The seminal Satisfiability Coding Lemma of Paturi, Pudlák, and Zane is a coding scheme for satisfying assignments of k-CNF formulas. We generalize it to give a coding scheme for implicants and use this generalized scheme to establish new structural and algorithmic properties of prime implicants of k-CNF formulas. Our first application is a near-optimal bound of n⋅ 3^{n(1-Ω(1/k))} on the number of prime implicants of any n-variable k-CNF formula. This resolves an open problem from the Ph.D. thesis of Talebanfard, who proved such a bound for the special case of constant-read k-CNF formulas. Our proof is algorithmic in nature, yielding an algorithm for computing the set of all prime implicants - the Blake Canonical Form - of a given k-CNF formula. The problem of computing the Blake Canonical Form of a given function is a classic one, dating back to Quine, and our work gives the first non-trivial algorithm for k-CNF formulas.
@InProceedings{mosse_et_al:LIPIcs.SAT.2022.9, author = {Moss\'{e}, Milan and Sha, Harry and Tan, Li-Yang}, title = {{A Generalization of the Satisfiability Coding Lemma and Its Applications}}, booktitle = {25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)}, pages = {9:1--9:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-242-6}, ISSN = {1868-8969}, year = {2022}, volume = {236}, editor = {Meel, Kuldeep S. and Strichman, Ofer}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2022.9}, URN = {urn:nbn:de:0030-drops-166837}, doi = {10.4230/LIPIcs.SAT.2022.9}, annote = {Keywords: Prime Implicants, Satisfiability Coding Lemma, Blake Canonical Form, k-SAT} }
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