FoCusNet: Modular Constraint Filtering

• FoCusNet is a modular framework that isolates and filters essential constraints using reusable modules across domains like constraint programming, neural parsing, and Boolean network control.
• It employs techniques such as interval concentration, SCC decomposition, and dynamic programming to optimize efficiency and manage complex constraint sets.
• Empirical results reveal that FoCusNet outperforms traditional methods by significantly boosting accuracy and scalability in constraint filtering applications.

Modular Constraint Filtering (FoCusNet) comprises a suite of principled methodologies for efficient selection, concentration, and filtering of constraints within large-scale constraint satisfaction and control frameworks. The unifying objective is to enable tractable reasoning and solution search within systems burdened by massive, fine-grained, or highly structured constraint sets. FoCusNet is instantiated in domains ranging from classical constraint programming to neural parsing for LLMs and modular control of Boolean networks, consistently leveraging modularity to isolate, filter, and focus on essential constraints. Architecturally, FoCusNet decomposes the constraint processing workflow into reusable modules—either algorithmic propagators, neural selectors, or graph-theoretic filters—allowing efficient scaling and adaptation across different problem families (Boffa et al., 28 Sep 2025, Murrugarra et al., 23 Jan 2024, Narodytska et al., 2013).

1. Foundational Principles: Concentration and Modularity

The foundational concept underlying FoCusNet is the concentration of constraint violations or control influences within compact or salient subsets drawn from a larger universe of constraints or network components. The classic FOCUS constraint expresses the principle of grouping high-value variables into a small number of short intervals (Narodytska et al., 2013). In Boolean networks, modular decomposition via strongly connected components (SCCs) underpins the semidirect product framework, permitting control and filtering decisions to be localized to modules and edges (Murrugarra et al., 23 Jan 2024). Within LLM pipelines, FoCusNet operationalizes modular filtering by selecting only those constraints relevant to a given query, offloading heavy parsing from large models (Boffa et al., 28 Sep 2025).

2. Architectural Realizations and Algorithmic Modules

FoCusNet materializes as a network of modular propagators or filtering mechanisms tailored to the problem domain:

• Constraint Programming: The FoCusNet approach decomposes interval-concentration constraints (FOCUS, SpringyFOCUS, WeightedFOCUS, WeightedSpringyFOCUS) into modules comprising Boolean channeling constraints, global cardinality constraints, and among/sum modules. These form a constraint graph with edges representing the propagation dependencies (Narodytska et al., 2013).
• Neural Parsing for LLMs: In large-scale constraint generation, FoCusNet is instantiated as a neural filtering model. It receives all candidate constraints (e.g., forbidden-word lists) and input text, computes embeddings via a frozen sentence encoder (all-mpnet-base-v2) and word encoders, applies learnable projection and aggregation layers, undertakes contrastive InfoNCE training to refine semantic discrimination, and ultimately predicts a filtered subset of constraints via a Random Forest classifier (Boffa et al., 28 Sep 2025).
• Modular Control in Boolean Networks: FoCusNet exploits topological modularity (SCC decomposition), canalizing function analysis (He–Kauffman stratification), and a filtering loop to eliminate upstream modules whose influence can be blocked by targeted node or edge-control interventions. The algorithmic procedure computes canalizing layers, records controls, and isolates the minimal subset of modules requiring intervention (Murrugarra et al., 23 Jan 2024).

3. Mathematical Formulations and Filtering Criteria

Mathematical formalization in FoCusNet varies by context but adheres to a modular principle:

• Constraint Filtering Model (LLM Context): Given a set of constraints $c = \{c_1, \dots, c_C\}$ and an input $S$, FoCusNet learns a mapping $f_\phi: C \to \{0,1\}^C$ with parameters $\phi = \{\chi, \gamma, \lambda, \text{RF-params}\}$, outputting a binary mask and compact constraint subset $k$. The training objective combines InfoNCE loss for the encoder module and a binary classification loss for the control (Boffa et al., 28 Sep 2025).
• Module Exclusion in Boolean Networks: The filtering criterion relies on the canalizing depth and layer dominance in the Boolean update functions. If each upstream module has only one output edge to a downstream module and its variables appear outside the most dominant canalizing layer, node control suffices for exclusion. If in the dominant layer, edge control suffices. This enables linear-time reduction of the control search space under standard connectivity assumptions (Murrugarra et al., 23 Jan 2024).
• Compositional Constraint Networks (CP Context): Constraints such as SpringyFOCUS and WeightedFOCUS are decomposed by introducing binary channeling variables, sum/among interval constraints, and GCC modules, with edge connections defining the modular filtering structure (Narodytska et al., 2013).

4. Filtering Algorithms, Propagators, and Complexity

FoCusNet enacts efficient filtering through specialized or modular algorithms:

• SpringyFOCUS Propagator: Bounds consistency (BC) is enforced using forward/backward sweeping, with per-position dynamic programming tracking minimal interval counts, interval lengths, and hole usage. Complexity is $O(n)$ time and space, and correctness follows by recurrence tightness (Narodytska et al., 2013).
• WeightedFOCUS and WeightedSpringyFOCUS: Filtering uses dynamic programming tables with states representing (# intervals, last interval length, holes if applicable), with optimality enforced under lexicographic cost order. Complexity scales as $O(n\,z_c^{\max})$ per cost-bound setting (Narodytska et al., 2013).
• Neural Filtering (FoCusNet for LLMs): Filtering involves sentence and word embeddings, learnable transformation, attention aggregation, and Random Forest inference. Training is conducted on $220$ K (sentence, word subset) pairs, InfoNCE with $d=128$, $\tau=0.05$, $24$ epochs. Inference aggregates the relevant constraints for downstream LLM prompt construction (Boffa et al., 28 Sep 2025).
• Modular Filtering in Boolean Networks: SCC decomposition and topological ordering require $O(n+|E|)$ time. Canalizing layer computation is $O(2^d)$ per function, with $d$ typically small; overall filtering is linear in the size of intermodule connectivity (Murrugarra et al., 23 Jan 2024).

5. Empirical Results, Benchmarks, and Impact

When deployed in large-scale constraint environments, FoCusNet demonstrates robust improvements:

• LLM Constraint Filtering: On the Words Checker LSCG benchmark with DeepSeek-R1-8B and constraint sets of size $|F|=500,1000$, FoCusNet achieves an $8$–$13$ point accuracy boost over concatenation-based steering (Simple Prompt, Chain-of-Thought, Best-of-N). Precision increases by $11$–$20$ points, driven by reduced false alarms, while recall remains comparable. In invalid sentence cases, parsing precision for FoCusNet hits $100\%$ in $68\%$ of instances, compared to $30\%$ for baselines (Boffa et al., 28 Sep 2025).

Strategy	500 words	Acc	Prec
Simple Prompt	70.51	66.33
Chain of Thought	68.20	63.03
Best of 3	62.70	58.81
FoCusNet	79.30	77.78

• Boolean Network Control: In the T-LGL leukemia model, modular filtering via canalizing-feature analysis results in exclusion of all upstream modules except for a single edge control (Ceramide $\to$ DISC), effecting phenotype lock-in with minimal intervention (Murrugarra et al., 23 Jan 2024).
• Constraint Programming Architectures: Empirical evaluation confirms that the modular decomposition network for FOCUS-type constraints matches the propagation strength of dedicated DP algorithms while providing improved solver architecture clarity and flexible modeling capability (Narodytska et al., 2013).

6. Limitations and Prospects for Extension

FoCusNet operates under certain structural and data-related limitations:

• In LLM-based LSCG, FoCusNet presupposes access to task-specific supervised data; low-resource scenarios may suffer, partially mitigated by synthetic augmentation (Boffa et al., 28 Sep 2025).
• Current implementations are tailored to text-based, interval-based, or Boolean constraint representations. Extension to richer semantic domains (e.g., multi-modal constraints—images, tables) and evolving constraint sets (via RAG) is an open direction (Boffa et al., 28 Sep 2025).
• In modular Boolean control, filtering is limited to cases with unique intermodule links and canalizing update functions. For highly entangled networks or non-canalizing interactions, alternate graph-theoretic or neural modules may be necessary (Murrugarra et al., 23 Jan 2024).
• Modular constraint networks in CP cannot always match propagation strength if nontrivial global reasoning is required, but empirical propagation is often competitive (Narodytska et al., 2013).

This suggests that the modular filtering paradigm embodied by FoCusNet is effective yet sensitive to problem structure and data availability. A plausible implication is that future FoCusNet architectures will require integration with dynamic retrieval, unsupervised learning, and generalized modular composition.

7. Summary and Scientific Significance

FoCusNet provides a scalable, modular, and adaptable framework for constraint filtering, propagation, and control across diverse scientific domains—including constraint programming, large-scale LLM inference, and modular Boolean network control. By distilling bulky or complex constraint sets into computationally manageable, task-relevant subsets, FoCusNet advances the tractability and reliability of large-scale reasoning systems. The modularity intrinsic to FoCusNet yields architectures that are reusable, efficient, and flexible; propagation and control strength is retained while modeling and solver implementation is simplified. These features position FoCusNet as a reference methodology for concentrated constraint processing in both classical and neural pipeline contexts (Boffa et al., 28 Sep 2025, Murrugarra et al., 23 Jan 2024, Narodytska et al., 2013).