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Extend neural network reasoning capabilities through recurrence (repeated computation cycles), external memory (intermediate state storage), and test-time compute scaling for multi-step reasoning.
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npx skills add https://github.com/ADu2021/skillXiv --skill recurrence-memory-reasoning-depthコマンドは1行のまま表示されます。コピー前に横へスクロールして全体を確認してください。
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Convert arXiv papers into ready-to-use agent skills using category-aware extraction. First classifies the paper into one or more of 11 research categories, then applies a specialized extraction pipeline for each category — because different types of papers produce different types of usable knowledge. A single paper can yield multiple skills if it spans categories. Use this skill whenever the user wants to turn a paper into a skill, extract practical techniques from research, build a skill library from papers, convert arXiv papers into reusable agent instructions, or batch-process multiple papers into skills. Also trigger when someone asks about extracting actionable knowledge from papers, making research practical for LLM agents, or systematically converting academic contributions into structured agent capabilities.
Establish regret bounds for behavior cloning with discretized actions combining statistical error and quantization error terms. Prove smoothness requirements for safe quantizer design, show that learning-based quantizers fail these requirements, and propose model-based augmentation to reduce error dependence from H² to H.
Dynamically allocate LoRA ranks per-layer during fine-tuning instead of using fixed uniform ranks. Learn optimal rank for each layer and subject via variational framework with discretized exponential distribution, reducing memory footprint while maintaining fidelity and text-alignment.
| name | recurrence-memory-reasoning-depth |
| title | Extending Reasoning Depth with Recurrence, Memory and Test-Time Compute |
| version | 0.0.2 |
| engine | skillxiv-v0.0.2-claude-opus-4.6 |
| license | MIT |
| url | https://arxiv.org/abs/2508.16745 |
| keywords | ["reasoning-depth","recurrent-computation","external-memory","test-time-scaling","sequential-reasoning"] |
| description | Extend neural network reasoning capabilities through recurrence (repeated computation cycles), external memory (intermediate state storage), and test-time compute scaling for multi-step reasoning. |
Multi-step reasoning requires computational depth beyond what fixed-depth architectures provide. This skill combines three mechanisms: recurrence (allowing repeated computation passes), external memory (storing intermediate states), and test-time compute scaling (allocating more cycles during inference) to extend effective reasoning depth. Studies on cellular automata and Boolean functions show that while models can memorize next-step predictions, multi-step reasoning requires explicit recurrent computation.
Create iterative computation mechanism:
import torch
import torch.nn as nn
from typing import Tuple, List, Dict
class RecurrentReasoningUnit(nn.Module):
"""Iterative computation for multi-step reasoning."""
def __init__(
self,
state_dim: int = 256,
hidden_dim: int = 512,
num_recurrence_steps: int = 5
):
super().__init__()
self.state_dim = state_dim
self.num_recurrence_steps = num_recurrence_steps
# Core recurrent computation
self.recurrent_cell = nn.GRUCell(state_dim, hidden_dim)
.state_proj = nn.Linear(hidden_dim, state_dim)
.input_gate = nn.Linear(state_dim, hidden_dim)
.forget_gate = nn.Linear(state_dim, hidden_dim)
.output_gate = nn.Linear(state_dim, hidden_dim)
() -> [torch.Tensor, [torch.Tensor]]:
num_steps :
num_steps = .num_recurrence_steps
batch_size = initial_state.shape[]
hidden_state = torch.zeros(batch_size, .recurrent_cell.hidden_size)
current_state = initial_state
state_trajectory = [current_state]
step (num_steps):
input_g = torch.sigmoid(.input_gate(current_state))
forget_g = torch.sigmoid(.forget_gate(current_state))
output_g = torch.sigmoid(.output_gate(current_state))
hidden_state = .recurrent_cell(current_state * input_g, hidden_state)
new_state = .state_proj(hidden_state * output_g)
current_state = current_state * forget_g + new_state * ( - forget_g)
state_trajectory.append(current_state)
current_state, state_trajectory
() -> :
num_layers =
effective_depth = num_layers * num_steps
effective_depth
Store and retrieve intermediate states:
class ExternalMemory(nn.Module):
"""External memory for storing reasoning states."""
def __init__(
self,
state_dim: int = 256,
memory_size: int = 32,
num_read_heads: int = 4
):
super().__init__()
self.state_dim = state_dim
self.memory_size = memory_size
self.num_read_heads = num_read_heads
# Memory storage
self.register_buffer("memory", torch.randn(memory_size, state_dim) * 0.01)
# Read/write mechanisms
self.write_controller = nn.Linear(state_dim, memory_size)
self.read_query = nn.Linear(state_dim, state_dim)
self.key_proj = nn.Linear(state_dim, state_dim)
def write_to_memory(
self,
state: torch.Tensor, # (batch, state_dim)
write_addr: torch.Tensor = None # Which locations to write
) -> torch.Tensor:
"""
Write state to memory with soft addressing.
"""
batch_size = state.shape[0]
if write_addr is None:
# Compute write addresses using attention
write_logits = self.write_controller(state) # (batch, memory_size)
write_weights = torch.softmax(write_logits, dim=-1)
else:
write_weights = write_addr
# Soft write: weighted update
for i in range(batch_size):
for j in range(self.memory_size):
# Gated write: preserve old + add new
self.memory[j] = 0.9 * self.memory[j] + 0.1 * write_weights[i, j] * state[i]
return write_weights
def read_from_memory(
self,
query: torch.Tensor, # (batch, state_dim)
num_read_heads: int = None
) -> torch.Tensor:
"""
Read from memory using content-based addressing.
"""
if num_read_heads is None:
num_read_heads = self.num_read_heads
batch_size = query.shape[0]
# Compute read weights using similarity
query_proj = self.read_query(query) # (batch, state_dim)
key_proj = self.key_proj(self.memory) # (memory_size, state_dim)
# Content-based addressing with multiple heads
read_output = torch.zeros(batch_size, self.state_dim)
for head in range(num_read_heads):
# Partition query and keys for this head
head_query = query_proj[:, :self.state_dim // num_read_heads]
head_keys = key_proj[:, :self.state_dim // num_read_heads]
# Compute attention
scores = torch.matmul(head_query, head_keys.t()) # (batch, memory_size)
weights = torch.softmax(scores / (self.state_dim ** 0.5), dim=-1)
# Read: weighted sum of memory
head_read = torch.matmul(weights, self.memory)
read_output[:, head * self.state_dim // num_read_heads:(head + 1) * self.state_dim // num_read_heads] = head_read[:, :self.state_dim // num_read_heads]
return read_output
def reset_memory(self):
"""Clear memory for new reasoning episode."""
self.memory.zero_()
Allocate variable computation budget during inference:
class TestTimeComputeScaler:
"""Dynamically allocate compute budget at test time."""
def __init__(
self,
base_steps: int = 5,
max_steps: int = 50,
budget_per_example: float = 1.0 # Relative budget
):
self.base_steps = base_steps
self.max_steps = max_steps
self.budget_per_example = budget_per_example
def get_step_budget(
self,
task_description: str,
task_complexity: float = 0.5
) -> int:
"""
Compute number of recurrence steps based on task complexity.
"""
# Complexity-based scaling
complexity_budget = int(self.base_steps * (1.0 + task_complexity * 5.0))
# Cap at maximum
budget = min(complexity_budget, self.max_steps)
return budget
def adaptive_compute(
self,
model: "ReasoningModel",
state: torch.Tensor,
task: str,
target_accuracy: float = 0.95,
max_total_steps: int = 50
) -> Tuple[torch.Tensor, Dict[str, any]]:
"""
Iteratively increase compute until convergence or budget exhausted.
"""
current_state = state
step = 0
previous_output = None
convergence_reached = False
metrics = {"total_steps": 0, "convergence_step": None, "final_output_stable": False}
while step < max_total_steps and not convergence_reached:
# Run recurrence step
current_state, trajectory = model.recurrence_unit(current_state, num_steps=1)
# Check for convergence: output stability
current_output = model.output_head(current_state)
if previous_output is not None:
similarity = torch.nn.functional.cosine_similarity(
current_output.unsqueeze(0),
previous_output.unsqueeze(0)
).item()
if similarity > target_accuracy:
convergence_reached = True
metrics["convergence_step"] = step
metrics["final_output_stable"] = True
previous_output = current_output
step += 1
metrics["total_steps"] = step
return current_state, metrics
def measure_reasoning_quality(
self,
trajectories: List[List[torch.Tensor]],
ground_truth: torch.Tensor
) -> Dict[str, float]:
"""
Measure quality of multi-step reasoning.
"""
metrics = {
"next_step_accuracy": 0.0,
"multistep_accuracy": 0.0,
"trajectory_stability": 0.0
}
for trajectory in trajectories:
# Next-step prediction accuracy
for t in range(len(trajectory) - 1):
pred_next = trajectory[t + 1]
actual_next = ground_truth[t + 1] if t + 1 < len(ground_truth) else None
if actual_next is not None:
match = torch.allclose(pred_next, actual_next, atol=1e-5)
metrics["next_step_accuracy"] += match
# Multi-step accuracy
final_pred = trajectory[-1]
final_actual = ground_truth[-1] if len(ground_truth) > 0 else None
if final_actual is not None:
match = torch.allclose(final_pred, final_actual, atol=1e-5)
metrics["multistep_accuracy"] += match
# Trajectory stability: variance across steps
stacked = torch.stack(trajectory)
variance = torch.var(stacked, dim=0).mean().item()
stability = 1.0 / (1.0 + variance)
metrics["trajectory_stability"] += stability
# Average
n = len(trajectories)
for key in metrics:
metrics[key] = metrics[key] / n if n > 0 else 0.0
return metrics
Combine all components:
class DeepReasoningModel(nn.Module):
"""Complete model with recurrence, memory, and test-time scaling."""
def __init__(
self,
input_dim: int = 256,
state_dim: int = 256,
hidden_dim: int = 512,
memory_size: int = 32
):
super().__init__()
self.input_proj = nn.Linear(input_dim, state_dim)
self.recurrence_unit = RecurrentReasoningUnit(state_dim, hidden_dim)
self.memory = ExternalMemory(state_dim, memory_size)
self.output_head = nn.Linear(state_dim, input_dim)
self.compute_scaler = TestTimeComputeScaler()
def forward(
self,
input_state: torch.Tensor,
num_steps: int = 5,
use_memory: bool = True
) -> torch.Tensor:
"""Forward pass with recurrence and memory."""
# Project input
state = self.input_proj(input_state)
# Recurrent reasoning
for step in range(num_steps):
# Update state through recurrence
state, _ = self.recurrence_unit(state, num_steps=1)
# Write to memory
if use_memory:
self.memory.write_to_memory(state)
# Read from memory for enrichment
if use_memory:
memory_context = self.memory.read_from_memory(state)
state = state + 0.3 * memory_context # Blend with memory
# Output
output = self.output_head(state)
return output
def solve_with_adaptive_compute(
self,
input_state: torch.Tensor,
task_description: str
) -> Tuple[torch.Tensor, Dict]:
"""Solve with test-time compute scaling."""
# Estimate task complexity
complexity = self._estimate_complexity(task_description)
# Get adaptive budget
budget = self.compute_scaler.get_step_budget(task_description, complexity)
# Solve with adaptive steps
output, metrics = self.compute_scaler.adaptive_compute(
self,
input_state,
task_description,
max_total_steps=budget
)
return output, metrics
def _estimate_complexity(self, task: str) -> float:
"""Estimate task complexity from description."""
complexity_indicators = ["recursive", "nested", "multi-step", "chain"]
complexity = sum(1 for ind in complexity_indicators if ind in task.lower()) / len(complexity_indicators)
return min(1.0, complexity)
Researchers. (2024). Beyond Memorization: Extending Reasoning Depth with Recurrence Memory and Test-Time Compute Scaling. arXiv preprint arXiv:2508.16745.