Cook-Levin Theorem

1
Cook-Levin Theorem

• Circuit Satisfiability
• A first NP-complete problem
• Reduction overview
• Example reduction

2
Circuit Satisfiability

• 5 types of gates
• Constant (T/F)
• And
• Or
• Not
• Variable
• Input size is total number of gates
• One output
• Question
• Is there an assignment of truth values to the
  variable gates that causes the output of the
  circuit to be true?

3
Reduction Overview

• We need to develop a reduction that can be
  applied to ANY problem in NP
• This leads to two key issues
• How do we represent the input for an arbitrary
  problem in NP?
• What do we know about an arbitrary problem in NP?

4
Encoding Inputs

• We assume the input is encoded as a string of 0s
  and 1s
• Example input is an undirected graph
• First specify the number of nodes in binary
• Then use an adjacency matrix to represent edges

5
Arbitrary problem in NP

• If ? belongs to NP, then there exists a
  verification algorithm A along with a set of
  certificates C
• Given any input x to ?, we construct a circuit to
  simulate the computation of A on x with the
  certificate C represented as variable gates

Circuit representing computation of A on x with
certificate C
R
6
Example

• We will show an example with the Hamiltonian Path
  Problem
• Key concept
• Representation of computation of A on x as a
  sequence of configurations
• Configuration With a configuration and the code
  for A, you should be able to complete the
  computation of A on x

7
Verification algorithm, certificate for HP

• bool HPV(graph G(V,E), path P)
• int i 1
• bool answer true
• bool usedV false / initialized to all false
  /
• if (i V) goto 6
• if (usedPi true) answer false else
  usedPi true
• if (!E(Pi,Pi1)) answer false
• i
• goto 1
• if (usedPi true) answer false
• return answer
• Certificate A path which is a permutation of the
  V nodes in the graph

8
Configurations of HPV computation

• What information should be recorded in a
  configuration of this program?
• Current instruction (PC)
• value of all variables
• Input x
• Certificate

• bool HPV(graph G(V,E), path P)
• int i 1
• bool answer true
• bool usedV false / initialized to all false
  /
• if (i V) goto 6
• if (usedPi true) answer false else
  usedPi true
• if (!E(Pi,Pi1)) answer false
• i
• goto 1
• if (usedPi true) answer false
• return answer
• Certificate A path which is a permutation of the
  V nodes in the graph

9
One computation of HPV
PC
G(V,E)
Path
i
answer
used

• bool HPV(graph G(V,E), path P)
• int i 1
• bool answer true
• bool usedV false / initialized to all false
  /
• if (i V) goto 6
• if (usedPi true) answer false else
  usedPi true
• if (!E(Pi,Pi1)) answer false
• i
• goto 1
• if (usedPi true) answer false
• return answer
• Certificate A path which is a permutation of the
  V nodes in the graph

Path P 1, 2, 3
10
One computation of HPV contd
PC
G(V,E)
Path
i
answer
used

• bool HPV(graph G(V,E), path P)
• int i 1
• bool answer true
• bool usedV false / initialized to all false
  /
• if (i V) goto 6
• if (usedPi true) answer false else
  usedPi true
• if (!E(Pi,Pi1)) answer false
• i
• goto 1
• if (usedPi true) answer false
• return answer
• Certificate A path which is a permutation of the
  V nodes in the graph

Path P 1, 2, 3
11
Second computation of HPV
PC
G(V,E)
Path
i
answer
used

• bool HPV(graph G(V,E), path P)
• int i 1
• bool answer true
• bool usedV false / initialized to all false
  /
• if (i V) goto 6
• if (usedPi true) answer false else
  usedPi true
• if (!E(Pi,Pi1)) answer false
• i
• goto 1
• if (usedPi true) answer false
• return answer
• Certificate A path which is a permutation of the
  V nodes in the graph

Path P 2, 3, 1
12
Second computation of HPV contd
PC
G(V,E)
Path
i
answer
used

• bool HPV(graph G(V,E), path P)
• int i 1
• bool answer true
• bool usedV false / initialized to all false
  /
• if (i V) goto 6
• if (usedPi true) answer false else
  usedPi true
• if (!E(Pi,Pi1)) answer false
• i
• goto 1
• if (usedPi true) answer false
• return answer
• Certificate A path which is a permutation of the
  V nodes in the graph

Path P 2, 3, 1
13
Computation represented as a Boolean Circuit

• bool HPV(graph G(V,E), path P)
• int i 1
• bool answer true
• bool usedV false / initialized to all false
  /
• if (i V) goto 6
• if (usedPi true) answer false else
  usedPi true
• if (!E(Pi,Pi1)) answer false
• i
• goto 1
• if (usedPi true) answer false
• return answer
• Certificate A path which is a permutation of the
  V nodes in the graph

1
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
1
1
1
1
0
0
1
1
1
1
Path P 2, 3, 1
14
Output of Circuit

• bool HPV(graph G(V,E), path P)
• int i 1
• bool answer true
• bool usedV false / initialized to all false
  /
• if (i V) goto 6
• if (usedPi true) answer false else
  usedPi true
• if (!E(Pi,Pi1)) answer false
• i
• goto 1
• if (usedPi true) answer false
• return answer
• Certificate A path which is a permutation of the
  V nodes in the graph

GATES
GATES
GATES
GATES
1
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
1
1
1
1
0
0
1
1
1
1
Path P 2, 3, 1
15
Constant Input Gates

• bool HPV(graph G(V,E), path P)
• int i 1
• bool answer true
• bool usedV false / initialized to all false
  /
• if (i V) goto 6
• if (usedPi true) answer false else
  usedPi true
• if (!E(Pi,Pi1)) answer false
• i
• goto 1
• if (usedPi true) answer false
• return answer
• Certificate A path which is a permutation of the
  V nodes in the graph

1
0
1
1
0
1
GATES
GATES
GATES
GATES
1
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
1
1
1
1
0
0
1
1
1
1
Path P 2, 3, 1
16
Variable Input Gates

• bool HPV(graph G(V,E), path P)
• int i 1
• bool answer true
• bool usedV false / initialized to all false
  /
• if (i V) goto 6
• if (usedPi true) answer false else
  usedPi true
• if (!E(Pi,Pi1)) answer false
• i
• goto 1
• if (usedPi true) answer false
• return answer
• Certificate A path which is a permutation of the
  V nodes in the graph

GATES
GATES
GATES
GATES
1
1
1
0
1
0
1
0
1
0
1
0
?
?
?
?
?
?
1
1
0
0
1
1
1
1
17
Answer-Preserving Nature

• The HP input below has an HP if and only if there
  exists an assignment of PATH variables in red
  such that the CIRCUIT will output 1.

GATES
GATES
1
1
1
0
1
0
1
0
1
0
1
0
?
?
?
?
?
?
1
1
1
1
18
Polynomial Time

• The number of rows is polynomial in the input
  size since A is assumed to be a polynomial time
  verification algorithm
• The number of columns is polynomial in the input
  size since the certificate must have polynomial
  size and the variables used must be polynomial in
  number
• GATES is essentially constant-sized as GATES is
  essentially independent of the input GATES
  depends mostly on the verification algorithm

GATES
GATES
1
1
1
0
1
0
1
0
1
0
1
0
?
?
?
?
?
?
1
1
1
1
19
Different sized inputs

• There is a minor issue with size
• The number of columns does change depending on
  the size of the input

20
General Case
Cert.
PC
Input
Vars

• In the general case, the input represented as a
  string of bits is transformed in an
  answer-preserving fashion into a polynomial-sized
  Boolean circuit.

GATES
Based on Verification Algorithm
GATES
0
1
0
1
0
1
0
1
0
1
1
0
1
0
1
0
1
0
1
0
?
?
?
?
?
?
1
1