Chain-of-Thought Prompting

At its core, CoT prompting modifies the prompt fed to the model so that the expected output includes a reasoning chain, not just a bare answer. There are two main flavors:

Few-Shot CoT — You provide several worked examples in the prompt, each showing question → reasoning steps → answer. The model pattern-matches to this structure.

Zero-Shot CoT — You add a trigger phrase like "Let's think step by step" and rely on the model having seen enough reasoning text during pretraining to self-generate steps. The probability the model assigns to the correct final answer $A$ given question $Q$ can be written as:

Explanation: The model's probability of the right answer improves when it first generates an intermediate reasoning chain $z$, because conditioning on $z$ makes the final answer much easier to predict. Here, $z$ is the intermediate reasoning trace (the "chain of thought"), $Q$ is the question, and $A$ is the final answer. Without CoT, the model collapses this sum by going directly from $Q$ to $A$, which is a much harder mapping. A practical self-consistency extension averages over multiple sampled reasoning chains:

Explanation: Generate $k$ different reasoning chains, collect the answer each one produces, and pick the answer that appears most often — the majority vote across chains tends to be more reliable than any single chain. Here $k$ is the number of sampled chains (typically 10–40), and $\mathbf{1}[\cdot]$ is 1 if chain $i$ concludes with answer $A$ and 0 otherwise. Below is a minimal Python example showing both zero-shot and few-shot CoT using the OpenAI API: ```python import openai client = openai.OpenAI() # uses OPENAI_API_KEY env variable QUESTION = "A store sells apples for $0.50 each and oranges for$0.75 each. " \ "If I buy 4 apples and 3 oranges, how much do I spend in total?"

# ── Zero-Shot CoT ───────────────────────────────────────────────────────────── zero_shot_prompt = QUESTION + "\n\nLet's think step by step." zero_shot = client.chat.completions.create( model="gpt-4o", messages=[{"role": "user", "content": zero_shot_prompt}] ) print("=== Zero-Shot CoT ===") print(zero_shot.choices[0].message.content)

# ── Few-Shot CoT ────────────────────────────────────────────────────────────── few_shot_example = """ Q: A bakery makes 12 muffins per batch. They bake 3 batches in the morning and 2 batches in the afternoon. How many muffins did they make? A: Let's think step by step.

• Morning: 3 batches × 12 muffins = 36 muffins
• Afternoon: 2 batches × 12 muffins = 24 muffins
• Total: 36 + 24 = 60 muffins

The answer is 60. Q: {question} A: Let's think step by step. """.format(question=QUESTION) few_shot = client.chat.completions.create( model="gpt-4o", messages=[{"role": "user", "content": few_shot_example}] ) print("\n=== Few-Shot CoT ===") print(few_shot.choices[0].message.content) ``` ```python

# ── Self-Consistency: sample k chains, majority-vote the answer ─────────────── from collections import Counter import re def extract_final_number(text: str) -> str | None: """Pull out the last number mentioned in a CoT response.""" matches = re.findall(r"$?[\d,]+\.?\d*", text) return matches[-1].replace(",", "") if matches else None def self_consistent_answer(question: str, k: int = 5) -> str: prompt = question + "\n\nLet's think step by step." answers = [] for _ in range(k): resp = client.chat.completions.create( model="gpt-4o", messages=[{"role": "user", "content": prompt}], temperature=0.7, # non-zero so chains vary ) ans = extract_final_number(resp.choices[0].message.content) if ans: answers.append(ans) majority, count = Counter(answers).most_common(1)[0] print(f"Votes: {dict(Counter(answers))}") print(f"Majority answer: {majority} (agreed by {count}/{len(answers)} chains)") return majority self_consistent_answer(QUESTION, k=5) ```

Prompt Engineering · In-Context Learning · Emergent Abilities in LLMs · ReAct Prompting · Tree of Thoughts