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

Revision as of 21:09, 18 February 2015

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 1: / Line 1: @@
 __NOTOC__
-<div class="section-container auto" data-section>
+Consider the lexical analyzer '''<nowiki>G = { ab, ab*, a|b }</nowiki>''', defined for the alphabet '''Σ = { a, b }'''.
-  <div class="section">
-    <p class="title" data-section-title>Problem</p>
-    <div class="content" data-section-content>
-<!-- ====================== START OF PROBLEM ====================== -->
-Consider the lexical analyzer '''<nowiki>G = { ab, ab*, a|b }</nowiki>''', 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.
-<!-- ====================== END OF PROBLEM ====================== -->
-    </div>
-  </div>
-  <div class="section">
-    <p class="title" data-section-title>Solution</p>
-    <div class="content" data-section-content>
-<!-- ====================== START OF SOLUTION ====================== -->
-Consider the lexical analyzer '''<nowiki>G = { ab, ab*, a|b }</nowiki>''', 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).
+== NFA ==
-Indicate the number of processing steps for the '''abaabb''' input string.
 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'''.
 <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>';
+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 ==
 Applying the determination algorithm to the above NFA, the following determination table is obtained:
@@ Line 38: / Line 25: @@
 ! style="padding-left: 20px; padding-right: 20px; background: wheat;" | I<sub>n+1</sub> = ε-closure(move(I<sub>n</sub>, α))
 |-
 ! style="font-weight: normal; align: center; background: #ffffcc;" | -
 ! style="font-weight: normal; align: center; background: #ffffcc;" | -
 ! style="font-weight: normal; align: center; background: #ffffcc;" | 0
 ! style="font-weight: normal; align: left;   background: #ffffcc;" | 0, 1, 4, 9, 10, 12
@@ Line 45: / Line 32: @@
 |-
 ! style="font-weight: normal; align: center; background: #e6e6e6;" | 0
 ! style="font-weight: normal; align: center; background: #e6e6e6;" | a
 ! style="font-weight: normal; align: center; background: #e6e6e6;" | 2, 5, 11
 ! style="font-weight: normal; align: left;   background: #e6e6e6;" | 2, 5, 6, '''8''', 11, '''14'''
@@ Line 51: / Line 38: @@
 |-
 ! style="font-weight: normal; align: center; background: #e6e6e6;" | 0
 ! style="font-weight: normal; align: center; background: #e6e6e6;" | b
 ! style="font-weight: normal; align: center; background: #e6e6e6;" | 13
 ! style="font-weight: normal; align: left;   background: #e6e6e6;" | 13, '''14'''
@@ Line 57: / Line 44: @@
 |-
 ! style="font-weight: normal; align: center; background: #ffffcc;" | 1
 ! style="font-weight: normal; align: center; background: #ffffcc;" | a
 ! style="font-weight: normal; align: center; background: #ffffcc;" | -
 ! style="font-weight: normal; align: left;   background: #ffffcc;" | -
 ! style="font-weight: normal; align: center; background: #ffffcc;" | -
 |-
 ! style="font-weight: normal; align: center; background: #ffffcc;" | 1
 ! style="font-weight: normal; align: center; background: #ffffcc;" | b
 ! style="font-weight: normal; align: center; background: #ffffcc;" | '''3''', 7
 ! style="font-weight: normal; align: left;   background: #ffffcc;" | '''3''', 6, 7, '''8'''
@@ Line 75: / Line 62: @@
 |-
 ! style="font-weight: normal; align: center; background: #e6e6e6;" | 2
 ! style="font-weight: normal; align: center; background: #e6e6e6;" | b
 ! style="font-weight: normal; align: center; background: #e6e6e6;" | -
 ! style="font-weight: normal; align: left;   background: #e6e6e6;" | -
@@ Line 87: / Line 74: @@
 |-
 ! style="font-weight: normal; align: center; background: #ffffcc;" | 3
 ! style="font-weight: normal; align: center; background: #ffffcc;" | b
 ! style="font-weight: normal; align: center; background: #ffffcc;" | 7
 ! style="font-weight: normal; align: left;   background: #ffffcc;" | 6, 7, '''8'''
@@ Line 99: / Line 86: @@
 |-
 ! style="font-weight: normal; align: center; background: #e6e6e6;" | 4
 ! style="font-weight: normal; align: center; background: #e6e6e6;" | b
 ! style="font-weight: normal; align: center; background: #e6e6e6;" | 7
 ! style="font-weight: normal; align: left;   background: #e6e6e6;" | 6, 7, '''8'''
@@ Line 114: / Line 101: @@
 <graph>
 digraph mintree {
    node [shape=none,fixedsize=true,width=0.3,fontsize=10]
    "{0, 1, 2, 3, 4}" -> "{0}" [label="NF",fontsize=10]
@@ Line 140: / Line 127: @@
 |-
 ! style="font-weight: normal; align: center; background: #ffffcc;" | 0
 ! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>abaabb$</tt>
 ! style="font-weight: normal; align: center; background: #ffffcc;" | 1
 |-
 ! style="font-weight: normal; align: center; background: #ffffcc;" | 1
 ! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>baabb$</tt>
 ! style="font-weight: normal; align: center; background: #ffffcc;" | 3
 |-
 ! style="font-weight: normal; align: center; background: #ffffcc;" | 3
 ! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>aabb$</tt>
 ! style="font-weight: normal; align: center; background: #ffffcc;" | '''T1'''
 |-
@@ Line 177: / Line 164: @@
 The input string ''abaabb'' is, after 9 steps, split into three tokens: '''T1''' (corresponding to lexeme ''ab''), '''T2''' (''a''), and '''T2''' (''abb'').
-<!-- ====================== END OF SOLUTION ====================== -->
-    </div>
-  </div>
-</div>
 [[category:Teaching]]
 [[category:Compilers]]
 [[en:Theoretical Aspects of Lexical Analysis]]