CENTRAL CIRCULATION BOOKSTACKS The person charging this material is re- sponsible for its renewal or its return to the library from which it was borrowed on or before the Latest Date stamped below. You may be charged a minimum fee of $75.00 for each lost book. Theft, mutilation, and underlining of books aro reason* for disciplinary action and may result in dismissal from the University. TO RENEW CALL TELEPHONE CENTER, 333-8400 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN DEC 141998 JUL 5 2000 When renewing by phone, write new due date below previous due date. L162 Digitized by the Internet Archive in 2013 http://archive.org/details/exponentiallower941plai fX/m 1/ / UIUCDCS-R-78-941 for An Exponential Lower Bound a Restricted Class of Monotone Formulae for 2-Unsatisfiabil i ty UILU-ENG 78 1734 NSF-MCS-22830 by David A. Plaisted September 1978 UIUCDCS-R-78-941 An Exponential Lower Bound for a Restricted Class of Monotone Formulae for 2-Unsatisfi ability by David A. Plaisted Department of Computer Science University of Illinois at Urbana-Champaign Urbana, Illinois 61801 September 1978 This research was supported in part by the National Science Foundation under Grant MCS 77-22830. r 6/0, &H 2-&C?n^ -2- 1. Introduction The complexity of minimal Boolean circuits for computing Boolean functions is polynomial ly related to a lower bound on the Turing complexity of the function [3]. Proving lower bounds on the Boolean circuit complexity of various functions is one potential method of obtaining lower bounds on their Turing complexity. As a small step in this direction, we prove an exponential lower bound on the monotone Boolean formula complexity of a specific function, if the formulae are restricted in certain ways. The function we consider is the satisfiability of a set of 2-literal clauses. Each clause is represented as an input to the Boolean formula. An input of TRUE indicates that the clause is in the set of clauses being considered, and FALSE indicated the clause is not in the set of clauses. We want the formula to have a value of TRUE if the set of clauses is inconsistent and FALSE if it is consistent. Although satisfiability of sets of 2-literal clauses can be decided in polynomial time [ 1 ], so that polynomial size circuits exist, the results presented here may suggest approaches for obtaining lower bounds for the Boolean circuit complexity of 3-satisfi ability. 2. The lower Bound Definition : A Boolean formula (or expression ) over the propositional variables x, , ..., x. is a well-formed expression composed of the variables x , ..., x. together with the unary con- nective 1, the binary connectives a and v, and the constants TRUE and FALSE. The connectives "I, a , v represent negation, conjunction, and disjunction, respectively. The value of a Boolean expression, -3- when the values of x,...x, are given, is determined in the usual way, We write x. as an abbreviation for~lx.. Definition : A Boolean function is a mapping from {TRUE, FALSE} 11 into {TRUE, FALSE} for some integer n. Definition : If E is a Boolean expression over the varia- bles X-, , . . . , x, then the function computed by_ E is the Boolean function f(x-,, ..., x. ) whose value is the value of E when the values of x, , ..., x. are specified. We write fr for the function computed by E. Definition : Two Boolean expressions El and E2 are equivalent if fr-. = fro- Definition : A Boolean formula E is monotone if the only connectives in E are a and v. Definition : Suppose f is a Boolean function of the pro- positional variables x, , x ? , ..., x.. Suppose sets a,, a. ? , ..., a are subsets of {x, , x ? , ..., x,} u {x-, , x ? , ..., x.L We say a-, a a~ a . . . a a is a conjunctive normal form of f if all of the following are true: 1 . f (x-, , . . . , x, ) is true iff for all i , 1 <_ i <_ m, either there exists j such that x. is TRUE and x. e a. or there exists j such that x. is FALSE and x. e a. . 2. There does not exist i , 1 <_ i <_ m, such that x. c a- and x. e a. for some j. 3. There do not exist i , j, 1 <_ i , j ± m, i f j, such that a- c a... J ■4- Thus {x,, x«} a {x, , x~} is a conjunctive normal form of (x, v Xp) a (x v xJ. Note that each a. represents the disjunction of its elements. If E is a Boolean formula, a conjunctive normal form of E is defined to be a conjunctive normal form of the function com- puted by E. If E is monotone, we require all elements of the a. to be (uncomplemented) propositional variables. (Expressions of the form x- are called complemented propositional variables.) If E is monotone, then its conjunctive normal form is unique (up to permuta- tions of the a.) . Definition : If the conjunctive normal form of E is a, a ... a a , then cnf(E) is the set {a,, a ? , . .., a. }. We call elements of cnf(E) disjunctions of cnf(E). Sometimes we write cnf(E) as a n a a a ... a a . v ' 1 2 n Definition : An interpretation I of a set {P-,, ..., P } of predicate symbols is an assignment of truth-values TRUE or FALSE to the predicate symbols. Thus there are 2 interpretations of {P, , ..., P }. We say P. is true in I if I assigns the value TRUE to P. and P. is false in I otherwise. Definition : A literal over {P, , . . . , P } is an element of the set {P r ? v ..., P R } u {P r ? v ..., P^}. Definition : A literal L is true in interpretation I if L is of the form P. and P. is true in I or if L is of the form P. and P. is false in I. ■5- Definition : A clause C over {P-j , . . . , P n > is a set of literals over (P, P }. A clause represents the disjunction of its elements. Thus C is true in interpretation I if at least one literal of C is true in I; otherwise C is false in I. Example : Let I be the interpretation making all predicate symbols true. Then {P^ P 2 , P3} is true in I but {P-j, P 2 , P3} is false in I. Definition : The complement L of a literal L over {P, , ..., P } is defined as follows: If L is of the form P. then L is P i . If L is of the form P. then L is Pj. Definition : Let 7 be the set of interpretations of {P , ..., P }. Fori el, let Contr(I) be the set of 2-literal clauses C over {P, , ..., P } such that C is false in I. From now on we consider 2-literal clauses over {P-j, ..., P n > to be propositional variables. An assignment of truth-values to the variables represents a set of clauses in the following way: If the clause C is assigned the value TRUE, then C is in the set of clauses, otherwise C is not in the set of clauses. Boolean functions of these i 2 ^) variables represent properties of sets of clauses. Definition : A set S of clauses over {P-j, ..., P R ) is inconsistent (or unsatisfiable ) iff for all I e I , there exists C e S such that is false in I. Note that the conjunctive normal form of the inconsistency function is A Contr(I). Example: The set {CI, C2, C3} of clauses is inconsistent, where CI = {P^, C2 = {P^, -P 2 > and C3= {P^, P 2 >. However, any proper subset of {CI, C2, C3} is consistent. We generally do not consider clauses C such that for some i, P. e C and P. e C. Such clauses are called tautologies and are true in all interpretations. There are only n 2-literal tautologies n 9 over {P, , ..., P }. There are 4( 2 ) or 2n - 2n 2-literal non-tautologies 2 over {P,, ..., P }. We are thus considering Boolean functions of 2n - 2n variables Definition : A monotone Boolean formula E has the subset property if for all sub-formulae El of E, the following is true: Suppose a-, a a« a . . . a a is the conjunctive normal form for El. Suppose B, a B ? a . . . a 8 is the conjunctive normal form for E. Then for all i, 1 <_ i £ m, there exists j, 1 ^ j < n such that a. eg.. Definition : Suppose E is a monotone Boolean formula. We say E satisfies the replacement condition if for all sub-formulae El of E, the following is true: Let E' be E with an occurrence of El replaced by FALSE. Then there exist disjunctions a and a' of cnf(E) and cnf(E'), res- pectively, such that a' is a proper subset of a. ■7- Theorem 1 . Let E be a monotone Boolean formula over the ( ^ )-n propositional variables representing 2-literal clauses over {P, , ..., P } which are not tautologies. Suppose that E represents the inconsistency property of sets of 2-literal clauses over {P, , ..., P }. (That is, E is true iff the set of clauses repre- sented by the values of its inputs, is inconsistent.) Suppose that E satisfies the replacement condition and has the subset property. Then E has at least 2 - 1 "a" connectives. An example of such an expression E for 1-literal clauses might be ({P,} a {P, }) v ({P-} a {P„}). This expression represents the set of clauses containing both {P,} and {P,} or both {P ? } and (Pp). This expression is monotone and logically implies inconsistency, but is not equivalent to inconsistency if n > 2. Fact : Suppose E is a monotone Boolean formula of the form El a E2. Then cnf(E) consists of the minimal elements of cnf(El) u cnf(E2) in the subset ordering. Suppose E is a monotone Boolean formula of the form El v E2. Then cnf(E) consists of the minimal elements of {a u B: a e cnf(El), 8 e cnf(E2)} in the subset ordering. Definition : Suppose E is a monotone Boolean formula. Then Interp(E) is {I e I : la e cnf(E) such that a c Contr(I)}. Fact : Suppose E is a monotone Boolean formula of the form El a E2. Then Interp(E) = Interp(El) u Interp(E2). Fact : Suppose E is a monotone Boolean formula of the form El v E2. Then Interp(E) = Interp(El) n Interp(E2). -8- Definition : Suppose a c Cbntr(I) for I e I . Suppose L is a set of literals that are false in I. We write a ^ I if the L following is true: For all literals L such that L is false in I, there exists a clause C in a. such that L e C. Also, if LI and L2 are literals in /., there must exist a clause CI in a. such that LI e CI and L2 e CI. (The set I need not include all the literals that are false in I.) Note that if a^ I for some L then u{C: C c a.) is the L set of literals that are false in I. Also, if there exists I e I such that a ^ I then I is unique. Lemma 1. Supoose a ^ I for some interoretation I e I . K ' L n Suppose B, c Contr(Il) and B ? c Contr(I2) for some interpretations II, 12 e I . Suppose that for all literals L such that L is true in II but false in 12 or false in II but true in 12, either L e L or L e L. Suppose that a c 8, u 8 ? . Then either I = II or I = 12. Proof : Not difficult. Lemma 2 . Suppose E is a monotone Boolean formula which satisfies the replacement condition. Suppose El is a sub-formula of E, Then El satisfies the replacement condition also. Proof : By induction on the depth of E. If E is a pro- positional variable, the result is immediate. Let E' be E with some sub-formula of El replaced by FALSE. Let El' be El with this sub- formula replaced by FALSE. We know that some disjunction of cnf(E') -9- is a proper subset of some disjunction of cnf(E). Suppose E is of the form Fl a F2. Suppose El is a sub-formula of Fl . Let Fl ' be Fl with El replaced by El 1 . Then some disjunction of cnf(Fl') must be a proper subset of some disjunction of cnf(Fl). Hence Fl satisfies the replacement condition. By induction, El does too. A similar argument applies if E is of the form Fl v F2. Definition : Suppose E and El are monotone Boolean formulae. Then we say a propositional variable x is diminished from E to El if there exist disjunctions a and B of cnf(E) and cnf(El), respec- tively, such that B c a and x e a - 8. Lemma 3 . Suppose E is a monotone Boolean formula satis- fying the reolacement condition. Suppose E is of the form Fl v F2. SuDpose x is a propositional variable of Fl . Then x must be diminished from E to F2. Proof : Let Fl ' be Fl with some occurrence of x replaced by FALSE. Let E' be E with this occurrence of x replaced by FALSE. Let a e cnf(E), a' e cnf(E') be disjunctions such that a' is a Droper subset of a. Then there exist disjunctions 5-,' e Fl ' , B« e F2 such that B, ' u Bp = a'. We cannot have B-, ' e cnf(E) since a' t cnf(E). Hence x I a' and there must exist 3, e cnf(El) such that x e B-, and B-i ' = 3-, - {x}. Since x I a', x i B ? . Also, a' is a proper subset of a, hence x e a and B-, u B ? is a subset of a. Hence a = B-, u B ? so a = a' u {x}. Hence x is diminished from E to F2, since B 2 c a and x e a - Sp. ■10- Definition : Suppose E is a monotone Boolean formula such that for all a e cnf(E), there exists I e I such that a c Contr(I). Then we say E has the contradiction property . Definition : Two clauses CI and C2 are resolvable if there exists a literal L such that L e CI and L e C2. Definition : Let E be the class of monotone Boolean formulae F satisfying the following conditions: 1. F satisfies the replacement condition. 2. Suppose a-| , a« e cnf(F) and II, 12 e I n - Suppose a, c Contr(Il) and a„ c Contr(I2). Suppose CI and C2 are two clauses such that CI and C2 are resolvable and such that CI e Contr(Il) and C2 e Contr(I2). Then CI e a, and C2 e a ? . 3. F and all its sub-formulae have the contradiction property. Lemma 4 . Suppose E is a Boolean formula in the class E. Then a) If E is of the form El v E2, then at least one of El and E2 are in the class E. b) If E is of the form El a E2, then both of El and E2 are in the class E. Proof : Part b) is easy. We prove part a). Suppose E e E and E is of the form El v E2. Suppose a, , a~ e Cp f(E) ancl H » I2 c * n - Suppose a, c Contr(Il) and a ? c Contr(I2). Suppose CI e Contr(Il) and C2 e Contr(I2). Suppose CI and C2 are resolvable. In particular, suppose L e CI and L e C2 for literal L. •11- We know by definition of E that CI £ a-, and C2 e a.*. Since E satisfies the replacement condition, CI must be diminished from E to either El or E2. Similarly for C2. Suppose CI is not diminished from E to El . We claim that C2 is not diminished from E to El . Assume this is not true. Let 8, be an element of cnf(El) such that 8, c Contr(Il) and such that CI e 8, . Let B ? be an element of cnf(El) such that B 2 c Contr(I2) and C2 £ B^. Let P, be an element of cnf(E2) such that V^ c Contr(Il) and CI £ Vy (We know that CI e B ] since CI is not diminished from E to El . Also, B ? exists because C2 is diminished from E to El . Furthermore, P, exists because CI is diminished from E to E2). Now, B ? u P must have some element of cnf(E) as a subset, since E is El v E2. Suppose a e cnf(E) and a c Bp u Pi . Let I e I be such that a. c Contr(I). Let L be the set of literals that are false in I but whose truth-values in II and 12 differ. It is not difficult to show that aYl using the fact that E is in the class E. Hence by Lemma 1 we have the I = II or I = 12. However, we cannot have I = II since CI £ P-, and CI £ B-. We cannot have I = 12 since C2 £ 8 2 and C2 £ B, . Thus we have a contradiction. Hence for CI and C2 as given, if CI is not diminished from E to El , neither is C2. Let P, and EL be elements of cnf(E). Suppose V, c Contr(Jl) and V c Contr(J2) for Jl , J2 e I . Let M be a literal such that M 2 — ' n is true in Jl and false in J2. Recall that L is a literal such that L e CI and L e C2. Now, the clause {L, M} or the clause {L, M} must occur in some disjunction of cnf(E) by definition of E. Suppose -12- {L, M} occurs in some disjunction of cnf(E). Then {L, M} cannot be diminished from E to El , by reasoning as above with clauses C2 and {L, M}. Some clause containing M must occur in some disjunction of cnf(E), by definition of the class E. But no such clause can be diminished from E to El , by reasoning as above. Extending this argument, no clause containing M can be diminished from E to El. It follows that El e E also. If CI is not diminished from E to E2, then E2 e E by similar reasoning. We now show that CI cannot be diminished from E to both El and E2, and so either El e E or E2 e E. Suppose CI is diminished from E to both El and E2. Then there exist a,' e cnf(El), a ? ' e cnf(E2) and interpretations II', 12' e I having the following properties: 1. a-i' c Contr(Il') and a^ c Contr(I2'). 2. CI e Contr(IT) and CI e Contr(I2'). 3. CI t a] ' and CI t a ' . 4. For some a e cnf(E) , a c a, ' u a ' . Suppose ac Contr(I); then CI e Contr(I). This is because U{C: C c a} must include all literals that are false in I, hence U{C: C c a, ' or C e a 2 '} includes all these literals, hence all literals of CI must occur in U{C: C e a, ' or C e a^'}. However, no complement of a literal in CI can occur in U{C: C g a^} or in U{C: C e a 2 '}. Since CI ^ a, ' and CI ^ a^\ therefore CI i a. But this contradicts the definition of E, since CI and C2 are resolvable and C2 e dp for cu e cnf (E) . -13- Therefore CI cannot be diminished from E to both El and E2. Hence El e E or E2 z E, and the theorem is proven. Theorem 2 . Suppose E is in the class E of Boolean formulae. Let N(E) be | Interp(E) | , the number of interpretations I e I such that there exists a. e cnf(E) such that a. c Contr(I). Then E has at least N(E)-1 " A " connectives. Proof : By induction on the depth of the formula. Suppose E is of the form El v E2. Then either El e E or E2 e E. Also, Interps(E) = Interps(El) n Interps(E2). Hence N(E) >_N(E1) and N(E) > N(E2). If El is in E, we can assume inductively that El (hence E) has at least N(E1) " A " connectives, and similarly for E2. Suppose E is of the form El a E2. Then El e E and E2 e E. Also, Interps(E) = Interps(El) u Interps(E2) so N(E) <_ N(E1) + N(E2). We can assume inductively that El has at least N(E1) - 1 "a" connectives and that E2 has at least N(E2)-1 "a" connectives. Hence E has at least N(E1) + N(E2)-1 "a" connectives, which is not less than N( E) -1 "a" connectives. Suppose E is a propositional variable. Then either N(E)=1, in which case the result is immediate, or else E I E. This completes the proof. Corollary (Theorem 1): Suopose E is a monotone Boolean formula for inconsistency of 2-literal clauses over the predicate symbols {P,, P^, ..., PL Suppose E has the subset oroperty and satisfies the replacement condition. Then E has at least 2 - 1 "a" connectives. Proof : E is in the class E, and N(E) = 2 n . -14- Note that the number of propositi onal variables in E is 4(~), which is 0(n ). Thus we have an exponential lower bound for this restricted class of formulae. A similar proof will work for 3-literal clauses, 4-literal clauses, et cetera. However, we know that inconsistency of 2-literal clauses can be decided in polynomial time, and so polynomial size Boolean circuits exist for inconsistency of 2-literal clauses. If such polynomial size circuits exist for 3-literal clauses, then P = MP relative to a slowly utilized oracle [2]. This seems un- likely. In fact, if no such polynomial size circuits exist for 3-literal clauses, then P f NP. In order to prove P f NP, we have to remove the subset and replacement conditions, allow negations, and transfer the results to circuits instead of formulae. This would require a method of distinguishing between the 2-literal and the 3-literal case. 3. Conclusions We have shown an exponential lower bound on the complexity of a monotone Boolean formula for 2-unsatisfiability , under certain restrictions on the structure of the formulae. The argument is a top-down combinatorial argument. It is conceivable that extensions of this result to the Boolean formula complexity of 3-unsatisfiabil ity might prove P f NP. -15- References [1] S. A. Cook, The complexity of theorem-proving procedures. Proceedings of the Third ACM Symposium on Theory of Computing (1971), pp. 151-158. [2] D. A. Plaisted, New NP-hard and NP-complete polynomial and integer divisibility problems. Proceedings of the 18th Annual SymDosium on Foundations of Computer Science (1977), pp. 241-253. [3] J. E. Savage, The Complexity of Computinq. Wiley and Sons, New York, 1976, pp. 202-210. BIBLIOGRAPHIC DATA SHEET 1. Report No. UIUCDCS-R-78-941 2. 3. Recipient's Accession No. 4. Title and Subtitle "An Exponential Lower Bound for a Restricted Class of Monotone Formulae for 2-Unsatisfiability" 5- Report Date September 1978 6. 7. Author(s) -. •in m • j David A. Plaisted 8- Performing Organization Rept. No. 9. Performing Organization Name and Address Department of Computer Science University of Illinois Urbana, Illinois 61801 10. Project/Task/Work Unit No. 11. Contract /Grant No. NSF MCS 77-22830 12. Sponsoring Organization Name and Address National Science Foundation Washington, D. C. 13. Type of Report & Period Covered 14. 15. Supplementary Notes 16. Abstracts Every monotone Boolean formula for 2unsatisfiability that satisfies certain restrictions must have exponential size. The oroof makes use of a top-down argument. Relationships with the P=NP question are discussed. 17. Key Words and Document Analysis. 17o. Descriptors Boolean formula, monotone formula, 2-satisfiability. computational complexity, satisfiability. 17b. ldentif iers /Open-Hnded Terms 17c. ( OSATI Held/Group 18. Availability Statement FORM NTIS-3S ( 10-70) 19. Security Class (This Report) UNCLASSIFIED 20. Security ("lass (Thb Page UNCLASSIFIED 21. No. of Pages 22. Price USCOMM-DC 40329-P71 ACo :