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Winter 2007SEG2101 Chapter 81 Chapter 8 Lexical Analysis

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Winter 2007SEG2101 Chapter 82 Contents The role of the lexical analyzer Specification of tokens Finite state machines From a regular expressions to an NFA Convert NFA to DFA Transforming grammars and regular expressions Transforming automata to grammars Language for specifying lexical analyzers

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Winter 2007SEG2101 Chapter 83 The Role of Lexical Analyzer Lexical analyzer is the first phase of a compiler. Its main task is to read input characters and produce as output a sequence of tokens that parser uses for syntax analysis.

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Winter 2007SEG2101 Chapter 84 Issues in Lexical Analysis There are several reasons for separating the analysis phase of compiling into lexical analysis and parsing: –Simpler design –Compiler efficiency –Compiler portability Specialized tools have been designed to help automate the construction of lexical analyzer and parser when they are separated.

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Winter 2007SEG2101 Chapter 85 Tokens, Patterns, Lexemes A lexeme is a sequence of characters in the source program that is matched by the pattern for a token. A lexeme is a basic lexical unit of a language comprising one or several words, the elements of which do not separately convey the meaning of the whole. The lexemes of a programming language include its identifier, literals, operators, and special words. A token of a language is a category of its lexemes. A pattern is a rule describing the set of lexemes that can represent as particular token in source program.

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Winter 2007SEG2101 Chapter 86 Examples of Tokens const pi = 3.1416; The substring pi is a lexeme for the token “identifier.”

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Winter 2007SEG2101 Chapter 87 Lexeme and Token semicolon; int_literal17 plus_op+ identifierCount multi_op* int_literal2 equal_sign= IdentifierIndex TokensLexemes Index = 2 * count +17;

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Winter 2007SEG2101 Chapter 88 Lexical Errors Few errors are discernible at the lexical level alone, because a lexical analyzer has a very localized view of a source program. Let some other phase of compiler handle any error. Panic mode Error recovery

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Winter 2007SEG2101 Chapter 89 Specification of Tokens Regular expressions are an important notation for specifying patterns. Operation on languages Regular expressions Regular definitions Notational shorthands

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Winter 2007SEG2101 Chapter 810 Operations on Languages

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Winter 2007SEG2101 Chapter 811 Regular Expressions Regular expression is a compact notation for describing string. In Pascal, an identifier is a letter followed by zero or more letter or digits letter(letter|digit)* | : or * : zero or more instance of a(a|d) *

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Winter 2007SEG2101 Chapter 812 Rules is a regular expression that denotes { }, the set containing empty string. If a is a symbol in , then a is a regular expression that denotes {a}, the set containing the string a. Suppose r and s are regular expressions denoting the language L(r) and L(s), then –(r) |(s) is a regular expression denoting L(r) L(s). –(r)(s) is regular expression denoting L (r) L(s). –(r) * is a regular expression denoting (L (r) )*. –(r) is a regular expression denoting L (r).

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Winter 2007SEG2101 Chapter 813 Precedence Conventions The unary operator * has the highest precedence and is left associative. Concatenation has the second highest precedence and is left associative. | has the lowest precedence and is left associative. (a)|(b)*(c) a|b*c

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Winter 2007SEG2101 Chapter 814 Example of Regular Expressions

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Winter 2007SEG2101 Chapter 815 Properties of Regular Expression

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Winter 2007SEG2101 Chapter 816 Regular Definitions If is an alphabet of basic symbols, then a regular definition is a sequence of definitions of the form: d 1 r 1 d 2 r 2... d n r n where each d i is a distinct name, and each r i is a regular expression over the symbols in {d 1,d 2,…,d i-1 }, i.e., the basic symbols and the previously defined names.

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Winter 2007SEG2101 Chapter 817 Examples of Regular Definitions Example 3.5. Unsigned numbers

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Winter 2007SEG2101 Chapter 818 Notational Shorthands

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Winter 2007SEG2101 Chapter 819 Finite Automata A recognizer for a language is a program that takes as input a string x and answer “yes” if x is a sentence of the language and “no” otherwise. We compile a regular expression into a recognizer by constructing a generalized transition diagram called a finite automaton. A finite automaton can be deterministic or nondeterministic, where nondeterministic means that more than one transition out of a state may be possible on the same input symbol.

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Winter 2007SEG2101 Chapter 820 Nondeterministic Finite Automata (NFA) A set of states S A set of input symbols A transition function move that maps state- symbol pairs to sets of states A state s 0 that is distinguished as the start (initial) state A set of states F distinguished as accepting (final) states.

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Winter 2007SEG2101 Chapter 821 NFA An NFA can be represented diagrammatically by a labeled directed graph, called a transition graph, in which the nodes are the states and the labeled edges represent the transition function. (a|b)*abb

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Winter 2007SEG2101 Chapter 822 NFA Transition Table The easiest implementation is a transition table in which there is a row for each state and a column for each input symbol and , if necessary.

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Winter 2007SEG2101 Chapter 823 Example of NFA

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Winter 2007SEG2101 Chapter 824 Deterministic Finite Automata (DFA) A DFA is a special case of a NFA in which –no state has an -transition –for each state s and input symbol a, there is at most one edge labeled a leaving s.

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Winter 2007SEG2101 Chapter 825 Simulating a DFA

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Winter 2007SEG2101 Chapter 826 Example of DFA

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Winter 2007SEG2101 Chapter 827 Conversion of an NFA into DFA Subset construction algorithm is useful for simulating an NFA by a computer program. In the transition table of an NFA, each entry is a set of states; in the transition table of a DFA, each entry is just a single state. The general idea behind the NFA-to-DFA construction is that each DFA state corresponds to a set of NFA states. The DFA uses its state to keep track of all possible states the NFA can be in after reading each input symbol.

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Winter 2007SEG2101 Chapter 828 Subset Construction - constructing a DFA from an NFA Input: An NFA N. Output: A DFA D accepting the same language. Method: We construct a transition table Dtran for D. Each DFA state is a set of NFA states and we construct Dtran so that D will simulate “in parallel” all possible moves N can make on a given input string.

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Winter 2007SEG2101 Chapter 829 Subset Construction (II) s represents an NFA state T represents a set of NFA states.

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Winter 2007SEG2101 Chapter 830 Subset Construction (III)

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Winter 2007SEG2101 Chapter 831 Subset Construction (IV) (ε-closure computation)

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Winter 2007SEG2101 Chapter 832 Example

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Winter 2007SEG2101 Chapter 833 Example (II)

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Winter 2007SEG2101 Chapter 834 Example (III)

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Winter 2007SEG2101 Chapter 835 Minimizing the number of states in DFA Minimize the number of states of a DFA by finding all groups of states that can be distinguished by some input string. Each group of states that cannot be distinguished is then merged into a single state. Algorithm 3.6 Aho page 142

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Winter 2007SEG2101 Chapter 836 Minimizing the number of states in DFA (II)

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Winter 2007SEG2101 Chapter 837 Construct New Partition An example in class

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Winter 2007SEG2101 Chapter 838 From Regular Expression to NFA Thompson’s construction - an NFA from a regular expression Input: a regular expression r over an alphabet . Output: an NFA N accepting L(r)

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Winter 2007SEG2101 Chapter 839 Method First parse r into its constituent subexpressions. Construct NFA’s for each of the basic symbols in r. –for –for a in

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Winter 2007SEG2101 Chapter 840 Method (II) For the regular expression s|t, For the regular expression st,

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Winter 2007SEG2101 Chapter 841 Method (III) For the regular expression s*, For the parenthesized regular expression (s), use N(s) itself as the NFA. Every time we construct a new state, we give it a distinct name.

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Winter 2007SEG2101 Chapter 842 Example - construct N(r) for r=(a|b)*abb

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Winter 2007SEG2101 Chapter 843 Example (II)

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Winter 2007SEG2101 Chapter 844 Example (III)

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Winter 2007SEG2101 Chapter 845 Regular Expressions Grammars More in class …

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Winter 2007SEG2101 Chapter 846 Grammars Regular Expressions More in class …

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Winter 2007SEG2101 Chapter 847 Automata Grammars More in class …

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Winter 2007SEG2101 Chapter 848 A Language for Specifying Lexical Analyzer yylex()

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Winter 2007SEG2101 Chapter 849 Simple Lex Example int num_lines = 0, num_chars = 0; % \n ++num_lines; ++num_chars;. ++num_chars; % main() { yylex(); printf( "# of lines = %d, # of chars = %d\n", num_lines, num_chars ); }

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Winter 2007SEG2101 Chapter 850 %{ #include /* need this for the call to atof() below */ #include /* need this for printf(), fopen() and stdin below */ %} DIGIT [0-9] ID [a-z][a-z0-9]* % {DIGIT}+ { printf("An integer: %s (%d)\n", yytext, atoi(yytext)); } {DIGIT}+"."{DIGIT}* { printf("A float: %s (%g)\n", yytext, atof(yytext)); } if|then|begin|end|procedure|function { printf("A keyword: %s\n", yytext); } {ID} printf("An identifier: %s\n", yytext); "+"|"-"|"*"|"/" printf("An operator: %s\n", yytext); "{"[^}\n]*"}" /* eat up one-line comments */ [ \t\n]+ /* eat up white space */. printf("Unrecognized character: %s\n", yytext); % int main(int argc, char *argv[]){ ++argv, --argc; /* skip over program name */ if (argc > 0) yyin = fopen(argv[0], "r"); else yyin = stdin; yylex(); }

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