Generating clause sequences of a CNF formula

Kristóf Bérczi, Endre Boros, Ondřej Čepek, Khaled Elbassioni, Petr Kučera, Kazuhisa Makino

Research output: Contribution to journalArticlepeer-review


Given a CNF formula Φ with clauses C1,…,Cm and variables V={x1,…,xn}, a truth assignment a:V→{0,1} of Φ leads to a clause sequence σΦ(a)=(C1(a),…,Cm(a))∈{0,1}m where Ci(a)=1 if clause Ci evaluates to 1 under assignment a, otherwise Ci(a)=0. The set of all possible clause sequences carries a lot of information on the formula, e.g. SAT, MAX-SAT and MIN-SAT can be encoded in terms of finding a clause sequence with extremal properties. We consider a problem posed at Dagstuhl Seminar 19211 “Enumeration in Data Management” (2019) about the generation of all possible clause sequences of a given CNF with bounded dimension. We prove that the problem can be solved in incremental polynomial time. We further give an algorithm with polynomial delay for the class of tractable CNF formulas. We also consider the generation of maximal and minimal clause sequences, and show that generating maximal clause sequences is NP-hard, while minimal clause sequences can be generated with polynomial delay.

Original languageBritish English
Pages (from-to)68-74
Number of pages7
JournalTheoretical Computer Science
StatePublished - 8 Feb 2021


  • Clause sequences
  • CNF formulas
  • Enumeration
  • Generation


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