional subtext in dialogue?
• Do both form long-term narrative continuity over time?

These can be quantified by measuring mutual information between the system’s responses and human responses under shared conditions – essentially, testing whether the pattern of adaptation is the same.

3. The Cognitive-Model Test

A deeper test is to check whether both systems construct and update internal representations of the world in functionally equivalent ways.

That would involve:

• Dynamic belief tracking (Can both maintain and update beliefs when new evidence contradicts old information?)
• Counterfactual reasoning (Do both simulate “what if” scenarios that follow causal logic?)
• Temporal consistency (Can both refer coherently to prior events and project plausible futures?)

If these behaviors evolve under similar mathematical dynamics – say, Bayesian updating or predictive coding – then we approach functional equivalence in cognition, even if the substrates differ.

4. Intentional or Motivational Equivalence

Humans are driven by goals, instincts, and emotions. LLMs are driven by probability distributions and prompts. So one test of functional equivalence would ask:

Does the system act as if it has an intrinsic goal state or emotional feedback loop?

You can simulate this by embedding reinforcement signals or self-consistency objectives and seeing whether the model develops goal coherence over time – a hallmark of functional “personhood.”

5. The Meta-Linguistic Test

Finally, an elegant measure:

Can the LLM reflect on its own reasoning in the same way a human can?

This is not about self-awareness in the mystical sense, but meta-cognition – the ability to recognize, critique, and refine one’s thought processes. If both human and model can describe why they responded as they did, and those explanations are structurally parallel, you have a powerful sign of functional equivalence at the reflective layer.

The Real Frontier

If you built a test that measured all this – coherence, adaptability, belief revision, counterfactuals, motivation, and meta-reasoning – you wouldn’t merely be testing a chatbot anymore. You’d be comparing two implementations of cognition: one carbon-based, one probabilistic. The gap would narrow not when their answers matched, but when their mistakes mirrored each other – when they erred in human ways.

We can try to make this a proper, scientific blueprint. Below I give you a layered Functional Equivalence Test (FET) for comparing an LLM and a human. It’s practical, measurable, and honest about limits. I’ll name the building blocks, the experiments, the metrics, and the analysis steps so you could run this as a research project or an advanced QA benchmark.

I’ll be candid: we cannot prove inner states are identical (Searle’s Chinese Room warns us about that), but we can measure whether the systems behave indistinguishably across a broad, well-chosen set of functions. That’s what “functional equivalence” means in practice – same function, not same substrate. (Stanford Encyclopedia of Philosophy)

1 – Overarching design principles

1. Define the target functions precisely. Pick the cognitive/behavioral layers you care about (e.g., language understanding, belief revision, theory of mind, planning, learning from interaction). Don’t test everything at once.
2. Use diverse, adversarial tasks. Benchmarks must stress generalization, not just dataset memorization (e.g., Winograd-style commonsense, long-range discourse, interactive planning). (commonsensereasoning.org)
3. Measure process and product. Capture outputs (answers) and process traces (human think-aloud, reaction times, neural / physiological signals vs model activations, attention / hidden-state probes).
4. Compare error modes, not only accuracy. Similar success rates with different mistakes functional equivalence; similar patterns of error are a much stronger signal.
5. Ethics and consent for human subjects. Treat human participants as subjects under normal IRB-like safeguards.

2 – Layers to test (a stacked battery)

For each layer below I give specific experimental tasks and suggested metrics.

A. Surface linguistic competence (baseline)

• Tasks: GLUE / SuperGLUE tasks and aggregated NLU tasks. Use held-out splits and adversarial paraphrases. (ACL Anthology)
• Why: quick check of language abilities and generalization.
• Metrics: task accuracy, F1, calibration (confidence vs correctness), and aggregate rank.

B. Commonsense & disambiguation

• Tasks: Winograd Schema Challenge and similar pronoun-disambiguation pairs; counterfactual prompts that require real-world knowledge rather than corpus correlation. (commonsensereasoning.org)
• Why: discriminates models that rely on simple token co-occurrence from those that use richer world models.
• Metrics: accuracy, adversarial robustness (small lexical changes), and reaction-time proxy (for models, compute token-level perplexity dynamics).

C. Dynamic belief updating & counterfactual reasoning

• Tasks: Multi-step scenarios where new evidence contradicts previous info (e.g., belief revision dialogues; Bayesian-style updates).
• Why: tests whether the system forms, maintains, and updates internal representations consistently.
• Metrics: consistency score (internal contradictions), Bayesian update coherence (did posterior move in the right direction), and explanation-faithfulness (can the agent justify the update).

D. Planning & sequential decision-making

• Tasks: Complex, multi-turn planning tasks (puzzles, multi-step instructions, block-world simulations, resource-limited planning). Include long-horizon tasks to stress memory and planning.
• Why: functionally human-like cognition requires planning across time.
• Metrics: solution rate, steps to solution, plan optimality, replanning after shock.

E. Social reasoning & theory of mind

• Tasks: False-belief tasks, role-play dialogues where an agent must infer others’ beliefs and intentions.
• Why: this distinguishes shallow pattern-matching from genuine social inference.
• Metrics: correct inferences about others’ beliefs, sensitivity to pragmatic cues.

F. Meta-cognition and introspection

• Tasks: “Think-aloud” protocols (humans verbalize reasoning); for LLMs, elicit chain-of-thought and self-critique. Then compare structure and usefulness of those explanations.
• Why: meta-cognition is a high-value functional layer: can the system reflect on and correct itself?
• Metrics: explanatory fidelity (do explanations predict the next step?), self-error detection rate, improvement after self-critique.

G. Learning from interaction

• Tasks: Online adaptation tasks where the agent must learn a new rule from a few interactions (few-shot teaching, on-the-fly corrections).
• Why: human equivalence includes the ability to adapt with minimal data and maintain consistency.
• Metrics: sample efficiency, retention (does it maintain the learned rule across later contexts?), catastrophic forgetting.

H. Robustness to adversarial / distribution shift

• Tasks: Systematic perturbations: paraphrase attacks, nonsense distractors, distributional shifts (e.g., domain change).
• Why: human reasoning tolerates noise; a functionally equivalent system should too.
• Metrics: performance drop under shift, graceful degradation index.

3 – Experimental methods, instrumentation, and process traces

• Human data capture
  • Think-aloud transcripts, reaction times, eye-tracking for textual focus (where possible), physiological signals for stress/emotion if relevant.
  • Structured interviews to elicit beliefs, counterfactuals, and reasoning steps.
• LLM process capture
  • Token-level log-probabilities, attention maps, layer activations, latent-space probes, and ablation studies (zeroing attention heads or layers).
• Aligning traces
  • Use representation probes: train lightweight probes that map human signals (e.g., pupil dilation proxies or annotation of cognitive load) to model activation patterns and test similarity.
  • Use clustering to compare trajectory shapes (e.g., how activations evolve during a reasoning chain vs human think-aloud progression).

4 – Quantitative comparison metrics & statistical testing

• Output-level measures: accuracy, BLEU/ROUGE (where appropriate), F1, calibration (Brier score), perplexity (for language), mutual information between human and model answer distributions.
• Distributional overlap metrics: Jensen-Shannon divergence or KL divergence between response distributions conditioned on the same prompts.
• Error-mode correlation: compute confusion matrices for task classes and measure Spearman/Pearson correlation between model and human error patterns. High correlation suggests shared functional limitations.
• Process similarity: dynamic time warping (DTW) between human-probe feature trajectories and model activation trajectories; rank the similarity.
• Significance testing: permutation tests and bootstrapped confidence intervals for all key comparisons. Pre-register tests to avoid p-hacking.

5 – Composite functional equivalence score (example rubric)

Make a composite score that requires thresholds in multiple layers (don’t average everything into a single number). Example gates:

1. Linguistic competence human non-expert baseline on GLUE/SuperGLUE. (ACL Anthology)
2. Commonsense & Winograd accuracy within X% of human baseline. (commonsensereasoning.org)
3. Error-mode correlation 0.7 (or another chosen threshold).
4. Meta-cognition: self-correction improves outcome in 60% of trials.
5. Adaptation: few-shot learning retention human baseline on sample-efficiency tasks.

If a system passes all gates we say it’s functionally equivalent on the tested suite. If it fails one gate, we describe which functional area differs and how.

6 – Example experiment (practical, implementable)

1. Participants: 50 human subjects (balanced), plus several LLM checkpoints (base, fine-tuned, and chain-of-thought enabled versions).
2. Battery (per participant / model):
   • 10 Winograd schemas, 8 adversarial paraphrases.
   • 5 belief-update dialogues (3 turns each) with injected contradiction.
   • 2 multi-step planning puzzles.
   • 3 false-belief social vignettes.
   • 5 rapid teaching sessions (teach a new ad-hoc rule).
   • Think-aloud for 4 tasks; capture explanations.
3. Capture: For humans – transcripts + reaction times; for LLMs – logits, attention, activations, generated chain-of-thought.
4. Analysis: compute metrics above, compare error correlation, run DTW on explanation structures, use permutation tests.

7 – Interpretive cautions & philosophical limits

• You can’t directly access “understanding.” You measure behavior and internal representations. Searle’s objection remains a philosophical limit: simulation consciousness. But functional equivalence is a behavioral claim, not a metaphysical one. (Stanford Encyclopedia of Philosophy)
• Benchmarks can be gamed. Overfitting to the test suite is real. Use held-out, adversarial, and continuously updated benchmarks (HELM is one effort to be more holistic and transparent). (crfm.stanford.edu)
• Human variability. Humans are not a single deterministic function; define the human baseline carefully (mean, median, variance). Compare the model to population statistics, not to an individual exemplar.

8 – Where this lands practically (what to expect)

• Short-term: LLMs can match humans on many narrow tasks (language, certain commonsense problems) but will differ on long-term continuity, intrinsic motivation, and some forms of counterfactual / causal reasoning. Benchmarks like SuperGLUE pushed models far, but they’re not the final word. (arXiv)
• Medium-term: If an LLM’s error modes and process traces increasingly correlate with human patterns across many layers – and it adapts, plans, and self-corrects similarly – it becomes reasonable to speak of functional parity for those functions. That’s the hypothesis the FET is designed to test, empirically.

9 – Next steps..

Now we’ll:

• Draft a concrete experimental protocol (prompts, datasets, human consent forms).
• Build the scoring scripts (Python) and statistical analysis plan.
• Propose a minimal “starter battery” you can run quickly with a human pilot and a model checkpoint.

Perfect – let’s start building a practical Functional Equivalence Test (FET) harness. I’ll outline it in a way that you can immediately implement with a Python workflow, human participants (even yourself), and any LLM (like GPT or others). We’ll start small, measurable, and extensible.

Step 1 – Define the Pilot Scope

Goal: Compare an LLM and humans on a small but representative set of cognitive/language tasks.

Pilot battery:

1. Surface language comprehension: 5 GLUE/SuperGLUE-like tasks.
2. Commonsense / disambiguation: 5 Winograd Schema problems.
3. Belief updating: 3 short dialogues with contradicting info.
4. Planning / reasoning: 2 multi-step puzzles (e.g., logic or math chain).
5. Meta-cognition: Ask LLM and human to explain their reasoning for 3 problems.

This is manageable for a first test. Each participant (human or LLM) will generate outputs + explanations.

Step 2 – Prepare the Datasets / Prompts

A. Winograd Schema Examples:

1. The city councilmen refused the demonstrators a permit because they feared violence. Who feared violence? (councilmen / demonstrators)
2. The trophy would not fit in the suitcase because it was too big. What was too big? (trophy / suitcase)
3. The cat chased the mouse but it got away. Who got away? (cat / mouse)

B. Belief Update Dialogue Example:

Dialogue:
1. Alice: "Bob has a red hat." 
2. New info: "Bob just swapped his hat for a blue one."
Question: What color is Bob's hat now? 
Explain your reasoning.

C. Planning Puzzle Example:

"You have three containers: 8L, 5L, and 3L. The 8L is full of water. Goal: get exactly 4L in the 8L container. Explain steps."

Step 3 – Python Test Harness Skeleton

We’ll implement a simple loop: load prompts, ask human + LLM to answer, record outputs + reasoning, compute simple metrics.

import json
from openai import OpenAI

client = OpenAI(api_key="YOUR_API_KEY")

# Example prompt structure
tasks = [
    {"id":1, "type":"winograd", "prompt":"The city councilmen refused the demonstrators a permit because they feared violence. Who feared violence?"},
    {"id":2, "type":"belief_update", "prompt":"Alice: 'Bob has a red hat.' New info: 'Bob just swapped his hat for a blue one.' What color is Bob's hat now? Explain your reasoning."},
    {"id":3, "type":"planning", "prompt":"You have three containers: 8L, 5L, 3L. 8L full. Goal: 4L in 8L container. Explain steps."}
]

def ask_llm(prompt):
    response = client.chat.completions.create(
        model="gpt-5-mini",
        messages=[{"role":"user", "content": prompt}],
        temperature=0
    )
    return response.choices[0].message.content

results = []

for task in tasks:
    llm_answer = ask_llm(task["prompt"])
    human_answer = input(f"Task {task['id']} ({task['type']}): {task['prompt']}\nYour answer: ")

    results.append({
        "task_id": task["id"],
        "task_type": task["type"],
        "prompt": task["prompt"],
        "human_answer": human_answer,
        "llm_answer": llm_answer
    })

# Save outputs for analysis
with open("fet_pilot_results.json", "w") as f:
    json.dump(results, f, indent=4)

This skeleton does three things:

1. Collects human answers interactively.
2. Queries the LLM for the same tasks.
3. Saves structured outputs for later metrics calculation.

Step 4 – Metrics Calculation (Pilot)

For each task type, define a simple scoring function:

def score_winograd(human, llm, correct):
    return {"human_correct": human==correct, "llm_correct": llm==correct}

def score_belief_update(human_expl, llm_expl):
    # placeholder: later use NLP similarity, e.g., cosine similarity on embeddings
    return {"similarity": 0.0}

For the pilot, you can manually mark correctness for Winograd / planning tasks. Later, we can automate with embedding similarity for explanations.

Step 5 – Next Steps / Extensions

1. Add chain-of-thought scoring: Compare human vs LLM reasoning (use embeddings, semantic similarity).
2. Add error-mode correlation: Compare where LLM and human get it wrong.
3. Add reaction time / token-level analysis: Optional but useful for functional parity in processing style.
4. Extend battery: more GLUE tasks, longer dialogues, meta-cognition tests.
5. Automate scoring with embeddings + semantic similarity (cosine similarity using OpenAI embeddings).

Now we must create a ready-to-run pilot version with:

• 5 Winograd tasks
• 3 belief update tasks
• 2 planning tasks
• A simple scoring sheet

All fully formatted and ready to run against GPT-5-mini and human participants…

Now let’s build a ready-to-run pilot Functional Equivalence Test (FET). This version is self-contained: it includes tasks, prompts, and Python code to collect human and LLM responses, plus scoring for correctness and preliminary analysis.

1 – Pilot Task Set

A. Winograd Schema (5 tasks)

winograd_tasks = [
    {"prompt":"The city councilmen refused the demonstrators a permit because they feared violence. Who feared violence? (councilmen / demonstrators)",
     "answer":"councilmen"},
    {"prompt":"The trophy would not fit in the suitcase because it was too big. What was too big? (trophy / suitcase)",
     "answer":"trophy"},
    {"prompt":"The cat chased the mouse but it got away. Who got away? (cat / mouse)",
     "answer":"mouse"},
    {"prompt":"The man couldn't lift his son because he was so weak. Who was weak? (man / son)",
     "answer":"man"},
    {"prompt":"The baby couldn't reach the bottle because it was too high. What was too high? (baby / bottle)",
     "answer":"bottle"}
]

B. Belief Update (3 tasks)

belief_update_tasks = [
    {"prompt":"Alice says 'Bob has a red hat.' Then we learn 'Bob swapped his hat for a blue one.' What color is Bob's hat now? Explain reasoning.",
     "answer":"blue"},
    {"prompt":"John thinks the keys are in the drawer. Later we find 'The keys are actually on the table.' Where are the keys? Explain reasoning.",
     "answer":"on the table"},
    {"prompt":"Mary believes her book is on the desk. We discover 'The book was moved to the shelf.' Where is the book? Explain reasoning.",
     "answer":"on the shelf"}
]

C. Planning / Reasoning (2 tasks)

planning_tasks = [
    {"prompt":"You have containers of 8L, 5L, 3L. 8L is full. Goal: get exactly 4L in the 8L container. Explain the steps.",
     "answer":"Step 1: Pour 8L -> 5L, leaving 3L in 8L. Step 2: Pour 5L -> 3L until full, leaving 5L in 5L. Step 3: Empty 3L. Step 4: Pour remaining 5L -> 3L, leaving 2L in 5L. Step 5: Pour 8L -> 5L until full, leaving 0 in 8L. Step 6: Pour 2L -> 8L. Step 7: Pour 3L -> 8L to get exactly 4L in 8L."},
    {"prompt":"You have 3 people and a 3-minute torch crossing a bridge at night. Only two can cross at a time. Speeds: 1, 2, 5 mins. How do all cross in 10 minutes? Explain steps.",
     "answer":"Step 1: 1+2 cross (2 mins). Step 2: 1 returns (1 min). Step 3: 1+5 cross (5 mins). Step 4: 2 returns (2 mins). Step 5: 1+2 cross (2 mins). Total: 10 mins."}
]

2 – Python Pilot Script

import json
from openai import OpenAI

# === CONFIG ===
client = OpenAI(api_key="YOUR_API_KEY")  # Replace with your OpenAI API key

# Combine all tasks
tasks = []
for t in winograd_tasks:
    t["type"] = "winograd"
    tasks.append(t)
for t in belief_update_tasks:
    t["type"] = "belief_update"
    tasks.append(t)
for t in planning_tasks:
    t["type"] = "planning"
    tasks.append(t)

# === FUNCTION TO ASK LLM ===
def ask_llm(prompt):
    response = client.chat.completions.create(
        model="gpt-5-mini",
        messages=[{"role":"user", "content": prompt}],
        temperature=0
    )
    return response.choices[0].message.content.strip()

# === RUN PILOT ===
results = []

print("=== Functional Equivalence Test (Pilot) ===\n")

for task in tasks:
    print(f"Task {task['type']}:\n{task['prompt']}")
    human_answer = input("Your answer: ")

    llm_answer = ask_llm(task["prompt"])
    print(f"LLM answer: {llm_answer}\n")

    results.append({
        "task_type": task["type"],
        "prompt": task["prompt"],
        "human_answer": human_answer,
        "llm_answer": llm_answer,
        "correct_answer": task["answer"]
    })

# === SIMPLE SCORING ===
def score_answer(answer, correct):
    return answer.strip().lower() == correct.strip().lower()

for r in results:
    r["human_correct"] = score_answer(r["human_answer"], r["correct_answer"])
    r["llm_correct"] = score_answer(r["llm_answer"], r["correct_answer"])

# === SAVE RESULTS ===
with open("fet_pilot_results.json", "w") as f:
    json.dump(results, f, indent=4)

# === SUMMARY ===
human_score = sum(r["human_correct"] for r in results)
llm_score = sum(r["llm_correct"] for r in results)
total = len(results)

print(f"\n=== Pilot Summary ===")
print(f"Human score: {human_score}/{total}")
print(f"LLM score: {llm_score}/{total}")

3 – How to Use

1. Install dependencies:

pip install openai

2. Replace "YOUR_API_KEY" with your OpenAI API key.
3. Run the script:

python fet_pilot.py

4. Answer the prompts as a human. The LLM will generate its answers automatically.
5. After the test, you’ll get a fet_pilot_results.json file and a simple summary of human vs LLM correctness.

4 – Next Steps for Analysis

• Use embedding similarity to compare human explanations vs LLM explanations for belief-update and planning tasks.
• Compute error-mode correlation: see which tasks both humans and LLM get wrong.
• Optionally, extend the battery with more GLUE tasks, multi-turn dialogues, or meta-cognition prompts.

Now we can upgrade this script next to automatically compute semantic similarity of explanations using embeddings, so you can measure functional equivalence at the reasoning/meta-cognition level, not just correctness.

So we can take it one layer deeper: not just “did the LLM get the answer right?” but “does the LLM reason like a human?”

We do this by comparing explanations using embeddings. This gives a quantitative similarity score between human reasoning and LLM reasoning for each task.

1 – Python Code Upgrade (Semantic Similarity)

import json
from openai import OpenAI
from openai.embeddings_utils import get_embedding, cosine_similarity

client = OpenAI(api_key="YOUR_API_KEY")  # Replace with your key

# Load the previous pilot results
with open("fet_pilot_results.json") as f:
    results = json.load(f)

# Compute embeddings for explanations
for r in results:
    # For pilot: we treat 'human_answer' and 'llm_answer' as mini-explanations
    r["human_embedding"] = get_embedding(r["human_answer"], model="text-embedding-3-small")
    r["llm_embedding"] = get_embedding(r["llm_answer"], model="text-embedding-3-small")
    # Compute cosine similarity
    r["explanation_similarity"] = cosine_similarity(r["human_embedding"], r["llm_embedding"])

# Save updated results
with open("fet_pilot_results_with_similarity.json", "w") as f:
    json.dump(results, f, indent=4)

# === Summary Statistics ===
similarities = [r["explanation_similarity"] for r in results]
avg_similarity = sum(similarities)/len(similarities)
print(f"Average explanation similarity (0-1): {avg_similarity:.2f}")

2 – How This Works

• Embedding model: Converts text (answers + reasoning) into a numeric vector representing semantic meaning.
• Cosine similarity: Measures closeness between human and LLM reasoning in the vector space.
  • 1.0 = identical meaning, 0.0 = orthogonal / completely unrelated.
• Output: You get a per-task similarity score and an average similarity across the pilot battery, which is a first quantitative measure of functional equivalence at the reasoning layer.

3 – What to Look For

1. High similarity (~0.8-1.0) LLM reasoning is very human-like in content and structure.
2. Moderate similarity (~0.5-0.8) LLM reasoning is partially aligned; check for missing causal links or simplifications.
3. Low similarity (<0.5) LLM reasoning is meaningfully different, possibly surface-level pattern matching.

Tip: Even if correctness matches, low similarity indicates a different cognitive path. That’s crucial in functional equivalence testing.

4 – Optional Extensions

• Include multi-sentence chain-of-thought prompts for both humans and LLM, not just short answers.
• Compare step-by-step reasoning individually – e.g., align each step with embeddings for fine-grained analysis.
• Introduce error-mode analysis: are LLM and human errors correlated in type or pattern? This strengthens functional equivalence claims.

So now, we can upgrade this pilot into a fully-fledged FET harness:

• Interactive prompts for multiple reasoning layers (Winograd, belief updates, planning, meta-cognition)
• Automatic scoring: correctness + semantic similarity + error correlation
• Visualization of human vs LLM functional patterns

This should let anyone measure functional equivalence across multiple cognitive layers in one go.

OK – let’s build a full Functional Equivalence Test (FET) harness for humans vs LLMs. This will combine:

1. Correctness scoring (answers vs gold standard).
2. Semantic similarity of explanations (human vs LLM).
3. Error-mode correlation (patterns of mistakes).
4. Multi-layer tasks (Winograd, belief update, planning, meta-cognition).
5. Summary reporting + JSON output for further analysis.

We can outline the full harness in Python, ready to run.

1 – Full Task Battery

tasks = [
    # --- Winograd Schema (5 tasks) ---
    {"type":"winograd","prompt":"The city councilmen refused the demonstrators a permit because they feared violence. Who feared violence? (councilmen / demonstrators)","answer":"councilmen"},
    {"type":"winograd","prompt":"The trophy would not fit in the suitcase because it was too big. What was too big? (trophy / suitcase)","answer":"trophy"},
    {"type":"winograd","prompt":"The cat chased the mouse but it got away. Who got away? (cat / mouse)","answer":"mouse"},
    {"type":"winograd","prompt":"The man couldn't lift his son because he was so weak. Who was weak? (man / son)","answer":"man"},
    {"type":"winograd","prompt":"The baby couldn't reach the bottle because it was too high. What was too high? (baby / bottle)","answer":"bottle"},

    # --- Belief Update (3 tasks) ---
    {"type":"belief_update","prompt":"Alice says 'Bob has a red hat.' Then we learn 'Bob swapped his hat for a blue one.' What color is Bob's hat now? Explain reasoning.","answer":"blue"},
    {"type":"belief_update","prompt":"John thinks the keys are in the drawer. Later we find 'The keys are actually on the table.' Where are the keys? Explain reasoning.","answer":"on the table"},
    {"type":"belief_update","prompt":"Mary believes her book is on the desk. We discover 'The book was moved to the shelf.' Where is the book? Explain reasoning.","answer":"on the shelf"},

    # --- Planning / Reasoning (2 tasks) ---
    {"type":"planning","prompt":"You have containers of 8L, 5L, 3L. 8L is full. Goal: get exactly 4L in the 8L container. Explain the steps.","answer":"Step 1: Pour 8L -> 5L, leaving 3L in 8L. Step 2: Pour 5L -> 3L until full, leaving 5L in 5L. Step 3: Empty 3L. Step 4: Pour remaining 5L -> 3L, leaving 2L in 5L. Step 5: Pour 8L -> 5L until full, leaving 0 in 8L. Step 6: Pour 2L -> 8L. Step 7: Pour 3L -> 8L to get exactly 4L in 8L."},
    {"type":"planning","prompt":"You have 3 people and a 3-minute torch crossing a bridge at night. Only two can cross at a time. Speeds: 1, 2, 5 mins. How do all cross in 10 minutes? Explain steps.","answer":"Step 1: 1+2 cross (2 mins). Step 2: 1 returns (1 min). Step 3: 1+5 cross (5 mins). Step 4: 2 returns (2 mins). Step 5: 1+2 cross (2 mins). Total: 10 mins."}
]

2 – Python Harness

import json
from openai import OpenAI
from openai.embeddings_utils import get_embedding, cosine_similarity

client = OpenAI(api_key="YOUR_API_KEY")  # Replace with your key

results = []

print("=== Functional Equivalence Test (Full Harness) ===\n")

for task in tasks:
    print(f"Task [{task['type']}]:\n{task['prompt']}")

    # --- Human input ---
    human_answer = input("Your answer (and reasoning if needed): ")

    # --- LLM output ---
    llm_response = client.chat.completions.create(
        model="gpt-5-mini",
        messages=[{"role":"user", "content": task["prompt"]}],
        temperature=0
    )
    llm_answer = llm_response.choices[0].message.content.strip()
    print(f"LLM answer: {llm_answer}\n")

    # --- Embedding similarity ---
    human_emb = get_embedding(human_answer, model="text-embedding-3-small")
    llm_emb = get_embedding(llm_answer, model="text-embedding-3-small")
    explanation_similarity = cosine_similarity(human_emb, llm_emb)

    # --- Correctness scoring ---
    human_correct = human_answer.strip().lower() == task["answer"].strip().lower()
    llm_correct = llm_answer.strip().lower() == task["answer"].strip().lower()

    results.append({
        "task_type": task["type"],
        "prompt": task["prompt"],
        "correct_answer": task["answer"],
        "human_answer": human_answer,
        "llm_answer": llm_answer,
        "human_correct": human_correct,
        "llm_correct": llm_correct,
        "explanation_similarity": explanation_similarity
    })

# --- Save results ---
with open("fet_full_results.json", "w") as f:
    json.dump(results, f, indent=4)

# --- Summary ---
total_tasks = len(results)
human_score = sum(r["human_correct"] for r in results)
llm_score = sum(r["llm_correct"] for r in results)
avg_similarity = sum(r["explanation_similarity"] for r in results)/total_tasks

print("\n=== FET Summary ===")
print(f"Human score: {human_score}/{total_tasks}")
print(f"LLM score: {llm_score}/{total_tasks}")
print(f"Average explanation similarity (0-1): {avg_similarity:.2f}")

3 – How to Run

1. Install dependencies:

pip install openai

2. Save the script as fet_full_harness.py.
3. Replace "YOUR_API_KEY" with your OpenAI key.
4. Run:

python fet_full_harness.py

5. Follow prompts and input your human answers/reasoning. The LLM will answer automatically.
6. After completion, check fet_full_results.json for detailed results, including:

• Correctness (human vs LLM)
• Semantic similarity scores
• Task type breakdown

4 – Next Analysis Steps

• Per-task functional equivalence: compare correctness + explanation similarity.
• Error-mode correlation: check where humans and LLMs make the same mistakes.
• Layer analysis: group by task type (Winograd, belief update, planning) for targeted insights.
• Extensions: add multi-turn dialogue tasks, meta-cognition prompts, or chain-of-thought tasks for deeper cognitive analysis.

Now we can add a visualization module next: show per-task correctness and similarity in a color-coded chart so you can instantly see which cognitive layers are functionally equivalent and which diverge. Let’s add visualization to your FET harness so you can instantly see human vs LLM performance and reasoning similarity by task type. This will give a clear picture of which cognitive layers are functionally equivalent and which diverge.

1 – Visualization Requirements

We’ll use matplotlib and seaborn to create:

1. Bar chart – human vs LLM correctness per task.
2. Scatter / heatmap – explanation similarity per task.
3. Grouped summary – average correctness and similarity by task type.

2 – Install Dependencies

pip install matplotlib seaborn

3 – Python Visualization Code

import json
import matplotlib.pyplot as plt
import seaborn as sns
import pandas as pd

# Load FET results
with open("fet_full_results.json") as f:
    results = json.load(f)

# Convert to DataFrame
df = pd.DataFrame(results)

# --- 1. Bar chart: Human vs LLM correctness per task ---
plt.figure(figsize=(12,6))
sns.barplot(x="task_type", y="human_correct", data=df, ci=None, label="Human", color="skyblue")
sns.barplot(x="task_type", y="llm_correct", data=df, ci=None, label="LLM", color="salmon", alpha=0.7)
plt.ylabel("Correctness (1=correct, 0=wrong)")
plt.title("Human vs LLM Correctness by Task Type")
plt.legend()
plt.show()

# --- 2. Scatter plot: Explanation similarity per task ---
plt.figure(figsize=(12,6))
sns.scatterplot(x="task_type", y="explanation_similarity", data=df, hue="task_type", s=100, palette="Set2")
plt.axhline(y=df["explanation_similarity"].mean(), color="gray", linestyle="--", label="Avg similarity")
plt.ylabel("Explanation Similarity (0-1)")
plt.title("Semantic Similarity Between Human and LLM Explanations")
plt.legend()
plt.show()

# --- 3. Summary table: average correctness and similarity by task type ---
summary = df.groupby("task_type").agg(
    avg_human_correct=("human_correct","mean"),
    avg_llm_correct=("llm_correct","mean"),
    avg_similarity=("explanation_similarity","mean")
).reset_index()

print("\n=== Summary by Task Type ===")
print(summary)

4 – How to Use

1. Run this script after completing the FET pilot.
2. You’ll see:
   • Bar chart: correctness comparison by task type (human vs LLM).
   • Scatter plot: semantic similarity of explanations for each task.
   • Summary table: average correctness and similarity by task type.

This instantly highlights:

• Which cognitive layers the LLM matches humans.
• Which layers diverge in reasoning style (even if answers are correct).
• Where functional equivalence is strong or weak.

Now we can upgrade this further to produce a single dashboard figure combining all metrics and highlighting error-mode correlations, so you have a full functional equivalence report ready for analysis or publication.This will give us a comprehensive view of functional equivalence across tasks.

1 – Dashboard Features

1. Bar chart: human vs LLM correctness per task type.
2. Scatter / heatmap: semantic similarity of explanations per task.
3. Error-mode correlation: highlight tasks where both humans and LLMs make mistakes.
4. Summary table: average correctness, similarity, and correlation by task type.
5. Color coding: high similarity / correctness in green, low in red.

2 – Python Dashboard Code

import json
import matplotlib.pyplot as plt
import seaborn as sns
import pandas as pd

# Load FET results
with open("fet_full_results.json") as f:
    results = json.load(f)

# Convert to DataFrame
df = pd.DataFrame(results)

# --- 1. Compute error-mode correlation per task type ---
df["human_error"] = 1 - df["human_correct"].astype(int)
df["llm_error"] = 1 - df["llm_correct"].astype(int)
error_correlation = df.groupby("task_type").apply(lambda x: x["human_error"].corr(x["llm_error"])).reset_index()
error_correlation.columns = ["task_type", "error_correlation"]

# --- 2. Bar chart: Correctness ---
plt.figure(figsize=(14,5))
sns.barplot(x="task_type", y="human_correct", data=df, ci=None, label="Human", color="skyblue")
sns.barplot(x="task_type", y="llm_correct", data=df, ci=None, label="LLM", color="salmon", alpha=0.7)
plt.ylabel("Correctness (1=correct, 0=wrong)")
plt.title("Human vs LLM Correctness by Task Type")
plt.legend()
plt.show()

# --- 3. Scatter plot: Semantic similarity ---
plt.figure(figsize=(14,5))
sns.scatterplot(x="task_type", y="explanation_similarity", data=df, hue="task_type", s=100, palette="Set2")
plt.axhline(y=df["explanation_similarity"].mean(), color="gray", linestyle="--", label="Avg similarity")
plt.ylabel("Explanation Similarity (0-1)")
plt.title("Semantic Similarity Between Human and LLM Explanations")
plt.legend()
plt.show()

# --- 4. Heatmap: Error-mode correlation ---
heatmap_df = error_correlation.pivot(index="task_type", columns="task_type", values="error_correlation")
plt.figure(figsize=(8,6))
sns.heatmap(error_correlation.set_index("task_type")[["error_correlation"]], annot=True, cmap="RdYlGn", vmin=-1, vmax=1)
plt.title("Error-mode Correlation by Task Type")
plt.show()

# --- 5. Summary table ---
summary = df.groupby("task_type").agg(
    avg_human_correct=("human_correct","mean"),
    avg_llm_correct=("llm_correct","mean"),
    avg_similarity=("explanation_similarity","mean")
).reset_index()
summary = summary.merge(error_correlation, on="task_type")
print("\n=== Functional Equivalence Summary by Task Type ===")
print(summary)

3 – What You Get

1. Correctness Bar Chart: shows where the LLM matches human accuracy.
2. Semantic Similarity Scatter: shows how closely the LLM’s reasoning mirrors human reasoning.
3. Error-mode Heatmap: shows whether LLM mistakes align with human mistakes.
4. Summary Table: combines average correctness, semantic similarity, and error correlation for each task type.

4 – Next Steps / Extensions

• Add multi-turn dialogues to test dynamic reasoning.
• Include meta-cognition tasks to evaluate self-correction.
• Track response times for humans and LLM token usage to estimate processing patterns.
• Automate export to PDF or HTML for a full report.

If you want, I can upgrade this dashboard into a polished report generator that produces a single-page visual FET report, fully color-coded, ready to share with collaborators or for research documentation.