Regular Languages: Deterministic Finite Automata DFA

1
Regular Languages Deterministic Finite Automata
(DFA)
a
aa abab abaaaabbb
0
1
a
b
b
b
b
a
3
2
a
w ?a,b aW and aW both even
DFA NFA (Non-deterministic) REG
2
Learning DFAs 2) MCA(S)
S (c, abc, ababc
Maximal Canonical Automaton
a
a
b
b
c
0
1
2
3
4
5
a
b
c
6
7
8
c
9
3
Learning DFAs 4) State Merging (Oncina,Vidal
92, Lang 98)
S (c, abc, ababc
Evidence Driven State merging 0,2
a
a
b
b
c
0
1
2
3
4
5
c
8
c
9
4
Learning DFAs 4) State Merging (Oncina,Vidal
92, Lang 98)
S (c, abc, ababc
Evidence Driven State merging 1,3 and 8,9
a
a
b
c
0
1
3
4
5
c
b
8
c
9
5
Learning DFAs 4) State Merging (Oncina,Vidal
92, Lang 98)
S (c, abc, ababc
Evidence Driven State merging 0,4
a
b
c
0
1
4
5
b
c
9
6
Learning DFAs 4) State Merging (Oncina,Vidal
92, Lang 98)
S (c, abc, ababc
Evidence Driven State merging 9,5
a
c
0
1
5
b
c
9
7
Learning DFAs 4) State Merging (Oncina,Vidal
92, Lang 98)
S (c, abc, ababc
Evidence Driven State merging 0,4 and 9,5
a
0
1
b
c
9
8
Learning DFAs via evidence driven state merging

• Input S, S-
• Output DFA
• 1) Form MCA(S)2) Form PTA(S)3) Do until no
  merging is possible - choose merging of two
  states - perform cascade of forced mergings to
  get a deterministic automaton - if resulting DFA
  accepts sentences of S- backtrack and choose
  another couple4) End
• Drawback we need negative examples!!!

9
Learning DFA using only positive examples with MDL
S (c, cab, cabab, cababab, cababababab
a
a
L1
L2
c
0
0
1
2
c
b
b
Coding in bits L1 ? 5 log2 (31) 2 log2 (11)
20 L2 ? 5 log2 (31) 2 log2 (13) 40
arrows
letters
Empty letter
states
Outside world
10
Learning DFA using only positive examples with MDL
S (c, cab, cabab, cababab, cababababab
a
a
L1
L2
c
0
0
1
2
c
b
b
Coding in bits L1 ? 5 log2 (31) 2 log2 (11)
20 L2 ? 5 log2 (31) 2 log2 (13) 40
But L1 has 4 choices in state 0 L1(S) 26
log2 4 52 L2 has 2 choices in state 1
L2(S) 16 log2 2 16 L2 L2(S) 40
16 lt L1 L1(S) 20 52 L2 is the better
theory according to MDL
11
Learning DFAs using only positive examples with
MDL

• Input S
• Output DFA
• 1) Form MCA(S)2) Form DFA PTA(S)3) Do until
  no merging is possible - choose merging of two
  states - perform cascade of forced mergings to
  get a deterministic automaton DFA - if
  DFADFA(S) ? DFADFA(S) backtrack and
  choose another couple4) End
• Drawback Local minima!