The difference between Recognizable and decidable

1
The difference -- between Recognizable and
decidable

• If L is decidable, then L is recognized by a TM M
  that halts on all inputs. Note that, L might be
  recognized by other TM M that does not always
  halt.
• If L is recognizable, then there might be such TM
  M that recognizes L but run forever, rather than
  rejecting, some inputs not in L.
• Simply, Decidable ---- always halt
• Recognizable ---- halt or loop

2

• Let INFINITEDFAltAgt A is a DFA and L(A) is an
  infinite language .
• Show that INFINITEDFA is decidable.

3

• IOn input ltAgt where A is a DFA
• 1.Let k be the number of states of A
• 2.Construct a DFA D that accepts all strings of
  length k or more
• 3Construct a DFA M such that L(M)L(A) intersect
  L(D)
• 4Test L(M)null,using the EDFA decider T from
  Theorem 4.4
• 5 If T accepts,rejectif Trejects,accept

4

• The DFA accepts some string of length k or
  more,where k is the number of states of the
  DFA,This string may be pumped in the pumping
  lemma for regular language to obtain infinitely
  many accepted string,

5

• Let AltMgt M is a DFA which doesnt accept any
  string containing an odd numbers of 1s .
• Show that A is decidable.

6

• IOn input ltMgt where M is a DFA
• 2.Construct a DFA O that accepts every string
  containing an odd numbers of 1s
• 3Construct a DFA B such that L(B)L(M) intersect
  L(O)
• 4Test L(B)null, using the EDFA decider T from
  Theorem 4.4
• 5 If T accepts,acceptif T rejects,reject

7

• EDFA ltAgtA is a DFA and L(A)null
• EDFA is a decidable language
• AltM,wgtM is a TM and M accepts w
• A is undecidable
• The halting problem