2-CNFS and Logical Embeddings Robert Cowen April 2, 2009

Abstract The expressive power of 2-cnfs, conjunctive normal forms with two literals per clause, is shown to be severly limited compared to 3-cnfs.

1 Introduction If , then 2-cnfs and 3-cnfs are strongly differentiated since satisfiability of 2-cnfs is polynomial while satisfiability of 3-cnfs is NP-Complete and would not be polynomial assuming (See [3]). However there are properties that differentiate these two classes without any assumptions at all. Piotr Wojtylak has shown in [5], that the Boolean Prime Ideal Theorem or BPI is stronger than the compactness theorem for propostional logic in ZF when all formulas are 2-cnfs; however the compactness theorem for 3-cnfs is known to be equivalent to BPI in ZF (see [2]). Here a much simpler property will be shown to distinguish between these classes of cnfs, the capacity to embed all propositional formulas.

2 The Logic of 2-cnfs A 2-cnf is a conjunction of disjuctions, each of which is of size two. Suppose is a 2-cnf. A literal is a propositional variable or a negated propositional variable; if is a literal will denote its opposite. and write

Definition 2.1 We say there is an X-sequence for or there is a sequence of clauses of , . Lemma 2.2 Proof. An

if and only if -sequence for

Lemma 2.3 If Proof. An

and

-sequence for

Lemma 2.4 If

and

if either is a clause of . We shall write for

.

when reversed is an -sequence for , then followed by an

.

. -sequence for

and

, then

Department of Mathematics, Emeritus, Queens College, CUNY

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will be an .

-sequence for

.

Proof. Since -sequence for

, , by Lemma 2.1. Then, an -sequence for , followed by an -sequence for is an -sequence for and

Lemma 2.5 If Proof. Since

, the

and

.

.

, the result follows from the previous lemmas.

and every clause of

Lemma 2.6 If

followed by an

is a clause of

, then

Proof. Obvious. Theorem 2.7 If

and

Proof. First, suppose

for some literal and

of

, then

, for some literal

is unsatisfiable.

of

. Then there are

-sequences,

, (1) . (2) If were satisfiable, all the clauses in (1) and (2) would be true under some interpretation of . If were true under this interpretation, would be false; but the clauses of (2) would all be true, and this implies must be true which means that is false! Similarly is false leads to contradiction using (1). Thus the assumption that is satisfiable leads us to contradiction. Therefore must be unsatisfiable if and , for some literal of . Theorem 2.8 If

, then

is unsatisfiable.

Proof. If is an -sequence for would have to make true and thus , false. Corollary 2.9 If Proof. satisfying

, then

, then any interpretation which satisfies

.

implies, by the Theorem, that will make false and so true.

is unsatisfiable; hence any interpretation

The following Theorem will play a central role in what follows. Theorem 2.10 Let , for

be a satisfiable 2-cnf and suppose that . Then , for some ,

is unsatisfiable, where .

be the statement of Proof. We proceed by induction on the number of clauses in X, that is, let . If and the Theorem when has clauses and is any integer, is unsatisfiable, then both and are unsatisfiable. Since , both , , for some , . Hence and holds. holds and , with clauses, is satisfiable but is unsatisfiable, Assume now where , for each , . Suppose, . Then both and are unsatisfiable. If , , then the induction hypothesis applied or ) and ( or ), . Equivalently, to yields, ( or ( and ). In the first case, , by Lemma 2.6; in the second case, again by Lemma 2.6, and ; however, we also have which implies 2

, by Lemma 2.4. or , for some . If both are true, and Assume then that . Suppose exactly one is true; say . Since . so, of course, and since is unsatisfiable, applying the induction hypothesis Then, since yields either or ; in the first case, ; while in the second case, ; this, together with , gives , by Lemma 2.3. Corollary 2.11 Let be a 2-cnf and suppose that is unsatisfiable, for some i,j, . Proof. If

is unsatisfiable. Then

is unsatisfiable or , for some , the result clearly follows; if not, , by Theorem 2.10. Theorem 2.8 then implies the desired conclusion.

3 Logical Embeddings If is a propositional formula, the set of propositional variables of will be denoted by . If are propositional formulas with then we say that is logically embeddable in and write if any satisfying interpretation of can be extended to a satisfying interpretation of and any satisfying interpretation of when restricted to , satisfies . We now can prove the following important theorem. Theorem 3.1 Let

be any literals. Then there is no 2-cnf

with

.

, for some 2-chf ; then is unProof. Suppose, on the contrary, that satisfiable, since, otherwise there would be an interpretation of whose restriction to , fails to . Therefore Corollary 2.11 implies that at least one of is satisfy unsatisfiable. Suppose, without loss of generality, is unsatisfiable. Since satisfies and there is an extension of which satisfies ; but this implies that is satisfiable! Therefore must be false. be sets of propositional formulas. Then Definition 3.2 Let if for every there is a with .

is logically embeddable in ,

,

The proof that 3-SAT (satisfiablility for 3-cnfs) is NP-Complete shows in essence that the set of all cnfs (equivalently, all propositional formulas) is logically embeddable in 3-CNF (the set of all 3-cnfs). This, however, is not the case with 2-CNF, the class of 2-cnfs, as Theorem 3.1 shows.

4 A graph theory application Although 2-CNF cannot embed all propositional formulas, it is not useless, as we show next. Let be a simple graph and for each vertex , let there be associated a list, . Then an L-list coloring of is a proper coloring of such that for all . (For more on list colorings, see [3].) Assume now that each is a pair, for each . We construct a 3

2-cnf which is satisfiable iff is -list colorable. For each vertex and , take a has List, L(v) = , add clause to . In addition, propositional variable . If vertex if vertices are connected by an edge of , , and , add the clause to .

u

(1,2)

v

w (1,3)

(2,3)

Figure 1 is , We give a small example. (For a more extensive example of the use of 2-sat see [1].) If the complete graph on three vertices; we label the vertices, and assign lists as follows: (See Figure 1). Then the associated 2-cnf is the following formula. . is satisfiable. However testing It is easy to show that a graph is -list colorable if and only if 2-cnfs for satisfiability is polynomial(see [3]; therefore algorithms used for testing 2-cnfs for satisfiability in polynomial time, can be used to determine list colorability for graphs with 2-element “color” lists in polynomial time.

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References [1] Bagchi, A., Servatius, B., Shi, W., 2-satisfiability and diagnosing faulty processors in massively parallel computing systems, Discrete Applied Math., 60(1995), 25-37. [2] Cowen, R., Two hypergraph theorems equivalent to BPI, Notre Dame J. Formal Logic, 31(1990), 232 -239. [3] Garey, M.R. and D.S. Johnson, Computers and Intractability. A Guide to the Theory of NPCompleteness, W. H. Freeman, San Francisco, 1979. [4] West, D.B., Introduction to Graph Theory, Prentice-Hall, Upper Saddle River, N.J., 1996. [5] Wojtylak, P., 2-SAT is not equivalent to the Boolean Prime Ideal Theorem, Logic at Work, Essays Dedicated to the Memory of Helena Rasiowa ed. Ewa Or/lowska, Springer-Verlag, 1998, pp. 580583

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