| paper | A Unifying Formal Approach to Importance Values in Boolean Functions |
| authors | Hans Harder, Simon Jantsch, Christel Baier, Clemens Dubslaff |
| venue | IJCAI 2023 |
| filed under | unclassified |
| judged by | gpt-5.6-luna / xhigh (triple__luna__xhigh__c2r1) |
| judge confidence | high |
| authors would recognise it | no |
The paper has no numbered theorem, lemma, corollary, or proposition asserting a computational complexity or algorithmic result. Its #P-completeness claim is unnumbered background, while Section 5 reports implementations and experiments. The proposed distributional influence is an outer distribution over Boolean assignments, not a continuized population under the programme's scope.
fails bit a — no named computational result to mirror
No eligible named computational result is covered; the axiomatic, structural, and empirical results remain outside the mirror.
The strongest honest case is limited by a formal problem: this paper contains no eligible named computational anchor.
Its numbered results are structural or axiomatic. Theorem 4 characterizes influence, Theorem 5 relates influence to Jeroslow–Wang values, and Theorems 6–7 establish properties of cooperative-game mappings. None asserts NP-hardness, membership in P, parameterized tractability, or another complexity classification. Section 5 presents implementation schemes and experiments, but no numbered worst-case algorithmic theorem. The introduction’s statement that influence and Shapley-value computation are \(\#\mathrm{P}\)-complete is unnumbered and cited from elsewhere, so it cannot serve as an anchor under the stated rules.
Thus, strictly speaking, there is no anchor and therefore no compliant “one continuous problem per anchor” case. Treating Theorem 4 or Proposition 2 as computational anchors would overstate what the paper proves.
The best substantive mirror, if the unnumbered influence claim were allowed, would be a follow-up called Distributional Influence Evaluation. A shared Boolean classifier or circuit \(f:\{0,1\}^X\to\{0,1\}\) is deployed over a large population of cases. A type is an input assignment \(u\in\{0,1\}^X\), and \(\mu_u\) is the fraction of cases of that type. For a variable \(x\), define
\[ \operatorname{Inf}_{\mu,x}(f) = \sum_{u\in\{0,1\}^X} \mu_u\, \mathbf{1}\!\left[f(u)\neq f(u^{\oplus x})\right]. \]
The instance consists of \(f\), a rational distribution \(\mu\) given succinctly—for example by rational independent-coordinate probabilities— and a target variable \(x\). The task is to compute \(\operatorname{Inf}_{\mu,x}(f)\), or decide whether it is at least a rational threshold \(q\). The objective is the expected fraction of population cases whose outcome changes when \(x\) is toggled.
The high-multiplicity regime is plausible for repeated executions, users, test cases, or operating contexts: millions of cases may share a relatively small collection of feature assignments, with \(N\gg\tau\). This is recognisably a population-weighted version of the paper’s influence, not merely fractionalising an outcome. With an explicitly listed support, evaluation is polynomial and arguably Class A; with succinct \(\mu\) and succinct \(f\), the original counting hardness remains, likely Class B or simply inherent Boolean-counting hardness. The weakness is that the paper itself studies uniform assignments to one Boolean function, not a population of cases, so the mirror is a natural extension rather than a result the authors have already formulated.
That is a credible research direction, but it cannot repair the absence of a qualifying named computational result in this paper.
The paper should be rejected as a ChoCo source. The proponent’s opening procedural point is correct: there is no eligible named computational anchor. Theorem 4 is an axiomatic characterization, Theorem 5 is a relation to Jeroslow–Wang values, and Theorems 2, 3, 6, and 7 establish properties of value functions and cooperative-game mappings. None gives a complexity classification or a worst-case algorithmic theorem. The \(\#\mathrm P\)-completeness statement in the introduction is unnumbered background attributed to other work, while Section 5 reports implementations and experiments rather than proved algorithms. Under the programme’s admission rule, that is decisive.
The proposed Distributional Influence Evaluation does not repair the problem. Its “types” are complete input assignments \(u\in\{0,1\}^X\), and \(\mu_u\) is a workload distribution over executions, test cases, or possible worlds. The quantity
\[ \operatorname{Inf}_{\mu,x}(f) = \sum_u \mu_u\, \mathbf 1[f(u)\ne f(u^{\oplus x})] \]
is therefore a weighted probability of pivotality. It does not make the population on which \(f\) operates continuous. The agents or actors remain the named variables \(X\); \(\mu\) is an outer distribution over their joint states. This is precisely a probability/noise model over discrete profiles, which the programme distinguishes from continuizing the society.
The natural “better” version does not escape this mismatch. One could introduce mass-transfer variables and ask for the cheapest redistribution of workload mass \(\mu\) that raises influence or blame above a threshold. But for fixed \(f\) and \(x\), the coefficient of each type is already fixed:
\[ a_x(u)=\mathbf 1[f(u)\ne f(u^{\oplus x})]. \]
The resulting problem is merely an optimal-transport wrapper around a weighted Boolean statistic. With explicit types it is an ordinary finite optimization problem; with succinct types, its difficulty comes from the representation and counting structure of \(f\), not from high population multiplicity. The campaigning or intervention semantics are newly invented and absent from the paper.
Trying instead to continuize the variables themselves is worse. For an arbitrary Boolean function, a variable’s importance depends on its entire interaction pattern with every other named variable. A type that preserves that information must encode the variable’s cofactors and global position in \(f\), which effectively restores identity. If the function is symmetric, Theorem 1 already makes all variables equally important, so the population model collapses to one number. If one creates cloned variables, there is no canonical clone operation: conjunction, disjunction, majority, and substitution produce different Boolean functions and different importance values. That would be a new theory of limits of Boolean functions, not a mirror of this paper.
The same obstruction is sharper for blame and responsibility. Their central object is a smallest finite critical set \(S\subseteq X\setminus\{x\}\), whose cardinality is measured by \(\rho(|S|)\). In a continuum of variables, an individual variable has measure zero and finite critical-set cardinality has no canonical limit. Normalizing or fractionalizing critical sets would define a new attribution measure rather than continuize the paper’s one.
There may be a worthwhile project on distribution-weighted influence, workload robustness, or population-level model debugging. But it would be an adjacent follow-up, not a continuous computational-social-choice mirror of a named result in this paper. The strongest honest verdict is therefore red under ChoCo’s strict standard; the universal claim that no related research could ever be useful is not provable, but this paper supplies no defensible continuous mirror.
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.