An improved version of width restricted resolution
An improved version of width restricted resolution Oliver Kullmann Department of Computer Science University of Toronto Toronto, Ontario M5S 3G4 e-mail: [email protected] http://www.cs.utoronto.ca/kullmann/
November 30, 1999 Abstract
A new form of width restricted resolution, called \CUBI resolution" (\closure under bounded input resolution") is presented. CUBI resolution allows to eliminate the dependence on the maximal input clause length in the general lower bound for resolution from [Ben-Sasson, Wigderson 99], and also the poly-time decidability problem about \k-resolution" ([Kleine Buning 93]) is bypassed (by slightly strengthening the calculus). Generalizing the well-known equivalence of the refutation power of Unit resolution and input resolution, CUBI resolution simulates \nested input resolution." 1
Introduction
1.1 Two forms of width restricted resolution Lower bounds for resolution complexity via width lower bounds 1975 Galil introduced bounded resolution in [9], using only clauses of bounded size, and he showed that certain Tseitin-type formulas in 3-CNF require very long clauses in any (full) resolution refutation, expressing hope, that this result might be useful for a super-polynomial lower bound for (full) resolution. Supported by the Natural Sciences and Engineering Research Council of Canada and by the Communications and Information Technology Ontario.
1
1981 Krishnamurthy/Moll made a precise general conjecture in [12] about a strong relation between the necessary size of clauses in resolution refutations and the necessary length of such refutations. Finally, 1999 Ben-Sasson/Wigderson justi ed in [2] these attempts by proving that if \long" clauses are necessary for a resolution refutation, then only \large" refutations are possible1) . Their proof is an adaption of an analogous result in [Clegg, Edmonds and Impagliazzo 96] ([7]) for \degree restricted Groebner bases proof systems"2) .
Generalizing Unit resolution Generalizing [Yamasaki and Doshita 83] ([19]), in [Gallo and Scutella 88] ([10]) a hierarchy of generalized Horn clause-sets has been introduced, which has been smoothened in [Kleine Buening 93] ([4]). Since Unit resolution is complete for refuting Horn clause-sets, it is natural to ask for a generalization w.r.t. that hierarchy. In [4] for that purpose k-resolution has been introduced, which is complete for \k-Horn clause-sets." While bounded resolution allows only resolution steps where both parent clauses have bounded length (or \width"), now only one parent clause has to ful ll this restriction.
1.2 A new form of width restricted resolution The main disadvantage of bounded resolution is its inability of even handling Horn formulas, which is also re ected in the dependence of the general lower bound for resolution in [2] on the maximal input clause length, preventing that bound from being used for clause-sets already containing \long clauses." On the other side, the main disadvantage of k-resolution is that it does not seem likely to be poly-time decidable for k 3. In [13] a new form of width restricted resolution has been introduced, called CUBI resolution in the present paper, combining the advantages of both approaches: - the dependence on the maximal input clause length in the general resolution lower bound has been removed, and thus this bound is now also applicable for clause-sets with large clauses (while no applications are lost); 1) thus from the width lower bound in Galil [9] in fact an exponential lower bound for (full) resolution follows; the precise form of the conjecture of Krishnamurthy/Moll has been refuted by [3] (the complexity of the Ramsey formulas for resolution is still open) 2) that the argumentation in [7] just needs the resolution calculus has been noted rst in the literature in [1]
2
- generalized Horn clause-sets can be handled; - for xed width derivability of the empty clause is poly-time decidable. The organization of this extended abstract is as follows: 1. In Section 3 the results on bounded resolution are outlined, 2. while in Section 4 k-resolution is discussed. 3. Then in Section 5 CUBI resolution is introduced and some main lemmas are stated. Also the notion of nested input resolution introduced in [13] is presented, which in fact suces to refute the generalized Horn formulas, and the relation to CUBI resolution is discussed. 4. Finally in Section 6 two methods for proving width lower bounds are given, and the basic ideas for two direct lower bound proofs (not possible when using bounded resolution) are given. 2
Notation
CLS is the set of all clause-sets (interpreted as CNF's) and p{CLS the set of all clause-sets with clauses of length at most p. HO is the set of all Horn clause-sets and HOk is the set of all generalized Horn clause-sets of level k
(HO 1 = HO ; see [10, 4] or Subsection 5.3 in [13]). n(F ) := j var(F)j is used for the number of variables in F 2 CLS , c(F ) is the number of clauses in F, `(F ) the number of literal occurrences, and p(F ) is the maximal clause length in F. By PHPm k the pigeonhole formulas with m pigeons and k holes are denoted, by GTm Krishnamurthy's graph formulas (expressing that a transitive simple directed graph with m nodes and without binary cycles must have a source; see [11] or Subsection 8.4 in [13]), by MGTm the translation of GTm into 3{CLS used in [3], and by Rm the \Ramsey formulas" introduced in [12]. For any undirected connected graph G the associated unsatis able Tseitin formula is denoted by T(G) (choosing any node labeling with odd sum; see [16] or [18]), and for a directed graph G which is acyclic and has exactly one sink, the \pebbling formulas" introduced in [2] are denoted by PF(G) (see also Subsection 6.2 in [13]). For the minimal number of leaves in a tree-like resolution refutation of an (unsatis able) clause-set F we use ComptR (F ), while the minimal number of (di erent) clauses needed in a resolution refutation of F is CompR (F ). 3
3
Bounded resolution
In [9] \bounded resolution" has been introduced, the most widely known form of width restricted resolution.
De nition 3.1 For k 2 N let bW k be the set of clause-sets refutable by resolution proofs T : F ` ? using only clauses of length k. The corresponding 0
hardness parameter is called bwid(F ) := inf f k 2 N0 : F 2 bW k g:
The following properties are known: 1. (a) For constant k the classes bW k are polynomially decidable. (b) All classes bW k are stable under partial assignments, renaming and formation of super-clause-sets, and furthermore bW k bW k+1 holds for all k 0. 2. Already in 1975, Galil ([9]) proved for a sequence (Gm ) of graphs with T(Gm ) 2 3{CLS , n(T(Gm )) m and c(T(Gm )) 8m the quasi-linear width lower bound b wid(T(Gm )) 6(logm m)2 : 2
Using the expander graphs (Gm ) from [17], in [2] a linear lower bound b wid(T(Gm )) (m) is obtained. 3. Krishnamurthy/Moll proved in [12] p b wid(Rm ) ( n(Rm )) and they conjectured (KM81) \log2 CompR(F) bwid(F)" for arbitrary clause-sets F. 4. In [2] (exploiting [7] for resolution) the general lower bound for full resolution ? b ? p(F))2 logCompR(F) ( wid(F) n(F) 0
0
for F 2 USAT has been shown. This lower bound is (nearly) tight as shown in [3]: CompR(MGTm ) O(m3 ) p b wid(MGTm ) (m) ( n(MGTm )) (thus falsifying the precise form of (KM81)). 4
5. For tree-like resolution in [2] the lower bound log2 ComptR(F) bwid(F) ? p(F) is shown, and using the graphs (Gm ) from [6] they get wid(PF(Gm )) 6 and log ComptR(PF(Gm )) (m= log m)
b
where `(PF(Gm )) O(m). A weakness of this concept of width restricted resolution is, that in order to be able to handle all clauses in a given F, the width parameter k must be at least p(F). So for example no bW k contains HO , and the general lower bounds for tree-like resolution (see Remark 5 in Section 3) becomes trivial in case of b wid(F) O(p(F)). 4
k-resolution
Generalizing Unit-resolution, in [4] \k-resolution" has been introduced in order to cope with generalized Horn formulas.
De nition 4.1 For k 2 N let uW k be the set of clause-sets refutable by resolution proofs T : F ` ? where for each resolution step at least one parent clause 0
has length at most k. The corresponding hardness parameter is called uwid(F ) := inf f k 2 N0 : F 2 uW k g:
1. All classes uW k are stable under partial assignments, renaming and formation of super-clause-sets, and furthermore uW k uW k+1 holds for all k 0. 2. F 2 uW 1 , F refutable by Unit resolution (, F refutable by input resolution). 3. uW 0 and uW 1 are decidable in poly-time, and also uW 2 as shown in [5]. 4. For all k 1 we have HO k uW k . 5. uwid(F) bwid(F) uwid(F)+p(F) (the second inequality follows from Lemma 5.3 in the next section; in [5] the weaker inequality bwid(F) max( uwid(F); 3) + max( uwid(F); p(F); 3) has been shown). The main drawback here is that it is not known whether the classes uW k are poly-time decidable for k 3. 5
5
CUBI resolution
In [13] (see De nition 8.3 there) a strengthening of k-resolution has been introduced, combining all the (positive) properties of bounded and k-resolution. The idea is simple: While for deciding \F 2 bW k ?" the closure of F under single resolution steps with resolvents of length at most k is computed, now we compute the closure of F under addition of clauses of length at most k derived by input resolution.
De nition 5.1 For k 2 N let iW k be the set of clause-sets such that by 0
iteratively adding clauses of length at most k derivable by input resolution (using also the new clauses) the empty clause can be generated. The corresponding hardness parameter is called iwid(F ) := inf f k 2 N0 : F 2 iW k g:
For the more general notions iW k (U ) and iwid (F), using a \partial oracle" U for (partial) decision of unsatis ability, as well as for the proofs of the following results, see [13]. U
Lemma 5.2 The classes W k are poly-time decidable for xed k 0. i
Lemma 5.3 For all unsatis able clause-sets F we have 1. bwid(F) ? p(F) iwid(F) bwid(F); 2. uwid(F) ? 1 iwid(F) uwid(F).
Theorem 5.4 For unsatis able clause-sets F with n(F) 6= 0 we have 1 iwid(F)2 < ln Comp (F) < (ln n(F)) ( iwid(F) + 1) + 2: R 8 n(F)
5.1 Simulation of nested input resolution In [13] (a forerunner is [14]) a hierarchy Gk (U ; S ) of clause-sets has been de ned, where U U0 is a set of unsatis able clause-sets stable under partial assignments, and S S0 is a set of satis able clause-sets stable under enforced assignments, that is, stable under such partial assignments ' which can be extended to a satisfying assignment, but where any deviation from ' would destroy satis ability. The basic cases are U = U0, where U0 consists of all clause-sets containing already the empty clause, and S = S0, where S0 is the singleton containing the empty clause-set. 6
The corresponding hardness parameter hU ;S (F ) for F 2 CLS is de ned (as usual) as the minimal k with F 2 Gk (U ; S ). For unsatis able F we often use h (F) instead since here S is not involved, and U in turn is omitted in case of U = U0 . Some main properties are (for an extensive treatment see [13]): U
1. For all xed k the classes Gk (U ; S ) are poly-time recognizable and SAT decidable (counting decisions \F 2 U ?" and \F 2 S ?" as one step). Searching through the hierarchy from below gives a (general) SAT algorithm with running time ?
O `(F) 2log2 (n(F )+1) 2hU S (F )
;
which for unsatis able F is bounded by ?
O `(F) ComptR(F)2 log2 (n(F )+1) ; \quasi-automatizing" tree-like resolution (answering a question in [2]), since (using U = U0 ) the algorithm produces as byproduct a resolution tree refutation with at most ComptR(F)log2 (n(F )+1) many leaves. 2. The \hardness parameter" h ; (F) has several natural characterizations, so for example for unsatis able F the quantity h(F) + 1 is just the space complexity of tree-like resolution as introduced in [8]. 3. HO k Gk (U ; S ) for all k and all U ; S . 4. h(F) log2 ComptR(F) log2 (n(F) + 1) h(F) for all unsatis able F (for the general relativized statement see [13]). U S
;k
;1
5. There is a natural generalization ` of input resolution (which is ` ) refuting exactly the unsatis able instances in Gk (U ; S ) = Gk (U ).3) U
U0
Theorem 5.5 For all unsatis able F with h(F) 1 we have wid(F) h(F) ? 1; i
that is, width restricted resolution simulates \nested input resolution." It follows log2 ComptR(F) iwid(F) + 1. 3)
The relativized classes
W k (U )
i
uses
U ;1
`
instead of
7
U0 ;1
`
.
5.2 Some remarks on "real world" applications The natural SAT decision algorithm associated with the hierarchy ( bW k ), testing for input F whether there is k 2 f0; : : : ; n(F)g with F 2 bW k (then F is unsatis able) or not (then F is satis able), has a very poor performance on satis able inputs, due to the fact, that the classes bW k contain only unsatis able instances (in contrast to the classes Gk ). Moreover, this algorithm has space requirements of b ? O `(F) + n(F) wid(F ) ; while the natural SAT decision algorithm associated with the hierarchy (Gk ) (see Remark 1 in Subsection 5.1) needs only space ? ? O `(F) + log n(F)h(F ) = O `(F) + h(F) log n(F) : Thus, for (most) practical purposes the use of the hierarchies (Gk (U ; S )) (see [13], Subsection 5.4 for improved satis ability handling) is strongly preferable to the use of ( iW k ) or ( bW k ). In fact already for medium size F the decision \F 2 bW 3 ?" is out of reach due to space limitations (considering, say \F 2 iW 2 ?" is of interest only in special cases), while for the hierarchy (Gk ) essentially only time is of importance, and k-values up to, say, k 5 can be handled. The algorithm searching through the classes Gk from below is the heart of the commercially successful SAT algorithm patented in [15]. However, width restricted resolution is (at least) a valuable tool to analyze full resolution (instead of tree-like resolution), and for special classes width restricted resolution can be exponentially better than tree-like resolution. For example, in [13], improving [2] (see Remark 5 in Section 3), for dag's G with in-degree 2 we show bwid(PF(G)) 6 and h(PF(G)) peb(G), where peb(G) is the minimal number of pebbles needed for G. 6
How to prove width lower bounds
In [2] a scheme has been extracted for proving lower bounds on bwid(F), which in fact is the basic method of all such proofs in the literature, starting with [9]. The basic observation here is as follows. We consider a xed unsatis able clause-set F in this section, and we use V := var(F) for the set of variables of F, while CL(V ) denotes the set of clauses C with var(C) var(F).
Lemma 6.1 Consider : CL(V ) ! N and k; m 1 with m (?)=2 (where ? is the empty clause), such that the following properties hold (for all clauses C; D 2 CL(V )): 0
8
- C 2 F ) (C) 1; - if C; D are resolvable with resolvent R, then (R) (C) + (D); - if m (C) 2m, then jC j k. Now bwid(F) k holds.
We do not consider this method here but we want to give examples where
direct lower bound proofs for iwid(F) are possible for minimally unsatis able F
with bwid(F) p(F) (and thus lower bounds for bwid(F) are senseless).
Lemma 6.2 Consider an unsatis able clause-set F , and assume that a \size
function"
s : CL(V ) ! N0
and N CL(V ) (\negligible clauses") are given ful lling - N is stable under resolution; - F n N is satis able;
- s(C) jC j for all C 2 CL(V ); - for all resolvable clauses C; D 2 CL(V ) with resolvent R we have - s(R) max(s(C); s(D)) ? 1, - C 2 N ) s(R) s(D).
Then iwid(F) min s(C) ? 1. C 2F nN
Two examples (for details see [13]): 1. For PHPmk let N be the set of the negative binary clauses in PHPmk , and let s(C) be the number of occupied columns in the (m k)-matrix representation of C. Then obviously the assumptions of Lemma 6.2 are ful lled, and we get iwid(PHPmk k ? 1. Since h(PHPmk ) k, in fact we obtain iwid(PHPmk ) = k ? 1 and h(PHPmk ) = k. 2. For GTm consider a variable vi;j 2 var(GTm ) as a directed edge from i to j, and literals vi;j ; vi;j as forward resp. backward edges from i to j resp. from j to i. Let N be the set of all clauses which represent a union of signed directed circuits, and let s(C) be the size of a spanning forest in the underlying undirected graph. Then the assumptions of Lemma 6.2 are ful lled, and we get iwid(PHPmk m ? 2. Since h(GTm ) m ? 1, in fact we have iwid(GTm ) = m ? 2 and h(GTm ) = m ? 1. 9
References
[1] Paul Beame and Toniann Pitassi. Simpli ed and improved resolution lower bounds. In 37th Symposium on Foundations of Computer Science (FOCS' 96), pages 274{282, 1996. [2] Eli Ben-Sasson and Avi Wigderson. Short proofs are narrow | resolution made simple. Technical Report TR99-022, ECCC Electronic Colloquium on Computational Complexity, July 1999. [3] Maria Luisa Bonet and Nicola Galesi. A study of proof search algorithms for resolution and polynomial calculus. In 40th Annual Symposium on Foundations of Computer Science (FOCS), pages 422{431, 1999. [4] Hans Kleine Buning. On generalized Horn formulas and k-resolution. Theoretical Computer Science, 116:405{413, 1993. [5] Michael Buro and Hans Kleine Buning. On resolution with short clauses. Annals of Mathematics and Arti cial Intelligence, 18(2-4):243{260, 1996. [6] J.R. Celoni, Wolfgang J. Paul, and Robert Endre Tarjan. Space bounds for a game on graphs. Mathematical Systems Theory, 10:239{251, 1977. [7] Matthew Clegg, Je ery Edmonds, and Russell Impagliazzo. Using the Groebner basis algorithm to nd proofs of unsatis ability. In Proceedings of the 28th ACM Symposium on Theory of Computation, pages 174{183, 1996. [8] Juan Luis Esteban and Jacobo Toran. Space bounds for resolution. In C. Meinel and S. Tison, editors, 16th Annual Symposium on Theoretical Aspects of Computer Science, volume 1563 of LNCS. Springer, 1999. [9] Zvi Galil. On the validity and complexity of bounded resolution. In Proceedings of seventh annual ACM Symposium on Theory of Computing, pages 72{82, May 1975. [10] Giorgio Gallo and Maria Grazia Scutella. Polynomially solvable satis ability problems. Information Processing Letters, 29:221{227, November 1988. [11] Balakrishnan Krishnamurthy. Short proofs for tricky formulas. Acta Informatica, 22:253{275, 1985. [12] Balakrishnan Krishnamurthy and Robert N. Moll. Examples of hard tautologies in the propositional calculus. In 13th ACM Symposium on Theory of Computing (STOC'81), pages 28{37, 1981. 10
[13] Oliver Kullmann. Investigating a general hierarchy of polynomially decidable classes of CNF's based on short tree-like resolution proofs. Technical Report TR99-041, Electronic Colloquium on Computational Complexity (ECCC), October 1999. [14] Daniele Pretolani. Hierarchies of polynomially solvable satis ability problems. Annals of Mathematics and Arti cial Intelligence, 17(3-4):339{357, 1996. [15] Gunnar Martin Natanael Stalmarck. Method and apparatus for checking propositional logic theorems in system analysis, 1990. European Patent Speci cation; application: December 19, 1990, Bulletin 90/51; publication: June 28, 1995, Bulletin 95/26. [16] G.S. Tseitin. On the complexity of derivation in propositional calculus. In Seminars in Mathematics, volume 8. V.A. Steklov Mathematical Institute, Leningrad, 1968. English translation: Studies in mathematics and mathematical logic, Part II (A.O. Slisenko, editor), 1970, pages 115-125. [17] Alasdair Urquhart. Hard examples for resolution. Journal of the ACM, 34:209{219, 1987. [18] Alasdair Urquhart. The complexity of propositional proofs. The Bulletin of Symbolic Logic, 1(4):425{467, 1995. [19] Susumu Yamasaki and Shuji Doshita. The satis ability problem for a class consisting of Horn sentences and some non-Horn sentences in propositional logic. Information and Control, 59:1{12, 1983.
11
November 30, 1999 Abstract
A new form of width restricted resolution, called \CUBI resolution" (\closure under bounded input resolution") is presented. CUBI resolution allows to eliminate the dependence on the maximal input clause length in the general lower bound for resolution from [Ben-Sasson, Wigderson 99], and also the poly-time decidability problem about \k-resolution" ([Kleine Buning 93]) is bypassed (by slightly strengthening the calculus). Generalizing the well-known equivalence of the refutation power of Unit resolution and input resolution, CUBI resolution simulates \nested input resolution." 1
Introduction
1.1 Two forms of width restricted resolution Lower bounds for resolution complexity via width lower bounds 1975 Galil introduced bounded resolution in [9], using only clauses of bounded size, and he showed that certain Tseitin-type formulas in 3-CNF require very long clauses in any (full) resolution refutation, expressing hope, that this result might be useful for a super-polynomial lower bound for (full) resolution. Supported by the Natural Sciences and Engineering Research Council of Canada and by the Communications and Information Technology Ontario.
1
1981 Krishnamurthy/Moll made a precise general conjecture in [12] about a strong relation between the necessary size of clauses in resolution refutations and the necessary length of such refutations. Finally, 1999 Ben-Sasson/Wigderson justi ed in [2] these attempts by proving that if \long" clauses are necessary for a resolution refutation, then only \large" refutations are possible1) . Their proof is an adaption of an analogous result in [Clegg, Edmonds and Impagliazzo 96] ([7]) for \degree restricted Groebner bases proof systems"2) .
Generalizing Unit resolution Generalizing [Yamasaki and Doshita 83] ([19]), in [Gallo and Scutella 88] ([10]) a hierarchy of generalized Horn clause-sets has been introduced, which has been smoothened in [Kleine Buening 93] ([4]). Since Unit resolution is complete for refuting Horn clause-sets, it is natural to ask for a generalization w.r.t. that hierarchy. In [4] for that purpose k-resolution has been introduced, which is complete for \k-Horn clause-sets." While bounded resolution allows only resolution steps where both parent clauses have bounded length (or \width"), now only one parent clause has to ful ll this restriction.
1.2 A new form of width restricted resolution The main disadvantage of bounded resolution is its inability of even handling Horn formulas, which is also re ected in the dependence of the general lower bound for resolution in [2] on the maximal input clause length, preventing that bound from being used for clause-sets already containing \long clauses." On the other side, the main disadvantage of k-resolution is that it does not seem likely to be poly-time decidable for k 3. In [13] a new form of width restricted resolution has been introduced, called CUBI resolution in the present paper, combining the advantages of both approaches: - the dependence on the maximal input clause length in the general resolution lower bound has been removed, and thus this bound is now also applicable for clause-sets with large clauses (while no applications are lost); 1) thus from the width lower bound in Galil [9] in fact an exponential lower bound for (full) resolution follows; the precise form of the conjecture of Krishnamurthy/Moll has been refuted by [3] (the complexity of the Ramsey formulas for resolution is still open) 2) that the argumentation in [7] just needs the resolution calculus has been noted rst in the literature in [1]
2
- generalized Horn clause-sets can be handled; - for xed width derivability of the empty clause is poly-time decidable. The organization of this extended abstract is as follows: 1. In Section 3 the results on bounded resolution are outlined, 2. while in Section 4 k-resolution is discussed. 3. Then in Section 5 CUBI resolution is introduced and some main lemmas are stated. Also the notion of nested input resolution introduced in [13] is presented, which in fact suces to refute the generalized Horn formulas, and the relation to CUBI resolution is discussed. 4. Finally in Section 6 two methods for proving width lower bounds are given, and the basic ideas for two direct lower bound proofs (not possible when using bounded resolution) are given. 2
Notation
CLS is the set of all clause-sets (interpreted as CNF's) and p{CLS the set of all clause-sets with clauses of length at most p. HO is the set of all Horn clause-sets and HOk is the set of all generalized Horn clause-sets of level k
(HO 1 = HO ; see [10, 4] or Subsection 5.3 in [13]). n(F ) := j var(F)j is used for the number of variables in F 2 CLS , c(F ) is the number of clauses in F, `(F ) the number of literal occurrences, and p(F ) is the maximal clause length in F. By PHPm k the pigeonhole formulas with m pigeons and k holes are denoted, by GTm Krishnamurthy's graph formulas (expressing that a transitive simple directed graph with m nodes and without binary cycles must have a source; see [11] or Subsection 8.4 in [13]), by MGTm the translation of GTm into 3{CLS used in [3], and by Rm the \Ramsey formulas" introduced in [12]. For any undirected connected graph G the associated unsatis able Tseitin formula is denoted by T(G) (choosing any node labeling with odd sum; see [16] or [18]), and for a directed graph G which is acyclic and has exactly one sink, the \pebbling formulas" introduced in [2] are denoted by PF(G) (see also Subsection 6.2 in [13]). For the minimal number of leaves in a tree-like resolution refutation of an (unsatis able) clause-set F we use ComptR (F ), while the minimal number of (di erent) clauses needed in a resolution refutation of F is CompR (F ). 3
3
Bounded resolution
In [9] \bounded resolution" has been introduced, the most widely known form of width restricted resolution.
De nition 3.1 For k 2 N let bW k be the set of clause-sets refutable by resolution proofs T : F ` ? using only clauses of length k. The corresponding 0
hardness parameter is called bwid(F ) := inf f k 2 N0 : F 2 bW k g:
The following properties are known: 1. (a) For constant k the classes bW k are polynomially decidable. (b) All classes bW k are stable under partial assignments, renaming and formation of super-clause-sets, and furthermore bW k bW k+1 holds for all k 0. 2. Already in 1975, Galil ([9]) proved for a sequence (Gm ) of graphs with T(Gm ) 2 3{CLS , n(T(Gm )) m and c(T(Gm )) 8m the quasi-linear width lower bound b wid(T(Gm )) 6(logm m)2 : 2
Using the expander graphs (Gm ) from [17], in [2] a linear lower bound b wid(T(Gm )) (m) is obtained. 3. Krishnamurthy/Moll proved in [12] p b wid(Rm ) ( n(Rm )) and they conjectured (KM81) \log2 CompR(F) bwid(F)" for arbitrary clause-sets F. 4. In [2] (exploiting [7] for resolution) the general lower bound for full resolution ? b ? p(F))2 logCompR(F) ( wid(F) n(F) 0
0
for F 2 USAT has been shown. This lower bound is (nearly) tight as shown in [3]: CompR(MGTm ) O(m3 ) p b wid(MGTm ) (m) ( n(MGTm )) (thus falsifying the precise form of (KM81)). 4
5. For tree-like resolution in [2] the lower bound log2 ComptR(F) bwid(F) ? p(F) is shown, and using the graphs (Gm ) from [6] they get wid(PF(Gm )) 6 and log ComptR(PF(Gm )) (m= log m)
b
where `(PF(Gm )) O(m). A weakness of this concept of width restricted resolution is, that in order to be able to handle all clauses in a given F, the width parameter k must be at least p(F). So for example no bW k contains HO , and the general lower bounds for tree-like resolution (see Remark 5 in Section 3) becomes trivial in case of b wid(F) O(p(F)). 4
k-resolution
Generalizing Unit-resolution, in [4] \k-resolution" has been introduced in order to cope with generalized Horn formulas.
De nition 4.1 For k 2 N let uW k be the set of clause-sets refutable by resolution proofs T : F ` ? where for each resolution step at least one parent clause 0
has length at most k. The corresponding hardness parameter is called uwid(F ) := inf f k 2 N0 : F 2 uW k g:
1. All classes uW k are stable under partial assignments, renaming and formation of super-clause-sets, and furthermore uW k uW k+1 holds for all k 0. 2. F 2 uW 1 , F refutable by Unit resolution (, F refutable by input resolution). 3. uW 0 and uW 1 are decidable in poly-time, and also uW 2 as shown in [5]. 4. For all k 1 we have HO k uW k . 5. uwid(F) bwid(F) uwid(F)+p(F) (the second inequality follows from Lemma 5.3 in the next section; in [5] the weaker inequality bwid(F) max( uwid(F); 3) + max( uwid(F); p(F); 3) has been shown). The main drawback here is that it is not known whether the classes uW k are poly-time decidable for k 3. 5
5
CUBI resolution
In [13] (see De nition 8.3 there) a strengthening of k-resolution has been introduced, combining all the (positive) properties of bounded and k-resolution. The idea is simple: While for deciding \F 2 bW k ?" the closure of F under single resolution steps with resolvents of length at most k is computed, now we compute the closure of F under addition of clauses of length at most k derived by input resolution.
De nition 5.1 For k 2 N let iW k be the set of clause-sets such that by 0
iteratively adding clauses of length at most k derivable by input resolution (using also the new clauses) the empty clause can be generated. The corresponding hardness parameter is called iwid(F ) := inf f k 2 N0 : F 2 iW k g:
For the more general notions iW k (U ) and iwid (F), using a \partial oracle" U for (partial) decision of unsatis ability, as well as for the proofs of the following results, see [13]. U
Lemma 5.2 The classes W k are poly-time decidable for xed k 0. i
Lemma 5.3 For all unsatis able clause-sets F we have 1. bwid(F) ? p(F) iwid(F) bwid(F); 2. uwid(F) ? 1 iwid(F) uwid(F).
Theorem 5.4 For unsatis able clause-sets F with n(F) 6= 0 we have 1 iwid(F)2 < ln Comp (F) < (ln n(F)) ( iwid(F) + 1) + 2: R 8 n(F)
5.1 Simulation of nested input resolution In [13] (a forerunner is [14]) a hierarchy Gk (U ; S ) of clause-sets has been de ned, where U U0 is a set of unsatis able clause-sets stable under partial assignments, and S S0 is a set of satis able clause-sets stable under enforced assignments, that is, stable under such partial assignments ' which can be extended to a satisfying assignment, but where any deviation from ' would destroy satis ability. The basic cases are U = U0, where U0 consists of all clause-sets containing already the empty clause, and S = S0, where S0 is the singleton containing the empty clause-set. 6
The corresponding hardness parameter hU ;S (F ) for F 2 CLS is de ned (as usual) as the minimal k with F 2 Gk (U ; S ). For unsatis able F we often use h (F) instead since here S is not involved, and U in turn is omitted in case of U = U0 . Some main properties are (for an extensive treatment see [13]): U
1. For all xed k the classes Gk (U ; S ) are poly-time recognizable and SAT decidable (counting decisions \F 2 U ?" and \F 2 S ?" as one step). Searching through the hierarchy from below gives a (general) SAT algorithm with running time ?
O `(F) 2log2 (n(F )+1) 2hU S (F )
;
which for unsatis able F is bounded by ?
O `(F) ComptR(F)2 log2 (n(F )+1) ; \quasi-automatizing" tree-like resolution (answering a question in [2]), since (using U = U0 ) the algorithm produces as byproduct a resolution tree refutation with at most ComptR(F)log2 (n(F )+1) many leaves. 2. The \hardness parameter" h ; (F) has several natural characterizations, so for example for unsatis able F the quantity h(F) + 1 is just the space complexity of tree-like resolution as introduced in [8]. 3. HO k Gk (U ; S ) for all k and all U ; S . 4. h(F) log2 ComptR(F) log2 (n(F) + 1) h(F) for all unsatis able F (for the general relativized statement see [13]). U S
;k
;1
5. There is a natural generalization ` of input resolution (which is ` ) refuting exactly the unsatis able instances in Gk (U ; S ) = Gk (U ).3) U
U0
Theorem 5.5 For all unsatis able F with h(F) 1 we have wid(F) h(F) ? 1; i
that is, width restricted resolution simulates \nested input resolution." It follows log2 ComptR(F) iwid(F) + 1. 3)
The relativized classes
W k (U )
i
uses
U ;1
`
instead of
7
U0 ;1
`
.
5.2 Some remarks on "real world" applications The natural SAT decision algorithm associated with the hierarchy ( bW k ), testing for input F whether there is k 2 f0; : : : ; n(F)g with F 2 bW k (then F is unsatis able) or not (then F is satis able), has a very poor performance on satis able inputs, due to the fact, that the classes bW k contain only unsatis able instances (in contrast to the classes Gk ). Moreover, this algorithm has space requirements of b ? O `(F) + n(F) wid(F ) ; while the natural SAT decision algorithm associated with the hierarchy (Gk ) (see Remark 1 in Subsection 5.1) needs only space ? ? O `(F) + log n(F)h(F ) = O `(F) + h(F) log n(F) : Thus, for (most) practical purposes the use of the hierarchies (Gk (U ; S )) (see [13], Subsection 5.4 for improved satis ability handling) is strongly preferable to the use of ( iW k ) or ( bW k ). In fact already for medium size F the decision \F 2 bW 3 ?" is out of reach due to space limitations (considering, say \F 2 iW 2 ?" is of interest only in special cases), while for the hierarchy (Gk ) essentially only time is of importance, and k-values up to, say, k 5 can be handled. The algorithm searching through the classes Gk from below is the heart of the commercially successful SAT algorithm patented in [15]. However, width restricted resolution is (at least) a valuable tool to analyze full resolution (instead of tree-like resolution), and for special classes width restricted resolution can be exponentially better than tree-like resolution. For example, in [13], improving [2] (see Remark 5 in Section 3), for dag's G with in-degree 2 we show bwid(PF(G)) 6 and h(PF(G)) peb(G), where peb(G) is the minimal number of pebbles needed for G. 6
How to prove width lower bounds
In [2] a scheme has been extracted for proving lower bounds on bwid(F), which in fact is the basic method of all such proofs in the literature, starting with [9]. The basic observation here is as follows. We consider a xed unsatis able clause-set F in this section, and we use V := var(F) for the set of variables of F, while CL(V ) denotes the set of clauses C with var(C) var(F).
Lemma 6.1 Consider : CL(V ) ! N and k; m 1 with m (?)=2 (where ? is the empty clause), such that the following properties hold (for all clauses C; D 2 CL(V )): 0
8
- C 2 F ) (C) 1; - if C; D are resolvable with resolvent R, then (R) (C) + (D); - if m (C) 2m, then jC j k. Now bwid(F) k holds.
We do not consider this method here but we want to give examples where
direct lower bound proofs for iwid(F) are possible for minimally unsatis able F
with bwid(F) p(F) (and thus lower bounds for bwid(F) are senseless).
Lemma 6.2 Consider an unsatis able clause-set F , and assume that a \size
function"
s : CL(V ) ! N0
and N CL(V ) (\negligible clauses") are given ful lling - N is stable under resolution; - F n N is satis able;
- s(C) jC j for all C 2 CL(V ); - for all resolvable clauses C; D 2 CL(V ) with resolvent R we have - s(R) max(s(C); s(D)) ? 1, - C 2 N ) s(R) s(D).
Then iwid(F) min s(C) ? 1. C 2F nN
Two examples (for details see [13]): 1. For PHPmk let N be the set of the negative binary clauses in PHPmk , and let s(C) be the number of occupied columns in the (m k)-matrix representation of C. Then obviously the assumptions of Lemma 6.2 are ful lled, and we get iwid(PHPmk k ? 1. Since h(PHPmk ) k, in fact we obtain iwid(PHPmk ) = k ? 1 and h(PHPmk ) = k. 2. For GTm consider a variable vi;j 2 var(GTm ) as a directed edge from i to j, and literals vi;j ; vi;j as forward resp. backward edges from i to j resp. from j to i. Let N be the set of all clauses which represent a union of signed directed circuits, and let s(C) be the size of a spanning forest in the underlying undirected graph. Then the assumptions of Lemma 6.2 are ful lled, and we get iwid(PHPmk m ? 2. Since h(GTm ) m ? 1, in fact we have iwid(GTm ) = m ? 2 and h(GTm ) = m ? 1. 9
References
[1] Paul Beame and Toniann Pitassi. Simpli ed and improved resolution lower bounds. In 37th Symposium on Foundations of Computer Science (FOCS' 96), pages 274{282, 1996. [2] Eli Ben-Sasson and Avi Wigderson. Short proofs are narrow | resolution made simple. Technical Report TR99-022, ECCC Electronic Colloquium on Computational Complexity, July 1999. [3] Maria Luisa Bonet and Nicola Galesi. A study of proof search algorithms for resolution and polynomial calculus. In 40th Annual Symposium on Foundations of Computer Science (FOCS), pages 422{431, 1999. [4] Hans Kleine Buning. On generalized Horn formulas and k-resolution. Theoretical Computer Science, 116:405{413, 1993. [5] Michael Buro and Hans Kleine Buning. On resolution with short clauses. Annals of Mathematics and Arti cial Intelligence, 18(2-4):243{260, 1996. [6] J.R. Celoni, Wolfgang J. Paul, and Robert Endre Tarjan. Space bounds for a game on graphs. Mathematical Systems Theory, 10:239{251, 1977. [7] Matthew Clegg, Je ery Edmonds, and Russell Impagliazzo. Using the Groebner basis algorithm to nd proofs of unsatis ability. In Proceedings of the 28th ACM Symposium on Theory of Computation, pages 174{183, 1996. [8] Juan Luis Esteban and Jacobo Toran. Space bounds for resolution. In C. Meinel and S. Tison, editors, 16th Annual Symposium on Theoretical Aspects of Computer Science, volume 1563 of LNCS. Springer, 1999. [9] Zvi Galil. On the validity and complexity of bounded resolution. In Proceedings of seventh annual ACM Symposium on Theory of Computing, pages 72{82, May 1975. [10] Giorgio Gallo and Maria Grazia Scutella. Polynomially solvable satis ability problems. Information Processing Letters, 29:221{227, November 1988. [11] Balakrishnan Krishnamurthy. Short proofs for tricky formulas. Acta Informatica, 22:253{275, 1985. [12] Balakrishnan Krishnamurthy and Robert N. Moll. Examples of hard tautologies in the propositional calculus. In 13th ACM Symposium on Theory of Computing (STOC'81), pages 28{37, 1981. 10
[13] Oliver Kullmann. Investigating a general hierarchy of polynomially decidable classes of CNF's based on short tree-like resolution proofs. Technical Report TR99-041, Electronic Colloquium on Computational Complexity (ECCC), October 1999. [14] Daniele Pretolani. Hierarchies of polynomially solvable satis ability problems. Annals of Mathematics and Arti cial Intelligence, 17(3-4):339{357, 1996. [15] Gunnar Martin Natanael Stalmarck. Method and apparatus for checking propositional logic theorems in system analysis, 1990. European Patent Speci cation; application: December 19, 1990, Bulletin 90/51; publication: June 28, 1995, Bulletin 95/26. [16] G.S. Tseitin. On the complexity of derivation in propositional calculus. In Seminars in Mathematics, volume 8. V.A. Steklov Mathematical Institute, Leningrad, 1968. English translation: Studies in mathematics and mathematical logic, Part II (A.O. Slisenko, editor), 1970, pages 115-125. [17] Alasdair Urquhart. Hard examples for resolution. Journal of the ACM, 34:209{219, 1987. [18] Alasdair Urquhart. The complexity of propositional proofs. The Bulletin of Symbolic Logic, 1(4):425{467, 1995. [19] Susumu Yamasaki and Shuji Doshita. The satis ability problem for a class consisting of Horn sentences and some non-Horn sentences in propositional logic. Information and Control, 59:1{12, 1983.
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