BCP: Budget-Constrained Perception

A discrete attention allocator for resource-constrained decision making. Given a budget, BCP tells you what to attend to and what to ignore.

The BCP Equation

Implementation: bcp/core.py:156-222

The allocation maximizes total value over the attended set:

TotalScore = Σ_attended [Gain(a) - λ(B) × Cost(a)] - γ × SetComplexity

Note: In earlier DUALITY-ZERO documentation, this equation is written as V(a) = G(a) - λ(B)·C(a). Here we use Gain(a) and Cost(a) for clarity.

Where:

• Gain(a): Expected benefit of attending to item a (0.0 - 1.0)
• λ(B) = k / (ε + B): Metabolic pressure (inverse of budget)
• Cost(a): Resource cost to attend to item a
• SetComplexity = log(1 + N): Global overhead from evaluating N options

For ranking items, we use the per-item score:

Score(a) = Gain(a) - λ(B) × Cost(a)

Items with positive Score are candidates for attention. The SetComplexity term is a global penalty applied once per decision, not per item—it does not affect relative ranking.

Sign convention: BCP defines Score as a value to maximize. Positive Score → attend. Negative Score → ignore.

Installation

pip install bcp-perception

Or install from source:

git clone https://github.com/mrdirno/nested-resonance-memory-archive.git
cd nested-resonance-memory-archive/bcp_lib
pip install -e .

Quick Start

from bcp import AttentionItem, BCPModel

# Create items with gain (benefit) and cost (effort)
items = [
    AttentionItem("Critical bug", gain=1.0, cost=0.3),
    AttentionItem("New feature", gain=0.7, cost=0.6),
    AttentionItem("Refactor", gain=0.5, cost=0.8),
    AttentionItem("Documentation", gain=0.4, cost=0.4),
]

# Create model and allocate with budget
model = BCPModel()
result = model.allocate(items, budget=1.0)

print(f"Phase: {result.phase.value}")
print(f"Attend to: {result.attended}")
print(f"Ignore: {result.ignored}")

Output:

Phase: scarcity
Attend to: ['Critical bug', 'New feature']
Ignore: ['Refactor', 'Documentation']

Key Concepts

Phase Transitions

Implementation: bcp/core.py:147-154 | Tests: tests/test_core.py:87-103

BCP predicts three distinct phases based on resource availability:

Phase	Default Threshold	Behavior
ABUNDANCE	B > 2.0	Attend to all positive-Score items
SCARCITY	0.5 < B < 2.0	Triage begins—ignore low-value items
CRISIS	B < 0.5	Focus on one or few highest-Score items

Note: Thresholds (2.0 and 0.5) are configurable defaults, not universal constants. Override via BCPModel(abundance_threshold=..., crisis_threshold=...) to match your domain's budget scale.

Metabolic Pressure (λ)

Lambda controls cost sensitivity. The relationship λ(B) = k / (ε + B) means:

• Low λ (high budget): Costs matter less, attend broadly
• High λ (low budget): Costs dominate, strict triage

# Defaults: lambda_scale=10.0, lambda_epsilon=0.1
# These produce:
#   budget = 5.0  → lambda ≈ 1.96  (low pressure)
#   budget = 0.5  → lambda ≈ 16.67 (high pressure)

model = BCPModel(lambda_scale=10.0, lambda_epsilon=0.1)

print(model.compute_lambda(5.0))   # ~1.96
print(model.compute_lambda(0.5))   # ~16.67

Set Complexity

The γ × SetComplexity term models the cognitive/computational overhead of evaluating many options:

SetComplexity = log(1 + N)

Where N is the number of items under consideration. This term:

• Is applied globally to the decision, not per-item
• Does not affect relative ranking between items
• Captures that overhead scales sublinearly with option count
• Default γ = 0.1 makes this a minor adjustment; increase for domains where option count matters more

Domain Presets

Implementation: bcp/domains.py | Tests: tests/test_core.py:200-227

Pre-configured scenarios inspired by use cases studied in the DUALITY-ZERO program:

from bcp import DOMAIN_PRESETS, BCPModel

model = BCPModel()

# Available domains
domains = ["finance", "medical", "education", "diplomacy",
           "ecosystem", "software", "emergency", "moderation",
           "manufacturing"]

# Use a preset
items = DOMAIN_PRESETS["medical"]()
result = model.allocate(items, budget=0.5)

Advanced Usage

Budget Sweep

Implementation: bcp/core.py:224-278 | Tests: tests/test_core.py:258-294

Analyze behavior across budget range:

import numpy as np
from bcp import BCPModel, AttentionItem

def create_items():
    return [
        AttentionItem("A", gain=0.9, cost=0.3),
        AttentionItem("B", gain=0.7, cost=0.5),
        AttentionItem("C", gain=0.5, cost=0.4),
    ]

model = BCPModel()
budgets = np.linspace(0.1, 3.0, 30)
results = model.sweep_budgets(create_items, budgets)

# Find phase transition thresholds
triage_threshold, crisis_threshold = model.find_phase_thresholds(
    create_items, budgets
)

Real-Time Monitoring

Implementation: bcp/monitor.py

Monitor system resources with BCP-based triage:

from bcp import BCPMonitor

monitor = BCPMonitor()

# Add monitoring tasks
monitor.add_task("cpu", gain=0.9, cost=0.1,
                 collector=lambda: get_cpu_usage())
monitor.add_task("memory", gain=0.8, cost=0.2,
                 collector=lambda: get_memory_usage())

# Sample with current budget
sample = monitor.sample(budget=1.0)
print(f"Attended: {sample.attended_tasks}")
print(f"Metrics: {sample.metrics}")

Custom Model Parameters

model = BCPModel(
    lambda_scale=10.0,       # k in λ = k/(ε+B). Higher = sharper transitions
    lambda_epsilon=0.1,      # ε in λ = k/(ε+B). Prevents division by zero
    gamma=0.1,               # Global set complexity coefficient
    abundance_threshold=2.0, # Budget above this = ABUNDANCE phase
    crisis_threshold=0.5,    # Budget below this = CRISIS phase
)

Research Background

BCP was developed through the DUALITY-ZERO research program, tested across 10 stylized scenarios:

1. Finance — Portfolio triage under capital constraints
2. Medical — Emergency triage under staff/time limits
3. Education — Student attention allocation
4. Diplomacy — Negotiation focus under time pressure
5. Ecosystem — Conservation priorities under funding limits
6. Software — Bug triage under engineer capacity
7. Emergency — Disaster response resource allocation
8. Moderation — Content prioritization under queue volume
9. Manufacturing — Quality control sampling
10. Systems — Real-time monitoring task scheduling

Scenarios were implemented as synthetic task sets with domain-inspired gain and cost distributions. In all cases, the same three-phase structure emerged (Abundance → Scarcity → Crisis), with similar threshold behavior up to scaling of the budget axis.

API Reference

Full source: bcp/ | Tests: tests/test_core.py

Classes

AttentionItem(name, gain, cost)

Source: bcp/core.py:20-54

Represents an item that can receive attention.

• name: Identifier
• gain: Expected benefit (0.0 - 1.0)
• cost: Resource cost
• compute_priority(lambda_): Returns per-item Score = Gain - λ × Cost. Used for ranking items.

BCPModel(lambda_scale, lambda_epsilon, gamma, abundance_threshold, crisis_threshold)

Source: bcp/core.py:100-278

The core BCP allocation model.

• compute_lambda(budget): Calculate metabolic pressure λ(B) = k/(ε+B)
• determine_phase(budget): Get current phase based on thresholds
• allocate(items, budget): Perform attention allocation, returns BCPResult
• sweep_budgets(items_fn, budget_range): Test across budget values
• find_phase_thresholds(items_fn, budget_range): Find empirical transition points

BCPResult

Source: bcp/core.py:57-97

Result of allocation containing:

• attended: List of attended item names
• ignored: List of ignored item names
• total_cost: Resources consumed
• total_gain: Sum of gains from attended items
• set_complexity: Global overhead term (γ × log(1+N))
• total_score: TotalScore = Σ(Gain - λ×Cost) - γ×log(1+N)
• phase: Current Phase enum
• lambda_: Computed metabolic pressure
• budget: Input budget value
• n_attended, n_ignored, attention_fraction: Computed properties

BCPMonitor

Source: bcp/monitor.py

Real-time monitoring with BCP triage.

• add_task(name, gain, cost, collector): Register metric
• remove_task(name): Unregister metric
• sample(budget): Take single sample
• run(budget_fn, interval, duration, callback): Continuous monitoring

Enums

Phase

Source: bcp/core.py:13-17

• Phase.ABUNDANCE: High budget, attend to all positive-Score items
• Phase.SCARCITY: Moderate budget, triage active
• Phase.CRISIS: Low budget, focus on highest-Score items

Relationship to Other Work

BCP is the discrete, decision-theoretic formulation of metabolic pressure concepts from the broader DUALITY-ZERO framework. It provides:

• A standalone allocator usable without knowledge of the underlying research
• The same λ(B) structure used in continuous control (Starving Philosopher experiment)
• A bridge between theory and practical engineering use cases

License

GPL-3.0

Author

Aldrin Payopay aldrin.gdf@gmail.com

Repository

https://github.com/mrdirno/nested-resonance-memory-archive