SDDs are Exponentially More Succinct than OBDDs (1601.00501v1)

Abstract: Introduced by Darwiche (2011), sentential decision diagrams (SDDs) are essentially as tractable as ordered binary decision diagrams (OBDDs), but tend to be more succinct in practice. This makes SDDs a prominent representation language, with many applications in artificial intelligence and knowledge compilation. We prove that SDDs are more succinct than OBDDs also in theory, by constructing a family of boolean functions where each member has polynomial SDD size but exponential OBDD size. This exponential separation improves a quasipolynomial separation recently established by Razgon (2013), and settles an open problem in knowledge compilation.

Review of "SDDs are Exponentially More Succinct than OBDDs"

Simone Bova's paper, "SDDs are Exponentially More Succinct than OBDDs," addresses the critical question within knowledge compilation: the relative succinctness of Sentential Decision Diagrams (SDDs) compared to Ordered Binary Decision Diagrams (OBDDs). The findings establish that SDDs are theoretically and practically more efficient than OBDDs, providing significant implications for fields such as artificial intelligence and knowledge compilation.

Core Findings

The paper introduces a family of boolean functions that showcases a polynomial size for SDDs while requiring exponential size for OBDDs. This result significantly exceeds the previously established quasipolynomial separation by Razgon, conclusively resolving an extant open problem within the domain.

Theoretical Implications

Through rigorous construction, it is proven that compressed SDDs are exponentially more succinct than OBDDs, utilizing the generalized hidden weighted bit function for demonstration. Theoretical implications extend beyond SDDs and OBDDs, touching on broader topics within structured deterministic NNFs. The paper speculates that the enhanced succinctness capabilities of SDDs make them a promising candidate for replacing traditional OBDDs in complex computational tasks where efficiency and scalability are pivotal.

Practical Applications

SDDs maintain tractability for key computational operations similar to OBDDs, while asserting depth in applicability across various domains. These applications include but are not limited to, verification processes and synthesis tasks within AI systems and probabilistic reasoning frameworks.

Numerical Results

The methodology relies on demonstrating the succinctness gap through exhaustive manipulations of boolean functions under distinct computational representations. Strong numerical evidence is presented detailing the size disparities in models, reinforcing the claim that SDDs have polynomial size while OBDDs scale exponentially for specific boolean functions.

Future Outlook

The insights from this paper set a pathway for expanding the use of SDDs, with potential refinement in algorithms favoring variable tree flexibility as opposed to linear variable ordering as seen in OBDDs. The investigation into compressed SDDs suggests that a comprehensive paper regarding canonical and non-canonical forms could yield further efficiency improvements and adaptations suitable for dynamic contexts in AI.

Conclusion

In summary, the paper provides an authoritative evaluation of the efficiency of SDDs compared to OBDDs, offering innovative perspectives that could potentially revolutionize knowledge compilation methodologies. Future research focused on the interaction between compression schemes in SDDs and computational complexity may uncover untapped potential, optimizing boolean function representations, and advancing artificial intelligence systems' performance at scale.