Lecture 4 Concepts of Programming Languages Arne Kutzner Hanyang University / Seoul Korea

Concepts of Programming Languages Topics Lexical Analysis The Parsing Problem Recursive-Descent Parsing Bottom-Up Parsing Concepts of Programming Languages

Concepts of Programming Languages Introduction Language implementation systems must analyze source code, regardless of the specific implementation approach Nearly all syntax analysis is based on a formal description of the syntax of the source language Concepts of Programming Languages

Advantages of Using CFG/BNF to Describe Syntax Provides a clear and concise syntax description A parser can be constructed on foundation of CFG/BNF Concepts of Programming Languages

Concepts of Programming Languages Syntax Analysis The syntax analysis portion of a language processor nearly always consists of two parts: A low-level part called a lexical analyzer (mathematically, a finite automaton based on a regular grammar) A high-level part called a syntax analyzer, or parser (mathematically, a push-down automaton based on a context-free grammar, or BNF) Concepts of Programming Languages

Concepts of Programming Languages Lexical Analysis A lexical analyzer is a “front-end” for the parser pattern matcher for character strings Identifies substrings of the source program that belong together - lexemes Lexemes match a character pattern, which is associated with a lexical category called a token sum is a lexeme; its token may be IDENT Concepts of Programming Languages

Reasons to Separate Lexical and Syntax Analysis Simplicity - less complex approaches can be used for lexical analysis (no need for the use of grammars for token extraction) Efficiency - separation allows significant less complex parsers Concepts of Programming Languages

We need first some theory…

Concepts of Programming Languages Regular Expressions Given a finite alphabet Σ, the following constants are defined as regular expressions: (empty set) Ø denoting the set Ø. (empty string) ε denoting the set containing only the "empty" string, which has no characters at all. (literal character) a in Σ denoting the set containing only the character a. Concepts of Programming Languages

Regular Expressions (cont.) Given regular expressions R and S, the following operations over them are defined to produce regular expressions: (concatenation) RS denotes the set of strings that can be obtained by concatenating a string in R and a string in S. For example {"ab", "c"}{"d", "ef"} = {"abd", "abef", "cd", "cef"}. (alternation) R | S denotes the set union of sets described by R and S. For example, if R describes {"ab", "c"} and S describes {"ab", "d", "ef"}, expression R | S describes {"ab", "c", "d", "ef"}. Alternation is sometimes denoted by + Concepts of Programming Languages

Regular Expressions (cont.) 3. (Kleene star) R* denotes the smallest superset of set described by R that contains ε and is closed under string concatenation. This is the set of all strings that can be made by concatenating any finite number (including zero) of strings from set described by R. For example, {"0","1"}* is the set of all finite binary strings (including the empty string), and {"ab", "c"}* = {ε, "ab", "c", "abab", "abc", "cab", "cc", "ababab", "abcab", ... }. Concepts of Programming Languages

Concepts of Programming Languages Regular Languages The collection of regular languages over an alphabet Σ is defined recursively as follows: The empty language Ø is a regular language. For each a ∈ Σ (a belongs to Σ), the singleton language {a} is a regular language. If A and B are regular languages, then A ∪ B (union), A • B (concatenation), and A* (Kleene star) are regular languages. No other languages over Σ are regular. Concepts of Programming Languages

Regular Expressions and Regular Languages The family of languages defined by regular expressions are the regular languages. Regular expressions can be used for lexeme/token description/specification. E.g.: description of a token Identifier as Letter (Digit | Letter)* Regular expressions are generators like grammars In fact, you can describe every regular expressions by means of a grammar Concepts of Programming Languages

Examples of regular expressions What are the words of the following expressions? (0 | 1)(00 | 01 | 10 | 11)* (0 | 1)(0 | 1)(0 | 1)(0 | 1)(0 | 1) Are languages of the following 3 expressions 0* 1 (0 | 1)* (0 | 1)* 1 (0 | 1)* (0 | 1)* 1 0* equal? Concepts of Programming Languages

Recognizer for Regular Expressions A deterministic finite automaton (DFA) M is a 5-tuple, (Q, Σ, δ, q0, F), consisting of a finite set of states (Q) a finite set of input symbols called the alphabet (Σ) a transition function (δ : Q × Σ → Q) a start state (q0 ∈ Q) a set of accept states (F ⊆ Q) Concepts of Programming Languages

Language accepted by a DFA Let w = a1a2 ... an be a string over the alphabet Σ. The automaton M accepts the string w if a sequence of states, r0,r1, ..., rn, exists in Q with the following conditions: r0 = q0 ri+1 = δ(ri, ai+1), for i = 0, ..., n−1 rn ∈ F. Concepts of Programming Languages

Concepts of Programming Languages DFA Example M = (Q, Σ, δ, q0, F) where Q = {S1, S2}, Σ = {0, 1}, q0 = S1, F = {S1}, δ is the following state transition table: corresponding state diagram for M Concepts of Programming Languages

Concepts of Programming Languages DFA Example (cont.) The Language recognized by M is the regular language given by the regular expression 1*( 0 1* 0 1* )*, The accepted language consists of all words that contains an even number of 0s. Concepts of Programming Languages

Concepts of Programming Languages Kleene’s Theorem Part 1: If R is regular expression over the alphabet Σ, and L is the language in Σ* corresponding to R, then there is a (deterministic) finite automaton M recognizing L. Part 2: If M = (Q, Σ, δ, q0, F) is a (deterministic) finite automaton recognizing the language L, then there is a regular expression over Σ corresponding to L. So, DFAs recognize exactly the set of regular languages/expressions. Concepts of Programming Languages

Limitations of regular languages There is no regular expression for the language 1n 0n , n ≥ 0 (n ones followed by n zeros) But you can easily give a CFG for the above language: <A> -> 1 <A> 0 | ε Other example: Dyck language; balanced strings of parentheses (e.g. [ [ ] [ [ ] ] ] [ ] ) Grammar ? (-> Exercise) Concepts of Programming Languages

Practical implementation of lexical analyzers DFAs and regular expressions are the foundations of lexical analyzer construction Possible approaches for implementing a lexical analyzer: Write a formal description of the tokens and use a software tool that constructs table-driven lexical analyzers given such a description Design a state diagram that describes the tokens and write a program that implements the state diagram Design a state diagram that describes the tokens and hand-construct a table-driven implementation of the state diagram Concepts of Programming Languages

Lexical Analysis (cont.) In many cases, symbols of transitions are “combined/grouped” in order to simplify the state diagram When recognizing an identifier, all uppercase and lowercase letters are equivalent Use a character class that includes all letters When recognizing an integer literal, all digits are equivalent - use a digit class Concepts of Programming Languages

Lexical Analysis (cont.) Reserved words can be recognized in the context of identifier recognition Use a table lookup to determine whether a possible identifier is in fact a reserved word Concepts of Programming Languages

Lexical Analysis (cont.) Example Program … The proposed lexical analyzer is a function that should be called by the parser when it request a fresh token/lexems Utility subprograms: getChar - gets the next character of input, puts it in nextChar, determines its class and puts the class in charClass addChar - puts the character from nextChar into the place the lexeme is being accumulated, lexeme lookup - determines whether the string in lexeme is a reserved word (returns a code) Concepts of Programming Languages

State diagram for recognizing identifiers and integer numbers Concepts of Programming Languages

Lexical Analysis / Example Prg. int lex() { lexLen = 0; static int first = 1; /* If it is the first call to lex, initialize by calling getChar */ if (first) { getChar(); first = 0; } getNonBlank(); switch (charClass) { /* Parse identifiers and reserved words */ case LETTER: addChar(); while (charClass == LETTER || charClass == DIGIT){ return lookup(lexeme); break; … Concepts of Programming Languages

Lexical Analysis / Example Prg. … /* Parse integer literals */ case DIGIT: addChar(); getChar(); while (charClass == DIGIT) { } return INT_LIT; break; } /* End of switch */ } /* End of function lex */ Concepts of Programming Languages

Parsing Problem…

Concepts of Programming Languages The Parsing Problem Goals of the parser, given an input program: Produce a parse tree Find all syntax errors; for each, produce an appropriate diagnostic message and recover quickly Concepts of Programming Languages

The Parsing Problem (cont.) Two categories of parsers Top down - produce the parse tree, beginning at the root Order is that of a leftmost derivation Traces or builds the parse tree in preorder Bottom up - produce the parse tree, beginning at the leaves Order is that of the reverse of a rightmost derivation Useful parsers look only one token ahead in the input Concepts of Programming Languages

The Parsing Problem (cont.) Top-down Parsers Given a sentential form, xA , the parser must choose the correct A-rule to get the next sentential form in the leftmost derivation, using only the first token produced by A The most common top-down parsing algorithms: Recursive descent - a coded implementation LL parsers - table driven implementation Concepts of Programming Languages

The Parsing Problem (cont.) Bottom-up parsers Special form of push down automata Given a right sentential form, , determine what substring of  is the right-hand side of the rule in the grammar that must be reduced to produce the previous sentential form in the right derivation The most common bottom-up parsing algorithms are in the LR family Concepts of Programming Languages

Recursive-Descent Parsing (cont.) Example grammar in BNF(CFG): <expr>  <term> - <expr> | <term1> <term>  a + <term> | b <term1>  c | ( <expr> ) a + b – (a + b – c) Concepts of Programming Languages

Recursive-Descent Parsing Approach - Coded parser: Subprogram for each nonterminal in the grammar, which can parse sentences that can be generated by that nonterminal EBNF well suited for being the basis of a recursive-descent parser, because EBNF minimizes the number of nonterminals Concepts of Programming Languages

Recursive-Descent Parsing (cont.) A grammar for simple expressions: <expr>  <term> {(+ | -) <term>} <term>  <factor> {(* | /) <factor>} <factor>  id | ( <expr> ) Concepts of Programming Languages

Recursive-Descent Parsing (cont.) Assume we have a lexical analyzer named lex, which puts the next token code in nextToken The coding process when there is only one RHS: For each terminal symbol in the RHS, compare it with the next input token; if they match, continue, else there is an error For each nonterminal symbol in the RHS, call its associated parsing subprogram Concepts of Programming Languages

Recursive-Descent Parsing (cont.) /* Function expr Parses strings in the language generated by the rule: <expr> → <term> {(+ | -) <term>} */ void expr() { /* Parse the first term */   term(); … Concepts of Programming Languages

Recursive-Descent Parsing /* As long as the next token is + or -, call lex to get the next token, and parse the next term */   while (nextToken == PLUS_CODE || nextToken == MINUS_CODE){     lex();     term();   } } This particular routine does not detect errors Convention: Every parsing routine leaves the next token in nextToken Concepts of Programming Languages

Recursive-Descent Parsing (cont.) A nonterminal that has more than one RHS requires an initial process to determine which RHS it is to parse The correct RHS is chosen on the basis of the next token of input (the look-ahead) The next token is compared with the first token that can be generated by each RHS until a match is found If no match is found, it is a syntax error Concepts of Programming Languages

Recursive-Descent Parsing (cont.) /* Function factor Parses strings in the language generated by the rule: <factor> -> id | (<expr>) */ void factor() { /* Determine which RHS */    if (nextToken) == ID_CODE) /* For the RHS id, just call lex */      lex(); Concepts of Programming Languages

Recursive-Descent Parsing (cont.) /* If the RHS is (<expr>) – call lex to pass over the left parenthesis, call expr, and check for the right parenthesis */    else if (nextToken == LEFT_PAREN_CODE) {      lex(); expr();     if (nextToken == RIGHT_PAREN_CODE) lex(); else error(); } /* End of else if (nextToken == ... */ else error(); /* Neither RHS matches */ } Concepts of Programming Languages

Recursive-Descent Parsing (cont.) The Left Recursion Problem: If a grammar comprises left recursion, either direct or indirect, it cannot be the basis of a top-down (recursive-decent) parser A grammar can be modified, so that it becomes free of left recursion LL Grammar Class = Class of grammars without left recursion Concepts of Programming Languages

Elimination of left recursion Direct recursion: For each nonterminal A, Group the A-rules as A → Aα1 | … | Aαm | β1 | β2 | … | βn where none of the β‘s begins with A 2. Replace the original A-rules with A → β1A’ | β2A’ | … | βnA’ A’ → α1A’ | α2A’ | … | αmA’ | ε Indirect recursion: See separated PDF-document Concepts of Programming Languages

Recursive-Descent Parsing (cont.) The other characteristic of grammars that disallows top-down parsing is the lack of pairwise disjointness The inability to determine the correct RHS on the basis of one token of lookahead Def: FIRST() = {a |  =>* a } (If  =>* ,  is in FIRST()) Concepts of Programming Languages

Recursive-Descent Parsing (cont.) Pairwise Disjointness Test: For each nonterminal, A, in the grammar that has more than one RHS, for each pair of rules, A  i and A  j, it must be true that FIRST(i) ⋂ FIRST(j) =  Examples: A  a | bB | cAb A  a | aB Concepts of Programming Languages

Recursive-Descent Parsing (cont.) Left factoring can be used for removing pairwise disjointness. Example: <variable>ident | ident'('<expression>')' left factor to: <variable>  ident <new> <new>   | '('<expression>')' or in EBNF: <variable>  ident [ '('<expression>')' ] Problem with first transformation: Introduction of  rule. (Troublemaker in the context of the elimination of left recursion) Concepts of Programming Languages

Concepts of Programming Languages Bottom-up Parsing The parsing problem is finding the correct RHS in a right-sentential form to reduce to get the previous right-sentential form in the derivation Bottom-up parser represent an extended form of push down automata. Concepts of Programming Languages

Definition PushDown Automaton A PDA is formally defined as a 7-tuple (Q, Σ, Γ, δ, q0, Z, F), where Q is a finite set of states Σ is a finite set which is called the input alphabet Γ is a finite set which is called the stack alphabet δ : Q × (Σ{ε}) × Γ → Q × Γ* , the transition function q0 ∈ Q is the start state Z ∈ Γ is the initial stack symbol F ⊆ Q is the set of accepting states Concepts of Programming Languages

Concepts of Programming Languages PDA computation Assume δ of M maps (p,a,A) to (q,α) and that M is in state p∈Q, with a ∈(Σ{ε}) on input and A∈ Γ as topmost stack symbol, Then M performs the following actions: may read a (move one position right on input) change the state to q pop A, replacing it by α IMPORTANT: The (Σ{ε}) component of the transition relation is used to formalize that the PDA can either read a letter from the input, or proceed leaving the input untouched. Concepts of Programming Languages

PDA computation graphically Concepts of Programming Languages

Concepts of Programming Languages Example PDA M=(Q, Σ, Γ, δ, p, Z, F), where states: Q = { p,q,r } input alphabet: Σ = {0, 1} stack alphabet: Γ = {A, Z} start state: q0 = p start stack symbol: Z accepting states: F = {r} Move number State Input Stack symbol Moves 1 p Z p, AZ 2 A p, AA 3 ε q, Z 4 q, A 5 q q, ε 6 r, Z Concepts of Programming Languages

Language of example PDA PDA for language {0n1n | n ≥ 0} Corresponding grammar: <A> -> 0 <A> 1 | ε Concepts of Programming Languages

Concepts of Programming Languages Important Lemmas For every grammar G there is a pushdown automaton M, so that the language generated by G is recognized by the automaton M. For very PDA M there is a grammar G, so that language recognized by M is generated by the grammar G. PDA and context free grammars are equal concepts with respect to its recognized/generated languages. Concepts of Programming Languages

Bottom-up Parsing / Handles Definitions of Handle / Phrase / Simple Phrase:  is the handle of the right sentential form  = w if and only if S =>*rm Aw =>rm w  is a phrase of the right sentential form  if and only if S =>*  = 1A2 =>+ 12  is a simple phrase of the right sentential form  if and only if S =>*  = 1A2 => 12 Concepts of Programming Languages

Bottom-up Parsing (cont.) Shift-Reduce Algorithms Reduce is the action of replacing the handle on the top of the parse stack with its corresponding LHS Shift is the action of moving the next token to the top of the parse stack Concepts of Programming Languages

Bottom-up Parsing (cont.) Advantages of LR parsers: They will work for nearly all grammars that describe programming languages. They can detect syntax errors as soon as it is possible. The LR class of grammars is a superset of the class parsable by LL parsers. Concepts of Programming Languages

Bottom-up Parsing (cont.) LR parsers must be constructed with a tool Knuth’s insight: A bottom-up parser could use the entire history of the parse, up to the current point, to make parsing decisions There were only a finite and relatively small number of different parse situations that could have occurred, so the history could be stored in a parser state, on the parse stack Concepts of Programming Languages

Bottom-up Parsing (cont.) An LR configuration stores the state of an LR parser (S0X1S1X2S2…XmSm, aiai+1…an$) Concepts of Programming Languages

Bottom-up Parsing (cont.) LR parsers are table driven, where the table has two components, an ACTION table and a GOTO table The ACTION table specifies the action of the parser, given the parser state and the next token Rows are state names; columns are terminals The GOTO table specifies which state to put on top of the parse stack after a reduction action is done Rows are state names; columns are nonterminals Concepts of Programming Languages

Structure of An LR Parser Concepts of Programming Languages

Bottom-up Parsing (cont.) Initial configuration: (S0, a1…an$) Parser actions: If ACTION[Sm, ai] = Shift S, the next configuration is: (S0X1S1X2S2…XmSmaiS, ai+1…an$) If ACTION[Sm, ai] = Reduce A   and S = GOTO[Sm-r, A], where r = the length of , the next configuration is (S0X1S1X2S2…Xm-rSm-rAS, aiai+1…an$) Concepts of Programming Languages

Bottom-up Parsing (cont.) Parser actions (continued): If ACTION[Sm, ai] = Accept, the parse is complete and no errors were found. If ACTION[Sm, ai] = Error, the parser calls an error-handling routine. Concepts of Programming Languages

Concepts of Programming Languages LR Parsing Table S4 Concepts of Programming Languages

Bottom-up Parsing (cont.) A parser table can be generated from a given grammar with a tool, e.g., yacc Concepts of Programming Languages