Theoretical Aspects of Lexical Analysis/Exercise 2

Problem

Use Thompson's algorithm to build the NFA for the following regular expression. Build the corresponding DFA and minimize it.

• (a*|b*)*

Solution

The non-deterministic finite automaton (NFA), built by applying Thompson's algorithm to the regular expression (a*|b*)* is the following: <graph> digraph nfa {

    { node [shape=circle style=invis] start }
 rankdir=LR; ratio=0.5
 node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 11
 node [shape=circle,fixedsize=true,width=0.2,fontsize=10];
 start -> 0
 0 -> 1; 0 -> 11 
 1 -> 2; 1 -> 6
 2 -> 3; 2 -> 5
 3 -> 4 [label="a",fontsize=10]
 4 -> 3; 4 -> 5
 5 -> 10
 6 -> 7; 6 -> 9
 7 -> 8 [label="b",fontsize=10]
 8 -> 7; 8 -> 9
 9 -> 10
 10 -> 1; 10 -> 11
 fontsize=10
 //label="NFA for (a*|b*)*"

} </graph>

Applying the determination algorithm to the above NFA, the following determination table is obtained:

In	α∈Σ	move(In, α)	ε-closure(move(In, α))	In+1 = ε-closure(move(In, α))
-	-	0	0, 1, 2, 3, 5, 6, 7, 9, 10, 11	0
0	a	4	1, 2, 3, 4, 5, 6, 7, 9, 10, 11	1
0	b	8	1, 2, 3, 5, 6, 7, 8, 9, 10, 11	2
1	a	4	1, 2, 3, 4, 5, 6, 7, 9, 10, 11	1
1	b	8	1, 2, 3, 5, 6, 7, 8, 9, 10, 11	2
2	a	4	1, 2, 3, 4, 5, 6, 7, 9, 10, 11	1
2	b	8	1, 2, 3, 5, 6, 7, 8, 9, 10, 11	2

Graphically, the DFA is represented as follows: <graph> digraph dfa { { node [shape=circle style=invis] start } rankdir=LR; ratio=0.5 node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 0 1 2 node [shape=circle,fixedsize=true,width=0.2,fontsize=10]; start -> 0 0 -> 1 [label="a"] 0 -> 2 [label="b"] 1 -> 1 [label="a"] 1 -> 2 [label="b"] 2 -> 1 [label="a"] 2 -> 2 [label="b"] fontsize=10 //label="DFA for (a|b)*" } </graph> Given the minimization tree to the right, the final minimal DFA is: <graph> digraph dfamin { { node [shape=circle style=invis] start } rankdir=LR; ratio=0.5 node [shape=doublecircle,fixedsize=true,width=0.4,fontsize=10]; 012 node [shape=circle,fixedsize=true,width=0.2,fontsize=10]; start -> 012 012 -> 012 [label="a"] 012 -> 012 [label="b"] fontsize=10 //label="DFA for (a|b)*" } </graph>

The minimization tree is as follows. As can be seen, the states are indistinguishable. <graph> digraph mintree { node [shape=none,fixedsize=true,width=0.2,fontsize=10] " {0, 1, 2}" -> "{}" [label="NF",fontsize=10] " {0, 1, 2}" -> "{0, 1, 2}" [label=" F",fontsize=10] "{0, 1, 2}" -> "{0, 1, 2} " [label=" a,b",fontsize=10] fontsize=10 //label="Minimization tree" } </graph>