Sentential Decision Diagrams

Updated 6 April 2026

Sentential Decision Diagrams (SDDs) are canonical, tractable representations of Boolean functions that generalize OBDDs, offering efficient operations like equivalence checking and model counting.
SDDs achieve succinctness through compression and unique normalization via vtrees, though reduced SDDs may encounter exponential worst-case costs during certain Apply operations.
SDDs are practically applied in probabilistic reasoning, database query processing, and belief revision, leveraging their structure for scalable symbolic reasoning.

Sentential Decision Diagrams (SDDs) are canonical, tractable circuit representations of Boolean functions that generalize Ordered Binary Decision Diagrams (OBDDs) while providing often exponentially greater succinctness. SDDs are central constructs in knowledge compilation, model counting, probabilistic reasoning, and various AI tasks requiring both semantic compactness and support for efficient operations such as equivalence checking, conditioning, and systematic model enumeration.

1. Formal Definition and Structure

SDDs are defined recursively over a fixed, full binary tree called a vtree whose leaves are in bijective correspondence with the underlying Boolean variables. Each SDD is normalized for a vtree, with the following inductive grammar:

Leaf Case: If the vtree node $v$ is a leaf labelled by variable $X$ , then the SDD $\alpha$ respecting $v$ is a terminal: $\top$ , $\bot$ , $X$ , or $\neg X$ .
Internal Node (Decision Case): If $v$ $v$ is internal with left child $v_L$ $v_{L}$ and right child $X$ $X$ 0, then $X$ $X$ 1 is a decision node of the form $X$ $X$ 2, where:
- Each $X$ 3 is an SDD respecting $X$ 4 (prime), and each $X$ 5 an SDD respecting $X$ 6 (sub).
- The primes $X$ 7 form a partition over the assignments of $X$ 8:
- Pairwise disjoint: $X$ 9 for $\alpha$ 0.
- Cover: $\alpha$ 1.
- No $\alpha$ 2.
Semantics: $\alpha$ 3.

By enforcing compression (merging elements with identical subs) and trimming (eliminating trivial decision nodes), SDDs are rendered canonical with respect to the vtree: each Boolean function admits a unique reduced SDD representation for each vtree (Broeck et al., 2014).

2. Key Theoretical Properties

2.1 Canonicity and Compression

Canonicity: For a fixed vtree, the reduced (compressed and trimmed) SDD representing any Boolean function is unique. This enables constant-time equivalence checks between Boolean functions.
Compression: At each decision node, elements sharing the same sub are merged. This is crucial for canonical form and for practical caching during bottom-up compilation.

2.2 Tractability and the Apply Operator

Apply Function: For unreduced SDDs, the Apply algorithm (for computing binary operations $\alpha$ 4) runs in $\alpha$ 5 time. On reduced (canonical) SDDs, the Apply operator can incur exponential time/space in the worst case (Broeck et al., 2014). However, empirical evidence shows practical efficiency when using dynamic vtree search and caching.
Bounded Vtrees: Polytime Apply can be retained for reduced SDDs if all left vtree subtrees have bounded size, thus unifying canonicity and polynomial-time combination for an important subclass.

Formulation	Guarantee	Complexity
Unreduced SDD	Polytime Apply	$\alpha$ 6
Reduced SDD	Canonicity, sometimes	Worst-case exponential
Reduced + bounded Vtree	Canonicity + Polytime	$\alpha$ 7

2.3 SDD width and Circuit Treewidth

Width: For each internal vtree node $\alpha$ 8, the SDD width at $\alpha$ 9 is the number of $v$ 0-gates structured at $v$ 1. The width of $v$ 2 is the minimal maximum ( $v$ 3) over all vtrees for $v$ 4.
Main Theorem: For any Boolean function $v$ 5, bounded circuit treewidth is equivalent to bounded SDD width. Formally, $v$ 6 (Bova et al., 2017). This equivalence stratifies function classes supporting compact SDD representations in query compilation.

3. Succinctness and Hierarchical Expressivity

3.1 Comparison with OBDDs and FBDDs

Succinctness: SDDs strictly generalize OBDDs, coinciding exactly when the vtree is right-linear. There exist Boolean functions (e.g., Hidden Weighted Bit) with polynomial-size SDDs but exponential-size OBDDs; this separation is both theoretical and empirical (Bova, 2016, Bollig et al., 2018).
Comparison with FBDDs: SDDs and Free BDDs (FBDDs) are incomparable: certain functions admit polynomial-size SDDs but exponential FBDDs and vice versa (Bollig et al., 2018).

3.2 Simulation Results

Any SDD of size $v$ 7 on $v$ 8 variables is simulated by an unambiguous nondeterministic OBDD of size $v$ 9 (Bollig et al., 2018).
Structured deterministic DNNFs encompass SDDs. However, a Boolean function $\top$ 0 can be SDD-representable in polynomial size if both $\top$ 1 and $\top$ 2 have polynomial-size structured $\top$ 3-DNNFs on the same vtree (Bollig et al., 2019).

4. Naturality and Categorical Semantics

SDDs uniquely capture Boolean functions as natural diagrams under the contravariant power-set functor (pre-image semantics). The semantics of SDDs is given as a natural transformation $\top$ 4, i.e., assignment commutes with the diagram action, making SDDs the canonical representations for propositional logic (as opposed to combination sets in ZSDDs) (Kojima, 2018). This functorial perspective excludes the possibility of mixed-natural semantics for Boolean functions, paralleling the distinction between BDDs and ZDDs.

5. Variants, Extensions, and Applications

5.1 Variable-Shift SDDs (VS-SDDs)

Variable Shift SDDs exploit symmetries under variable renaming in isomorphic vtree subtrees (Nakamura et al., 2020).
There exist families of Boolean functions where the VS-SDD is exponentially smaller than the ordinary SDD, while preserving tractable operations (Apply, model counting, conditioning) in $\top$ 5 or $\top$ 6 time depending on function size.

5.2 Probabilistic and Credal SDDs

Probabilistic SDDs (PSDDs) annotate decision nodes with local probability parameters, providing a compact representation and inducing a full joint PMF constrained by the logic (Mattei et al., 2020).
Credal SDDs generalize this by allowing sets of parameters (polytopes) at each node, supporting inference of lower and upper probabilities, robust MAP, and conditional probabilities using bottom-up dynamic programming with embedded LPs. All core inferences remain tractable in the circuit size.

5.3 Knowledge Compilation and Database Query Processing

SDDs are used in query lineage computation, allowing efficient evaluation of probabilistic queries when the circuit treewidth (hence SDD width) is bounded. For unions of conjunctive queries (UCQs), the dichotomy is: inversion-free queries admit linear-size SDDs, UCQs with inversions require exponential SDD size (Bova et al., 2017).

5.4 Belief Revision

SDDs enable direct implementation of Dalal revision and other belief update operations, exploiting their canonicity and structure; operations are implemented compositionally using local substitutions and partitioned combination (Mattei et al., 2022).

Application Domain	Role of SDDs	Key Properties
Query Compilation	Lineage, tractable PQE	Bound via treewidth/SDD width
Probabilistic Reasoning	PSDDs/CSDDs	Compact PMF/imprecise inference
Explanations/Feature Attribution	FMP in NP, SAT encodings	Polytime for SDDs
Belief Revision	Revision operator	Empirically smaller/faster

6. Open Problems and Future Directions

SDDs continue to serve as a rigorous benchmark in the study of representation languages for Boolean functions and tractable model counting.

Current open directions include:

Improved heuristics and algorithms for near-optimal vtree selection to maximize SDD compactness (Bollig et al., 2018).
Precise delineation of function classes permitting polynomial-size reduced SDDs under canonical vtrees.
Further exploration of the relationship between SDD classes and unambiguous nondeterministic OBDDs.
Extension of variable-shift and other symmetry-driven SDD variants for broader classes of renaming and structural equivalences.
Tighter complexity bounds for transformation operations on reduced SDDs, especially under bounded or dynamically constructed vtrees.
Deeper categorical and functorial understanding of knowledge compilation languages, informed by the naturality theorems established for SDDs and related structures (Kojima, 2018).

SDDs retain central importance in both theory and deployed systems for scalable symbolic reasoning, probabilistic logic, and interpretable machine learning (Bova, 2016, Mattei et al., 2020, Huang et al., 2022).