Compiler Construction

1
Compiler Construction

• Sohail Aslam
• Lecture 7

2
Table Encoding of FA
b

• Transition table

a
a
0
1
2
a b
0 1 err
1 2 1
2 err err
3
Simulating FA

• trans_tableNSTATESNCHARS accept_statesNSTATE
  S
• state INITIAL
• while(state ! err)
• c input.read()
• if(c EOF ) break
• statetrans_tablestatec
• return accept_statesstate

4
Simulating FA

• trans_tableNSTATESNCHARS accept_statesNSTATE
  S
• state INITIAL
• while(state ! err)
• c input.read()
• if(c EOF ) break
• statetrans_tablestatec
• return accept_statesstate

5
Simulating FA

• trans_tableNSTATESNCHARS accept_statesNSTATE
  S
• state INITIAL
• while(state ! err)
• c input.read()
• if(c EOF ) break
• statetrans_tablestatec
• return accept_statesstate

6
Simulating FA

• trans_tableNSTATESNCHARS accept_statesNSTATE
  S
• state INITIAL
• while(state ! err)
• c input.read()
• if(c EOF ) break
• statetrans_tablestatec
• return accept_statesstate

7
Simulating FA

• trans_tableNSTATESNCHARS accept_statesNSTATE
  S
• state INITIAL
• while(state ! err)
• c input.read()
• if(c EOF ) break
• statetrans_tablestatec
• return accept_statesstate

8
Simulating FA

• trans_tableNSTATESNCHARS accept_statesNSTATE
  S
• state INITIAL
• while(state ! err)
• c input.read()
• if(c EOF ) break
• statetrans_tablestatec
• return accept_statesstate

9
RE ? Finite Automata

• Can we build a finite automaton for every regular
  expression?
• Yes, build FA inductively based on the
  definition of Regular Expression

10
NFA

• Nondeterministic Finite Automaton (NFA)
• Can have multiple transitions for one input in a
  given state
• Can have e - moves

11
Epsilon Moves

• e movesmachine can move from state A to state
  B without consuming input

e
A
B
12
NFA

• operation of the automaton is not completely
  defined by input

1
1
0
A
B
C
On input 11, automaton could be in either state
13
Execution of FA

• A NFA can choose
• Whether to make e-moves.
• Which of multiple transitions to take for a
  single input.

14
Acceptance of NFA

• NFA can get into multiple states
• Rule NFA accepts if it can get in a final state

1
1
0
A
B
C
0
15
DFA and NFA

• Deterministic Finite Automata (DFA)
• One transition per input per state.
• No e - moves

16
Execution of FA

• A DFA
• can take only one path through the state graph.
• Completely determined by input.

17
NFA vs DFA

• NFAs and DFAs recognize the same set of languages
  (regular languages)
• DFAs are easier to implement table driven.

18
NFA vs DFA

• For a given language, the NFA can be simpler than
  the DFA.
• DFA can be exponentially larger than NFA.

19
NFA vs DFA

• NFAs are the key to automating RE ? DFA
  construction.

20
RE ? NFA Construction

• Thompsons construction (CACM 1968)
• Build an NFA for each RE term.
• Combine NFAs with e-moves.

21
RE ? NFA Construction

• Subset construction NFA ? DFA
• Build the simulation.
• Minimize number of states in DFA (Hopcrofts
  algorithm)

22
RE ? NFA Construction

• Key idea
• NFA pattern for each symbol and each operator.
• Join them with e-moves in precedence order.

23
RE ? NFA Construction
a
s0
s1
NFA for a
e
a
b
s0
s1
s3
s4
NFA for ab
24
RE ? NFA Construction
a
s0
s1
NFA for a
25
RE ? NFA Construction
a
s0
s1
NFA for a
b
s3
s4
NFA for b
26
RE ? NFA Construction
a
s0
s1
NFA for a
b
s3
s4
NFA for b
a
b
s0
s1
s3
s4
27
RE ? NFA Construction
a
s0
s1
NFA for a
b
s3
s4
NFA for b
e
a
b
s0
s1
s3
s4
NFA for ab
28
RE ? NFA Construction
a
s1
s2
e
e
s0
s5
b
e
s3
s4
e
NFA for a b
29
RE ? NFA Construction
a
s1
s2
NFA for a
30
RE ? NFA Construction
a
s1
s2
b
s3
s4
NFA for a and b
31
RE ? NFA Construction
a
s1
s2
e
e
s0
s5
b
e
s3
s4
e
NFA for a b
32
RE ? NFA Construction
e
a
e
e
s1
s0
s4
s2
e
NFA for a
33
RE ? NFA Construction
a
s1
s2
NFA for a
34
RE ? NFA Construction
e
a
e
e
s1
s0
s4
s2
e
NFA for a
35
Example RE ? NFA
NFA for a ( bc )
e
b
s4
s5
e
e
e
e
a
e
s8
s0
s1
s2
s3
s9
c
e
s6
s7
e
e
36
Example RE ? NFA
building NFA for a ( bc )
a
s0
s1
37
Example RE ? NFA
NFA for a, b and c
b
s4
s5
a
s0
s1
c
s6
s7
38
Example RE ? NFA
NFA for a and bc
b
s4
s5
e
e
a
s8
s0
s1
s3
c
e
s6
s7
e
39
Example RE ? NFA
NFA for a and ( bc )
e
b
s4
s5
e
e
e
a
e
s8
s0
s1
s2
s3
s9
c
e
s6
s7
e
e
40
Example RE ? NFA
NFA for a ( bc )
e
b
s4
s5
e
e
e
e
a
e
s8
s0
s1
s2
s3
s9
c
e
s6
s7
e
e