2 research outputs found
Propagation complete encodings of smooth DNNF theories
We investigate conjunctive normal form (CNF) encodings of a function
represented with a smooth decomposable negation normal form (DNNF). Several
encodings of DNNFs and decision diagrams were considered by (Abio et al. 2016).
The authors differentiate between encodings which implement consistency or
domain consistency from encodings which implement unit refutation completeness
or propagation completeness (in both cases implements means by unit
propagation). The difference is that in the former case we do not care about
properties of the encoding with respect to the auxiliary variables while in the
latter case we treat all variables (the input ones and the auxiliary ones) in
the same way. The latter case is useful if a DNNF is a part of a problem
containing also other constraints and a SAT solver is used to test
satisfiability. The currently known encodings of smooth DNNF theories implement
domain consistency. Building on this and the result of (Abio et al. 2016) on an
encoding of decision diagrams which implements propagation completeness, we
present a new encoding of a smooth DNNF which implements propagation
completeness. This closes the gap left open in the literature on encodings of
DNNFs.Comment: 29 pages, correction of the example in Section 2.
Bounds on the size of PC and URC formulas
In this paper we investigate CNF formulas, for which the unit propagation is
strong enough to derive a contradiction if the formula together with a partial
assignment of the variables is unsatisfiable (unit refutation complete or URC
formulas) or additionally to derive all implied literals if the formula is
satisfiable (propagation complete or PC formulas). If a formula represents a
function using existentially quantified auxiliary variables, it is called an
encoding of the function. We prove several results on the sizes of PC and URC
formulas and encodings. One of them are separations between the sizes of
formulas of different types. Namely, we prove an exponential separation between
the size of URC formulas and PC formulas and between the size of PC encodings
using auxiliary variables and URC formulas. Besides of this, we prove that the
sizes of any two irredundant PC formulas for 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.Comment: 24 pages, minor corrections and improvements of the tex