### Revision(s):

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Revision #1 to TR07-009 | 15th August 2007 00:00
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#### Nearly-Exponential Size Lower Bounds for Symbolic Quantifier Elimination Algorithms and OBDD-Based Proofs of Unsatisfiability

**Abstract:**
We demonstrate a family of propositional formulas in conjunctive normal form so that a formula of size $N$ requires size $2^{Omega(sqrt[7]{N/logN})}$ to refute using the tree-like OBDD refutation system of Atserias, Kolaitis and Vardi with respect to all variable orderings. All known symbolic quantifier elimination algorithms for satisfiability generate tree-like proofs when run on unsatisfiable CNFs, so this lower bound applies to the run-times of these algorithms. Furthermore, the lower bound generalizes earlier results on OBDD-based proofs of unsatisfiability in that it applies for all variable orderings, it applies when the clauses are processed according to an arbitrary schedule, and it applies when variables are eliminated via quantification.

This version differs from the earlier version in that some of the proofs have been cleaned up and made shorter.

### Paper:

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TR07-009 | 8th January 2007 00:00
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#### Nearly-Exponential Size Lower Bounds for Symbolic Quantifier Elimination Algorithms and OBDD-Based Proofs of Unsatisfiability

**Abstract:**
We demonstrate a family of propositional formulas in conjunctive normal form

so that a formula of size $N$ requires size $2^{\Omega(\sqrt[7]{N/logN})}$

to refute using the tree-like OBDD refutation system of

Atserias, Kolaitis and Vardi

with respect to all variable orderings.

All known symbolic quantifier elimination algorithms for satisfiability

generate tree-like proofs when run on unsatisfiable CNFs, so this lower bound

applies to the run-times of these algorithms.

Furthermore, the lower bound generalizes earlier

results on OBDD-based

proofs of unsatisfiability in that it applies for all variable

orderings, it applies when the clauses are processed according to

an arbitrary schedule, and it applies when variables are eliminated via

quantification.

### Comment(s):

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Comment #1 to TR07-009 | 15th February 2007 06:56
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#### Brief summary of differences between TR07-007 and TR07-009 Comment on: TR07-009

**Abstract:**
A brief summary the differences between the main results in TR07-007 and TR07-009