Possible Independence (PIA)

• Possible Independence (PIA) is a concept that defines independence under incomplete or underspecified data by requiring that some completion of the data satisfies classical independence.
• The formal definition of PIA employs a grounding procedure that imputes missing values, thereby allowing the evaluation of independence under a possible extension of the present dataset.
• PIA influences various fields by shaping inference systems and computational frameworks, with complexity results ranging from tractable cases in unary settings to NP-complete scenarios for larger attribute sets.

Possible Independence (PIA) is a formal notion of independence adapted to settings where information is incomplete, underspecified, or physically constrained. The PIA concept fundamentally addresses whether independence between sets of variables, attributes, or statements could possibly exist under some extension, completion, or grounding of the present data, structure, or theory. PIA has seen rigorous treatment in database theory, finite axiom systems, nonmonotonic reasoning, possibility theory, and quantum foundations, where its distinction from stronger (certain, universal) independence is essential to both technical results and foundational interpretation.

1. Formal Definition in Incomplete Information Contexts

Classically, an independence atom $X\perp\!\!\!\perp Y$ over a relation schema $R$ asserts that, for any tuples $t_1, t_2$ in relation $r$, there exists a tuple $t$ in $r$ combining the values from $t_1$ on $X$ and $t_2$ on $Y$, assuming all attributes are fully specified. When data is incomplete—represented using null markers (“$R$0”)—this condition becomes ambiguous.

PIA provides a relaxation by introducing the possible independence atom $R$1, defined as:

$R$2

Here, a grounding of $R$3 is a completion where every null in $R$4 is replaced arbitrarily from the domain, maintaining agreement on non-null positions. Consequently, possible independence means that there is at least one way to impute missing data such that the independence condition is satisfied. In contrast, certain independence ($R$5) requires that all groundings satisfy the independence atom (Hannula et al., 9 May 2025).

2. Axiomatics and Inference Systems for PIA

The inference system for possible independence adapts selected rules from the classical Geiger–Paz–Pearl system, omitting the exchange rule:

Rule	Description
Triviality ($R$6)	Any set is possibly independent of the empty set
Symmetry ($R$7)	Independence is symmetric ($R$8)
Constancy ($R$9)	Extending the independent set with itself does not matter
Decomposition ($t_1, t_2$0)	Independence from a superset implies independence from a subset

Soundness holds for all possible implications among PIAs; completeness is obtained only for restricted cases, namely when the size of one of the involved attribute sets is 1, or their sizes differ by at most 1. The exchange rule is not sound for possible independence, and thus the full axiomatisation of the implication problem remains open beyond these fragments (Hannula et al., 9 May 2025).

3. Computational Complexity

The model checking problem for possible independence (PIA) in relational settings is NP-complete when checking atoms with three or more distinct attributes. For unary PIAs ($t_1, t_2$1 or $t_1, t_2$2), the problem is tractable (in $t_1, t_2$3), reducing in the latter case to a bipartite matching problem between non-null projections. Certain independence always remains tractable (in polynomial time), as it is definable in first-order logic and reduces to the classical implication problem for complete relations (Hannula et al., 9 May 2025).

For the implication problem (i.e., deciding whether a set of PIA constraints entails another), polynomial-time procedures exist for the restricted fragments where completeness holds. The general case remains unsettled: no complete axiom system nor a definitive complexity classification is currently known (Hannula et al., 9 May 2025).

4. PIA in Possibility Theory and Uncertainty Reasoning

In possibility theory, a spectrum of independence notions is studied, but the qualitative (ordinal) form of independence aligns conceptually with PIA: it encodes whether, under maximal extension by specificity, an event remains accepted. Weak independence ($t_1, t_2$4), defined as $t_1, t_2$5 and $t_1, t_2$6, mirrors the possible independence idea—plausibility is checked under some scenario, not universally (Dubois et al., 2013).

This weak form underpins minimal change in belief revision, corresponding to Gärdenfors's criterion that only beliefs relevant to new information should be discarded; those irrelevant (i.e., weakly independent) are retained under at least one extension. Practically, this notion allows for the unblocking of property inheritance in nonmonotonic reasoning—for instance, ensuring that exceptions (like penguins not flying) do not globally block inheritance of unrelated properties (“penguins have legs”) (Dubois et al., 2013).

5. Possible Independence in Quantum Foundations

PIA appears within discussions of the statistical independence assumption in Bell-type hidden variable models. Here, “possible independence” corresponds to the requirement that the preparation distribution over hidden variables ($t_1, t_2$7) is independent of measurement settings. Supermeasured models violate Bell's statistical independence (by letting the state-space measure $t_1, t_2$8 depend on setting), but preserve "possible independence" in the sense that all physically preparable distributions remain setting-independent. The independence pertains to the lack of a physical correlation under any conceivable preparation, rather than stronger requirements on all mathematical representations (Hance et al., 2021).

6. PIA in Axiom Systems and Proof Theory

In the finite schematization of first-order logic (e.g., Megill’s TMM system), possible independence emerges in distinguishing between scheme-level and instance-level independence. An axiom scheme can be independent (not derivable from the remaining schemes) even when every instance (“object-level” formula) is derivable from instances of others. The notion of supertruth and its variants, such as semisupertruth, provides a semantic instrument for analyzing possible independence at the scheme level, which does not reduce to the classical independence of specific theorems or formulas (Jubin, 2022).

7. Practical Significance and Open Problems

PIA enables more nuanced reasoning about independence in incomplete, underdetermined, or physically restricted settings. In databases, it informs query optimization, update validation, and schema normalization by allowing for independence constraints under possible completions of data. In uncertainty and nonmonotonic reasoning, it supports robust belief revision and property inheritance. In logic, it sharpens the analysis of axiomatizability and proof-theoretic redundancy.

Open problems include the full axiomatization of the possible independence implication problem beyond size-restricted fragments, the existence and structure of Armstrong relations for PIAs, and the extension of PIA concepts to probabilistic databases and nonclassical dependency frameworks (Hannula et al., 9 May 2025). The computational boundaries outside the known polynomial time and NP-complete fragments are also subject to ongoing research.