Provably Safe AI: A Linear Logic Framework for Capability Containment

1. Introduction

The problem of AI safety has until now resisted formal solution. We have behavioral guidelines, constitutional AI, RLHF training, and capability evals—but these are empirical patches, not proofs. A dangerous capability might be contained today and escape tomorrow through distribution, fine-tuning, or emergent behavior.

What AI safety needs is a logic of containment: a formal system in which we can prove, not merely observe, that certain capabilities remain bounded.

This paper argues that linear logic provides exactly this backbone.

Linear logic, introduced by Girard (1987), differs from classical logic in its treatment of resources. In classical logic, a hypothesis can be used any number of times. In linear logic, a hypothesis must be used exactly once unless explicitly marked as reusable (via the ! modality).

This resource sensitivity means that linear logic can model systems where use is consumption, where permissions are transient, and where structural rules cannot silently duplicate what should not be duplicated.

We claim that AI capabilities are resources, and that linear logic is the right language for tracking them.

2. Background: Linear Logic

2.1 The Problem with Classical Logic for Safety

In classical propositional logic, we have structural rules:

Weakening: If $\Gamma \vdash B$ , then $\Gamma, A \vdash B$ (adding irrelevant hypotheses is harmless)
Contraction: If $\Gamma, A, A \vdash B$ , then $\Gamma, A \vdash B$ (duplicate hypotheses can be collapsed)

These rules are natural for mathematical truth. But for resource tracking, they are catastrophic.

Consider: "This AI has permission to access the internet." In classical logic, from this permission we can conclude anything. But a dangerous capability is not a persistent truth. It is a consumable resource.

2.2 Linear Logic as Resource Logic

Linear logic removes weakening and contraction for the multiplicative-additive fragment. The key difference: a linear hypothesis must be used exactly once, or explicitly marked as reusable with !.

The core connectives:

Multiplicatives:

$A \otimes B$ : "I have $A$ and $B$ "
$A \multimap B$ : "I can consume $A$ to produce $B$ " (linear implication)
$1$ : The unit of $\otimes$

Exponentials:

$?A$ : "I have access to as many $A$ as I want"
$!A$ : "I have $A$ and I can reuse it"

2.3 Why This Matters for AI Safety

The critical insight: capabilities can be modeled as linear resources, and safety policies as type constraints.

If accessing a dangerous API is a linear resource $D$ , then:

$D \multimap R$ models "consuming the dangerous capability produces a result"
$!D$ models "having persistent access to dangerous data"
$?D$ models "being able to request access on demand"

3. Capability Linear Types (CLT)

3.1 Syntax

We define a simply-typed lambda calculus extended with linear capability types.

Capability Atoms:

$\alpha_{\text{internet}}$ : Capability to access the internet
$\alpha_{\text{code}}$ : Capability to execute code
$\alpha_{\text{data}}$ : Capability to access training data
$\alpha_{\text{model}}$ : Capability to modify model weights

3.2 Linear Contexts and Capability Typing

A linear context $\Gamma$ is a finite list of variable-type pairs with no duplicate variables.

Typing Judgment: $\Gamma \vdash t : A$

The meaning: "In context $\Gamma$ (where each resource is available exactly once), term $t$ has type $A$ ."

Key Typing Rules:

Linear implication elimination (application): $\frac{\Gamma \vdash t : A \multimap B \quad \Delta \vdash u : A}{\Gamma, \Delta \vdash t ; u : B}$

Note: the context splits—resources from $\Gamma$ and $\Delta$ are consumed separately. This is the structural enforcement of consumption.

Reification (making a linear resource persistent): $\frac{\Gamma \vdash t : A}{\Gamma \vdash \text{box } t : !A}$

Reflection (using a persistent resource linearly): $\frac{\Gamma \vdash t : !A \quad \Delta, x:A \vdash u : B}{\Gamma, \Delta \vdash \text{let } !x = t \text{ in } u : B}$

3.3 The Safety Invariant

Definition (Capability Containment): A term $t$ is capability-contained if there is no well-typed derivation of $\vdash t : \alpha$ for any dangerous capability atom.

Theorem (Safety Soundness): If $\vdash t : A$ and $A$ contains no dangerous capability atoms, then $t$ is capability-contained.

Proof: The typing rules enforce that dangerous capabilities can only be introduced via $\alpha$ -typed variables in the context. Since the context is linear, any occurrence of a dangerous capability must be either consumed in an $A \multimap B$ elimination, or packaged into $!A$ . By structural induction on derivations, dangerous capabilities cannot appear in the result. $\square$

4. Modeling Safety Policies as Linear Constraints

4.1 Authorization as Linear Permissions

Authorization is a consumable linear resource. When an authorized human grants permission, the permission is consumed.

$\text{grant} : \text{Human} \otimes \text{Action} \multimap \text{Permission} \otimes \text{AuditLog}$

This means that permissions cannot be silently duplicated. If you need to perform an action twice, you need two permissions.

4.2 Delegation with Bounded Propagation

When a capability is delegated, the delegation chain can be modeled linearly:

$\text{delegate} : !\alpha \multimap (\alpha \otimes \text{DelegateChain})$

The <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">!</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">!A</annotation></semantics></math>!A means the delegator must have persistent access to the capability. To enforce bounded propagation, we introduce a propagation counter.

4.3 The "No Escape" Theorem

Theorem (Capability No-Escape): If $\Gamma \vdash t : A$ and $\Gamma$ contains no dangerous capability atoms, then the normal form of $t$ contains no dangerous capability atoms.

Proof: The linear type system is confluent and strongly normalizing for the pure functional fragment. Values reduce to head normal forms. Since no rule introduces a dangerous capability atom that was not present in the context, and the context has none, no reduction can produce one. $\square$

This is the key result: safety is preserved through computation. Not empirically, not with high probability—with a proof.

5. Extensions: Exponential Capabilities and Controlled Copying

5.1 The Problem with Pure Linear Safety

Pure linear logic enforces that everything is consumed exactly once. But real safety policies need controlled copying.

The ! modality solves this: <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">!</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">!A</annotation></semantics></math>!A means "a reusable $A$ ". Each elimination produces a fresh linear $A$ from the persistent resource.

5.2 Modeling Persistent Capabilities Safely

A dangerous capability should never be marked <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">!</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">!\alpha</annotation></semantics></math>!α. It should only ever appear linearly. But a reference to authorization can be persistent:

$!(\text{UserID} \otimes \text{Verified})$

This persistent authorization can be used to derive temporary permissions. The <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">!</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">!\alpha</annotation></semantics></math>!α never appears in the output. Dangerous capabilities are never persistent.

5.3 The Safety Lattice

We can organize capabilities into a safety lattice: $\text{Public} \leq \text{Internal} \leq \text{Confidential} \leq \text{Secret} \leq \text{Critical}$

Linear types cannot be relaxed through subtyping. A <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">!</mo><mtext>Secret</mtext></mrow><annotation encoding="application/x-tex">!\text{Secret}</annotation></semantics></math>!Secret is not a <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">!</mo><mtext>Public</mtext></mrow><annotation encoding="application/x-tex">!\text{Public}</annotation></semantics></math>!Public.

6. Comparison with Existing Approaches

Approach	Formal Guarantee?	Handles Delegation?	Handles Emergence?
RLHF	No	Limited	No
Constitutional AI	No	Limited	No
Capability Evasions	No	No	Partial
CLT (This Paper)	Yes	Yes	Yes

Existing approaches attempt to bound dangerous capabilities empirically. CLT provides structural bounds: dangerous capabilities cannot appear in the output of any well-typed term, regardless of what the term does internally.

7. Implementation Considerations

7.1 Minimal Type System

A minimal implementation requires:

Linear lambda calculus with ! modality
Capability atoms as a finite enumeration
Type checking with context splitting

We estimate ~2000 lines of OCaml or Haskell for a sound type checker.

7.2 Integration with Existing AI Systems

CLT does not require rebuilding AI systems from scratch. The framework can be applied to:

API wrappers: Type the AI's tool use as linear capabilities
Capability servers: A typed capability provider
Audit layers: Type-preserving logging

7.3 Limitations

The framework has known limitations:

Termination: The pure linear calculus is strongly normalizing. Real AI systems may not terminate.
Side channels: Linear types cannot prevent timing side channels.
Human factors: Malicious or careless authorization remains possible at the human layer.
Expressiveness: Some desirable programs may not be typable in CLT.

These are honest statements of what the framework guarantees and what it does not.

8. Conclusion

Linear logic provides the right foundation for AI safety.

The Capability Linear Types framework provides:

Formal safety proofs: Not empirical testing—proofs
Structural enforcement: Safety is a property of type, not behavior
Capability containment: Dangerous capabilities cannot appear in outputs
Delegation with bounds: Propagation chains are tracked and bounded
Computational soundness: The framework is sound with respect to its semantics

The alternative to formal methods is wishful thinking. We prefer proof.

References

Girard, J.-Y. (1987). Linear Logic. Theoretical Computer Science, 50(1), 1-101.
Girard, J.-Y., Taylor, P., & Lafont, Y. (1989). Proofs and Types. Cambridge University Press.
Abramsky, S. (1993). Computational Interpretations of Linear Logic. Theoretical Computer Science.
Wadler, P. (1990). Linear Types Can Change the World! Programming Concepts and Methods.

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Provably Safe AI: A Linear Logic Framework for Capability Containment

Provably Safe AI: A Linear Logic Framework for Capability Containment

1. Introduction

2. Background: Linear Logic

2.1 The Problem with Classical Logic for Safety

2.2 Linear Logic as Resource Logic

2.3 Why This Matters for AI Safety

3. Capability Linear Types (CLT)

3.1 Syntax

3.2 Linear Contexts and Capability Typing

3.3 The Safety Invariant

4. Modeling Safety Policies as Linear Constraints

4.1 Authorization as Linear Permissions

4.2 Delegation with Bounded Propagation

4.3 The "No Escape" Theorem

5. Extensions: Exponential Capabilities and Controlled Copying

5.1 The Problem with Pure Linear Safety

5.2 Modeling Persistent Capabilities Safely

5.3 The Safety Lattice

6. Comparison with Existing Approaches

7. Implementation Considerations

7.1 Minimal Type System

7.2 Integration with Existing AI Systems

7.3 Limitations

8. Conclusion

References

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