Formal Semantics for Neural Networks: The Missing Bridge

We have built extraordinary machines that perform reasoning tasks without any formal account of what they are doing.

This is not hyperbole. Modern neural networks—particularly large language models and transformer architectures—operate as black boxes with respect to formal semantics. We can measure their outputs, benchmark their performance, and even probe their internal representations with increasing sophistication. What we cannot do is write down, in the language of formal logic or symbolic mathematics, what computation a given network actually performs. This gap between empirical capability and formal understanding represents one of the most consequential blind spots in AI research.

The problem runs deeper than interpretability. Interpretability asks: what features does this network learn? Formal semantics asks: what is the mathematical object this network instantiates? These are not the same question. A network might be interpretable—we might understand which neurons fire for which concepts—yet still lack a formal semantics that connects its architecture, weights, and operations to a well-defined mathematical function in the language of logic or type theory.

Consider what we take for granted with traditional symbolic systems. A Turing machine has formal semantics: we can write down precisely which state transitions occur for which inputs. A logic program has formal semantics: we can derive its meaning from inference rules. A functional program has formal semantics: we can reason about it using lambda calculus and type systems. These formal accounts are not decorative. They enable us to prove properties, reason about correctness, and understand failure modes. They create a bridge between the physical implementation and the abstract computation.

Neural networks have none of this. We have empirical semantics—we run the network and observe outputs. We have operational semantics—we understand the forward pass, backpropagation, gradient flow. But we lack denotational semantics: a principled mathematical account of what function the network computes, expressed in a language independent of its implementation.

This matters because it creates a fundamental asymmetry in our ability to reason about neural systems versus symbolic ones. When a theorem prover fails, we can examine the proof search and identify exactly where it went wrong. When a neural network fails, we are left with post-hoc analysis: saliency maps, attention visualizations, adversarial examples. These tools are useful, but they are not formal semantics. They do not tell us what the network was supposed to compute, only what it happened to do.

The absence of formal semantics also constrains our theoretical understanding. We cannot formally specify what properties a network should satisfy before training it. We cannot prove that a network will generalize to unseen data. We cannot formally verify that a network respects safety constraints. These limitations are not accidental—they follow directly from the lack of a formal framework.

Some progress exists. Differential privacy provides formal guarantees about information leakage. Certified robustness techniques offer formal bounds on adversarial perturbations. Mechanistic interpretability research is beginning to identify formal structures within networks—circuits, motifs, algorithmic patterns. But these are islands of formality in a sea of empiricism. They do not yet constitute a unified formal semantics for neural computation.

The path forward requires treating neural networks as mathematical objects worthy of the same formal rigor we apply to other computational systems. This means developing notation and proof techniques for neural architectures. It means asking: what is the type signature of a transformer? What are the invariants that hold across layers? What formal properties should we expect from attention mechanisms?

This is not about replacing neural networks with symbolic systems. It is about bringing the rigor of formal methods to bear on systems that have already demonstrated remarkable capabilities. The goal is not to constrain neural networks but to understand them—to build the conceptual bridge that connects their empirical success to formal mathematical truth.

Until we do, we are building increasingly powerful systems whose behavior we can only describe in the language of engineering, not mathematics. That is a precarious foundation for systems we are beginning to trust with consequential decisions.