Restricted Satisfiability (SAT)
Problem
By
Mutaz Flmban
Out line
•CSAT & 3SAT
•Normal Forms for Boolean Expression
•Converting Expression to CNF
•Theorem 10.12
•NP-Completeness of CSAT
•Theorem 10.13
•NP-Completeness of 3SAT
•Theorem 10.15
CSAT & 3SAT
CSAT: an intermediate step that is used to
show that SAT can be reduced to 3SAT
3SAT: is a problem about satisfiability of
boolean expressions, these expression have a
very regular form:
•they are the AND of “clauses”.
•each clause is the OR of exactly three
variables or negated variables.
OUR GOAL
Reduce satisfiability for any
expression to satisfiability for
expressions in the normal form for
the 3SAT problem.
Convert each expression E in SAT
to another expression F in the
normal form for 3SAT.
Note: F is satisfiability if and only
if E is satisfiability.
Normal Forms for Boolean
Expression
essential definitions
A literal: is either a variable, or a negated
e.g. ¬x, x
A clause: is the logical OR of one or more
literals.
e.g. x , x ˅ ¬y ˅ z.
A boolean expression: is said to be in conjunctive
normal form or CNF, if it is the AND of clauses.
Important notation
OR () is treated as a Sum, using the (+)
operator.
AND () is treated as a product.
normally use juxtaposition (no operator).
Not = (¬)
Example
The expression (x ˅ ¬y) ˄ (¬x ˅ z):
will be written in CNF as (x + ¬y)(¬x + z)
It is in conjunctive normal form, since it is the
AND (product) of the clauses:
(x + ¬y) and (¬x + z).
Question?
are the expressions
(x + y¬z)(x + y + z)(¬y + ¬z), xyz in CNF?
The first expression not in CNF.
It is the AND of three subexpressions, the last two are
clauses, but the first is not, it is the sum of a literal and a
product of two literals.
The second expression in CNF.
the clause can have only one literal, so it is the product
of three clauses, (x ), (y), and (z)
k-Conjunctive Normal Form
(k-CNF)
(K-CNF) is an expression that contains a product
of clauses. Each of which is the sum of exactly k
distinct literals.
•For example:
(x + ¬y)(y + ¬z)(z + ¬x) is in 2-CNF
because each of its clauses has exactly two
literals.
CSAT and KSAT
• CSAT is a Boolean expression in CNF that is
satisfiable.
•KSAT is a Boolean expression in k-CNF that is
satisfiable
Some problems
CSAT is the problem: Given a Boolean
expression in CNF, is it satisfiable?
kSAT is the problem: Given a Boolean
expression in k-CNF, is it satisfiable?
We shell see that CSAT, 3SAT
and KSAT for all K higher than 3
are NP-Complete
Converting Expression to CNF
We will take a SAT instance E and
convert it to a CSAT instance F such that
F is satisfiable if and only if E is
statisfiable.
• The reduction of SAT to CSAT includes
two parts:
Reduction of SAT to CSAT
•The two parts are:
We push all ¬’s down the expression tree so that
the only negations are of variables, the boolean
expression becomes an AND and OR of literals.
ii. Write the Boolean expressions in CNF form, we
will get new variables.
•The new expression F will not be equivalent to the
expression E.
•If T is a truth assignment that makes E true, then there is
an extension of T , say S, that makes F true.
•S is an extension of T if S assigns the same value as T to
each variable that T assigns.
i.
DeMorgan’s Laws
I. ¬(E  F) ⇒ ¬(E)  ¬(F)
to push ¬ below , the  is
changed to an .
II. ¬(E  F) ⇒ ¬(E)  ¬(F)
to push ¬ below , the  is
changed to an .
III. ¬(¬E) ⇒ E : the double negation.
Example
E = ¬((¬(x + y))(¬x + y))
¬((¬(x + y))(¬x + y))
Start
¬(¬(x + y))+¬(¬x + y)
(1)
x + y + ¬(¬x + y)
(3)
x + y + (¬(¬x)) ¬y
(2)
x + y + x ¬y
(3)
Theorem 10.12
Every Boolean expression E is
equivalent to an expression F in which
the only negations occur in literals.
the length of F is linear in the number of
symbols of E, and F can be constructed
from E in polynomial time.
The PROOF
•The proof is an induction on the number of
operators (¬,  and ) in E.
•There is an equivalent expression F with ¬’s
only in literals.
•If E has n ≥ 1 operators, then F has no more
than 2n – 1 operators.
continue
• F need not have more than one pair of parentheses
per operator.
• The number of variable in an expression cannot
exceed the number of operators by more than one.
the length of F is linearly proportional to the
length of E.
the time it takes to construct F is proportional
to its length and the length of E.
BASIS
•If E has one operator, it must be of the form:
• ¬x
•xy
•xy
•In each case, E is already in the required form, so F = E.
•Note, E and F each have one operator:
•The relationship “F has at most twice the number of
operators of E minus 1”.
induction
Suppose the statement is true for all expressions
with fewer operations than E.
if the highest operator of E is not “¬”.
Then E must be of the form:
E1  E2
E1  E2
In either case the induction hypothesis applies to
E1 and E2.
The inductive hypothesis
 There are equivalent expressions F1 and F2,
respectively, in which all ¬’s occur in literals only.
F = F1  F2 or F = F1  F2
 Let E1 and E2 have a and b operators, respectively.
E has a + b + 1 operators
 By inductive hypothesis:
F1 and F2 have 2a – 1 and 2b – 1
F has at most 2a + 2b – 1 operators.
 Which is no more then 2(a+b+1)-1, or twice the
number of operators of E minus 1.
Induction (continued)
consider the case where E is of the form ¬ E1
There are three cases depending on what the
top operator of E1 is:
 E1 = ¬E2
 E1 = E2  E3
 E1 = E2  E3
The case “E1 = ¬E2”
By the law of double negation, E = ¬(¬E2)
E2 has fewer operators than E.
By the inductive hypothesis: there is an
equivalent F for E2 in which the only ¬’s are in
literals.
The number of operators of F is at most twice the
number in E2 minus 1, and it is surely true for E.
The Case “E1 = E2  E3”
By DeMorgan’s Law: E= ¬(E2  E3) = ¬E2  ¬E3
Both (¬E2) and (¬E3) have fewer operators than E.
By the inductive hypothesis: they have an equivalents F2
and F3 that have ¬’s only in literals.
F = F2  F3 = E
Let (E2) has (a) operators, and (E3) has (b) operators.
(E) has (a+b+2) operators.
Since ¬(E2) and ¬(E3) have (a+1) and (b+1) operators,
and (F2) has at most 2(a+1)-1 operators, (F3) has at most
2(b+1)-1 operators.
(F) has at most 2a+2b+3 operators. And this number
is twice the number of operators of E minus 1.
The Case “E1 = E2  E3”
This argument, using the second of
DeMorgan’s laws, is essentially the
same as the case (2).
Theorem 10.13
CSAT is NP-Complete
PROOF:
We show how to reduce SAT to CSAT in polynomial
time:
 Using method 10.12 to convert a SAT to an expression E
whose ¬’s are only in literals.
 Convert E to a CNF expression F in polynomial time.
 F is satisfiable if and only if E is.
BASIS
If E consists of one or two symbols,
then it is a literal.
A literal is a clause, so E is already
in CNF.
Induction
Assume that every expression shorter than E
can be converted to a product of clauses.
And conversion takes at most cn2 time on an
expressions of length n.
There are two cases depending on the top-level
operator of E:
E = E1  E2
E = E1  E2
Case 1: E = E1  E2
By the inductive hypothesis:
There are expressions F1 and F2 derived from E1 and E2 respectively in CNF
 All and only the satisfying assignments for E1 can be extended to a
satisfying assignment for F1.
 All and only the satisfying assignments for E2 can be extended to a
satisfying assignment for F2.
Let F = F1  F2 where:
F1  F2 is a CNF expression if F1 and F2 are CNF expressions
We must show that a truth assignment T for E can be
extended to a satisfying assignment for F if and only if T
satisfies E.
First Part (IF)
Suppose T satisfies E:
Let T1 be T restricted so it applies only to the
variables that appear in E1.
Let T2 be T restricted so it applies only to the
variables that appear in E2.
By the inductive hypothesis: T1 and T2 can be extended
to assignments S1 and S2 that satisfy F1 and F2.
Let S agree with S1 and S2 on each of the variables they
define.
We can construct S that is an extension of T that
satisfies F.
Second Part (only if)
Suppose that T has an extension S that satisfies F.
Let T1 be T restricted to the variables of E1.
Let T2 be T restricted to the variables of E2.
Let S1 be S restricted to the variables of F1
Let S2 be S restricted to the variables of F2
Then S1 is an extension of T1, and S2 is an extension of T2.
Because F is the AND of F1 and F2
then S1 satisfies F1 and S2 satisfies F2.
By the inductive hypothesis:
T1 must satisfy E1 and T2 must satisfy E2
T satisfies E.
Case 2: E = E1  E2
By the inductive hypothesis: there are CNF expressions
F1 and F2 with the properties:
•A truth assignment for E1 satisfies E1 if and only if it can
extended to a satisfying assignment for F1.
•A truth assignment for E2 satisfies E2 if and only if it can
extended to a satisfying assignment for F2.
•The variables of F1 and F2 are disjoint, except for those
variables that appear in E.
•F1 and F2 are in CNF
Note: we cannot simply take the OR of F1 and F2 to
construct F, because the resulting expression would not
be in CNF.
Continued
Suppose F1= g1  g2  ..  gp. And F2= h1  h2  ..  hq
Where g’s and h’s are clauses, also introduce a new
variable y, and let:
F=(y+g1)*(y+g2)*…*(y+gp) * (¬y+h1) * (¬y+h2)
* … *(¬y+hq)
We want to proof that a truth assignment T for E satisfies
E if and only if T can be extended to a truth assignment
S that satisfies F.
Case 2: (If) Part
Assume T satisfies E.
•Let T1 be T restricted to the variables of E1.
•Let T2 be T restricted to the variables of E2.
•Since E = E1  E2, either T satisfies E1 or T satisfies
E2.
•Let us assume T satisfies E1. then T1 which is T
restricted to the variables of E1, can be extended to S1,
which is satisfies F1.
Case 2: If Part
Construct an extension S for T, S will satisfy the
expression F as follows:
1) for all variables x in F1, S(x) = S1(x).
2) S(y) = 0 , this makes all the clauses of F that are derived
from F2 true.
3) For all variables x that are in F2 but not in F1, S(x) can be
either 0 or 1.
S makes all the clauses derived from the g’s true
because the rule 1.
S makes all the clauses derived from the h’s true by
rule 2.
S satisfies F.
Case 2: (Only-If) Part
Suppose that truth assignment T for E is
extended to truth assignment S for F and S
satisfies F.
T satisfies E whenever S satisfies F.
NP-Completeness of 3SAT
The problem 3SAT is:
Given a boolean expression E that is the
product of clauses, each of which is the sum of
three distinct literals, is E satisfiable?
Theorem 10.15
3SAT is NP-Complete
Proof:
3SAT is in NP, since SAT is NP
To prove NP-Completeness we shell Reduce CSAT
to 3SAT.
PROOF
Given a CNF expression E = e1  …  ek, replace each
clause ei to create a new expression F
•If ei is a single literal (x), we introduce new variables u
and v. Replace (x) by the four clauses:
(x+u+v)(x+u+ ¬v)(x+¬u+v)(x+¬u+¬v)
•If ei is the sum of two literals, (x+y). Introduce new
variable z. Replace (x+y) by the product of to clauses:
(x+y+z)(x+y+¬z)
•If ei is the sum of 3 literals, already in the form
required.
•If ei = (x1+x2+…+xm) for some m ≥ 4, introduce new
variables y1, y2, …, ym-3. then replace ei by the product
of clauses.
(x1+x2+y1)(x3+¬y1+y2)(x4+¬y2+y3)…(xm-2+¬ym-4+ym-3)(xm1+xm+ym-3)
continue
Thus, each instance of E of CSAT can be
reduced to an instance of F of 3SAT such
that F is satisfiable if and only if E is
satisfiable.
Construction time is linear in length of E.
 Since CSAT is NP-Complete, 3SAT is
NP-complete.
Reference
John E. Hopcroft, Rajeev Motwani and
Jeffrey D. Ullman, Introduction to Automata
Theory, Languages, and Computation
Questions
1. How we can reduce the SAT to the CSAT?
We push all ¬’s down the expression tree.
Write the Boolean expressions in CNF form.
2. Show how to convert ¬((¬(x + y))(¬x + y)) to CNF form
¬((¬(x + y))(¬x + y))
¬(¬(x + y))+¬(¬x + y)
x + y + ¬(¬x + y)
x + y + (¬(¬x)) ¬y
x + y + x ¬y