Bounds on the Size of PC and URC Formulas

0536433 - ÚI 2021 RIV US eng J - Journal Article
Kučera, P. - Savický, Petr
Bounds on the Size of PC and URC Formulas.
Journal of Artificial Intelligence Research. Roč. 69, 24 December (2020), s. 1395-1420. ISSN 1076-9757. E-ISSN 1943-5037
R&D Projects: GA ČR(CZ) GA19-19463S
Institutional support: RVO:67985807
Keywords : automated reasoning * knowledge representation * satisfiability
OECD category: Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Impact factor: 2.776, year: 2020
Method of publishing: Open access

In this paper, we investigate CNF encodings, for which unit propagation is strong enough to derive a contradiction if the encoding is not consistent with a partial assignment of the variables (unit refutation complete or URC encoding) or additionally to derive all implied literals if the encoding is consistent with the partial assignment (propagation complete or PC encoding). We prove an exponential separation between the sizes of PC and URC encodings without auxiliary variables and strengthen the known results on their relationship to the PC and URC encodings that can use auxiliary variables. Besides of this, we prove that the sizes of any two irredundant PC formulas representing the same function differ at most by a factor polynomial in the number of the variables and present an example of a function demonstrating that a similar statement is not true for URC formulas. One of the separations above implies that a q-Horn formula may require an exponential number of additional clauses to become a URC formula. On the other hand, for every q-Horn formula, we present a polynomial size URC encoding of the same function using auxiliary variables. This encoding is not q-Horn in general.
Permanent Link: http://hdl.handle.net/11104/0314210