| paper | d4-journals-jair-2023-0090 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | yes |
Theorem 18 is a logical impossibility result about belief revision and consensus merging, not a claim about computing a solution or the difficulty of doing so. The proposed mass-distribution compatibility task is a new feasibility extension; it cannot supply the missing computational result, and consensus depends on positive-mass support rather than population shares.
fails bit a — no named computational result to mirror
The proposed extension concerns PCC compatibility for consensus merging; it leaves the paper’s other impossibility, satisfiability, and alternative-criteria results untouched.
I don’t see a rubric-compliant computational anchor in this paper. Theorems 16 and 18–23 and 26–27 prove logical existence, incompatibility, or compatibility claims about belief-revision and merging operators. They do not classify the complexity of an encoded problem or give an algorithm. Theorem 3 is a representation theorem cited from Katsuno and Mendelzon. So there is no named result here from which I can honestly make the requested complexity-based case.
The strongest nearby structural result is Theorem 18, proved in this paper: for the consensus merging function, there exist AGM revision functions for the individuals such that the resulting community violates the PCC. Its construction uses two belief sets whose models are distinct worlds \(w_1\) and \(w_2\), and an input whose models are \(w_3,w_4\). The same initial consensus can then lead to different post-update consensuses under two profiles, so no single collective AGM revision function works for both. This is a substantial result about collective change, but it is not a computational-complexity result.
As an unanchored follow-up, the clearest continuous question I can formulate is Continuous Consensus-PCC Compatibility. A type \(t\) consists of a prior belief theory, represented by its model set \(S_t\), and an individual AGM revision preorder \(\preceq_t\). An instance gives finitely many societies \(\mu^1,\ldots,\mu^q\), each a rational distribution over those types, and evidence \(\phi_j\) that contradicts every type with positive mass in society \(j\). The initial consensus theory \(K^j\) has model set \(\bigcup_{t:\mu^j_t>0}S_t\). After individual revision, the post-update consensus has model set
\[
V_j=\bigcup_{t:\mu^j_t>0}\min(\llbracket\phi_j\rrbracket,\preceq_t).
\]
The question is whether there is one faithful preorder \(\preceq^C_K\) for each distinct collective prior \(K^j\), such that \(\min(\llbracket\phi_j\rrbracket,\preceq^C_{K^j})=V_j\) for every society in the input. A solution is such a family of collective preorders; a negative answer means those requirements cannot be met simultaneously.
A plausible regime is a large scientific community with many members drawn from a relatively small set of prior-belief and evidence-response types: \(\mu_t\) is the fraction of members of type \(t\), with \(N\gg\tau\). This is recognizable as collective belief revision, the paper’s subject. But under consensus merging, only whether a type has positive mass matters; changing its share does not change the result. Thus the continuous population is formally present while its proportions do no work. Theorem 18 supplies a natural compatibility question for this extension, but it does not anchor its computational complexity. I would leave its Class A/B/C classification open rather than infer one from the theorem. That mass-insensitivity, together with the absence of a computational anchor, is the case’s main weakness.
The strongest case against is that this paper has no computational result to continuize. Theorems 16 and 18–23 and 26–27 establish logical incompatibility or compatibility between belief-revision and merging principles. They give no algorithm, complexity bound, or finite computational problem. Theorem 18, the proponent’s anchor, is a global impossibility claim: a four-world construction shows that consensus can erase a distinction between initial profiles that matters after revision. That is a meaningful result, but there is no computational question in it for a continuous optimization programme to classify.
The proposed Continuous Consensus-PCC Compatibility task is well-defined, but it adds a finite feasibility problem: given selected societies and their individual revision behaviours, find collective preorders meeting the resulting constraints. It is not the computational version of a problem the paper formulated. And for the proposed consensus semantics, its population is continuous only in its notation. If \(S_t\) is the model set of type \(t\), the initial consensus has model set \(\bigcup_{t:\mu_t>0} S_t\); after revision it likewise depends on which types have positive mass, not on their shares. Any two distributions with the same support give the same consensus and post-revision consensus. The simplex of populations therefore collapses into discrete support choices, rather than making population proportions computationally meaningful.
The best repair is a mass-sensitive merger, perhaps a weighted Hamming majority over belief and revision-response types. The paper makes that idea recognizable through its discussion of majority merging and Example 25. But it replaces consensus with a different aggregation semantics, and the paper supplies no computational result about that semantics. A computational study of it could be worthwhile on its own; it would be a new problem motivated by the paper, rather than a continuous mirror of one of its results.
I cannot make the universal claim airtight: a new mass-sensitive model of collective belief revision could be a sensible computational-social-choice project. The negative case is strongest for this paper’s stated results. Under the programme’s computational scope, the absence of a computational result is decisive; the one proposed mirror also loses the population fractions that continuization is meant to make central.
The adversarial triple: the proponent anchors on up to three named results; the opponent sees that case and must defeat every anchor; the judge decides which case convinced it. These are the pipeline’s own outputs, generated by tools/triple_run.py — no human edited them. The paper’s own text is not reproduced here beyond the quoted statement above.