FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY

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15-453
FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY
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(No Transcript)
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Theorem L ? NP ? if there exists a poly-time
Turing machine V(erifier) with
L x ?y(witness) y poly(x) and V(x,y)
accepts
Proof

• If L x ?y y poly(x) and V(x,y)
  accepts
• then L ? NP

Because we can guess y and then run V
(2) If L ? NP then L x ?y y poly(x)
and V(x,y) accepts
Let N be a non-deterministic poly-time TM that
decides L and define V(x,y) to accept if y is an
accepting computation history of N on x
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A language is in NP if and only if there exist
polynomial-length certificates for membership to
the language
SAT is in NP because a satisfying assignment is a
polynomial-length certificate that a formula is
satisfiable
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NP all the problems for which once you have the
answer it is easy (i.e. efficient) to verify
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P NP?
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If P NP
Mathematicians would be out of a job
Cryptography as we know it would not be possible
(e.g. RSA)
AI program become perfect as exhaustive search is
efficient
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If P NP
Generation is as easy as recognition
Being a chef is as easy as eating.
Writing symphonies is as easy as listening to
them.
Writing Shakespeare is as easy as recognizing
Shakespeare.
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POLY-TIME REDUCIBILITY
A language A is polynomial time reducible to
language B, written A ?P B, if there is a
polynomial time computable function f S ? S,
where for every w,
w ? A ? f(w) ? B
f is called a polynomial time reduction of A to B
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NP Complete Problem hardest problem in NP

• Intuitively, L is harder than L if L is
  polynomial reducible to L.
• NP Complete Problem If such a problem has an
  efficient algorithm (in P), then every other
  problem also has efficient algorithm.

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Theorem (Cook-Levin) SAT is NP-complete
Corollary SAT ? P if and only if P NP
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Any thing in NP ?P 3SAT
NP
P
SAT
You can think of -gt as easier than. SAT is the
hardest problem in NP.
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NP-Complete Problems

• 3SAT
• k-Clique
• Vertex Cover
• Independent Set

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3SAT

• Problem Given a CNF where each clause has 3
  variables, decide whether it is satisfiable or
  not.
• (x1 ? x1 ? x2) ? (?x1 ? ?x2 ? ?x2) ? (?x1 ? x2 ?
  x2)

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3SAT is NP Complete
3SAT
? ?y such that y is a satisfying assigment to
? and ? is in 3cnf
How about 2SAT?
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What we want to prove?
3SAT
NP
P
SAT
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How to prove?

• We can convert (in polynomial time) a given SAT
  instance S into a 3SAT instance S such that
• If S is satisfiable, then S is satisfiable.
• If S is satisfiable, then S is satisfiable.

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Key Observation
x1 ? x2 ? x3 ? x4 ? x5 is satisfiable if and
only if (x1 ? x2 ? z1)? ( ? z1 ? x3 ? z2 )? (?
z2 ? x4 ? z3)? (? z3 ? x4 ? x5) is satisfiable.
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Polynomial Time Reduction

• Clauses in 3SAT

• Clause in SAT

• x1 ? x1 ? x1
• x1 ? x1 ? ? x2
• x1 ? x2 ? x3
• (x1 ? x2 ? z1)? ( ? z1 ? x3 ? x4)
• (x1 ? x2 ? z1)? ( ? z1 ? x3 ? z2)?
• (? z2 ? x4 ? z3)? (? z3? x4 ? x5)

• x1
• x1 ? ? x2
• x1 ? x2 ? x3
• x1 ? x2 ? x3 ? x4
• x1 ? x2 ? x3 ? x4 ? x5

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CLIQUE
k-clique complete subgraph of k nodes
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K-Cliques
A K-clique is a set of K nodes with all K(K-1)/2
possible edges between them
This graph contains a 4-clique
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Assume a reasonable encoding of graphs (example
the adjacency matrix is reasonable)
CLIQUE (G,k) G is an undirected graph
with a k-clique
Theorem CLIQUE is NP-Complete
(1) CLIQUE ? NP
(2) 3SAT ?P CLIQUE
Brute Force Algorithm Try out all n choose k
possible locations for the k clique
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CLIQUE is NP-Complete
NP
P
3SAT
CLIQUE
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3SAT ?P CLIQUE
We transform a 3-cnf formula ? into (G,k) such
that
? ? 3SAT ? (G,k) ? CLIQUE
The transformation can be done in time that is
polynomial in the length of ?
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3SAT ?P CLIQUE
We transform a 3-cnf formula ? into (G,k) such
that
? ? 3SAT ? (G,k) ? CLIQUE
If ? has m clauses, we create a graph with m
clusters of 3 nodes each, and set km Each
cluster corresponds to a clause. Each node in a
cluster is labeled with a literal from the clause.
We do not connect any nodes in the same cluster
We connect nodes in different clusters whenever
they are not contradictory
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(x1 ? x1 ? x2) ? (?x1 ? ?x2 ? ?x2) ? (?x1 ? x2 ?
x2)
clause
k clauses
nodes 3( clauses)
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(x1 ? x1 ? x1) ? (?x1 ? ?x1 ? x2) ? (x2 ? x2 ?
x2) ? (?x2 ? ?x2 ? x1)
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Independent Set
An independent set is a set of nodes with no
edges between them
This graph contains an independent set of size 3
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Independent Set
G(V,E)
Problem Given a graph G and k, is there a size k
independent set?
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Complement of G
Given a graph G, let G, the complement of G, be
the graph such that two nodes are connected in
G if and only if the corresponding nodes are not
connected in G
G
G
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Key Observation

• For a graph G, vertex set S is an independent set
  if and only if S is clique in G.

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Let G be an n-node graph
Is there size clique in G?
Is there size k independent set in G?
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VERTEX COVER
vertex cover set of nodes that cover all edges
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Vertex Cover is NP Complete

• Given a Graph G(V,E), decide if there is k
  vertex such that every edge is covered by one of
  them?

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K-Indep Set ?P Vertex Cover
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Key Observation

• For a graph G(V,E), S is a independent set if and
  only if V-S is a vertex cover.

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Other NP-Complete Problems

• Travelling Salesman Problem
• Hamiltonian Path
• Max Cut
• Subset Sum
• Integer Programming
• .

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Other Problems in NP

• Graph Isomorphism
• Factoring Number
• We dont know if they are NP Complete or not.