Heuristics Based on Unit Propagation for Satis ability Problems
Chu Min Li & Anbulagan
LaRIA, Univ. de Picardie Jules Verne, 33, Rue St. Leu, 80039 Amiens Cedex, France
fax: (33) 3 22 82 75 02, e-mail: [email protected], [email protected]
having Maximum Occurrences in clauses of Minimum Abstract
[
Size Dub ois et al., 1993; Freeman, 1995; Pretolani, 1993;
The pap er studies new unit propagation based
]
Crawford and Auton, 1996; Jeroslow and Wang, 1990 .
heuristics for Davis-Putnam-Loveland (DPL)
Intuitively these variables allowtowell exploit the p ower
pro cedure. These are the novel combinations of
of unit propagation and to augment the chance to reach
unit propagation and the usual "Maximum Oc-
an empty clause. Recently another heuristic based on
currences in clauses of Minimum Size" heuris-
Unit Propagation (UP heuristic) has proven useful and
tics. Based on the exp erimental evaluations of
allows to exploit yet more the p ower of unit prop-
di erent alternatives a new simple unit prop-
[
agation Freeman, 1995; Crawford and Auton, 1996;
agation based heuristic is put forward. This
]
Li, 1996 . Givenavariable x, a UP heuristic examines x
compares favorably with the heuristics em-
by resp ectively adding the unit clause x andx to F and
ployed in the current state-of-the-art DPL im-
indep endently makes two unit propagations. The real
plementations (C-SAT, Tableau, POSIT).
e ect of the unit propagations is then used to weigh x.
1 Intro duction
pro cedure DPL(F)
Begin
Consider a prop ositional formula F in Conjunctive
if F is empty, return "satisfiable";
Normal Form (CNF) on a set of Bo olean variables
fx ;x ; :::; x g, the satis ability (SAT) problem consists
1 2 n
F:=UnitPropagation(F ); If F contains an empty
in testing whether clauses in F can all b e satis ed by
clause, return "unsatisfiable".
some consistent assignment of truth values (1 or 0) to
the variables. If it is the case, F is said satis able; oth-
/* branching rule */
erwise, F is said unsatis able. If each clause exactly
select a variable x in F according to a heuristic
contains r literals, the subproblem is called r -SAT prob-
H , if the calling of DPL(F [fxg) returns
lem.
"satisfiable" then return "satisfiable", otherwise
SAT problem is fundamental in many elds of com-
return the result of calling DPL(F [fx g).
puter science, electrical engineering and mathematics.
End.
[ ]
It is the rst NP-Complete problem Co ok, 1971 with
3-SAT as the smallest NP-Complete subproblem.
pro cedure UnitPropagation(F)
[
The Davis-Putnam-Loveland pro cedure (DPL) Davis
Begin
]
et al., 1962 is a well known complete metho d to solve
While there is no empty clause and a unit clause l
SAT problems, roughly sketched in Figure 1.
exists in F , assign a truth value to the variable
DPL pro cedure essentially constructs a binary search
containedin l to satisfy l and simplify F .
tree, each recursive call constituting a no de of the tree.
Return F .
Recall that all leaves (except eventually one for a sat-
End.
is able problem) of a search tree represent a dead end
Figure 1: DPL Pro cedure
where an empty clause is found. The branching vari-
However, since examining a variable bytwo unit prop- ables are generally selected to allowtoreach as early as
agations is time consuming, two ma jor problems remain p ossible a dead end, i.e. to minimize the length of the
op en: should one examine all free variables by unit prop- current path in the searchtree.
agation at every no de of a search tree? if not, what are The most p opular SAT heuristic actually is Mom's
the variables to b e examined at a search tree no de? heuristic, whichinvolves branching next on the variable
The essential reason to use UP heuristics instead of In this pap er we try to exp erimentally solve these two
Mom's one is that Mom's heuristic may not maximize the problems to obtain an optimal exploitation of UP heuris-
e ectiveness of unit propagation, b ecause it only takes tic. We de ne a PROP predicate at a search tree no de
binary clauses (if any) into accounttoweigh a variable, whose denotational semantics is the set of variables to b e
although some extensions try to also take longer clauses examined at a search tree no de, i.e. x is to b e examined
into account with exp onentially smaller weights (e.g. 5 if and only if PROP(x) is true. By appropriately chang-
ternary clauses are counted as 1 binary clauses). A UP ing PROP,we exp erimentally analyse the b ehaviour of
heuristic allows to take all clauses containing a variable di erent UP heuristics. We write 12 DPL pro cedures
and their relations into accountinavery e ectiveway which are di erentonlyinPROP and run these pro-
to weigh the variable. As a secondary e ect, it allows cedures on a very large sample of hard random 3-SAT
to detect the so-called failedliterals in F whichwhen problems.
satis ed falsify F in a single unit propagation. However
We b egin in section 2 by describing the 12 DPL pro ce-
since examining a variable bytwo unit propagations is
dures and summarizing the exp erimental results on these
time consuming, it is natural to try to restrict the vari-
programs. In section 3 we compare a pure UP heuris-
ables to b e examined. For this purp ose weuse PROP
tic and a pure Mom's heuristic and show the sup eriority
predicate de ned at a searchtreenode.
of UP heuristics. In section 4 we study di erent restric-
The success of Mom's heuristic suggests that the larger
tions of UP heuristics. In section 5 we discuss the related
the numb er of binary o ccurrences of a variable is, the
work and compare the b est DPL pro cedure in our exp er-
higher its probability of b eing a go o d branching vari-
imentation with three state-of-the-art DPL pro cedures.
able is, implying that if one should restrict UP heuris-
Section 6 concludes the pap er.
tics by means of PROP, he should restrict UP heuristics
to those variables having a sucientnumb er of binary
2 UP Heuristics Driven by PROP
o ccurrences. For this reason, all restrictions on PROP
Let dif f (F ;F ) b e a function whichgives the number
studied in this pap er are de ned according to the number
1 2
of clauses of minimum size in F but not in F ,we show
of binary o ccurrences of a variable, so that the resulted
1 2
a generic branching rule in Figure 2, where the equation
UP heuristics rely on combinations of unit propagation
[ ]
de ning H (x) is suggested in Freeman, 1995 and the
and Mom's heuristics.
weight 5 for uniformizing clauses of di erent length is
empirically optimal. PROP21
x such that PROP(x) is
For each free variable PROP10 PROP20 PROP31
true do
0 00
F and F be two copies of F
let PROP30 PROP41 PROP 42
Begin
0 0
:= UnitPropagation(F [fxg);
F PROP 00
00 40
:= UnitPropagation(F [fx g);
F inclusion relation
0 00
If both F and F contain an empty clause then
Figure 3: PROP predicate hierarchyby inclusion relation.
return "F is unsatisfiable".
0
PROP (x) is true i x has i binary o ccurrences of whichat
ij
If F contains an empty clause then x := 0,
00 00
least j negativeandj p ositive.
F := F else if F contains an empty
0
clause then x := 1, F := F ;
0 00
If neither F nor F contains an empty clause
The rst PROP predicate in our exp erimentation is
then let w (x) denote the weight of x
called PROP and has empty denotational semantics,
0 00
a
w (x):=dif f (F ;F) and w (x):=dif f (F ;F)
the resulted branching rule using a pure Mom's heuris-
End;
tic, and the second is called PROP whose denotational
0
If all variables examined above are valued or
semantics is the set of all free variables, the resulted UP
PROP(x) is false for every x then
heuristic is in its pure form and plays its full role:
For each free variable x in F do
PROP : PROP (x) is false for every free variable x;
let r be the length of the clause C
a a
i i
r r
i i
w (x):= 5 and w (x):= 5
x2C x2C
i i
PROP : PROP (x) is true for every free variable x.
0 0
For each variable x do
Between PROP and PROP , there are manyother
a 0
H (x):=w (x) w (x) 1024 + w (x)+ w (x)
p ossible PROP predicates. Figure 3 de nes 8 predicates
constituting a hierarchyby inclusion relation of their de-
Branching on the free variable x such that H (x) is
notational semantics.
the greatest.
PROP is de ned to b e PROP , but for no des
3141 31
under a xed depth of a search tree, it is de ned to b e
Figure 2: A Generic Branching Rule Driven by PROP
[
also similar to the exp onential factors in C-SAT Dub ois PROP .LetT b e a constant, the last PROP predicate
41
]
et al.,1993 where 5.71 ternary clauses are counted as 1 is named PROP and is de ned to b e the rst of the
z
binary clause. three predicates PROP , PROP and PROP (in this
41 31 0
h has at least T variables in its denotational order) whic 6 x 10
5
semantics.
h PROP predicate results in a DPL pro cedure,
Eac 4.5 sata
PROP , PROP , PROP and PROP resp ectively
0 3141 z a sat42
4
Sata, Sat0, Sat and Satz,andPROP giv-
giving sat40
3141 ij
sat41
ing Sat . These programs are di erent only in PROP
ij 3.5
sat3141
predicate, except Sat0 which need not count the o ccur-
3 sat30
rences of variables.
sat31 e run the 12 programs (compiled using gcc with op- W 2.5
sat21
entium CPU under timization) on a PC with a 133 Mhz P sat20
2
ux op erating system on a very large sample of ran- Lin satz
1.5 sat10
dom 3-SAT problems generated by using the metho d of
search tree size (number of nodes) sat0
[ ]
hell et al. Mitchell et al.,1992.Given a set V of n
Mitc 1
Bo olean variables fx ;x ; :::; x g,we randomly generate
1 2 n
0.5
m clauses of length 3. Each clause is pro duced by ran-
ho osing 3 variables from V and negating each domly c 0
140 160 180 200 220 240 260 280 300 320 340
y 0.5. Empirically, when the ratio m=n is
with probabilit nb. of variables
near 4.25 for a 3-SAT formula F , F is unsatis able with
Figure 4: Mean search tree size of each program as a function
a probability 0.5 and is the most dicult to solve. We
of n for hard random 3-SAT problems at the ratio m=n = 4.25
vary n from 140 variables to 340 variables incrementing
by 20, for each n the ratio clauses-to-variables (m=n)is
set to 4.0, 4.1, 4.2, 4.25, 4.3, 4.4, 4.5. Ateach ratio and
by each program, if n< 280 then 1000 problems are
2500
solved, if 280 n 300 then 500 problems are solved,
n = 320 then 300 problems are solved, and if n = 340 if sata
sat42
ed. A problem is solved suc- then 100 problems are solv 2000
sat10
ely by all the 12 DPL pro cedures b efore another
cessiv sat40
vironment to all programs. Due to ensure the same en sat20 1500
sat30
to the lackofspace,we only present the exp erimental sat0
time in sec.
m=n = 4.25 in Figures 4, 5, and 6,
results for the ratio sat21 es where the DPL pro cedures corresp onding to the curv 1000 sat41
sat3141 are listed in the same order from top to b ottom. The
sat31
tal results on the other ratios give exactly the exp erimen satz
500
same conclusions.
A Pure UP Heuristic Versus a Pure 3 0 140 160 180 200 220 240 260 280 300 320 340
nb. of variables
Mom's Heuristics: Sat0 vs S ata
Sat0 systematically examines all the variables by unit
Figure 5: Mean run time of each program as a function of n
for hard random 3-SAT problems at the ratio m=n = 4.25
propagation at all no des, using a pure UP heuristic,
while S ata do es not examine anyvariable so and em-
Sat0 actually is slower than ve other programs based
ploys a pure Mom's heuristic. One might b elieve that
on balanced restrictions of variables to b e examined by
Sat0would b e simply to o slow, but it is not the case.
unit propagation, but not substantially so (except Satz).
Sat0 is much faster than Sata. In fact from Figures 4
The surprisingly go o d p erformance of Sat0 con rms the
and 5, all DPL pro cedures using a UP heuristic in our
power of UP heuristics for selecting the next branching
exp erimentation are substantially b etter than S ata in
variable and suggests that its e ect for detecting failed
terms of search tree size and real run time.
literals is only secondary.
Note that Mom's heuristic used in S ata is similar to
[
the so-called two-sided Jeroslow-Wang rule Ho oker and
4 Restricted UP Heuristics
]
Vinay, 1995 , with the only di erence that a clause of
length i is counted as 5 clauses of length i+1 instead of Figure 6 illustrates the number of variables examined by
2. Our exp eriments suggest that 5 is b etter than 2. 5 is di erent restricted UP heuristics at a no de.
250
tional semantics of a PROP predicate suchasPROP
sat10 31 urthermore, the no des are
sat20 b ecomes larger and larger. F
umerous near the leaves and the branching vari- sat21 more n
200
y a less imp ortant role there. It app eared that
sat30 ables pla
ariables to b e examined by sat31 one could restrict more the v
sat3141
ves without imp ortant loss 150 unit propagation near the lea
sat40
on the search tree size so as to obtain some gain in terms
sat41 of real run time. sat42
100 POSIT's UP heuristic (called BCP-based heuristic)
[ ]
Freeman, 1995 realizes this idea: under the level9ofa
search tree, at most 10 variables are examined by unit
50 propagation.
Sat uses PROP from the top of a search tree,
3141 31
but under the depth empirically xed to n 4=70, it uses
average number of variables examined per search tree node
PROP , where n is the numberofvariables in the initial 0 41
0 10 20 30
T problem. Note that if n 160, n 4=70
search tree depth input 3-SA
9, so Sat generally strengthens the restriction later
3141
Figure 6: Average number of variables examined at a search
than POSIT.
tree no de in a given depth when solving hard random 3-SAT
problems of 300 variables and 1275 clauses (500 problems are
From Figures 4 and 5 Sat is not b etter than Sat ,
3141 31
solved) for 9 programs
although it makes many fewer unit propagations to ex-
amine variables (see Figure 6), suggesting that the search
4.1 Restriction by total numb er of binary
tree depth is rather irrelevant to the restriction of UP
o ccurrences of a variable
heuristics.
Four programs Sat , Sat , Sat and Sat realize this
10 20 30 40
4.4 Dynamic restriction bynumber of
typ e of restrictions. While a classical Mom's heuristic
variables to b e examined
selects the next branching variable having maximum bi-
The relatively p o or p erformance of Sat seems due to
42
nary o ccurrences, the restricted UP heuristics examine
the small numberofvariables examined at eachnode
a set of variables having more binary o ccurrences than
(see Figure 6), though these variables havemany binary
others, including the variable having maximum binary
o ccurrences. A careful analysis shows that even Sat ,
31
o ccurrences. From Figure 4, it is clear that the more
the b est one up to now, examines few or no variables
variables are examined, the smaller the search tree size
at some no des, esp ecially near the ro ot where there are
is.
few binary clauses, although these no des are more de-
4.2 Balanced restriction by total number
terminant for the nal search tree size. PROP is then
z
of binary o ccurrences of a variable intro duced to ensure that at least T variables are exam-
ined at each no de, T b eing empirically xed to 10. Near
Four programs Sat , Sat , Sat and Sat realize this
21 31 41 42
the ro ot, all free variables are examined to exploit the
typ e of restrictions. The PROP predicates require that
full p ower of UP heuristic. As so on as the number of
avariable o ccurs b oth p ositively and negatively in bi-
variables o ccurring b oth negatively and p ositively in bi-
nary clauses to balance the search tree. We compare the
nary clauses and having at least 4 (3) binary o ccurrences
duet Sat and Sat (i=2, 3, 4) and observethatSat
i0 i1 i1
is larger than T , only these variables are examined to se-
examines strictly fewer variables than Sat and is faster
i0
lect the next branching variable.
than it in spite of a slightly larger search tree. In partic-
ular, Sat and Sat examine almost the same number
41 40
5 Related Work
of variables (see Figure 6), but the balanced restriction
gives a faster DPL pro cedure. [ ]
C-SAT Dub ois et al., 1993 examines some variables by
Wepay sp ecial attention to PROP and PROP
unit propagations (called lo cal pro cessing) near the b ot-
31 41
since they seem to b e the b est balanced restrictions.
tom of a search tree to rapidly detect failed literals there.
Pretolani also uses a similar approach (called pruning
4.3 Dynamic restriction as a function of
[ ]
metho d) based on hyp ergraphs in H2R Pretolani, 1993 .
search tree depth
But the lo cal pro cessing and the pruning metho d as are
[ ] [
Sat realizes this restriction. A general observation resp ectively presented in Dub ois et al., 1993 and Pre-
3141
]
when solving 3-SAT problems using a DPL pro cedure tolani, 1993 do not contribute to the heuristic to select
is that there are more and more binary clauses when the next branching variable. We nd the rst e ective
[ ]
descending from the search tree ro ot and the denota- exploitation of UP heuristic in POSIT Freeman, 1995
[ ]
vided mak e command on the SUN Sparc 20 workstation and Tableau Crawford and Auton, 1996 which use a
1
from the sources named posit-1:0:tar:g z .Table 2 shows similar idea as in C-SAT to determine the variables to
the p erformances of the 4 DPL pro cedures on problems b e examined at a no de by unit propagation: x is to be
of 300, 350, and 400 variables, where time standing for examinedi x is among the k most weighted variables
the real mean run time is rep orted by the unix com- by a Mom's heuristic.
mand /usr/bin/time and t siz e standing for search tree
The main di erence of S atz with Tableau and POSIT
size (numb er of no des) is rep orted (or computed from
is that S atz do es not sp ecify a upp er b ound k of the
numberofbranches rep orted) by the DPL pro cedures.
number of variables to b e examined at a no de by unit
propagation. Instead, S atz sp eci es a lower b ound. In
300 vars 350 vars 400 vars
fact, S atz examines many more variables by an optimal
300 problems 250 problems 100 problems
combination of unit propagation and Mom's heuristics.
System time t siz e time t siz e time t siz e
C-SAT 77 49567 512 275303 3818 1624869
Given the depth of a no de, Table 1 illustrates the aver-
Tableau 79 43041 558 253366 4544 1524551
age numberofvariables examined (#examined vars)at
POSIT 57 61797 474 400588 3592 2751611
S atz 34 32780 203 174337 1207 916569
the no de by S atz , with the depth of the ro ot b eing 0. In
order to compare with C-SAT, Tableau and POSIT we
Table 2: Mean run time (in second) and mean search tree
also give the theoretical value of k (for C-SAT), k (for
C T
size of C-SAT, Tableau, POSIT and Satz on ratio m=n=4.25
Tableau) and k (for POSIT) at the no de, resp ectively
P
[
according to the de nitions of k in Dub ois et al., 1993;
Table 2 shows that S atz is faster than the ab ovecited
]
Crawford and Auton, 1996; Freeman, 1995 .
versions of C-SAT, Tableau and POSIT, S atz 's search
tree size is the smallest, and S atz 's run time and search
depth #free vars #examined vars k k k
C T P
tree size growmoreslowly.Table 3 shows the gain of
1 298.24 298.24 0 263 265
S atz compared with the cited version of C-SAT, Tableau
2 296.52 296.52 0 227 230
3 294.92 293.89 0 193 198
and POSIT at the ratio m=n=4.25. Each item is com-
4 292.44 292.21 0 141 149
puted from Table 2 using the following equation:
5 288.60 282.04 0 61 72
6 285.36 252.14 0 0 10 or 3
7 281.68 192.82 0 0 10 or 3
gain =(value(sy ste m)=v al ue(S atz ) 1) 100%
8 277.54 125.13 0 0 10 or 3
9 273.17 71.51 0 0 10 or 3
where value is real mean run time or real mean search
10 268.76 40.65 0 0 10 or 3
11 264.55 26.81 0 0 10 or 3
tree size and sy stem is C-SAT, Tableau or POSIT. From
12 260.53 21.55 0 0 10 or 3
Table 3, it is clear that the gain of S atz grows with the
13 256.79 19.80 0 0 10 or 3
14 253.28 19.24 0 0 10 or 3
size of the input formula.
15 249.96 19.16 0 0 10 or 3
16 246.77 19.28 0 0 10 or 3
17 243.68 19.57 0 0 10 or 3
300 vars 350 vars 400 vars
18 240.68 19.97 0 0 10 or 3
300 problems 250 problems 100 problems
19 237.73 20.46 0 0 10 or 3
System time t siz e time t siz e time t siz e
20 234.82 20.97 0 0 10 or 3
C-SAT 126% 51% 152% 58% 216% 77%
Tableau 132% 31% 175% 45% 276% 66%
Table 1: Average number of variables examined in S atz
POSIT 68% 89% 133% 130% 198% 200%
at a no de in a given depth when solving a hard random 3-
Table 3: The gain of Satz vs. C-SAT, Tableau and POSIT in
SAT problem of 300 variables and 1275 clauses (500 problems
terms of run time and search tree size on the ratio m=n=4.25
are solved) compared with theoretical value of k in C-SAT,
computed from Table 2
Tableau and POSIT
The central strategy of S atz is to try to reachan
It is clear that S atz examines many more variables at
empty clause as early as p ossible. Further along the line,
each no de than any of C-SAT, Tableau or POSIT. Near
we maketwo relatively small resolvents-driven improve-
the ro ot, S atz examines all free variables. Elsewhere
ments in S atz . The rst improvement is the prepro cess-
S atz examines a sucientnumber (T )ofvariables.
ing of the input formula by adding some resolvents of
We compare C-SAT, Tableau, POSIT and S atz on a
length 3, The second improvement consists in re n-
large sample of hard random 3-SAT problems on a SUN
ing yet more the heuristic H in the no des where all free
Sparc 20 workstation with a 125 MHz CPU. The 3-SAT
variables are examined by unit propagation. Refer to
problems are generated from 3 sets of n variables and
Figure 2, when PROP is equal to PROP wede ne
z 0
m clauses at the ratio m=n =4:25, n steping from 300
w (x)asthenumb er of resolvents the newly pro duced
variables to 400 variables by50.
0
binary clauses would result in in F by a single step of
We use an executable of C-SAT dated July 1996. The
resolution. w (x) is similarly de ned.
versionofTableau used here is called 3tab and is the
[
same used for the exp erimentation presented in Craw-
1
publicly available via anonymous ftp to ftp.cis.up enn.edu in
]
pub/freeman/ directory ford and Auton, 1996 . POSIT is compiled using the pro-
not successful and enhances the b elief that if a DPL pro- S atz improved in this way solves many real-world or
cedure is ecient for random SAT problems, it should structured SAT problems where previous heuristics were
b e also ecient for a lot of structured ones. not successful. For example, Table 4 shows the p erfor-
mance of the 4 DPL pro cedures on the well-known Bei-
2
Acknowledgments jing challenging problems , where a problem that can not
b e solved in less than 2 hours is marked by"> 7200"
We thank Olivier Dub ois, James M. Crawford and Jon
3
and the version of Tableau is called ntab . It is clear
W. Freeman for kindly providing us their DPL pro ce-
that S atz is much more ecient and solves many more
dures and anonymous referees for their comments which
problems in less than two hours.
help ed improve this pap er.
Problem Satz C-SAT Posit ntab
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0
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41 31
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3141
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31
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z
[ ]
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0
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Mitchell et al., 1992 D. Mitchell, B. Selman, H.
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Pretolani, 1993 Daniele Pretolani. Satis abilityand
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2
available from http://www.cirl.uoregon.edu/crawford/b eijing
3
available from http://www.cirl.uoregon.edu/crawford/