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Decomposing Constraint Networks for Calculating c-Representations
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Introduces constraint networks built from safe covers of conditional belief bases that let c-representations (Spohn ranking models) be computed locally and iteratively, and shows skeptical c-inference can be localized to safe sub-bases.
Problem
Computing c-representations (ranking models of conditional belief bases) and drawing skeptical c-inferences is computationally expensive; existing decomposition methods (syntax splitting, OCF-LEG networks) are too restrictive for overlapping sub-bases.
Approach
Decompose belief bases into safe covers of overlapping sub-bases (canonical safe covers) and organize their penalty-point constraint satisfaction problems in a directed acyclic constraint network, solving them iteratively and propagating penalty points so the c-consistency condition holds automatically.
Key results
- Proves that canonical safe covers yield a constraint network whose iterative solving (Algorithm 1) produces a valid global c-representation (Theorem 1, Theorem 3).
- Shows skeptical c-inference can be localized: for a query q, the inference holds iff it holds on the milieu of the sub-base of conditionals in its constraints (Theorem 2), reducing the possible worlds from 2^|Σ| to 2^|Σ(∆q)|.
- On the running example, the maximal residuum size is 2 versus |∆ex| = 5, illustrating reduced problem size.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Decompose belief base into safe sub-bases via canonical safe cover and solve local CSPs iteratively over a constraint network (Algorithm 1) | c-representation computation for conditional belief bases · running example Delta_ex (5 conditionals) | maximum residuum size of constraint problems: max |R| = 2 vs |Delta_ex| = 5 | Helped |
| Apply natural constraint reduction (rules R1-R4) to constraint sets before solving | CSP(Delta) for c-representations · running example Delta_ex | constraint form simplified (C(delta3) reduces to eta3 > 0) | No clear effect |
Surprising findings
- Syntax-based decomposition (OCF-LEG networks) cannot decompose the running example at all, whereas the constraint-driven safe cover approach can.
Limitations
- Evaluation is purely theoretical with a single running example; no experiments on real-world belief bases.
- Solving constraint satisfaction problems is NP-complete in general, so local solving still carries worst-case cost.
- Only propositional conditionals are treated; first-order extension is future work.
Experimental setup
Baselines: OCF-LEG networks constraint splittings syntax splitting
Builds on: c-representations syntax splitting OCF-LEG networks constraint splittings skeptical c-inference
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