This note introduces the Structured Inference Over Constraints framework and mental model that I came up with and have been experimenting with. This mental model is intriguing because it attempts to provide a unified perspective that treats search, optimization, learning, theorem proving, and even constraint discovery as instances of inference over structured domains with hard and soft constraints.

I developed this model because practitioners (myself included) often view classical search, optimization, machine learning, theorem proving, formal verirication, etc. as isolated fields. However, in reality, many algorithms and theoretical insights transfer directly when problems are approached using this mental model / are cast in this common schema. By adopting this lens, you can leverage existing solutions, recognize hidden connections across domains, and avoid reinventing the wheel.

This document also serves as a personal reference that I will use to study and internalize this framework. Please note that I will refine and update it as my understanding grows and would appreciate any constructive criticism via email.

The remainder of this note presents the purpose, structure, and practical boundaries of the “Structured Inference Over Constraints” framework, showing how diverse tasks from search to learning to theorem proving can be unified, and where this perspective may fall short.

Why “Structured Inference Over Constraints”?

I chose this name to emphasize two complementary dimensions of modern computational reasoning:

  1. Inference under Constraints: Performing search, optimization, and verification within fixed hard constraints (logical, safety, feasibility) and soft objectives (costs, losses, rewards).
  2. Meta-Inference over Constraints: Treating constraints themselves as variables to be learned, abstracted, relaxed, or synthesized—optimizing over the space of possible specifications, abstractions, and reward functions.

This dual perspective transcends the narrower notion of “constrained optimization,” which implies only minimizing an objective subject to fixed conditions. Our terminology captures both optimizing within constraints and optimizing the constraints themselves, ensuring the framework is both comprehensive and nuanced.

Unifying Perspective: Structured Inference Over Constraints

Core Schema

  • Domain ($\mathcal{S}$): states, proofs, parameters, programs, policies, etc.
  • Hard Constraints ($C(s)$): logical conditions, safety properties, specification adherence.
  • Soft Constraints / Objectives ($f(s)$): cost, loss, reward, simplicity, error metrics.
  • Constraint Space (optional, $\mathcal{C}$): for learning, abstracting, or optimizing the constraints themselves.

Inference Goals

  • Search: Find $s$ such that $C(s) = \text{true}$ (e.g., pathfinding, CSPs, SAT solving).
  • Optimization: Find $s^* = \arg\min_{C(s)} f(s)$ (e.g., numeric optimization, reinforcement learning).
  • Learning / Synthesis: Infer $\theta$ or $(s, C)$ to minimize loss and fit data or examples (e.g., supervised learning, inverse reinforcement learning).
  • Theorem Proving: Search over proof trees or derivation sequences $s$ such that $C(s)$ is a valid proof (e.g., Metamath, Coq, HOL).
  • Formal Verification: Show $\forall s, C(s)$ holds, or find $s$ that violates it (e.g., symbolic model checking of DTMCs against PCTL or LTL specifications).

Example Instantiations

  • Pathfinding and Classical Search
    • $\mathcal{S}$: possible paths or action sequences
    • $C(s)$: reaches goal state without violating constraints (e.g., obstacles, bounds)
    • $f(s)$: path length or cost
    • Goal: find an optimal or feasible path
  • Constraint Satisfaction (CSP/SAT)
    • $\mathcal{S}$: variable assignments
    • $C(s)$: formula or predicate satisfaction
    • $f$: none (pure feasibility search)
  • Constrained Optimization
    • $\mathcal{S} = \mathbb{R}^n$
    • $C$: algebraic constraints (e.g., $Ax \leq b$)
    • $f$: scalar cost or objective function
    • Goal: find a feasible $s$ that minimizes $f(s)$
  • Machine Learning (Classical Supervised Learning)
    • $\mathcal{S}$: model parameters or function classes
    • $C$: satisfies training or architectural constraints (e.g., parameter bounds, input-output type)
    • $f$: prediction error or generalization loss on data
    • Goal: learn a model that minimizes error while remaining feasible
  • Theorem Proving
    • $\mathcal{S}$: sequences of inference steps
    • $C$: well-formed derivation from axioms to goal
    • $f$: proof length, complexity, or heuristic score
  • Formal Verification (Model Checking)
    • $\mathcal{S}$: system executions or transition paths
    • $C(s)$: satisfaction of probabilistic temporal logic formula (e.g., $P_{\geq 0.95}(\text{F } \text{safe})$ in a DTMC)
    • $f$: counterexample length or probability of violation
  • Abstraction & Meta-Inference
    • $\mathcal{C}$: abstraction domains, specifications, or automata
    • Goal: choose a constraint representation $C$ that balances tractability and precision

Where It’s Useful & Where It’s NOT

As a high-level conceptual map, not a one-size-fits-all solution. Pair it with deep domain knowledge, careful representation design, and human-centered considerations to build real-world intelligent systems.

Key Limits & Pitfalls

  • Representation & Structure: Domain-specific grammars, inductive biases, and symbolic structures often lead to more efficient or meaningful solutions than generic formulations.
  • Interaction & Dynamics: Real-time systems, human-in-the-loop setups, and multi-agent environments involve feedback loops and adaptivity beyond a single static solution $s$.
  • Complexity Barriers: Many problems are undecidable or computationally hard (e.g., PSPACE-complete), which may not be obvious under this abstraction.
  • Normative & Semantic Depth: Values like safety, fairness, or ethics often resist crisp formulation as $f$ or $C$ and require interpretability, stakeholder input, or procedural guarantees.