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1 Lexical Analysis Cheng-Chia Chen

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2 Outline l Introduction to lexical analyzer l Tokens l Regular expressions (RE) l Finite automata (FA) »deterministic and nondeterministic finite automata (DFA and NFA) »from RE to NFA »from NFA to DFA »from DFA to optimized DFA

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3 Outline l Informal sketch of lexical analysis »Identifies tokens in input string l Issues in lexical analysis »Lookahead »Ambiguities l Specifying lexers »Regular expressions »Examples of regular expressions

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4 The Structure of a Compiler SourceTokensInterm. Language Lexical analysis Parsing Code Gen. Machine Code Optimization

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5 Lexical Analysis l What do we want to do? Example: if (i == j) Z = 0; else Z = 1; l The input is just a sequence of characters: \tif (i == j)\n\t\tz = 0;\n\telse\n\t\tz = 1; l Goal: Partition input string into substrings »And determine the categories (tokens) to which the substrings belong

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6 What ’ s a Token? l Output of lexical analysis is a stream of tokens l A token is a syntactic category »In English: noun, verb, adjective, … »In a programming language: Identifier, Integer, Keyword, Whitespace, … l Parser relies on the token distinctions: identifiers are treated differently than keywords

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7 Aspects of Token l language view: a set of strings (belonging to that token) »[while], [identifier], [arithOp] l Pattern (grammar): a rule defining a token »[while]: while »[identifier]: letter followed by letters and digits »[arithOp]: + or - or * or / l Lexeme (member): a string matched by the pattern of a token »while, var23, count, +, *

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8 Attributes of Tokens l Attributes are used to distinguish different lexemes in a token »[while,] »[identifier, var35] »[arithOp, +] »[integer, 26] »[string, “26”] l positional information: start/end line/position l Tokens affect syntax analysis and attributes affect semantic analysis and error handling

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9 Lexical Analyzer: Implementation l An implementation must do two things: 1.Recognize substrings corresponding to tokens 2.Return the attributes( lexeme and positional information) of the token –The lexeme is either the substring or some data object constructed from the substring.

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10 Example l input lines: \tif (i == j)\n\t\tz = 0;\n\telse\n\t\tz = 1; l Token-lexeme pairs returned by the lexer: »[Whitespace, “ \t ” ] »[if, ] »[OpenPar, “ ( “ ] »[Identifier, “ i ” ] »[Relation, “ == “ ] »[Identifier, “ j ” ] »…

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11 Lexical Analyzer: Implementation The lexer usually discards “ uninteresting ” tokens that don ’ t contribute to parsing. l Examples: Whitespace, Comments l Question: What happens if we remove all whitespace and all comments prior to lexing?

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12 Lexical Analysis in FORTRAN l FORTRAN rule: Whitespace is insignificant E.g., VAR1 is the same as VA R1 l Footnote: FORTRAN whitespace rule motivated by inaccuracy of punch card operators

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13 A terrible design! Example l Consider »DO 5 I = 1,25 »DO 5 I = 1.25 The first is DO 5 I = 1, 25 The second is DO5I = 1.25 Reading left-to-right, cannot tell if DO5I is a variable or DO stmt. until after “, ” is reached

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14 Lexical Analysis in FORTRAN. Lookahead. l Two important points: 1.The goal is to partition the string. This is implemented by reading left-to-right, recognizing one token at a time 2.“ Lookahead ” may be required to decide where one token ends and the next token begins »Even our simple example has lookahead issues i vs. if = vs. ==

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15 Lexical Analysis in PL/I l PL/I keywords are not reserved IF ELSE THEN THEN = ELSE; ELSE ELSE = THEN l PL/I Declarations: DECLARE (ARG1,..., ARGN) Can ’ t tell whether DECLARE is a keyword or array reference until after the ). »Requires arbitrary lookahead!

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16 Review l The goal of lexical analysis is to »Partition the input string into lexemes »Identify the token of each lexeme l Left-to-right scan => lookahead sometimes required

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17 Next l We still need »A way to describe the lexemes of each token »A way to resolve ambiguities –Is if two variables i and f ? –Is == two equal signs = = ?

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18 Regular Expressions and Regular Languages l There are several formalisms for specifying tokens l Regular languages are the most popular »Simple and useful theory »Easy to understand »Efficient implementations

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19 Languages Def. Let S be a set of characters. A language over S is a set of strings of characters drawn from S ( is called the alphabet )

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20 Examples of Languages l Alphabet = English characters l Language = English words l Not every string on English characters is an English word »likes, school, … »beee,yykk, … l Alphabet = ASCII l Language = C programs l Note: ASCII character set is different from English character set

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21 Notation l Languages are sets of strings. l Need some notation for specifying which sets we want l For lexical analysis we care about regular languages, which can be described using regular expressions.

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22 Regular Expressions and Regular Languages l Each regular expression is a notation for a regular language (a set of words) l If A is a regular expression then we write L(A) to refer to the language denoted by A

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23 Atomic Regular Expressions l Single character: c L( c) = { c } (for any c 2 ) Epsilon (empty string): L( ) = { }

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24 Compound Regular Expressions l Union (or choice) A | B = { s | s 2 A or s 2 B } l Concatenation: AB (where A and B are reg. exp.) L(A B) = { | 2 L(A) and 2 L(B) } l Note: »A B (set concatenation) and (string concatenation) will be abbreviated to AB and , respectively. l Examples: »if | then | else = { if, then, else} »0 | 1 | … | 9 = { 0, 1, …, 9 } »Another example: (0 | 1) (0 | 1) = { 00, 01, 10, 11 }

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25 More Compound Regular Expressions l So far we do not have a notation for infinite languages l Iteration: A * L(A * ) = { } [ L(A) [ L(AA) [ L(AAA) [ … l Examples: 0 * = { , 0, 00, 000, … } 1 0 * = { strings starting with 1 and followed by 0 ’ s }

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26 Example: Keyword »Keyword: else or if or begin … else | if | begin | …

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27 Example: Integers Integer: a non-empty string of digits ( 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 ) ( 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 )* l problem: reuse complicated expression l improvement: define intermediate reg. expr. digit = 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 number = digit digit * Abbreviation: A + = A A *

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28 Regular Definitions l Names for regular expressions »d 1 = r 1 »d 2 = r 2 »... »d n = r n where r i over alphabet {d 1, d 2,..., d i-1 } l note: Recursion is not allowed.

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29 Example »Identifier: strings of letters or digits, starting with a letter digit = 0 | 1 |... | 9 letter = A | … | Z | a | … | z identifier = letter (letter | digit) * »Is (letter* | digit*) the same ?

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30 Example: Whitespace Whitespace: a non-empty sequence of blanks, newlines, CRNL and tabs WS = (\ | \t | \n | \r\n ) +

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31 Example: Email Addresses l Consider chencc@cs.nccu.edu.tw = letters [ {., @ } name = letter + address = name ‘ @ ’ name ( ‘. ’ name) *

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32 Notational Shorthands l One or more instances »r+ = r r* »r* = r+ | l Zero or one instance »r? = r | l Character classes »[abc] = a | b | c »[a-z] = a | b |... | z »[ac-f] = a | c | d | e | f »[^ac-f] = – [ac-f]

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33 Summary l Regular expressions describe many useful languages l Regular languages are a language specification »We still need an implementation l problem: Given a string s and a rexp R, is

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34 Implementation of Lexical Analysis

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35 Outline l Specifying lexical structure using regular expressions l Finite automata »Deterministic Finite Automata (DFAs) »Non-deterministic Finite Automata (NFAs) l Implementation of regular expressions RegExp => NFA => DFA =>optimized DFA => Tables

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36 Regular Expressions in Lexical Specification l Last lecture: a specification for the predicate s L(R) l But a yes/no answer is not enough ! l Instead: partition the input into lexemes l We will adapt regular expressions to this goal

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37 Regular Expressions => Lexical Spec. (1) 1. Select a set of tokens Number, Keyword, Identifier,... 2. Write a rexp for the lexemes of each token Number = digit + Keyword = if | else | … Identifier = letter (letter | digit)* OpenPar = ‘ ( ‘ …

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38 Regular Expressions => Lexical Spec. (2) 3. Construct R, matching all lexemes for all tokens R = Keyword | Identifier | Number | … = R 1 | R 2 | R 3 + … Facts: If s 2 L(R) then s is a lexeme »Furthermore s 2 L(R i ) for some “ i ” »This “ i ” determines the token that is reported

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39 Regular Expressions => Lexical Spec. (3) 4. Let the input be x 1 … x n (x 1... x n are characters in the language alphabet) For 1 i n check x 1 … x i L(R) ? 5. It must be that x 1 … x i L(R j ) for some j 6. Remove x 1 … x i from input and go to (4)

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40 How to Handle Spaces and Comments? 1. We could create a token Whitespace Whitespace = ( ‘ ’ + ‘ \n ’ + ‘ \r ’ + ‘ \t ’ ) + »We could also add comments in there »An input “ \t\n 5555 “ is transformed into Whitespace Integer Whitespace 2. Lexer skips spaces (preferred) Modify step 5 from before as follows: It must be that x k... x i 2 L(R j ) for some j such that x 1... x k-1 2 L(Whitespace) Parser is not bothered with spaces

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41 Ambiguities (1) l There are ambiguities in the algorithm l How much input is used? What if – x 1 … x i L(R) and also – x 1 … x K L(R) »Rule: Pick the longest possible substring »The maximal match

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42 Ambiguities (2) l Which token is used? What if – x 1 … x i L(R j ) and also – x 1 … x i L(R k ) »Rule: use rule listed first (j if j < k) l Example: »R 1 = Keyword and R 2 = Identifier »“ if ” matches both. »Treats “ if ” as a keyword not an identifier

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43 Error Handling l What if No rule matches a prefix of input ? Problem: Can ’ t just get stuck … l Solution: »Write a rule matching all “ bad ” strings »Put it last l Lexer tools allow the writing of: R = R 1 |... | R n | Error »Token Error matches if nothing else matches

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44 Summary l Regular expressions provide a concise notation for string patterns l Use in lexical analysis requires small extensions »To resolve ambiguities »To handle errors l Good algorithms known (next) »Require only single pass over the input »Few operations per character (table lookup)

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45 Finite Automata l Regular expressions = specification l Finite automata = implementation l A finite automaton consists of »An input alphabet »A set of states S »A start state n »A set of accepting states F S »A set of transitions state input state

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46 Finite Automata l Transition s1 a s2s1 a s2 l Is read In state s 1 on input “ a ” go to state s 2 l If end of input (or no transition possible) »If in accepting state => accept »Otherwise => reject

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47 Finite Automata State Graphs l A state The start state An accepting state A transition a

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48 A Simple Example A finite automaton that accepts only “ 1 ” 1

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49 Another Simple Example A finite automaton accepting any number of 1 ’ s followed by a single 0 l Alphabet: {0,1} 0 1

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50 And Another Example l Alphabet {0,1} l What language does this recognize? 0 1 0 1 0 1

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51 And Another Example l Alphabet still { 0, 1 } l The operation of the automaton is not completely defined by the input »On input “ 11 ” the automaton could be in either state 1 1

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52 Epsilon Moves l Another kind of transition: -moves Machine can move from state A to state B without reading input A B

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53 Deterministic and Nondeterministic Automata l Deterministic Finite Automata (DFA) »One transition per input per state »No -moves l Nondeterministic Finite Automata (NFA) »Can have multiple transitions for one input in a given state »Can have -moves l Finite automata have finite memory »Need only to encode the current state

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54 Execution of Finite Automata l A DFA can take only one path through the state graph »Completely determined by input l NFAs can choose »Whether to make -moves »Which of multiple transitions for a single input to take

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55 Acceptance of NFAs l An NFA can get into multiple states Input: 0 1 1 0 101 Rule: NFA accepts it it can get in a final state

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56 Acceptance of a Finite Automata l A FA (DFA or NFA) accepts an input string s iff there is some path in the transition diagram from the start state to some final state such that the edge labels along this path spell out s

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57 NFA vs. DFA (1) l NFAs and DFAs recognize the same set of languages (regular languages) l DFAs are easier to implement »There are no choices to consider

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58 NFA vs. DFA (2) l For a given language the NFA can be simpler than the DFA 0 1 0 0 0 1 0 1 0 1 NFA DFA DFA can be exponentially larger than NFA

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59 Operations on NFA states -closure(s): set of NFA states reachable from NFA state s on -transitions alone -closure(S): set of NFA states reachable from some NFA state s in S on -transitions alone l move(S, c): set of NFA states to which there is a transition on input symbol c from some NFA state s in S notes: » -closure(S) = U s ∈ S -closure(S); » -closure(s) = U s ∈ S -closure({s}); » -closure(S) = ?

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60 Computing -closure l Input. An NFA and a set of NFA states S. Output. E = -closure(S). begin push all states in S onto stack; T := S; while stack is not empty do begin pop t, the top element, off of stack; for each state u with an edge from t to u labeled do if u is not in T do begin add u to T; push u onto stack end end; return T end.

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61 Simulating an NFA l Input. An input string ended with eof and an NFA with start state s0 and final states F. l Output. The answer “yes” if accepts, “no” otherwise. begin S := -closure({s0}); c := next_symbol(); while c != eof do begin S := -closure(move(S, c)); c := next_symbol(); end; if S F != then return “yes” else return “no” end.

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62 Regular Expressions to Finite Automata l High-level sketch Regular expressions NFADFA Lexical Specification Table-driven Implementation of DFA Optimized DFA

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63 Regular Expressions to NFA (1) l For each kind of rexp, define an NFA »Notation: NFA for rexp A A For For input a a

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64 Regular Expressions to NFA (2) l For AB A B For A + B A B

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65 Regular Expressions to NFA (3) l For A* A

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66 Example of RegExp -> NFA conversion l Consider the regular expression (1+0)*1 l The NFA is 1 C E 0 DF B G A H 1 IJ

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67 NFA to DFA

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68 Regular Expressions to Finite Automata l High-level sketch Regular expressions NFADFA Lexical Specification Table-driven Implementation of DFA Optimized DFA

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69 RegExp -> NFA : an Examlpe l Consider the regular expression (1+0)*1 l The NFA is 1 C E 0 DF B G A H 1 IJ

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70 NFA to DFA. The Trick l Simulate the NFA l Each state of DFA = a non-empty subset of states of the NFA l Start state = the set of NFA states reachable through - moves from NFA start state Add a transition S a S ’ to DFA iff »S ’ is the set of NFA states reachable from any state in S after seeing the input a –considering -moves as well

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71 NFA -> DFA Example 1 0 1 A B C D E F G H IJ ABCDHI FGABCDHI EJGABCDHI 0 1 0 1 0 1

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72 NFA to DFA. Remark l An NFA may be in many states at any time l How many different states ? l If there are N states, the NFA must be in some subset of those N states l How many non-empty subsets are there? »2 N - 1 = finitely many

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73 From an NFA to a DFA l Subset construction Algorithm. l Input. An NFA N. l Output. A DFA D with states S and trasition table mv. begin add -closure(s0) as an unmarked state to S; while there is an unmarked state T in S do begin mark T; for each input symbol a do begin U := -closure(move(T, a)); if U is not in S then add U as an unmarked state to S; mv[T, a] := U end end end.

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74 Implementation l A DFA can be implemented by a 2D table T »One dimension is “ states ” »Other dimension is “ input symbols ” »For every transition S i a S k define mv[i,a] = k DFA “ execution ” »If in state S i and input a, read mv[i,a] = k and skip to state S k »Very efficient

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75 Table Implementation of a DFA S T U 0 1 0 1 0 1 01 STU TTU UTU

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76 Simulation of a DFA l Input. An input string ended with eof and a DFA with start state s0 and final states F. Output. The answer “yes” if accepts, “no” otherwise. begin s := s0; c := next_symbol(); while c <> eof do begin s := mv(s, c); c := next_symbol() end; if s is in F then return “yes” else return “no” end.

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77 Implementation (Cont.) l NFA -> DFA conversion is at the heart of tools such as flex l But, DFAs can be huge »DFA => optimized DFA : try to decrease the number of states. »not always helpful! l In practice, flex-like tools trade off speed for space in the choice of NFA and DFA representations

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78 Time-Space Tradeoffs l RE to NFA, simulate NFA »time: O(|r| * |x|), space: O(|r|) l RE to NFA, NFA to DFA, simulate DFA »time: O(|x|), space: O(2|r|) l Lazy transition evaluation »transitions are computed as needed at run time; »computed transitions are stored in cache for later use

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79 DFA to optimized DFA

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80 Motivations Problems: 1. Given a DFA M with k states, is it possible to find an equivalent DFA M’ (I.e., L(M) = L(M’)) with state number fewer than k ? 2. Given a regular language A, how to find a machine with minimum number of states ? Ex: A = L((a+b)*aba(a+b)*) can be accepted by the following NFA: By applying the subset construction, we can construct a DFA M2 with 2 4 =16 states, of which only 6 are accessible from the initial state {s}. stuv ab a a,b

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81 Inaccessible states l A state p Q is said to be inaccessible (or unreachable) [from the initial state] if there exists no path from from the initial state to it. If a state is not inaccessible, it is accessible. l Inaccessible states can be removed from the DFA without affecting the behavior of the machine. l Problem: Given a DFA (or NFA), how to find all inaccessible states ?

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82 Finding all accessible states: (like e-closure) l Input. An FA (DFA or NFA) l Output. the set of all accessible states begin push all start states onto stack; Add all start states into A; while stack is not empty do begin pop t, the top element, off of stack; for each state u with an edge from t to u do if u is not in A do begin add u to A; push u onto stack end end; return A end.

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83 Minimization process l Minimization process for a DFA: »1. Remove all inaccessible states »2. Collapse all equivalent states l What does it mean that two states are equivalent? »both states have the same observable behaviors.i.e., »there is no way to distinguish their difference, or »more formally, we say p and q are not equivalent(or distinguishable) iff there is a string x * s.t. exactly one of (p,x) and (q,x) is a final state, »where (p,x) is the ending state of the path from p with x as the input. l Equivalents sates can be merged to form a simpler machine.

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84 0 1 24 3 a a a,b a b bb 5 0 5 1,23,4 a,b Example:

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85 Quotient Construction M=(Q, , ,s,F): a DFA. : a relation on Q defined by: p q for all x * (p,x) F iff (q,x) F Property: is an equivalence relation. l Hence it partitions Q into equivalence classes [p] = {q Q | p q} for p Q. and the quotient set Q/ = {[p] | p Q}. Every p Q belongs to exactly one class [p] and p q iff [p]=[q]. Define the quotient machine M/ = where »Q’=Q/ ; s’=[s]; F’={[p] | p F}; and ’([p],a)=[ (p,a)] for all p Q and a .

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86 Minimization algorithm l input: a DFA l output: a optimized DFA 1. Write down a table of all pairs {p,q}, initially unmarked. 2. mark {p,q} if p ∈ F and q F or vice versa. 3. Repeat until no more change: 3.1 if ∃ unmarked pair {p,q} s.t. {move(p,q), move(q,a)} is marked for some a ∈ S, then mark {p,q}. 4. When done, p q iff {p,q} is not marked. 5. merge all equivalent states into one class and return the resulting machine

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87 An Example: l The DFA: ab >012 1F1F34 2F2F43 355 455 5F5F55

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88 Initial Table 1- 2-- 3--- 4---- 5----- 01234

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89 After step 2 1M 2M- 3-MM 4-MM- 5M--MM 01234

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90 After first pass of step 3 1M 2M- 3-MM 4-MM- 5MMMMM 01234

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91 2nd pass of step 3. The result : 1 2 and 3 4. 1M 2M- 3M2MM 4 MM- 5MM1 MM 01234

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