On the Complexity and Properties of Preferential Propositional Dependence Logic

Abstract: This paper considers the complexity and properties of KLM-style preferential reasoning in the setting of propositional logic with team semantics and dependence atoms, also known as propositional dependence logic. Preferential team-based reasoning is shown to be cumulative, yet violates System~P. We give intuitive conditions that fully characterise those cases where preferential propositional dependence logic satisfies System~P. We show that these characterisations do, surprisingly, not carry over to preferential team-based propositional logic. Furthermore, we show how classical entailment and dependence logic entailment can be expressed in terms of non-trivial preferential models. Finally, we present the complexity of preferential team-based reasoning for two natural representations. This includes novel complexity results for classical (non-team-based) preferential reasoning.

• The paper establishes that preferential entailment in PDL is cumulative, satisfying System C but failing System P under general team semantics.
• It identifies two key properties, the star and triangle properties, as necessary and sufficient conditions for System P satisfaction in preferential models.
• The study provides a detailed computational complexity classification for preferential reasoning in classical, team-based, and dependence logic contexts.

Summary of "On the Complexity and Properties of Preferential Propositional Dependence Logic" (2505.08522)

This paper analyzes the complexity and structural properties of KLM-style preferential reasoning in propositional dependence logic (PDL), particularly focusing on team semantics. It explores how preferential entailment interacts with axiomatic systems (System~C and System~P), delineates conditions for System~P satisfaction, examines how classical entailment and dependence logic entailment can be captured by preferential models, and provides a granular classification of the computational complexity of preferential reasoning in classical, team-based, and dependence logic contexts.

Preferential Reasoning and Team Semantics

The KLM preferential reasoning framework relies on preferential models, which equip interpretations (models) with a strict partial order ($\prec$), indicating degrees of exceptionality. Preferential entailment holds when the $\prec$-minimal models of $\phi$ also satisfy $\psi$; i.e., this reflects non-monotonic inference: "if $\phi$ holds, one expects $\psi$."

Team semantics underpin logics such as PDL by interpreting formulas on sets of assignments (teams), enabling formal treatment of dependencies and plurality, as in database theory or conditional independence in statistics. Team semantics extend expressivity beyond classical semantics but do not—as shown here—straightforwardly inherit key properties of preferential reasoning from the classical setting.

Relationship to System~C and System~P

System~C comprises basic non-monotonic reasoning postulates: reflexivity, left logical equivalence, right weakening, cut, and cautious monotonicity. System~P augments System~C with the "Or" rule (reasoning by case), which is considered fundamental in preferential reasoning over classical propositional logic.

A major result established is that preferential entailment in PDL is cumulative (satisfies System~C) but fails to satisfy System~P. The central technical findings are:

• Characterization of System~P satisfaction: Two properties of preferential models, the $\star$-property ($\star$) and $\triangle$-property ($\triangle$), are shown to provide necessary and sufficient conditions for System~P to hold in preferential PDL.
  • $(\star)$: Minimal teams of a disjunction $(\phi\lor\psi)$ must be covered by minimal teams of $\phi$ or $\psi$ individually.
  • $(\triangle)$: For all non-singleton teams, there exists a proper subteam preferred over the original team.
• Flatness and team semantics: For formulas satisfying the flatness property, preferential reasoning in team semantics reduces to classical preferential entailment. Thus, System~P models for PDL essentially emulate their flat (classical) counterparts—a major limitation for leveraging team semantics in robust non-monotonic reasoning.

Expressivity and Axiomatic Limitations

The work demonstrates that System~P satisfaction in preferential team-based logics does not coincide with the criteria established for PDL, despite the latter being a fragment of PDL. Example constructions show that standard characterizations ($\star$ and $\triangle$) do not suffice for team logic fragments, highlighting the subtlety and complexity of designing axiomatic systems compatible with preferential entailment in non-flat logics.

A canonical construction is provided for expressing classical and dependence logic entailment within preferential models, making explicit the structural embeddings and bridging the expressivity gap between classical and team-based or dependence semantics.

Complexity Analysis of Preferential Entailment

The paper delivers a rigorous classification of the computational complexity associated with preferential entailment for various logics, including both "explicit" and "succinct" representations of preferential models:

Complexity Results:

• Preferential classical propositional logic (CPL) ($\mathsf{PL}$):
  • Explicit model (input enumeration): $\mathsf{P}$-complete, $NC^1$-hard.
  • Succinct model (circuit representation, lexicographic order): $\Delta_2^p$-complete.
  • Generic succinct model: $\Pi_2^p$, $\Delta_2^p$-hard.
• Preferential propositional dependence logic (PDL):
  • Explicit model: $\Theta_2^p$ (parallel $\mathsf{NP}$), $\mathsf{NP}$-hard.
  • Succinct model: $\Pi_2^p$, $\Delta_2^p$-hard.
• Preferential propositional logic with team semantics:
  • Explicit model: $\mathsf{P}$, $NC^1$-hard.
  • Succinct model: $\Pi_2^p$, $\Delta_2^p$-hard.

These results tightly bound the complexity of reasoning tasks in both classical and team-based settings, revealing a sharp divergence between explicit enumeration (tractable) and succinct/circuit-based representation (upper levels of the polynomial hierarchy).

Implications and Future Directions

From a theoretical perspective, the findings elucidate the limitations of team semantics in capturing non-monotonic reasoning principles, especially underscoring the failure of System~P in preferential PDL unless restrictive structural conditions are met. Practically, the complexity results inform the feasibility of implementing preferential reasoning in systems dealing with databases, epistemic scenarios, and answering queries under uncertainty and dependencies.

Future work should target:

• Classification of query and data complexity for preferential entailment problems in both explicit and succinct models.
• Tightening upper and lower bounds, exploring the complexity of variants such as team-based OLMS.
• Axiomatic characterization for other team-based logics (e.g., inclusion logic).
• Investigating the potential for richer preferential frameworks in natural language semantics, epistemic reasoning, and free-choice inferences.

Conclusion

This paper systematically extends KLM-style preferential reasoning into dependence logic with team semantics, proving that strong non-monotonic reasoning axioms (System~P) generally fail and providing precise structural and complexity criteria for preferential entailment. It advances the understanding of how non-monotonic inference interacts with logical dependencies and team-based models, setting a foundation for further research in non-classical logics and knowledge representation.