Ordered Binary Decision Diagrams, Pigeonhole Formulas and Beyond1

Tveretina, Olga; Sinz, Carsten; Zantema, Hans

doi:10.3233/sat190074

Cited by 10 publications

(5 citation statements)

References 20 publications

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“…Groote and Zantema proved that resolution and OBDDs do not simulate each other polynomially on arbitrary inputs for limited OBDD derivations [8]. Tveretina, Sinz and Zantema strengthened the above result and presented a class of CNFs hard for an arbitrary OBDD derivation and easy for resolution [17].…”

Section: Introductionmentioning

confidence: 88%

Resolution Simulates Ordered Binary Decision Diagrams for Formulas in Conjunctive Normal Form

Tveretina

2017

Preprint

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A classical question of propositional logic is one of the shortest proof of a tautology. A related fundamental problem is to determine the relative efficiency of standard proof systems, where the relative complexity is measured using the notion of polynomial simulation. Presently, the state-of-the-art satisfiability algorithms are based on resolution in combination with search. An Ordered Binary Decision Diagram (OBDD) is a data structure that is used to represent Boolean functions. Groote and Zantema have proved that there is exponential separation between resolution and a proof system based on limited OBDD derivations. However, formal comparison of these methods is not straightforward because OBDDs work on arbitrary formulas, whereas resolution can only be applied to formulas in Conjunctive Normal Form (CNFs). Contrary to popular belief, we argue that resolution simulates OBDDs polynomially if we limit both to CNFs and thus answer negatively the open question of Groote and Zantema whether there exist unsatisfiable CNFs having polynomial OBDD refutations and requiring exponentially long resolution refutations.

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Section: Introductionmentioning

confidence: 88%

Resolution Simulates Ordered Binary Decision Diagrams for Formulas in Conjunctive Normal Form

Tveretina

2017

Preprint

Self Cite

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“…This was mentioned for OBDD e.g. in (Narodytska and Walsh 2007;Huang and Darwiche 2004), and proved for specific bottomup algorithms compiling unsatisfiable CNF formulas into OBDD in (Krajícek 2008;Tveretina, Sinz, and Zantema 2010;Friedman and Xu 2013). As remarked by these works, large intermediate results are problematic because they may lead to failed compilation due to memory outs or very long runtime even for instances that have small representations.…”

Section: Introductionmentioning

confidence: 90%

“…Let us compare Corollary 1 with known exponential lower bounds on the size of intermediate results for similar refutation systems, see for instance (Krajícek 2008;Segerlind 2008;Tveretina, Sinz, and Zantema 2010;Friedman and Xu 2013). First, we are not aware of refutation systems using str-DNNF circuits that are not OBDD or branching programs.…”

Section: Tseitin Formulasmentioning

confidence: 99%

Lower Bounds on Intermediate Results in Bottom-Up Knowledge Compilation

Colnet¹,

Mengel²

2022

AAAI

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Bottom-up knowledge compilation is a paradigm for generating representations of functions by iteratively conjoining constraints using a so-called apply function. When the input is not efficiently compilable into a language - generally a class of circuits - because optimal compiled representations are provably large, the problem is not the compilation algorithm as much as the choice of a language too restrictive for the input. In contrast, in this paper, we look at CNF formulas for which very small circuits exists and look at the efficiency of their bottom-up compilation in one of the most general languages, namely that of structured decomposable negation normal forms (str-DNNF). We prove that, while the inputs have constant size representations as str-DNNF, any bottom-up compilation in the general setting where conjunction and structure modification are allowed takes exponential time and space, since large intermediate results have to be produced. This unconditionally proves that the inefficiency of bottom-up compilation resides in the bottom-up paradigm itself.

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“…Using that SAT solver we solve CNF formulas of the N-Queens problem and of the Pigeonhole Principle. The latter, as given in [36], expresses that there exists no isomorphism between the sets [N + 1] and [N ].…”

Section: Sat Solvermentioning

confidence: 99%

Efficient Binary Decision Diagram Manipulation in External Memory

Sølvsten¹,

Pol²,

Jakobsen³

et al. 2021

Preprint

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We follow up on the idea of Lars Arge to rephrase the Reduce and Apply algorithms of Binary Decision Diagrams (BDDs) as iterative I/Oefficient algorithms. We identify multiple avenues to improve the performance of his proposed algorithms and extend the technique to other basic BDD algorithms.These algorithms are implemented in a new BDD library, named Adiar. We see very promising results when comparing the performance of Adiar with conventional BDD libraries that use recursive depth-first algorithms. For instances of about 50 GiB, our algorithms, using external memory, are only up to 3.9 times slower compared to Sylvan, exclusively using internal memory. Yet, our proposed techniques are able to obtain this performance at a fraction of the internal memory needed by Sylvan to function. Furthermore, with Adiar we are able to manipulate BDDs that outgrow main memory and so surpass the limits of the other BDD libraries.

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Ordered Binary Decision Diagrams, Pigeonhole Formulas and Beyond1

Cited by 10 publications

References 20 publications

Resolution Simulates Ordered Binary Decision Diagrams for Formulas in Conjunctive Normal Form

Resolution Simulates Ordered Binary Decision Diagrams for Formulas in Conjunctive Normal Form

Lower Bounds on Intermediate Results in Bottom-Up Knowledge Compilation

Efficient Binary Decision Diagram Manipulation in External Memory

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