What Is a Signature in First-Order Logic?

What Is a Signature in First-Order Logic?

A signature in first-order logic is the formal vocabulary of a logical theory. It consists of three components: a set of constant symbols (e.g. Alice, 0), a set of function symbols each with a fixed arity (e.g. +/2, parent_of/1), and a set of relation symbols each with a fixed arity (e.g. =/2, Knows/2). The signature is the alphabet on which well-formed formulas are built; axioms come later and are stated over the signature. In AI it appears wherever symbolic reasoning, knowledge representation, neuro-symbolic systems, or formal verification meet LLMs that emit structured logical forms.

Why It Matters in Production LLM and Agent Systems

Pure neural systems are flexible but unauditable. Pure symbolic systems are auditable but brittle. Neuro-symbolic stacks pair LLMs with formal reasoners — and the bridge between them is almost always a signature. The LLM emits text; a parser turns text into a formula over a fixed signature; the reasoner proves, queries, or plans on that formula.

The pain of an undisciplined signature shows up across roles. A reasoning-system engineer ships a planner that emits logic over a free-form symbol set; the downstream solver fails on 30% of inputs because symbols are inconsistent (parent, parent_of, parentOf). A knowledge-graph team imports an LLM-extracted schema and finds three different relations that mean “isManagerOf” because no signature was fixed in advance. A formal-verification team tries to prove an agent’s safety properties and discovers the agent’s tool-call signatures drift across versions, invalidating prior proofs.

In 2026 agent stacks where LLMs increasingly emit structured outputs — tool calls, JSON Schema-conformant responses, planning traces in DSL-like syntax — the signature is the contract that makes outputs composable. The same discipline that drives a JSON Schema in modern APIs drives a first-order signature in formal AI: lock the vocabulary, constrain the parser, and only then evaluate the reasoning.

How FutureAGI Handles First-Order Logic Signatures

FutureAGI does not implement theorem provers or first-order solvers — that lives in symbolic libraries (Z3, Vampire, Prolog). We evaluate LLM outputs that target a fixed signature. At the schema level, the SchemaCompliance and JSONValidation evaluators score whether an LLM’s output conforms to a target structure; for a signature-typed output, you encode the signature as a JSON Schema with constants, function-arity rules, and relation-arity rules. At the reasoning level, ReasoningQuality and MultiHopReasoning score the chain-of-thought that produced the formal output, catching cases where the model picked the right symbols but reasoned incorrectly. At the dataset level, a Dataset of input-text-to-signature-conformant-formula pairs becomes a regression suite for any model upgrade.

Concretely: a neuro-symbolic team builds a planner where the LLM emits Action(robot, move, kitchen) -style atoms over a fixed signature with three function symbols and seven relations. They wire a traceAI-instrumented call into the planner, capture the LLM output as a span, and run SchemaCompliance against the signature schema; failures (a fourth function appearing in output, a wrong arity) are caught before the symbolic solver receives them. They also run ReasoningQuality on the chain-of-thought to surface cases where the model invents a new symbol because it could not solve with the existing vocabulary. FutureAGI does not own the logic, but it catches the bridge breaking.

How to Measure or Detect It

Signature-related signals to wire into evaluation:

• SchemaCompliance — boolean+score for whether the LLM output conforms to the signature schema (declared constants, function arities, relation arities).
• JSONValidation — strict schema conformance when the signature is encoded as JSON.
• ReasoningQuality — scores whether the reasoning steps respected the vocabulary; catches “model picked the wrong symbol”.
• Vocabulary-drift rate — count of out-of-signature symbols per N predictions; should be 0 in production.
• Arity-mismatch rate — function or relation called with wrong number of arguments; usually a parser-level fail.

Minimal Python:

from fi.evals import SchemaCompliance

compliance = SchemaCompliance()
result = compliance.evaluate(
    output=llm_output_formula,
    schema=signature_schema_json,
)
print(result.score, result.reason)

Common Mistakes

• Letting the LLM invent symbols. Without an enforced signature, models invent synonyms (father, parent, parent_of) and break composition.
• Loose arity checking. A relation declared Knows/2 accepting three arguments is silently parsed by lenient downstream tools and corrupts proofs.
• Mixing signature with ontology. A signature is syntactic. Stuffing axioms into the signature description confuses the model and the verifier.
• No regression eval on the signature. Model upgrades silently break the vocabulary. Run SchemaCompliance nightly against a versioned signature schema.
• Treating LLM-emitted logic as ground truth. The LLM is a translator, not a solver. Always pass the formula to a real first-order tool for proof or query.