Random CNF’s are Hard for the Polynomial Calculus Eli Ben-Sasson Institute of Computer Science Hebrew University Jerusalem, Israel
[email protected]
Abstract We show a general reduction that derives lower bounds on degrees of polynomial calculus proofs of tautologies over any field of characteristic other than 2 from lower bounds for resolution proofs of a related set of linear equations modulo 2. We apply this to derive linear lower bounds on the degrees of PC proofs of randomly generated tautologies.
Russell Impagliazzo Computer Science and Engineering Univ. of Calif., San Diego La Jolla, CA 92093-0114
[email protected]
A set of boolean functions is said to have large expansion if every “medium size” subset of it has a large number of variables that appear in only one function of the subset. Most lower bounds for Resolution follow straight forwardly from the high expansion of the input CNF [1]. The key observation of this paper, is that high input expansion yields high refutation degree, whenever the input is a non satisfiable set of linear equations mod , and the refutation is carried out in a field with characteristic p > . Notice that in a field with characteristic the system can be easily refuted with degree , by Gaussian elimination. This observation has several consequences.
2
1
1. Introduction The seminal paper of Chv´atal and Szemer´edi [3] showed that almost every unsatisfiable -CNF formula with n variables and n clauses (for a large enough constant), is extremely hard for Resolution to refute, i.e. the refutation size is exponential in n. This result followed lower bounds for concrete contradictions (the pigeonhole principle [5], Tseitin contradictions of expander graphs [10]), and showed that the weakness of Resolution is not limited to specially tailored formulas. On the contrary, the formulas that are easy to refute are the exception. In the past decade, several algebraic proof systems have entered the proof complexity scene, the Polynomial Calculus (PC) drawing most attention, because of its simplicity and partial automatizability [4, 8]. The main complexity measure of PC is the minimal refutation degree, which is closely related to the minimal proof size [4]. Over the past few years, several refutation degree lower bounds have been established for concrete contradictions such as the pigeonhole principle [9, 7], Tseitin contradictions and other counting principles [2]. A major open question has been whether PC displays a weakness for random inputs, similar to that of Resolution. In this paper we answer this question positively, for all fields of characteristic greater than .
3
2
Research Supported by NSF Award CCR-9734911, Sloan Research Fellowship BR-3311, grant of the joint US-Czechoslovak Science and Technology Program, and USA-Israel BSF Grant 97-00188
#93025
2
2
1. A simplification of the recent linear lower bounds of [2] for Tseitin tautologies and counting principles. 2. Lower bounds for random CNF’s, by the following reasoning. Any random CNF can be viewed as a subset of the defining CNF formula of a random system of linear equations mod . Moreover, there is a simple reduction of such systems to the set of its defining clauses. Hence, a lower bound on random systems of linear equations is also a lower bound on random CNF’s. Now we apply the observation, noticing that whp a random system of linear equations has high expansion.
2
3. Perhaps the main contribution of this paper is in pushing towards a more unified understanding of lower bounds for propositional proof systems, connecting the hardness of refutation to the expansion of the input formulas. Currently, the meta-theorem: High-expansion ) Refutation-hardness applies nicely only to Resolution, and to a lesser extent to PC. Perhaps some version of expansion may yet prove to be the main cause for refutation hardness, for other contradictions and for stronger propositional proof systems.
Finally, we also give some non-trivial upper bounds on such systems of linear systems mod . Any such system has a resolution refutation of size n=2+o(n) and a polynomial calculus degree of n= o n . This is in contrast to work of Pudl´ak and Implagliazzo [6], who show that tree-like resolution of such systems can require size (1 )n for arbitrary > , thus approaching exhaustive search. The paper is organized as follows. Section 2 presents the equality of refutation degree to Gaussian width, which is the maximal number of variables appearing in a line, during Gaussian elimination of a system of linear equations. Section 3 presents lower bounds for Gaussian width in terms of the expansion of the input system. Section 4 describes the reduction of random linear equations to random CNF’s and the consequent lower bound for random formulas. Section 5 concludes with presenting non trivial upper bounds for Resolution width and PC degree of refuting systems of linear equations.
4+ ( )
2
2
2
0
1.1. Notation P
[℄
S n a set of indices, the notation S xi will ForP mean i2S xi (similarly for products). The letter F will denote a field. Calligraphic letters denote sets of formulas: P will be a set of polynomials, and L will be reserved for sets of linear equations. 2. Gaussian Width and Refutation Degree We shall start by dealing with sets of non-satisfiable linear equations mod q , a prime (most of the paper sets q ). Let L f`i gmi=1 be such a set over n variables x1 ; : : : xn . For simplicity let us assume that L is minimal unsatisfiable, i.e. every proper subset of it is satisfiable. For reasons that will become clear later on, we allow variables to appear on both sides of the equality sign, so formally `i has the following structure:
=2
=
X
j 2Si1
aj x j = i +
01
X
j 2Si2
bj xj
(1)
1
g. where aj ; bj ; i 2 f ; : : : q Perhaps the most natural way to prove that L is unsatisfiable, is by Gaussian elimination in the field Fq . We start with the lines of L as axioms, and produce new linear equations by addition of existing lines and scalar multiplication. We prove that L is unsatisfiable by deriving the unsatisfiable linear equation . The number of lines needed to refute L is at most m2 . Let us call such a refutation the Gaussian Calculus Refutation, or simply, a Gaussian refutation. Let the Gaussian width of refuting L, denoted wG L , be the maximal number of variables appearing in a line of the refutation.
1=0
( )
Suppose we wish to show that L is unsatisfiable, but we work within a field F with characteristic p 6 q , which has a primitive q ’th root of unity called ! . The multiplicative group generated by ! is isomorphic to the additive group Zq , and we use this isomorphism to translate L into a set of polynomials in F :
=
Definition 2.1 (Linear Equations mod q = 6 p) Let L = f`i gmi=1 be a minimal unsatisfiable set of linear equations mod q , where `i is as described in (1). Let F be a field of characteristic p 6 q , having a primitive unity root of order q, denoted !. The polynomial translation of the equation `i in F , denoted PF `i is the following polynomial: Y a Y (2) yj j ! i yjbj Si1 Si2
= ( )
=
( )
The translation of L in F , denoted PF L is the union of the following sets of polynomials: 1. fyiq gni=1 . This set forces all variables to take on values in ; !; ! 2 ; : : : ; ! q 1 .
=1
2.
1
fPF (`i )gmi=1
Remarks: 1.
Lp is a set of binomials, i.e. polynomials with two monomials. This simple observation will play a crucial role in our investigation.
2. Notice that the transformation described in the previous definition is invertible, i.e. every binomial P which has structure as in (2) where aj ; bj ; i 2 f ; ; : : : q g, corresponds to a linear equation `. Moreover, ` is satisfied by an assignment f i gni=1 2 f ; : : : q gn iff PF ` is satisfied by the assignment f! i gni=1 .
01
1
0
( )
1
=
()
( ) ( )
3. Suppose PF `1 ; PF `2 are polynomial translations of `1 ; `2 , respectively. It is easy to see that PF `1 is equivalent to PF `2 modulo the ideal generated by fyiq gni=1 iff `1 `2 (mod q).
=1
( )
( )
Clearly, PF L is unsatisfiable in F , but Gaussian elimination is not sufficient for proving this fact (i.e. reaching a contradiction such as ). In order to reach a contradiction, one must allow multiplication of lines by variables, working within the framework of the Polynomial Calculus (PC).
1=0
Definition 2.2 (The Polynomial Calculus) Let F be a fixed field, and P a set of polynomials over F . A derivation of a polynomial Q from P is a sequence of polynomials fP1 ; : : : ; PS g such that PS Q, and each Pi is either an axiom from P or derived from previous lines by addition of existing lines, or by multiplication of existing lines by scalars and variables. . A refutation is a derivation of the polynomial
=
=
1=0
The natural complexity measure for the Polynomial Calculus is the degree: Definition 2.3 (Degree) The degree of a Polynomial Calculus proof is the maximal degree of a polynomial appearing as a line in the proof. The degree of refuting a contradictory set of equations P , denoted d P is the minimal degree over all refutations of P .
( )
=2
The following lemma, stated here for q , can be natu: rally generalized to systems of linear equations mod q 6
( )
( )
=2
Lemma 2.4 Let PF `1 ; PF `2 be the translations of `1; `2 , two equivalent linear equations mod , into the field F , with characteristic greater than . There is a PC derivation of PF `2 from PF `1 [ fyi2 gni=1 with degree at most fd PF `1 ; d PF `2 g.
2 ( ) ( ) =1 1 + max ( ( )) ( ( ))
2
The next theorem can be proven either by means of the Groebner Basis algorithm described in [4], or by the use of the Laurent relation described in [2]. A binomial refutation is a refutation in which all polynomials are binomials. Theorem 2.5 If P is a set of binomials, then there is a binomial refutation of P of minimal degree. Moreover, if all coefficients of P are in f ; g, then there is binomial refutation of P of minimal degree, in which all monomials have coefficients in f ; g.
1 1
1 1
Proof: In [4], it is shown that, if a system of equations has a degree d refutation, then such a refutation is generated by the Groebner basis algorithm (using any degree-respecting term order). In fact, the proof so generated has the following restriction: each time a line is generated as a linear combination of two previous lines, one of the co-efficients is 1 and the other is chosen to cancel a term. In any such step, if the linear combination is between binomials and cancels monomial m1 , then we can write the two binomials b m2 and m1 b0 m3 , and the result is as m1 0 b m2 b m3 . Unless m2 m3 , this is also a binomial with co-efficients in f ; g . If m2 m3 , we can immediately divide the result by 2 to get a single monomial with co-efficient 1. Also, if we multiply a binomial by a variable, we obtain a binomial with the same co-efficients. Thus, inductively, the Groebner basis algorithm produces a proof all lines of which are binomials with co-efficients in f ; g.
( 1)
+ ( 1) + ( 1)
+ ( 1) = 1 1
=
1 1
The main theorem of this section is: Theorem 2.6 For L a minimal contradictory set of linear equations mod of width at most k , and P its formulation in a field F with characteristic p 6 ,
2
=2 1 d(P ) = maxfk; wG (L) + (1)g 2
Proof: To see that d P d 21 wG L e, take any Gaussian proof of width w, and let d dw= e . Assume k < d. Inductively, for each line in the Gaussian proof, we will b S yi with S1 ; S2 show that any equation S1 yi 2 a disjoint partition of S with each of size at most dw= e, d. Assume the Gaussian has a PC proof of degree at mostP b and `2 proof adds the two lines ` 1 i2S xi P 0 to obtain P 0 . Let P1 ; P2 x b x b b i i i2T i2S T be corresponding polynomials, such that P1 is equivalent to PF `1 modulo fyi2 gni=1 , and similarly for P2 . Define R S \ T; r jRj; s jS Rj; t jT Rj. Note that s t jS T j w. Suppose r dw= e. Assume wlog s t. Let T 0 T R be an arbitrary set of size t0 ft; dw= eg, and let T 00 T R T 0 and t00 jT 00 j. Notice that t00 s dw= e, because s t0 t00 w, and s t. Use P2 to derive
( )
=
:=
= ( ) = = + = 2
=(
( ) 2 +1 = ( 1)
=
2
)
Y
P4 := Q
and multiply it by
P5 :=
yi = (
1)b
yi = (
1)b
R[T 00
Use P1 to derive
Y
R
= min
=
+ +
P3 :=
=
=
=
Y
0
T0
Y
S
2
+
yi
yi
T 00 yi to derive Y
R[T 00
yi = (
1)b
0
Y
S
Subtract P3 from P5 to derive
P6 :=
2 :=
Y
(S R)[T
yi = (
1)bb
0
00
yi Y
T0
yi
It is easy to verify that the maximal degree of fP1 ; : : : P6 g is at most d, and by use of lemma 2.4 this is enough. Next assume that r > dw= e. Let R0 R be an arbitrary set of size jR0 j dw= e, and set R00 R R0 . Use P1 to derive Y Y b P3 yi yi R0 S R0
=
2
:=
Use P2 to derive
P4 :=
2
=
= ( 1)
Y
R0
yi = (
1)b
Y
0
T R0
yi
Subtract the two to get the binomial
P5 :=
Y
S R0
=(
yi = (
) (
1)bb
0
Y
T R0
)
yi
Notice that R00 S R0 \ T R0 is exactly the index set of the variables appearing in both monomials of P5 , of
which we wish to dispose. For yj2 , derive
=1
Y
(S
R0 ) fj g
yi = (
j
1)bb
2 R00 , using the axiom Y
0
(T
R0 ) fj g
= ( 1) ( ) 1 2 ( ) + (1) 1
2
+
= +
1 = + +
( ) 2( )
3. High Expansion Yields High Degree We have seen that the PC refutation degree of a set of linear equations mod is equal to the Gaussian width of refut. ing the same set, whenever the field has characteristic 6 In this section we show that this width is directly connected to the expansion measure of the input set. This measure is a generalization of the graph-theoretic expansion, and is based on the following definition.
2
=2
Definition 3.1 (Boundary) For f a function and x a variable, we say that f is dependent on x if there is some assignment to all variables other than x, such that the value of f is not fixed. let V ars f be the set of all variables that f is dependent upon, and let the width of f be jV ars f j. For F a set of functions, the Boundary of F , denoted F , is the set of variables x such that there is exactly one function f 2 F that is dependent on x.
()
()
( )=
yi
This derivation will have degree at most d. Continue in this R00 have been 0removed, arriving manner till all variables Q in bb Q at the polynomial S R yi T R yi . Once again, by lemma 2.4 the derivation has degree at most d. Thus, we have d P = wG L O . In the other direction, assume Lp has a PC refutation of degree d, hence, by theorem 2.5 it has a refutation in which each line is a multi-linear binomial and all monomial . We may construct inductively a coefficients are or corresponding Gaussian refutation with Gaussian degree at most d. To see this, notice that the PC refutation has no . A mulscalar multiplications by constants other than P by the variable y corresponds to changing tiplication of j P P P P S2 xi , S1 xi b xj S2 xi into xj S1 xi b and for any addition of P1 and P2 , one of the monomials in P1 ; P2 must be identical (otherwise the output would not be a binomial), and hence adding the corresponding linear equations would yield the proper result . If we get a single monomial by adding two binomials in the PC proof, then both sides of the above equations involve the same sets S1 ; S2 of variables, but with different constants b. Then adding the corresponding linear equations will give us a non-zero constant equaling 0, and we have obtained a contradiction. Thus, wG L d P .
1
expansion, in this setting. Given a set of functions F that is minimal unsatisfiable (i.e. F n ff g is satisfiable, for any f 2 F ), define the following hypergraph. The vertices are S labeled by V ars F f 2F V ars f , and for each function f 2 F , a hyper-edge is added, consisting of V ars f . Such a hypergraph is said to be expanding, if every mediumsize subset of edges has large (linear) boundary size. Suppose the set F is derived by taking a d-regular graph, assigning a unique variable xe to every edge e, and assigning a function fv to every vertex v , such that V ars fv fxe v 2 eg. In this case F is expanding iff G is an expander graph, in the traditional sense. To see this notice that for V 0 V G , and FV 0 ffv v 2 V 0 g,
()
Before formally defining function expansion, we give an intuitive description of it, that will justify the use of the term
()
( )=
:
( )
= : FV = fxe : e 2 E (V 0 ; V n V 0 )g where E (V 0 ; V n V 0 )g is the set of edges between V 0 and 0
V n V 0.
Thus, our definition can be viewed as a generalization of the traditional one. And now for the formal definition. Definition 3.2 (Expansion) For F an unsatisfiable set of functions, let s be the minimal size of an unsatisfiable subset of F . The Expansion of F is
0 2s 3 jF j 3 g
e(F ) def = minfj F 0j : F 0 F ; s
The following theorem, and its proof, are an application of [1] to our present context. Theorem 3.3 For L a set of unsatisfiable linear equations mod , each of width at most k , and F a field of characteristic > , d PF L fk; 12 e L g.
2
2 ( ( )) max
( ) + (1)
Proof: We assume that k < 12 e L . By theorem 2.6 we only have to prove that wG L e L . Let s be the minimal size of an unsatisfiable subset of L. Define the following measure on linear equations:
( )
( ) ( )
(`) def = minfjL0j : L0 L; L0 j= `g It is easy to verify the following facts:
(`) 1.
1. For ` 2 L, 2. 3.
(1 = 0) = s.
(`1 + `2) (`1 ) + (`2 ). L1 j= `1 ; L2 L1 [ L2 j= `1 + `2 .
This is because if
j= `2 , then clearly
Hence, in every Gaussian refutation there must be a line ` 23s . Fix such a line `, and let L0 be a minimal subset of size ` implying `. We claim that ` depends on L0 . Assume this is not the case, i.e. assume xi 2 L0 n V ars ` , and let `0 be the unique equation in
` for which 3s
()
()
()
L0 which depends on xi .
By the minimality of L0 there is some assignment that satisfies all of L0 n f`0g but falsifies `0 and `. One can flip the assignment of on xi so as to satisfy `0 , without affecting ` or the rest of L0 , thus L0 6j `. Contradiction.
=
We note that the recent linear lower bounds on degree, by [2], are immediate corollaries of the previous theorem, as they have large expansion. We quote only one such result. Let T G be a Tseitin contradiction based on the graph G, and let e G be the expansion of G, defined formally as eG fjE S; V n S j jV3 j jS j 2j3V j g.
( ) ( ) ( ) = min (
): Corollary 3.4 [2] For T (G) a Tseitin contradiction based on the graph G, and F a field with characteristic greater than 2, d(T (G)) 12 e(G).
4. Random k -CNF’s In this section we consider the relative efficiency of PC over Resolution, as a refutation system for random k CNF’s. It is well-known [3] that whp (with high probability) a random k -CNF with n variables and n clauses, demands exponential size resolution refutations. Stated simply: Resolution is usually no better than exhaustively checking all possible n assignments. The simplest explanation for this inefficiency is that Resolution performs badly on inputs that have high expansion according to definition 3.2 [1]. In the previous discussion, we have already noticed that linear equations with high expansion are an “Achilles heel” of the polynomial calculus. We shall show a natural reduction from random CNF’s to random linear equations, which have high expansion. We start with a formal definition of random formulas.
2
Definition 4.1 (Random k -CNF’s) Let F Fkn; denote that F is a random k -CNF formula on n variables and m n clauses, chosen at random by picking n clauses i.i.d from the set of all nk k clauses, with repetitions. is called the clause density.
2
=
The standard Q formulation of a clause Q C as a polynomial is PC , x i P ositive Negative xi where P ositive is the set of positive literals appearing in C (the set Negative is similarly defined). A CNF formula C fCi gmi=1 , formulated in polynomial calculus, is the fol2 n lowing set of polynomials fPCi gm i=1 [fxi xi gi=1 . From now on, we shall think of all CNF formulas as formulated in this manner. A different method for producing the distribution Fkn; goes through random linear equations.
:=
=
(1
)
=0
=
Definition 4.2 (Random linear equations) Let L Ln; k denote that L is a random set of linear equations mod ,
2
each having k variables, chosen at random by picking n equation i.i.d from the set of all nk equations, with repetitions.
2
n; Given Ln; k , one can easily produce Fk , by replacing k 1 each ` 2 L with one of its defining clauses, picked at random, with equal probability. Let us denote by CL any CNF formula derived from L by this process. Notice that L j CL , but the converse is not necessarily true. In the other direction, for any clause C there is a unique linear equation mod `C such that `C j C . For C a CNF formula, let LC f`C C 2 Cg be the (unique) linear closure of C . The following definition and lemma are from [2].
2
=
=
2 :
=
() () (
Definition 4.3 [2] Let P x , Q y be two sets of polynomials over a field F . Then P is d1 ; d2 -reducible to Q if: (1) For every yi , there is a degree d1 definition of yi in terms of the x’s. That is, for every i, there exists a degree d1 polynomial ri where yi will be viewed as being defined by ri x1 ; : : : ; xn ; (2) there exists a degree d2 PC derivation of the polynomials Q r x1 ; : : : ; xn from the polynomials Px.
( ()
)
((
)
))
() ( )
)
Lemma 4.4 [2] Suppose that P x is d1 ; d2 -reducible to Q y . Then if there is a degree d3 PC refutation of Q y , then there is a degree max d2 ; d3 d1 PC refutation of P x .
()
(
() ()
Lemma 4.5 Let ` be a linear equation mod 2, over the variables x1 ; : : : ; xk , and let P PF ` [ fyi2 gki=1 be its formulation as a set of polynomials. Let C be a clause over x1 ; : : : xk such that ` j C . P is ; k -reducible to C .
= () =1 (1 + 1)
= Proof: Set xi = (1 yi )=2. From this definition and yi2 = 1 2
one can deduce xi xi . PC is implicationally complete, thus, there is a derivation of C from P , using only variables that appear either in P or in C . This derivation can be as, because every time a sumed to have degree at most k monomial with degree k appears in the derivation, some variable x has degree , and the monomial can be reduced to degree k by use of the axioms x2 x.
2
+1
+1
The following theorem is the main observation of this section. It says that taking the linear closure of a nonsatisfiable CNF which has high expansion, cannot help decrease the refutation degree substantially.
( )
Theorem 4.6 For C a k -CNF, and PF LC the formulation of the linear closure of C as a set of polynomials in the field F , of characteristic greater than , we have dC fd PF LC ; k g.
2 ( ) max ( ( )) + 1 Proof: Let C` be the k -CNF definition of the linear equation `, and let CL be the conjunction the C` ’s, for all ` 2 L (formulated as a set of polynomials). Clearly, C CL , hence any refutation of C is also a refutation of CL . The previous
( ) (1; k + 1)-reducible to
lemma 4.5 implies that PF LC is CL, and the theorem follows.
The following lemma appears originally in [3] for CNF formulas. It is easily proven using a union bound, and applies to sets of linear equations just as well (see e.g. [1], section 6.3). We state it for k , but it can be generalized to larger k as well.
=3
Lemma 4.7 For every > dependent of n, such that if whp e L n= 2 .
( )
0 there is some constant , in n 12 and L Ln;3 , then
Let us sum up: lemma 4.7 claims that almost all random linear equations L have high expansion, theorem 3.3 says that high expansion demands large degree refutations, over any field with characteristic p > , and theorem 4.6 concludes that whp any random CNF demands high degree, since its refutation degree is at least as large as the degree of refuting its linear closure.
2
Corollary 4.8 For any > , F F3n; , and F a field n with characteristic other than , whp d F 2+
0 2
( ) = (
)
The main theorem of this section is:
= (1)
2
wG (L) n=2 + o(n)
( )
=
=1
=1
1
2+ ( )
1
=
=
2+ ( )
=
=
1
=
Pr(Ri = 0) Pr(Ri = 1) = lX =ti
ti l q (1 q)ti l = l l=0 (1 2q)ti > 0
( )
Corollary 5.2 For F L the CNF formulation of a system of width O linear equations mod ,
Thus, for tautologies expressing systems of linear equations modulo 2, the proof complexity is significantly less
℄
0=1
( )
2 wR (F (L)) n=2 + o(n) Corollary 5.3 For F (L) the CNF formulation of a system of width O(1) linear equations mod 2, SR (F (L)) 2n=2+o(n) Corollary 5.4 For Pp a system of linear equations mod 2, in the field Fp , p > 2, d(Pp ) n=4 + o(n)
=
(1) ( )
Let SR F (wR F ) be the minimal size (width, resp.) of a Resolution refutation of the CNF F . The previous result and its proof have the following implications:
(1)
2
0
Pr[
5. Upper Bounds on Linear Equations
Theorem 5.1 For L a system of linear equations modulo of width k O with n variables,
than exhaustive search verification. This contrasts with recent work by Pudl´ak and Impagliazzo [6], showing that treelike resolution proofs of such systems can require (1 )n size for arbitrary > . Hence, resolution based methods with more memory could be faster than DLL algorithms for larger clause sizes. P bj for j Proof: Let the equations be i2Sj xi :::m, withjSj j k . Assume without loss of generality that the equations are minimally dependent, i.e., that j Sj ; and j bj . Let be a random permutation of ; ::m, and consider the proof obtained by always adding the equations one by one in the order ; :: m . We claim that with high probability the width of this proof is at most n= o n . Let j m ; we will show that with high probability, the width of the j ’th line, Wj j j0 j S(j0 ) j n= o n . Let q j=n. Consider the random experiment of picking equations with probability qPindependently and summing the picked equations to get i2S 0 xi b0 for some set S 0 and bit b0 . Let C be the number of chosen equations. Then if we subsequently subtract C j chosen equations if C > j or add j C random unchosen equations otherwise, the result is distributed as the j ’th equation. The difference in widths of the two processes is bounded by jC j jk . Since the expectation of C is j and it is normally distributed, jC j j n1=2 e 2 =2 . Thus, we concentrate on the width of S 0 . Let ti be the number of equations xi appears in; since the sum of equations gives , ti is even. Let Bad fijti kn1=4 g. Since m < n, we know that jBadj n3=4 . Look 0 at the random P indicator variable Ri which is if xi 2 S . Let R i62Bad Ri be the number of non-bad elements in S0.
This implies E
V (R)
= = =
So,
( 1)l
(Ri ) 1=2 for every i. Additionally,
E (R2 ) X
i;i0 62Bad
(E (R)2 ) (E (Ri Ri ) 0
X
i;i0 62Badj9 ;i;i0 2S kn5=4
E (Ri )E (Ri )) 0
(E (Ri Ri ) 0
E (Ri )E (Ri )) 0
References
Pr(R n=2 + kn3=4) Pr(jR
Hence, with probability
Wj
1 2n
E (R)j kn3=4 ) kn5=4 =(k2n3=2 ) n 1=4
1=4,
R + jBadj + jC j jk n=2 + kn3=4 + n3=4 + n1=2 ln n
=
n=2 + O(kn3=4 )
We only obtained a probability O n 1=4 bound on the above probability. However, this suffices to show that with probability 3/4, each j which is a multiple of Dn3=4 for a sufficiently large constant D has Wj n= O kn3=4 . However, if this is true, then since Wj Wj 0 k jj j 0 j, the same is true for any j , using j 0 the nearest multiple of Dn3=4 . Note that the process of adding the lines one by one ensures that the set of variables for each new equation involves at most k variables that do not appear in the old one. This is used to prove Corollary 5.3, since then the derivation of the j ’th line in the Gaussian proof is logically valid and depends on at most Wj k variables. It thus follows from the implicational completeness of resolution that the line can be simulated by a resolution proof of size Wj +k .
1
(
)
2+ ( +
)
+
2
6. Open Problems 6.1. Are random CNF’s hard for PC over F2 ? For F F3n;10 , is there a linear lower bound on the refutation degree when working in the simple, two-element field F2 ? We suspect the answer is positive, but it remains an interesting open problem. In particular, the lower bound for fields of higher characteristic, relies on the possibility of encoding a linear equation as a binomial. We do not know of a similar reduction from a set of “hard” binomials to a random CNF over F2 , or any other field of characteristic , and hence cannot obtain the lower bounds for such fields.
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7. Acknowledgments We would like to thank Avi Wigderson for many helpful discussions.
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