Compiler Construction Principles

1
Compiler Construction Principles Implementation
Techniques

• Dr. Ying JIN
• Associate Professor
• Oct. 2007

2
Implementation of DFA
3
Implementation of DFA

• Objective (meaning of implementing a DFA)
• Given a DFA which defines rules for a set of
  strings
• Develop a program, which
• Read a string
• Check whether this string is accepted by the DFA
• If a string is accepted by a DFA,
• Next state
• Stop in the final state
• If a string is not accepted by a DAF,
• No next state (?)
• Not stop in the final state

4
Implementation of DFA

• Two ways
• Basing on transforming table of DFA
• Basing on graphical representation of DFA

5
Transforming Table based Implementation

• Main idea
• Input a string
• Output true if acceptable, otherwise false
• Data structure
• Transforming table (two dimensional array T)
• Two variables
• CurrentState record current state
• CurrentChar record current character that is
  read in the string

6
Transforming Table based Implementation

• Main idea
• General Algorithm
• 1.CurrentState S0
• 2. read the first character as
  CurrentChar
• 3. if CurrentChar is not the end of the
  string,
• if T(CurrentState,CurrentChar)?
  error,
• CurrentState
  T(CurrentState,CurrentChar),
• read next character of
  the string as CurrentChar,
• goto 3
• 4. if CurrentChar is the end of the
  string and CurrentState is one of the
• terminal states, return true
  otherwise, return false.

7
Example
a b c d
S0 S1 ? S2 S
S1 ? S1 ? S2
S2 S ? ? ?
S ? ? S ?
1) abbacc true
2) cab
false
8
Program structure for Table-based Implementation
Transforming table for the DFA
Variables CurrentChar, CurrentState
Read the string that want to be checked
Checking process
9
Graph based Implementation of DFA

• each state corresponds to a case statement
• each edge corresponds to a goto statement
• for accept state, add one more branch, if current
  char is the end of the string then accept

Li case CurrentChar of a
goto Lj b goto Lk
other Error( )
Li case CurrentChar of a
goto Lj b goto Lk
return true
other Error( )
i
10
LS0 read character to CurrentChar case
CurrentChar of a goto LS1
c goto LS2 d goto LS3
other return false
LS1 read character to CurrentChar case
CurrentChar of b goto LS1 d
goto LS2 other return false
11
LS2 read character to CurrentChar case
CurrentChar of a goto LS3
other return false
LS3 read character to CurrentChar
case CurrentChar of c goto LS2
return true other return
false
12
Definition of NFA
13
Formal Definition

• (?,SS, SS0, ?, TS)
• ?(alphabet),set of allowed characters, each
  character can be called as input symbol
• SS S0, S1, S2, ,a finite set, each element
  is called state
• S0? SS, set of start states
• ? SS ? ? ? power set of SS ? ? , transforming
  function
• TS?SS, set of terminal (accept) states
• Note? is a function which accepts a state and a
  symbol and returns a set of states or ?(no
  definition)

14
Differences between DFA NFA
DFA NFA
Start state One start state Set of start states
? ? ?
T (S, a) S or ? S1, , Sn or ?
implementation easy Non-deterministic
15
Example of NFA

• a, b, c, d
• SS S0, S10, S2, S
• Set of Start state S0 , S10
  
• Set of terminal states S
• (S0,a)? S10, S,(S0,?)? S2,
• (S10,b)?S10, (S10, ?)?S2,
• (S2, ?)?S,
• (S, c)?S

S10
S2
16
From NFA to DFA
17
Main Idea

• Solve two problems
• ? edge
• ?-closure (SS)
  ???
• Merging those edges with the same symbol
• NextStates(SS, a)
• Conversion of NFA to DFA
• Using a set of states in NFA as one state in DFA
• Assuring accepting the same set of strings

18
The process to calculate ?-closure (???)

• For a given NFA A, and a set of states SS,
• ?-closure(SS) SS
• If there exists a state s in SS, which has a
  ?-edge referring to a state s and
  s??-closure(SS), add to s to ?-closure(SS)
• Repeat until there is no state having ?-edge to
  states that is not in ?-closure(SS)

19
?-closure (???) -- Example
?-closure(S0, S10) ? S0, S10 ? S0,
S10, S2 ? S0, S10, S2 ? S0, S10, S2,S
S10
S2
20
Moving States

• For a given set of states SS and a symbol a in a
  NFA A,
• NextStates(SS, a) s if there is a state
  s1?SS, and a
• edge s1 s in A

a
21
Moving States
NextStates(S0, S10, a) S10, S
S10
S2
NextStates(S0, S10, b) S2
22
Algorithm

• Given a NFA A ?, SS, SS0, ?, TS
• Generating an equivalent DFA A ?, SS,S0, ?,
  TS
• Steps
• (1) S0 ?-closure(SS0), add S0 to SS
• (2) select one state s from SS, for any symbol
  a??,
• let s NextStates(?-closure(s), a),
• add (s, a) ? s to ?,
• if s?SS, add s to SS
• (3) repeat (2) until all states are handled
• (4) for a state s in SS, s S1, .., Sn, if
  there exists Si?TS, then s is an accept state in
  A, add s to TS

23
Example

• a, b, c,

S0 ?-closure(S0, S10) S0, S10,
S2,S ,
S10
a b c
S0, S10, S2,S S10, S,S2 S10, S,S2 S
S10, S,S2 S10, S,S2 S
S S
S2
24
Minimizing DFA
25
Problem

• Equivalent of two DFAs
• If the set of strings accepted by two DFAs are
  the same
• Among those DFAs that accept the same set of
  strings, the minimal DFA refers to the one that
  has minimal number of states

How this happens?
26
Equivalent DFAs
S1
S1
S2
ad, bd
There are states that accepting the same set of
strings!
27
Main Idea

• Equivalent states(????)
• For two states s1 and s2 in a DFA, if treat s1
  and s2 as start states and they accept the same
  set of strings, s1 and s2 will be called
  equivalent states
• Two ways to minimizing DFA
• Merging equivalent states (????)
• Splitting non-equivalent states(????)

28
Algorithm

• Given a DFA A ?, SS, S0, ?, TS
• Generating an equivalent DFA A ?, SS,S0,
  ?, TS
• Splitting Steps
• (1) two groups non-terminal states, terminal
  states
• (2) select one group of states SSi Si1,,
  Sin,
• replace SSi with split(SSi)
• (3) repeat (2) until all groups are handled
• (4) SS set of groups
• (5) S0 is the group consisting of S0
• (6) if the group consisting of terminal states of
  A, it is terminal state of A
• (7) ? SSi SSj , if there is Si
  Sj in A, Si?SSi, Sj?SSj

a
a
29
Splitting a Set of States

• Given
• a NFA A ?, SS, S0, ?, TS
• Groups of states SS1, , SSm, SS1? ?SSm SS
  
• SSi Si1,, Sin,
• split(SSi) is to split SSi into two group G1 and
  G2,
• For j 1 to n
• for any a??,
• If (Si1,a ) ? Sk ? (Sij, a) ?Sl ? Sk and Sl
  belong to the same group SSp , add Sij to G1
• Otherwise, add Sij to G2

30
Simple Example
S0, S1, S2, S3, S4
S1
S0 , S1, S2, S3, S4
S2
31
Assignment

• Define a DFA for accepting a set of binary
  numbers, each binary number can be divided by 4
• (????DFA,???????????4???????)
• Implementation of the DFA above

32
Assignment

• From NFA to DFA, and minimize it

?
S1
S2