[Paper Review] On Compiling DNNFs without Determinism

Umut Öztok, Adnan Darwiche|arXiv (Cornell University)|Sep 20, 2017

Machine Learning and Algorithms28 references3 citations

TL;DR

This paper proposes a novel method to compile Boolean formulas into DNNFs without enforcing determinism, by introducing auxiliary variables, compiling the resulting formula into a deterministic DNNF, and then existentially quantifying out the auxiliary variables. The key contribution is that this approach can produce DNNFs exponentially more succinct than deterministic DNNFs, even with just one auxiliary variable, and leverages existing compilation techniques like bounded variable addition.

ABSTRACT

State-of-the-art knowledge compilers generate deterministic subsets of DNNF, which have been recently shown to be exponentially less succinct than DNNF. In this paper, we propose a new method to compile DNNFs without enforcing determinism necessarily. Our approach is based on compiling deterministic DNNFs with the addition of auxiliary variables to the input formula. These variables are then existentially quantified from the deterministic structure in linear time, which would lead to a DNNF that is equivalent to the input formula and not necessarily deterministic. On the theoretical side, we show that the new method could generate exponentially smaller DNNFs than deterministic ones, even by adding a single auxiliary variable. Further, we show that various existing techniques that introduce auxiliary variables to the input formulas can be employed in our framework. On the practical side, we empirically demonstrate that our new method can significantly advance DNNF compilation on certain benchmarks.

Motivation & Objective

• To address the inefficiency of current knowledge compilers that enforce determinism, leading to less succinct DNNF representations.
• To explore whether relaxing determinism in DNNF compilation can yield significantly more compact representations for certain formulas.
• To develop a practical framework that enables existing deterministic DNNF compilers to generate non-deterministic DNNFs via auxiliary variable introduction and forgetting.
• To identify and utilize existing variable introduction techniques (e.g., BVA, Tseitin) as valid preprocessing steps for non-deterministic DNNF compilation.

Proposed method

• Introduce auxiliary variables to the input formula to create a new formula that is amenable to deterministic DNNF compilation.
• Compile the modified formula into a deterministic DNNF using existing knowledge compilers.
• Apply existential quantification (forgetting) over the auxiliary variables to recover a DNNF equivalent to the original formula but not necessarily deterministic.
• Leverage the fact that forgetting variables from a DNNF preserves decomposability and can be done in linear time.
• Utilize known variable introduction techniques—such as bounded variable addition (BVA), Tseitin transformation, and extended resolution—as preprocessing steps that yield formulas equivalent modulo forgetting.
• Apply BVA to reduce treewidth, enabling efficient CNF-to-DNNF compilation and improving compactness of the final DNNF.

Experimental results

Research questions

• RQ1 Can DNNFs without determinism be exponentially more succinct than deterministic DNNFs for certain formulas?
• RQ2 To what extent can introducing and forgetting auxiliary variables lead to more compact DNNF representations?
• RQ3 Which existing variable introduction techniques (e.g., BVA, Tseitin) can be used to generate formulas equivalent modulo forgetting for use in non-deterministic DNNF compilation?
• RQ4 Can bounded variable addition (BVA) effectively reduce treewidth and thus improve DNNF compilation performance on hard instances?
• RQ5 How does the proposed method compare to state-of-the-art deterministic DNNF compilers in practice, especially on benchmarks where determinism is not required?

Key findings

• Forgetting a single auxiliary variable can lead to an exponential reduction in DNNF size compared to deterministic DNNFs, demonstrating that non-deterministic DNNFs can be exponentially more succinct.
• Bounded variable addition (BVA) can reduce the treewidth of a CNF from unbounded to a constant value, enabling efficient compilation of formulas that were previously intractable for deterministic DNNF compilers.
• On FPGA routing benchmarks, the proposed method (c2d_forget) compiled DNNFs up to 100× faster than the deterministic c2d compiler, with significantly smaller DNNF sizes, and succeeded where c2d failed due to memory limits.
• Theoretical analysis confirms that techniques like Tseitin transformation and extended resolution also produce formulas equivalent modulo forgetting, making them applicable in the proposed framework.
• Empirical results show that BVA, traditionally used for SAT solving, significantly advances DNNF compilation when used as a preprocessing step in the new framework.
• The method enables the use of existing deterministic DNNF compilers to generate non-deterministic DNNFs, avoiding the need for new compilation algorithms.