Difference between revisions of "Theoretical Aspects of Lexical Analysis/Exercise 5"

Latest revision as of 22:58, 11 February 2019

Consider the lexical analyzer G = { ab, ab*, a|b }, defined for the alphabet Σ = { a, b }.

Compute the non-deterministic finite automaton (NFA) by using Thompson's algorithm. Compute the minimal deterministic finite automaton (DFA).

Indicate the number of processing steps for the abaabb input string.

NFA

The following is the result of applying Thompson's algorithm. State 3 recognizes the first expression (token T1); state 8 recognizes token T2; and state 14 recognizes token T3.

DFA

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

I_n	α∈Σ	move(I_n, α)	ε-closure(move(I_n, α))	I_n+1 = ε-closure(move(I_n, α))
-	-	0	0, 1, 4, 9, 10, 12	0
0	a	2, 5, 11	2, 5, 6, 8, 11, 14	1 (T2)
0	b	13	13, 14	2 (T3)
1	a	-	-	-
1	b	3, 7	3, 6, 7, 8	3 (T1)
2	a	-	-	-
2	b	-	-	-
3	a	-	-	-
3	b	7	6, 7, 8	4 (T2)
4	a	-	-	-
4	b	7	6, 7, 8	4 (T2)

Graphically, the DFA is represented as follows:

The minimization tree is as follows. Note that before considering transition behavior, states are split according to the token they recognize.

The tree expansion for non-splitting sets has been omitted for simplicity ("a" transition for final states {1, 4}).

Given the minimization tree, the final minimal DFA is exactly the same as the original DFA (all leaf sets are singular).

Input Analysis

I_n	Input	I_n+1 / Token
0	`abaabb$`	1
1	`baabb$`	3
3	`aabb$`	T1
0	`aabb$`	1
1	`abb$`	T2
0	`abb$`	1
1	`bb$`	3
3	`b$`	4
4	`$`	T2

The input string abaabb is, after 9 steps, split into three tokens: T1 (corresponding to lexeme ab), T2 (a), and T2 (abb).

@@ Line 10: / Line 10: @@
 The following is the result of applying Thompson's algorithm. State '''3''' recognizes the first expression (token '''T1'''); state '''8''' recognizes token '''T2'''; and state '''14''' recognizes token '''T3'''.
+<dot-hack>
+digraph nfa {
+     { node [shape=circle style=invis] s }
+  rankdir=LR; ratio=0.5
+  node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 3 8 14
+  node [shape=circle,fixedsize=true,width=0.2,fontsize=10];
+  s -> 0
+-> 1
+-> 2 [label="a",fontsize=10]
+-> 3 [label="b",fontsize=10]
+-> 4
+-> 5 [label="a",fontsize=10]
+-> 6
+-> 8
+-> 7 [label="b",fontsize=10]
+-> 6
+-> 8
+-> 9
+-> 10
+-> 12
+-> 11 [label="a",fontsize=10]
+-> 13 [label="b",fontsize=10]
+-> 14
+-> 14
+  fontsize=10
+}
+</dot-hack>
+<!--
 <runphp>
 print '<div id="mynetwork2" style="height: 400px;"></div><script type="text/javascript">var container = document.getElementById("mynetwork2"); var nodes = [ {id: 0, label: "0", group: "all", level: 1}, {id: 1, label: "1", level: 2, group: "t1"}, {id: 2, label: "2", level: 3, group: "t1"}, {id: 3, label: "3", level: 4, borderWidth: 3, group: "t1"}, {id: 4, label: "4", level: 2, group: "t2"}, {id: 5, label: "5", level: 3, group: "t2"}, {id: 6, label: "6", level: 4, group: "t2"}, {id: 7, label: "7", level: 5, group: "t2"}, {id: 8, label: "8", level: 5, borderWidth: 3, group: "t2"}, {id: 9, label: "9", level: 2, group: "t3"}, {id: 10, label: "10", level: 3, group: "t3"}, {id: 11, label: "11", level: 4, group: "t3"}, {id: 12, label: "12", level: 3, group: "t3"}, {id: 13, label: "13", level: 4, group: "t3"}, {id: 14, label: "14", borderWidth: 3, level: 5, group: "t3"} ]; var edges = [  {from: 0, to: 1}, {from: 1, to: 2, label: "a" }, {from: 2, to: 3, label: "b" }, {from: 0, to: 4}, {from: 4, to: 5, label: "a" }, {from: 5, to: 6}, {from: 5, to: 8}, {from: 6, to: 7, label: "b" }, {from: 7, to: 6}, {from: 7, to: 8}, {from: 0, to: 9}, {from: 9, to: 10}, {from: 9, to: 12}, {from: 10, to: 11, label: "a" }, {from: 12, to: 13, label: "b" }, {from: 11, to: 14}, {from: 13, to: 14}, ]; var options = { groups: { t1: { color: { border: "#41a906", background: "#7be141", highlight: { border: "#41a906", background: "#7be141", } } }, t2: { color: { border: "#f31d22", background: "#fa8a8c", highlight: { border: "#f31d22", background: "#fa8a8c", } } }, t3: { color: { border: "#ffa500", background: "#ffff00", highlight: { border: "#ffa500", background: "#ffff00", } } } }, edges: { style: "arrow" }, nodes: { color: { background: "white", border: "#2B7CE9" }, radius: 30 },  hierarchicalLayout: { direction: "LR" }, zoomable: false }; var data = { nodes: nodes, edges: edges }; var network = new vis.Network(container, data, options);</script>';
 </runphp>
+-->
 == DFA ==
@@ Line 93: / Line 126: @@
 Graphically, the DFA is represented as follows:
+<dot-hack>
+digraph dfa {
+     { node [shape=circle style=invis] s }
+  rankdir=LR; ratio=0.5
+  node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 1 2 3 4
+  node [shape=circle,fixedsize=true,width=0.2,fontsize=10];
+  s -> 0
+-> 1 [label="a",fontsize=10]
+-> 2 [label="b",fontsize=10]
+-> 3 [label="b",fontsize=10]
+-> 4 [label="b",fontsize=10]
+-> 4 [label="b",fontsize=10]
+  fontsize=10
+}
+</dot-hack>
+<!--
 <runphp>
 echo<<<___EOT___
@@ Line 105: / Line 154: @@
 ___EOT___;
 </runphp>
+-->
 The minimization tree is as follows. Note that before considering transition behavior, states are split according to the token they recognize.
-<graph>
+<dot-hack>
 digraph mintree {
    node [shape=none,fixedsize=true,width=0.3,fontsize=10]
@@ Line 116: / Line 165: @@
    "{1, 2, 3, 4}" -> "{1, 4}" [label="  T2",fontsize=10]
    "{1, 2, 3, 4}" -> "{2}" [label="  T3",fontsize=10]
-   "{1, 4}" -> "{1}" //[label="  T3",fontsize=10]
+   "{1, 4}" -> "{1}" /*[label="  T3",fontsize=10]*/
    "{1, 4}" -> "{4}" [label="  b",fontsize=10]
    fontsize=10
-  //label="Minimization tree"
 }
-</graph>
+</dot-hack>
 The tree expansion for non-splitting sets has been omitted for simplicity ("a" transition for final states {1, 4}).
@@ Line 173: / Line 221: @@
 The input string ''abaabb'' is, after 9 steps, split into three tokens: '''T1''' (corresponding to lexeme ''ab''), '''T2''' (''a''), and '''T2''' (''abb'').
-[[category:Teaching]]
+[[category:Compiladores]]
-[[category:Compilers]]
+[[category:Ensino]]
 [[en:Theoretical Aspects of Lexical Analysis]]