A = { G : G is a connected undirected graph}
This is called an encoding
Typically for graphs G = (V, E)
A = { G : G is a connected undirected graph}
M=
“On input <G>
1. Mark an arbitrary vertex
2. Repeat until no new vertices are marked
• For each vertex
• Mark it if it is connected by an edge to a
marked vertex
3. If all vertices are marked then accept else reject
Decider
ADFA = { D, w : D is a DFA that accepts input w}
ANFA = { N, w : N is an NFA that accepts input w}
emptiness testing
EDFA = { D : D is an DFA and L(D) = ∅}
emptiness testing
EDFA = { D : D is an DFA and L(D) = ∅}
Try #1
MEDFA =
“On input D
1. If D has any accept states
reject else accept”
A = { G : G is connected undirected graph}
M=
“On input <G>
1. Mark an arbitrary vertex
2. Repeat until no new vertices are marked
• For each vertex
• Mark it if it is connected by an edge to a
marked vertex
3. If all vertices are marked then accept else reject
equivalence testing
EQDFA = { D1, D2 : D1, D2 are DFAs
and L(D1) = L(D2)}
equivalence testing
EQDFA = { D1, D2 : D1, D2 are DFAs
and L(D1) = L(D2)}
Try #1
MEQDFA =
“On input D1, D2
1. For every w ∈ Σ* If D1, D2 either both accept or both reject
accept else reject”
equivalence testing
EQDFA = { D1, D2 : D1, D2 are DFAs
and L(D1) = L(D2)}
idea: use emptiness testing of the
symmetric difference of L(D1) and L(D2)
ATM = { M, w : M is a TM that accepts input w}
U=
“On input M, w
1. Simulate M on w
2. If M ends in an accept state, accept
else if it ends in a reject state reject”
L(U) = ATM
Decider or Recognizer?
ATM = { M, w : M is a TM that accepts input w}
is undecidable
ATM = { M, w : M is a TM that accepts input w}
is undecidable
Proof by contradiction
ATM = { M, w : M is a TM that accepts input w}
is undecidable
Suppose ATM is decidable.
Then there is a TM H that decides ATM.
H( M, w ) =
{
accept
reject
iff M accepts w
iff M does not accept w
ATM = { M, w : M is a TM that accepts input w}
is undecidable
Construct TM D that uses TM H as a subroutine:
D=
“On input M where M is a TM
1. Run TM H on M, M
2. Output the opposite of H
a) If H ends in an accept state, reject
b) If H ends in a reject state, accept”
ATM = { M, w : M is a TM that accepts input w}
is undecidable
reject
iff M accepts
M
accept
iff M does not accept
M
{
D( M ) =
ATM = { M, w : M is a TM that accepts input w}
is undecidable
D( D ) =
{
reject
iff D accepts
D
accept
iff D does not accept
D
Contradiction!