[Research] Resurrect the categorical structure? Graded monads for belief composition
- Dominant language
- Typst
- Stars
- 0
- Forks
- 0
- PR merge metrics
- No merged PRs in 30d
Description
## Background
The archive contains `categorical-structure.md` with extensive formalization of CLAIR as a **graded monad** over the confidence monoid `([0,1], ×, 1)`.
Key insight: Belief isn't a standard monad (metadata accumulation breaks laws), but it IS a graded monad where the grading tracks confidence decay through derivation.
## Core Formalism
```
Belief : [0,1] → (Type → Type) -- graded by confidence
(>>=) : Belief_c → (A → Belief_d) → Belief_{c×d}
```
## Research Questions
1. **Can we implement this in Haskell/Idris?**
- Define `Belief c a` as a graded monad
- Implement confidence propagation as the grading operation
- Type-safe belief composition with confidence tracking
2. **Alternative gradings**: The doc explores `([0,1], min, 1)` and the semiring `([0,1], ⊕, ×, 0, 1)`
- Which is the "right" structure?
- Can agents choose their grading based on risk tolerance?
3. **Dependent belief types**: `Belief<Π(x:A).B(x)>` vs `Π(x:A).Belief`
- Distribution laws and their semantics
- Computational interpretation
4. **Linear beliefs**: `!Belief` for reusable axioms
- Only confidence-1 beliefs can be promoted to `!`
- Linear derivation consumes confidence
## Why This Matters Now
CLAIR v1 is "just a trace format"—but the categorical foundation suggests:
- **Composable beliefs**: Confidence propagation as type-level computation
- **Static verification**: Type checker ensures valid confidence combinations
- **Optimization**: Graded types enable fusion of belief operations
## Open Questions
- Is the graded monad structure useful for implementation or just theory?
- Can we extract executable code from well-typed CLAIR derivations?
- What happens to provenance/justification in the graded formulation?
---
*From archive: `categorical-structure.md` — this was well-developed but abandoned when CLAIR pivoted from programming language to trace format.*
Contributor guide
No contributing guide indexed for this repository
Assessment
This issue has not been assessed yet.