Theoretical Aspects of Lexical Analysis/Exercise 8 - Revision history

Root: /* DFA */

2025-08-06T17:06:01Z

DFA

← Older revision		Revision as of 17:06, 6 August 2025
Line 130:		Line 130:
	Graphically, the DFA is represented as follows:		Graphically, the DFA is represented as follows:

	<~~dot-hack~~>		<kroki lang="graphviz">
	digraph dfa {		digraph dfa {
	{ node [shape=circle style=invis] s }		{ node [shape=circle style=invis] s }
Line 145:		Line 145:
	fontsize=10		fontsize=10
	}		}
	</~~dot-hack~~>		</kroki>

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

	<~~dot-hack~~>		<kroki lang="graphviz">
	digraph mintree {		digraph mintree {
	node [shape=none,fixedsize=true,width=0.3,fontsize=10]		node [shape=none,fixedsize=true,width=0.3,fontsize=10]
Line 162:		Line 162:
	fontsize=10		fontsize=10
	}		}
	</~~dot-hack~~>		</kroki>

	Given the minimization tree, the final minimal DFA is as follows. Note that states 2 and 4 cannot be the same since they recognize different tokens.		Given the minimization tree, the final minimal DFA is as follows. Note that states 2 and 4 cannot be the same since they recognize different tokens.

	<~~dot-hack~~>		<kroki lang="graphviz">
	digraph mindfa {		digraph mindfa {
	{ node [shape=circle style=invis] s }		{ node [shape=circle style=invis] s }
Line 180:		Line 180:
	fontsize=10		fontsize=10
	}		}
	</~~dot-hack~~>		</kroki>

	== Input Analysis ==		== Input Analysis ==

Root: /* NFA */

2025-08-06T17:04:48Z

NFA

← Older revision		Revision as of 17:04, 6 August 2025
Line 7:		Line 7:
	The following is the result of applying Thompson's algorithm. State '''4''' recognizes the first expression (token '''T1'''); state '''12''' recognizes token '''T2'''; and state '''20''' recognizes token '''T3'''.		The following is the result of applying Thompson's algorithm. State '''4''' recognizes the first expression (token '''T1'''); state '''12''' recognizes token '''T2'''; and state '''20''' recognizes token '''T3'''.

	<kroki>		<kroki lang="graphviz">
	digraph nfa {		digraph nfa {
	{ node [shape=circle style=invis] s }		{ node [shape=circle style=invis] s }

Root: /* NFA */

2025-08-06T16:59:13Z

NFA

← Older revision		Revision as of 16:59, 6 August 2025
Line 7:		Line 7:
	The following is the result of applying Thompson's algorithm. State '''4''' recognizes the first expression (token '''T1'''); state '''12''' recognizes token '''T2'''; and state '''20''' recognizes token '''T3'''.		The following is the result of applying Thompson's algorithm. State '''4''' recognizes the first expression (token '''T1'''); state '''12''' recognizes token '''T2'''; and state '''20''' recognizes token '''T3'''.

	<~~dot-hack~~>		<kroki>
	digraph nfa {		digraph nfa {
	{ node [shape=circle style=invis] s }		{ node [shape=circle style=invis] s }
Line 48:		Line 48:
	fontsize=10		fontsize=10
	}		}
	</~~dot-hack~~>		</kroki>

	== DFA ==		== DFA ==

Root: /* DFA */

2023-05-09T17:26:13Z

DFA

← Older revision		Revision as of 17:26, 9 May 2023
Line 159:		Line 159:
	"{0, 1, 3}" -> "{0}" [label=" b",fontsize=10]		"{0, 1, 3}" -> "{0}" [label=" b",fontsize=10]
	"{0, 1, 3}" -> "{1,3}"		"{0, 1, 3}" -> "{1,3}"
			"{1,3}" -> "{1,3} " [label=" a, b",fontsize=10]
	fontsize=10		fontsize=10
	}		}
	</dot-hack>		</dot-hack>

	The tree expansion for non-splitting sets has been omitted for simplicity ("a" transitions for super-state {0, 1, 3}, and "a" and "b" transitions for super-state {1,3} -- note that these transitions would not split state {1,3}).

	Given the minimization tree, the final minimal DFA is as follows. Note that states 2 and 4 cannot be the same since they recognize different tokens.		Given the minimization tree, the final minimal DFA is as follows. Note that states 2 and 4 cannot be the same since they recognize different tokens.

Root at 17:18, 9 May 2023

2023-05-09T17:18:39Z

← Older revision		Revision as of 17:18, 9 May 2023
Line 163:		Line 163:
	</dot-hack>		</dot-hack>

	The tree expansion for non-splitting sets has been omitted for simplicity ("a" transitions for super-state {0, 1, 3}, and "a" and "b" transitions for super-state {1,3}).		The tree expansion for non-splitting sets has been omitted for simplicity ("a" transitions for super-state {0, 1, 3}, and "a" and "b" transitions for super-state {1,3} -- note that these transitions would not split state {1,3}).

	Given the minimization tree, the final minimal DFA is as follows. Note that states 2 and 4 cannot be the same since they recognize different tokens.		Given the minimization tree, the final minimal DFA is as follows. Note that states 2 and 4 cannot be the same since they recognize different tokens.

Root: /* DFA */

2019-03-07T15:30:34Z

DFA

← Older revision		Revision as of 15:30, 7 March 2019
Line 167:		Line 167:
	Given the minimization tree, the final minimal DFA is as follows. Note that states 2 and 4 cannot be the same since they recognize different tokens.		Given the minimization tree, the final minimal DFA is as follows. Note that states 2 and 4 cannot be the same since they recognize different tokens.

	<~~graph~~>		<dot-hack>
	digraph mindfa {		digraph mindfa {
	{ node [shape=circle style=invis] s }		{ node [shape=circle style=invis] s }
Line 181:		Line 181:
	fontsize=10		fontsize=10
	}		}
	</~~graph~~>		</dot-hack>

	== Input Analysis ==		== Input Analysis ==

Root at 10:17, 12 February 2019

2019-02-12T10:17:07Z

← Older revision		Revision as of 10:17, 12 February 2019
Line 7:		Line 7:
	The following is the result of applying Thompson's algorithm. State '''4''' recognizes the first expression (token '''T1'''); state '''12''' recognizes token '''T2'''; and state '''20''' recognizes token '''T3'''.		The following is the result of applying Thompson's algorithm. State '''4''' recognizes the first expression (token '''T1'''); state '''12''' recognizes token '''T2'''; and state '''20''' recognizes token '''T3'''.

	<~~graph~~>		<dot-hack>
	digraph nfa {		digraph nfa {
	{ node [shape=circle style=invis] s }		{ node [shape=circle style=invis] s }
Line 48:		Line 48:
	fontsize=10		fontsize=10
	}		}
	</~~graph~~>		</dot-hack>

	== DFA ==		== DFA ==
Line 130:		Line 130:
	Graphically, the DFA is represented as follows:		Graphically, the DFA is represented as follows:

	<~~graph~~>		<dot-hack>
	digraph dfa {		digraph dfa {
	{ node [shape=circle style=invis] s }		{ node [shape=circle style=invis] s }
Line 145:		Line 145:
	fontsize=10		fontsize=10
	}		}
	</~~graph~~>		</dot-hack>

	The minimization tree is as follows. Note that before considering transition behavior, states are split according to the token they recognize.		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 {		digraph mintree {
	node [shape=none,fixedsize=true,width=0.3,fontsize=10]		node [shape=none,fixedsize=true,width=0.3,fontsize=10]
Line 158:		Line 158:
	"{0, 1, 2, 3, 4} " -> "{4}" [label=" T3",fontsize=10]		"{0, 1, 2, 3, 4} " -> "{4}" [label=" T3",fontsize=10]
	"{0, 1, 3}" -> "{0}" [label=" b",fontsize=10]		"{0, 1, 3}" -> "{0}" [label=" b",fontsize=10]
	"{0, 1, 3}" -> "{1,3}" ~~//[label=" b",fontsize=10]~~		"{0, 1, 3}" -> "{1,3}"
	fontsize=10		fontsize=10
	~~//label="Minimization tree"~~
	}		}
	</~~graph~~>		</dot-hack>

	The tree expansion for non-splitting sets has been omitted for simplicity ("a" transitions for super-state {0, 1, 3}, and "a" and "b" transitions for super-state {1,3}).		The tree expansion for non-splitting sets has been omitted for simplicity ("a" transitions for super-state {0, 1, 3}, and "a" and "b" transitions for super-state {1,3}).

Root at 14:39, 6 April 2015

2015-04-06T14:39:16Z

← Older revision		Revision as of 14:39, 6 April 2015
Line 234:		Line 234:
	The input string ''aababb'' is, after 10 steps, split into three tokens: '''T1''' (corresponding to lexeme ''aa''), '''T2''' (''b''), '''T1''' (''a''), and '''T3''' (''bb'').		The input string ''aababb'' is, after 10 steps, split into three tokens: '''T1''' (corresponding to lexeme ''aa''), '''T2''' (''b''), '''T1''' (''a''), and '''T3''' (''bb'').

	[[category:~~Teaching~~]]		[[category:Compiladores]]
	[[category:~~Compilers~~]]		[[category:Ensino]]

	[[en:Theoretical Aspects of Lexical Analysis]]		[[en:Theoretical Aspects of Lexical Analysis]]

Root: New page: NOTOC Compute the non-deterministic finite automaton (NFA) by using Thompson's algorithm. Compute the minimal deterministic finite automaton (DFA).
The alphabet is Σ = { a, b }. I...

2009-03-28T20:43:33Z

New page: __NOTOC__ Compute the non-deterministic finite automaton (NFA) by using Thompson's algorithm. Compute the minimal deterministic finite automaton (DFA). The alphabet is Σ = { a, b }. I...

New page

__NOTOC__
Compute the non-deterministic finite automaton (NFA) by using Thompson's algorithm. Compute the minimal deterministic finite automaton (DFA). The alphabet is Σ = { a, b }. Indicate the number of processing steps for the given input string.
* <nowiki>G = { a*, a*|b, a|b* }</nowiki>, input string = aababb

== NFA ==

The following is the result of applying Thompson's algorithm. State '''4''' recognizes the first expression (token '''T1'''); state '''12''' recognizes token '''T2'''; and state '''20''' recognizes token '''T3'''.

<graph>
digraph nfa {
{ node [shape=circle style=invis] s }
rankdir=LR; ratio=0.5
node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 4 12 20
node [shape=circle,fixedsize=true,width=0.2,fontsize=10];

s -> 0

0 -> 1
1 -> 2
1 -> 4
2 -> 3 [label="a",fontsize=10]
3 -> 2
3 -> 4

0 -> 5
5 -> 6
5 -> 10
6 -> 7
6 -> 9
7 -> 8 [label="a",fontsize=10]
8 -> 7
8 -> 9
9 -> 12
10 -> 11 [label="b",fontsize=10]
11 -> 12

0 -> 13
13 -> 14
13 -> 16
14 -> 15 [label="a",fontsize=10]
15 -> 20
16 -> 17
16 -> 19
17 -> 18 [label="b",fontsize=10]
18 -> 17
18 -> 19
19 -> 20
fontsize=10
}
</graph>

== DFA ==

Determination table for the above NFA:

{| cellspacing="2"
! style="padding-left: 20px; padding-right: 20px; background: wheat;" | In
! style="padding-left: 20px; padding-right: 20px; background: wheat;" | α∈Σ
! style="padding-left: 20px; padding-right: 20px; background: wheat;" | move(In, α)
! style="padding-left: 20px; padding-right: 20px; background: wheat;" | ε-closure(move(In, α))
! style="padding-left: 20px; padding-right: 20px; background: wheat;" | In+1 = ε-closure(move(In, 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, 2, '''4''', 5, 6, 7, 9, 10, '''12''', 13, 14, 16, 17, 19, '''20'''
! style="font-weight: normal; align: center; background: #ffffcc;" | '''0''' (T1)
|-
! 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;" | 3, 8, 15
! style="font-weight: normal; align: left; background: #e6e6e6;" | 2, 3, '''4''', 7, 8, 9, '''12''', 15, '''20'''
! style="font-weight: normal; align: center; background: #e6e6e6;" | '''1''' (T1)
|-
! 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;" | 11, 18
! style="font-weight: normal; align: left; background: #e6e6e6;" | 11, '''12''', 17, 18, 19, '''20'''
! style="font-weight: normal; align: center; background: #e6e6e6;" | '''2''' (T2)
|-
! 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;" | 3, 8
! style="font-weight: normal; align: left; background: #ffffcc;" | 2, 3, '''4''', 7, 8, 9, '''12'''
! style="font-weight: normal; align: center; background: #ffffcc;" | '''3''' (T1)
|-
! 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;" | -
! style="font-weight: normal; align: left; background: #ffffcc;" | -
! style="font-weight: normal; align: center; background: #ffffcc;" | -
|-
! style="font-weight: normal; align: center; background: #e6e6e6;" | 2
! style="font-weight: normal; align: center; background: #e6e6e6;" | a
! style="font-weight: normal; align: center; background: #e6e6e6;" | -
! style="font-weight: normal; align: left; background: #e6e6e6;" | -
! style="font-weight: normal; align: center; background: #e6e6e6;" | -
|-
! 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;" | 18
! style="font-weight: normal; align: left; background: #e6e6e6;" | 17, 18, 19, '''20'''
! style="font-weight: normal; align: center; background: #e6e6e6;" | '''4''' (T3)
|-
! style="font-weight: normal; align: center; background: #ffffcc;" | 3
! style="font-weight: normal; align: center; background: #ffffcc;" | a
! style="font-weight: normal; align: center; background: #ffffcc;" | 3, 8
! style="font-weight: normal; align: left; background: #ffffcc;" | 2, 3, '''4''', 7, 8, 9, '''12'''
! style="font-weight: normal; align: center; background: #ffffcc;" | '''3''' (T1)
|-
! 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;" | -
! style="font-weight: normal; align: left; background: #ffffcc;" | -
! style="font-weight: normal; align: center; background: #ffffcc;" | -
|-
! style="font-weight: normal; align: center; background: #e6e6e6;" | 4
! style="font-weight: normal; align: center; background: #e6e6e6;" | a
! style="font-weight: normal; align: center; background: #e6e6e6;" | -
! style="font-weight: normal; align: left; background: #e6e6e6;" | -
! style="font-weight: normal; align: center; background: #e6e6e6;" | -
|-
! 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;" | 18
! style="font-weight: normal; align: left; background: #e6e6e6;" | 17, 18, 19, '''20'''
! style="font-weight: normal; align: center; background: #e6e6e6;" | '''4''' (T3)
|}

Graphically, the DFA is represented as follows:

<graph>
digraph dfa {
{ node [shape=circle style=invis] s }
rankdir=LR; ratio=0.5
node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 0 1 2 3 4
node [shape=circle,fixedsize=true,width=0.2,fontsize=10];
s -> 0
0 -> 1 [label="a",fontsize=10]
0 -> 2 [label="b",fontsize=10]
1 -> 3 [label="a",fontsize=10]
2 -> 4 [label="b",fontsize=10]
3 -> 3 [label="a",fontsize=10]
4 -> 4 [label="b",fontsize=10]
fontsize=10
}
</graph>

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

<graph>
digraph mintree {
node [shape=none,fixedsize=true,width=0.3,fontsize=10]
"{0, 1, 2, 3, 4}" -> "{}" [label="NF",fontsize=10]
"{0, 1, 2, 3, 4}" -> "{0, 1, 2, 3, 4} " [label=" F",fontsize=10]
"{0, 1, 2, 3, 4} " -> "{0, 1, 3}" [label=" T1",fontsize=10]
"{0, 1, 2, 3, 4} " -> "{2}" [label=" T2",fontsize=10]
"{0, 1, 2, 3, 4} " -> "{4}" [label=" T3",fontsize=10]
"{0, 1, 3}" -> "{0}" [label=" b",fontsize=10]
"{0, 1, 3}" -> "{1,3}" //[label=" b",fontsize=10]
fontsize=10
//label="Minimization tree"
}
</graph>

The tree expansion for non-splitting sets has been omitted for simplicity ("a" transitions for super-state {0, 1, 3}, and "a" and "b" transitions for super-state {1,3}).

Given the minimization tree, the final minimal DFA is as follows. Note that states 2 and 4 cannot be the same since they recognize different tokens.

<graph>
digraph mindfa {
{ node [shape=circle style=invis] s }
rankdir=LR; ratio=0.5
node [shape=doublecircle,fixedsize=true,width=0.2,fontsize=10]; 0 13 2 4
node [shape=circle,fixedsize=true,width=0.2,fontsize=10];
s -> 0
0 -> 13 [label="a",fontsize=10]
0 -> 2 [label="b",fontsize=10]
13 -> 13 [label="a",fontsize=10]
2 -> 4 [label="b",fontsize=10]
4 -> 4 [label="b",fontsize=10]
fontsize=10
}
</graph>

== Input Analysis ==

{| cellspacing="2"
! style="padding-left: 20px; padding-right: 20px; background: wheat;" | In
! style="padding-left: 20px; padding-right: 20px; background: wheat;" | Input
! style="padding-left: 20px; padding-right: 20px; background: wheat;" | In+1 / Token
|-
! style="font-weight: normal; align: center; background: #ffffcc;" | 0
! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>aababb$</tt>
! style="font-weight: normal; align: center; background: #ffffcc;" | 13
|-
! style="font-weight: normal; align: center; background: #ffffcc;" | 13
! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>ababb$</tt>
! style="font-weight: normal; align: center; background: #ffffcc;" | 13
|-
! style="font-weight: normal; align: center; background: #ffffcc;" | 13
! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>babb$</tt>
! style="font-weight: normal; align: center; background: #ffffcc;" | '''T1''' (aa)
|-
! style="font-weight: normal; align: center; background: #e6e6e6;" | 0
! style="font-weight: normal; text-align: right; background: #e6e6e6;" | <tt>babb$</tt>
! style="font-weight: normal; align: center; background: #e6e6e6;" | 2
|-
! style="font-weight: normal; align: center; background: #e6e6e6;" | 2
! style="font-weight: normal; text-align: right; background: #e6e6e6;" | <tt>abb$</tt>
! style="font-weight: normal; align: center; background: #e6e6e6;" | '''T2''' (b)
|-
! style="font-weight: normal; align: center; background: #ffffcc;" | 0
! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>abb$</tt>
! style="font-weight: normal; align: center; background: #ffffcc;" | 13
|-
! style="font-weight: normal; align: center; background: #ffffcc;" | 13
! style="font-weight: normal; text-align: right; background: #ffffcc;" | <tt>bb$</tt>
! style="font-weight: normal; align: center; background: #ffffcc;" | '''T1''' (a)
|-
! style="font-weight: normal; align: center; background: #e6e6e6;" | 0
! style="font-weight: normal; text-align: right; background: #e6e6e6;" | <tt>bb$</tt>
! style="font-weight: normal; align: center; background: #e6e6e6;" | 2
|-
! style="font-weight: normal; align: center; background: #e6e6e6;" | 2
! style="font-weight: normal; text-align: right; background: #e6e6e6;" | <tt>b$</tt>
! style="font-weight: normal; align: center; background: #e6e6e6;" | 4
|-
! style="font-weight: normal; align: center; background: #e6e6e6;" | 4
! style="font-weight: normal; text-align: right; background: #e6e6e6;" | <tt>$</tt>
! style="font-weight: normal; align: center; background: #e6e6e6;" | '''T3''' (bb)
|}

The input string ''aababb'' is, after 10 steps, split into three tokens: '''T1''' (corresponding to lexeme ''aa''), '''T2''' (''b''), '''T1''' (''a''), and '''T3''' (''bb'').

[[category:Teaching]]
[[category:Compilers]]
[[en:Theoretical Aspects of Lexical Analysis]]

Theoretical Aspects of Lexical Analysis/Exercise 8 - Revision history

Root: /* DFA */

Root: /* NFA */

Root: /* NFA */

Root: /* DFA */

Root at 17:18, 9 May 2023

Root: /* DFA */

Root at 10:17, 12 February 2019

Root at 14:39, 6 April 2015

Root: New page: __NOTOC__ Compute the non-deterministic finite automaton (NFA) by using Thompson's algorithm. Compute the minimal deterministic finite automaton (DFA).The alphabet is Σ = { a, b }. I...

Root: New page: NOTOC Compute the non-deterministic finite automaton (NFA) by using Thompson's algorithm. Compute the minimal deterministic finite automaton (DFA).
The alphabet is Σ = { a, b }. I...