Abstract
Boolean formulas often have what are known as hidden structures. We study the complexity of whether such structures (of certain sizes) exist in a formula, the complexity of getting one’s hands on the structure’s information, and whether even when in hand the information can be efficiently exploited. In particular, backdoors and backbones of Boolean formulas are important hidden structural properties. A natural goal, already in part realized, is that solver algorithms seek better performance by exploiting these structures. However, the present paper is not intended to improve the performance of SAT solvers, but rather is a cautionary tale. The theme of this paper is that there is a potential chasm between the existence of such structures in the Boolean formula and being able to effectively exploit them. This does not mean that these structures are not useful to solvers. It does mean that one must be very careful not to assume that it is computationally easy to go from the existence of information to being able to get one’s hands on it and/or being able to exploit it. We construct backdoor- and backbone-based cases where, if \(\mathrm {P}\ne \mathrm {NP}\), such assumptions fail. For example, if \(\mathrm {P}\ne \mathrm {NP}\), then (a) there are easily recognizable families of Boolean formulas with strong backdoors that are easy to find, yet for which it is hard to determine whether the formulas are satisfiable, and (b) there are easily recognizable sets of Boolean formulas for which it is hard to determine whether they have a large backbone.
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Notes
Recall that if F is unsatisfiable, then “A determines F” means, perhaps somewhat confusingly as regards the plain English meaning of “determines,” that A proclaims F’s unsatisfiability.
Here we assume that a clause \(\{x\}\) precedes a clause \(\{y\}\) in lexicographical order if x precedes y in lexicographical order.
We have not been able to find Corollary (to the Proof) 4.2 in the literature. Certainly, two things that on their surface might seem to be the claim we are making in Corollary (to the Proof) 4.2 are either trivially true or are in the literature. However, upon closer inspection they turn out to be quite different from our claim.
In particular, if one removes the word “nontrivial” from Corollary (to the Proof) 4.2’s statement, and one is in the model in which every satisfiable formula is considered to have the empty collection of variables as a backbone and every unsatisfiable formula is considered to have no backbones, then the thus-altered version of Corollary (to the Proof) 4.2 is clearly true, since if one with those changes takes A to be the set of all Boolean formulas, then the theorem degenerates to the statement that if \(\mathrm {P}\ne \mathrm {NP}\), then SAT is (NP-complete, and) not in \(\mathrm {P}\).
Also, it is stated in Kilby et al. [25] that finding a backbone of CNF formulas is “NP-hard.” However, though this might seem to be our result, their claim and model differ from ours in many ways, making this a quite different issue. First, their hardness refers to Turing reductions (and in contrast our paper is about many-one reductions and many-one completeness). Second, they are not even speaking of NP-Turing-hardness—much less NP-Turing-completeness—in the standard sense since their model is assuming a function reply from the oracle rather than having a set as the oracle. Third, even their notion of backbones is quite different as it (unlike the influential Williams, Gomes, and Selman 2003 paper [37] and our paper) in effect requires that the function-oracle gives back both a variable and its setting. Fourth, our claim is about nontrivial backbones.
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Acknowledgements
An earlier version [21] appeared in the SOFSEM 2019 conference. We are extremely grateful to the anonymous conference and journal referees for valuable comments and suggestions that much improved this paper.
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Supported in part by NSF grant CCF-2006496, NSF grant CCF-2030859 to the Computing Research Association for the CIFellows Project, a Renewed Research Stay award from the Alexander von Humboldt Foundation, and sabbatical visit support from ETH Zürich’s Department of Computer Science.
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Hemaspaandra, L.A., Narváez, D.E. Existence versus exploitation: the opacity of backdoors and backbones. Prog Artif Intell 10, 297–308 (2021). https://doi.org/10.1007/s13748-021-00234-6
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DOI: https://doi.org/10.1007/s13748-021-00234-6