In this section we look deeper into the soundness and completeness of proof systems and refutation systems.
Let us consider a combination proof/refutation system S for a problem P. This means that given an instance I of P, S either generates a certificate of feasibility or a refutation. We can use S to examine the relationship between soundness and completeness of proof systems and refutation systems.
Theorem 5.3.1. S is sound as a proof system for P if and only if it is complete as a refutation system for P
Proof. First assume that S is a sound proof system for the problem P. Let Iu ∈ Uu be
an infeasible instance of P. Since S is sound as a proof system, it will not generate a
CHAPTER 5. PROOF SYSTEMS AND REFUTATION SYSTEMS 51
every infeasible instance, thus S is a complete refutation system.
Now assume that S is a complete refutation system for the problem P. Let Iu∈ Uube an
infeasible instance of P. Since S is complete as a proof system, it will generate a refutation
for Iu. Thus, S will not generate a certificate of feasibility for Iu. This is true for every
infeasible instance, thus S is a sound proof system.
Theorem 5.3.2. S is complete as a proof system for P if and only if it is sound as a refutation system for P
Proof. First assume that S is a complete proof system for the problem P. Let Is∈ Us be a
feasible instance of P. Since S is complete as a proof system, it will generate a certificate of
feasibility for Is. Thus, S will not generate a refutation for Is. This is true for every feasible
instance, thus S is a sound refutation system.
Now assume that S is a sound refutation system for the problem P. Let Is ∈ Us be a
feasible instance of P. Since S is sound as a proof system, it will not generate a refutation
for Is. Thus, S must generate a certificate of feasibility for Is. This is true for every feasible
52
Part II
53
Chapter 6
2-CNF Clausal Formulas
6.1
Motivation and Related Work
In this section, we briefly enumerate the motivation for our work and some related approaches in the literature.
1. Monotone NAE-SAT is equivalent to Hyper-graph bicolorability (set splitting). [PSSW14, GJ79, RS06]
2. Monotone NAE 2-SAT is equivalent to graph bicolorability (bipartite). The shortest (weighted) ROR NAE-resolution refutation corresponds to the shortest proof that a graph is not bipartite (the shortest (weighted) cycle with an odd number of edges) [SG11].
3. Let F be a CNF formula. The dual formula D(F) is constructed by replacing all disjunctions in f with conjunctions, and all conjunctions with disjunctions. The resultant formula is clearly in disjunctive normal form.
We have that F is NAE unsatisfiable if and only if F |= D(F).
Let Fcbe the CNF formula obtained by complementing each literal in F. We see that
CHAPTER 6. 2-CNF CLAUSAL FORMULAS 54
We have that F is NAE unsatisfiable if and only if F ∧ Fc is unsatisfiable. This can
happen if and only if F |= ¬Fcwhich is equivalent to saying that F |= D(F).
Thus, the question of whether a CNF formula F entails its dual formula D(F) is equivalent to the problem of NAE unsatisfiablity.
Resolution is a refutation procedure that was introduced in [Rob65] to establish the un- satisfiability of clausal boolean formulas. Resolution is a sound and complete procedure, although it is not efficient in general [Hak85]. Resolution is one among many proof systems (refutation systems) that have been discussed in the literature [Urq95]; indeed it is among the weaker proof systems [BP97] in that there exist propositional formulas for which short proofs exist (in powerful proof systems) but resolution proofs of unsatisfiability are ex- ponentially long. Resolution remains an attractive option for studying the complexity of constraint classes on account of its simplicity and wide applicability; it is important to note that resolution is the backbone of a range of automated theorem provers [BP96].
Resolution refutation techniques often arise in proof complexity. Research in proof complexity is primarily concerned with the establishment of non-trivial lower bounds on the proof lengths of propositional tautologies (alternatively refutation lengths of propo- sitional contradictions). An essential aspect of establishing a lower bound is the proof system used to establish the bound. For instance, super-polynomial bounds for tautologies have been established for weak proof systems such as resolution [Hak85]. Establishing that there exist short refutations for all contradictions in a given proof system causes the classes NP and coNP to coincide [CR73].
There are a number of different types of resolution refutation that have been discussed in the literature [RV01]. The most important types of resolution refutation are tree-like, dag- like and read-once. Each type of resolution is characterized by a restriction on input clause combination. One of the simplest types of resolution is Read-once Resolution (ROR). In an ROR refutation, each input clause and each derived clause may be used at most once. There are several reasons to prefer a ROR proof over a generalized resolution proof, not the least of which is that ROR proofs must necessarily be of length polynomial (actually, linear) in
CHAPTER 6. 2-CNF CLAUSAL FORMULAS 55
the size of the input. It follows that ROR cannot be a complete proof system unless NP = coNP. That does not preclude the possibility that we could check in polynomial time whether or not a given CNF formula has a ROR refutation. Iwama [IM95] showed that even in case of 3CNF formulas, the problem of checking ROR existence (henceforth, ROR decidability) is NP-complete.
It is well-known that 2CNF satisfiability is decidable in polynomial time. There are several algorithms for 2CNF satisfiability, most of which convert the clausal formula into a directed graph and then exploit the connection between the existence of labeled paths in the digraph and the satisfiability of the input formula. A natural progression of this research is to establish the ROR complexity of 2CNF formulas.