Chain-of-Thought Reasoning

# Chain-of-Thought Reasoning

## Introduction & Motivation

Chain-of-thought: explicit intermediate reasoning steps. Improve reasoning on complex tasks. Applications: improved task performance, interpretability.

Motivation: Enable step-by-step reasoning in language models.

Applications: Mathematical reasoning, logical inference, multi-step problems.

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## Core Concepts & Theory

### Reasoning Steps

Decompose into substeps.

### Intermediate States

Maintain partial solutions.

### Error Correction

Verify steps incrementally.

### Interpretability

Make reasoning transparent.

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## Mathematical Formulation

Reasoning Trajectory:
$$t_1 ightarrow t_2 ightarrow ... ightarrow t_n$$

Performance:
$$ ext{Accuracy}( ext{CoT}) > ext{Accuracy}( ext{direct})$$

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## Advanced Theory & Extensions

### Self-Consistency

Sample multiple paths.

### Verification

Verify reasoning correctness.

### Program-Aided Reasoning

Leverage external tools.

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## Computational Considerations

Extra tokens: Longer sequences.

Inference: Multiple forward passes.

Complexity: Depends on problem.

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## Practical Implementation Strategies

### Prompt Template

Define reasoning format.

### Example Selection

Choose good demonstrations.

### Verification Methods

Check step validity.

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## Benchmark Datasets & Evaluation

Math Word Problems: GSM8K benchmark.

Logical Reasoning: SVAMP.

Complex QA: HotpotQA.

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## Key Challenges & Limitations

### Hallucination

Incorrect reasoning steps.

### Verification Cost

Requires validation.

### Task Dependency

Not effective for all tasks.

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## Hyperparameter Tuning

Number of steps: Task-dependent.

Verification strength: 0.5-1.0.

Sampling temperature: 0.7-1.0.

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## Real-World Applications & Case Studies

Math Solving: Step-by-step solutions.

Code Generation: Decompose into functions.

Planning: Multi-step plans.

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## Integration with Other Methods

Chain-of-thought + program-aided reasoning; + self-consistency.

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## Summary & Key Takeaways

Chain-of-thought improves reasoning capability.

Principles:
1. Explicit steps: Show reasoning.
2. Decomposition: Break into substeps.
3. Verification: Check correctness.
4. Interpretability: Understand reasoning.
5. Scalability: Works with scale.

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## Appendix: Practical Labs

### Lab 1: Decompose Problem

def decompose_reasoning(problem, num_steps=5):
 """Break problem into reasoning steps"""
 steps = []
 for i in range(num_steps):
 step = f"Step {i+1}: {problem}"
 steps.append(step)
 return steps

problem = "Solve 5+3*2"
steps = decompose_reasoning(problem, 3)
assert len(steps) == 3
print(f"✓ Decomposition: {len(steps)} steps")

### Lab 2: Verify Reasoning

def verify_step(step, expected_type):
 """Verify correctness of reasoning step"""
 valid_types = ["calculation", "logic", "deduction"]
 is_valid = expected_type in valid_types
 return is_valid

assert verify_step("5+3", "calculation")
assert not verify_step("5+3", "invalid")
print("✓ Step verification working")

### Lab 3: Multi-Path Sampling

import numpy as np

def sample_reasoning_paths(num_paths=5, num_steps=3):
 """Sample multiple reasoning paths"""
 paths = []
 for p in range(num_paths):
 path = [f"Step{s}" for s in range(num_steps)]
 paths.append(path)
 return paths

paths = sample_reasoning_paths(5, 3)
assert len(paths) == 5
print(f"✓ Sampled {len(paths)} reasoning paths")

### Lab 4: Self-Consistency

import numpy as np

def self_consistency_check(paths, answers):
 """Check consistency across paths"""
 counts = {}
 for answer in answers:
 counts[answer] = counts.get(answer, 0) + 1
 
 # Majority vote
 majority = max(counts, key=counts.get)
 confidence = counts[majority] / len(answers)
 
 return majority, confidence

answers = ["42", "42", "40", "42"]
majority, conf = self_consistency_check([], answers)
assert majority == "42"
assert conf > 0.5
print(f"✓ Self-consistency: {majority} with {conf:.1%} confidence")

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