Dependence Logic and Probabilistic Semantics

Dependence logic emerges from extending classical logics to capture explicit variable dependencies within a unified semantic framework. Rather than evaluating formulae on individual assignments, this approach employs sets of assignments, known as teams, enabling the direct expression of functional dependencies through specialised atoms. Probabilistic semantics enriches this foundation by assigning weights or distributions over these teams, thereby modelling uncertainty, statistical properties and causal influences within logical languages. Together, these developments unify logical and probabilistic reasoning, offering tools for analysing data dependencies, formalising causal queries and supporting applications in database theory, artificial intelligence and the foundations of causal inference. Recent conceptual advances have refined expressive power, normal forms and decidability boundaries, while practical extensions now address probabilistic interventions and multiteam structures that encode causal-probabilistic statements without presupposing a fixed probability interpretation.

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Latest studies have advanced the theoretical underpinnings of dependence logic and its computational aspects. A foundational logic of functional dependence has been formulated within a decidable modal-style extension of propositional logic, introducing dependence quantifiers as modalities and establishing complete proof calculi and meta-properties. This work demonstrates how continuous, linear and temporal dependencies can be uniformly captured, broadening applications from dynamical systems to vector spaces. Concurrent investigations into inclusion logic—a fragment of dependence logic—have revealed sharp complexity thresholds: quantifier-free formulae with minimal inclusion atoms generate model-checking and membership problems complete for non-deterministic logarithmic space and polynomial time. These findings chart the landscape of tractability and connect directly to consistent query answering in database systems. More recently, the advent of multiteam semantics has bridged probabilistic and causal reasoning. By interpreting teams as multisets endowed with probabilistic weights, researchers have shown that standard causal-probabilistic constructs, including average causal effect and do-calculus interventions, admit natural characterisations. A normal form theorem further elucidates the layered structure of causal inference, demonstrating that finitary languages suffice to express a wide array of empirical queries without committing to a particular probability distribution.

Dependence Logic and Probabilistic Semantics publication trend

The graph below shows the total number of articles in dependence logic and probabilistic semantics across all publications each year (not limited to Nature Index journals).

Technical terms

Team semantics: A semantic framework evaluating formulae over sets of variable assignments rather than single assignments.

Dependence atom: A construct expressing that the value of one tuple of variables functionally determines another variable within a team.

Inclusion atom: A relation stating that every assignment in one subteam projects to an assignment in another subteam, capturing data inclusion constraints.

Multiteam semantics: An extension of team semantics in which assignments carry multiplicities or probabilities, enabling probabilistic and causal interpretations.

References

1. A Simple Logic of Functional Dependence. Journal of Philosophical Logic (2021).
2. Complexity thresholds in inclusion logic. Information and Computation (2022).
3. Multiteam Semantics for Interventionist Counterfactuals: Probabilities and Causation. Journal of Philosophical Logic (2024).