CS252: Systems Programming

Ninghui Li

Topic 5: Parsing

Prepared by Evan Hanau

ehanau@purdue.edu

Introduction to Parsing with Yacc

An Introduction to Parsing with Yacc

• Context-Free Grammars• Yacc Parsing• An example Infix Calculator Program

Context-Free Grammar

Background: The Context-Free Grammar

• By CS252 you are already somewhat familiar with Regular Expressions.

• Regular expressions can be used to describe regular languages, which belong to a larger classification of language types.

In CS, we classify languages on the Chomsky Hierarchy.

Type-0 Recursively EnumerableType-1 Context-SensitiveType-2 Context-Free Type-3 Regular

Context-Free Grammar

Type-(i) is a superset of Type-(i+1)

• Languages generated by regular expressions belong to type 3.

• Note: Your specific regular expression engine (e.g. POSIX extended RE) is likely capable of more complex productions.

• In any case, we need more than regular expressions to parse computer programming languages and shell scripts.

Context-Free Grammar

• You can do a great deal with regular expressions.

• Exercise: Create a regular expression that matches on any English phrase that is a palindrome, for instance the string “some men interpret nine memos”.

Context-Free Grammar

• This is in fact not possible with regex (by its strict CS definition!). You would be limited to palindromes of a finite length only.• RE cannot express “anbn”, a string with some number of

a’s followed by equal number of b’s• The expression a*b* does not require number of a’s

equal that of b’s

• We must use a context-free grammar to describe palindromes and other constructs.

• More powerful than a regular expression, and useful when some notion of “what came before” is required.

Context-Free Grammar

• BNF or Backus-Naur form is used in CS to describe context-free grammars. It is often used to describe the syntax of programming languages.consists of one or more of the following:

<nonterminal> ::= __expression__

• Where __expression__ consists of one or more terminals and nonterminals or nothing (epsilon).

Backus-Naur form

<postal-address> ::= <name-part> <street-address> <zip-part>

<name-part> ::= <personal-part> <last-name> <opt-suffix-part> <EOL>

| <personal-part> <name-part>

<personal-part> ::= <first-name> | <initial> "."

<street-address> ::= <house-num> <street-name> <opt-apt-num> <EOL>

<zip-part> ::= <town-name> "," <state-code> <ZIP-code> <EOL>

<opt-suffix-part> ::= "Sr." | "Jr." | <roman-numeral> | ""

<opt-apt-num> ::= <apt-num> | ""

US Post Address in Backus-Naur form (from wikipedia)

Let’s define a grammar for a primitive add or multiply expression:

<expr> ::= <expr> * <expr> | <expr> + <expr>

| number

In this case, <expr> is a terminal and number, *, and + are the nonterminals.

Context-Free Grammar for Simple Expressions

• Clearly, there is some ambiguity here, because operator precedence (sometimes referred to as binding) is not defined.

• The grammar does not distinguish between 2+2*2+2 = 16 (incorrect under normal rules) or 2+2*2+2 = 8 (correct).

Context-Free Grammar

One Solution: Define expressions of different levels:<expr> ::= <add_expr><add_expr> ::= <add_expr> + <mul_expr> | <mul_expr><mul_expr> ::= <mul_expr> * number | number

Now, multiplication will bind tighter than addition (this may require a few sample expressions to wrap your head around!)

Context-Free Grammar

Associativity follows from the above example (Hint: What side of the multiply and add operation did we have the “deeper” production on?)

Context-Free Grammar

<palindrome> ::= letter

| // empty string

<palindrome> ::= “a” <palindrome> “a”

| “b” <palindrome> “b”

….

Or, <palindrome> ::= letter <palindrome> letter

However, we need to check the two letters are the same.

CFG for anbn:

<anbn> = // empty string | “a” <anbn> “b”

CFG for palindrome

Chomsky Hierarchy (From Wikipedia Page)

Grammar Languages AutomatonProduction rules

(constraints)

Type-0 Recursively enumerable

Turing machine  (no restrictions)

Type-1 Context-sensitive Linear-bounded non-deterministic Turing machine

(equivalently, Right side no shorter than left)

Type-2 Context-freeNon-deterministic pushdown automaton

Type-3 Regular Finite state automaton

and

Type-0 Recursively EnumerableType-1 Context-Sensitive

Cannot encode all strings r1r2 such that r1 and r2 are two regular expressions that are equivalentType-2 Context-Free (Pushdown Automaton, i.e, Finite State Automaton with a Stack)

Can encode anbn, but not anbncn

Type-3 Regular (Finite State Automaton)

Can encode a*b*, but not anbn

Chomsky Hierarchy Revisited.

Why is Context-Free Grammar Called Context Free?

In a CFG, the left hand of each production is a single non-terminal, e.g., <palindrome> ::= “a” <palindrome> “a”

This means that “a”, followed by a <palindrome>, and by “a” will always be considered as <palindrome>, no matter what is the context, hence context free.

In a Context-Sensitive Grammar, left hand of production rules can include other things

An Example Context-Sensitive Grammar for anbncn

1. S a B C

2. S a S B C

3. B C C B

4. a B a b

5. b B b b

6. b C b c

7. c C c c

S→2 aSBC →1 aaBCBC

→3 aaBBCC

→4 aabBCC

→5 aabbCC

→6 aabbcC

→7 aabbcc

Yacc & Parsing

There are many ways to parse BNF grammars, most of which are discussed in a compilers course.• Recall: A finite state automaton (FSA) is

used for regular expressions. (CS182).• For a context-free grammar, we use a

pushdown automaton, which combines features of a FSA with a stack.

Yacc generates what is known as a LALR parser, which is generated from the BNF grammar in your Yacc file. This parser is defined in the C source file that Yacc generates.

We use Lex to make a lexer to generate our terminals, which are matched with regular expressions before being fed into the parser.

Yacc & Parsing

Yacc is capable of generating a powerful parser that will handle many different grammars.

LEX Lexer

YACC Parser

Rule-Based

Behavior

terminals

Input Characters

Yacc & Parsing

• Recall that parsing combines a state machine with a stack. States go on a stack to keep track of where parsing is. Yacc uses a parse table which defines possible states.

• Yacc’s parser operates using two primary actions, shift and reduce.

• shift puts a state on the stack, reduce pops state(s) off the stack and reduces combinations of nonterminals and terminals to a single nonterminal. After a reduction to a rule, Yacc’s parser will optionally run some user-defined code.

Yacc & Parsing

A very basic example:

<rule> := “hello” “world” “\n”

The parser would shift each word, successively pushing each state (.”hello”, .”world”, .”\n”) onto the stack. Then at the end of the rule, reduce everything to <rule> and pop the three states.

Yacc & Parsing

A Lex/Yacc Infix Calculator

• Yacc’s parser is powerful, but is not capable of parsing all grammars.

• Certain ambiguous grammars may produce what is known as a shift/reduce or reduce/reduce conflict. Yacc will, by default, shift instead of reduce.

Consider the classic shift/reduce conflict example:

<ifexp> ::= IF <expr> THEN <stmt> ELSE <stmt> | IF <expr> THEN <stmt>

Yacc will have a shift/reduce conflict here, but will go with shift (the top option) by default. It’s greedy!

Yacc & Parsing

A Lex/Yacc Infix Calculator

• To demonstrate the utility of Lex and Yacc (or in our case, Flex and Bison) we provide an example infix calculator.

• Similar to several of the examples provided on the Lex and Yacc manpage at http://dinosaur.compilertools.net, but with added features

A Lex/Yacc Infix Calculator

• Make sure to read ALL source code comments, particularly those that describe source file organization.

• Lex definition file: calculator.l• Yacc grammar file: calculator.y• AST Classes: ast.cc• Symbol table: symtab.cc

A Lex/Yacc Infix Calculator

The example calculator application uses Lex and Yacc to parse mathematical expressions and produce an Abstract Syntax Tree, which is then used to evaluate those expressions.

It allows the =, +, *, -, +, ^, () and unary minus operators, with appropriate levels of binding and precedence. Examine calculator.y, because it is heavily commented.

A Lex/Yacc Infix Calculator

• The symbol table (implemented here in simple O(n) access time) maps variables to values.

• Print the AST after every expression evaluation by running calculator with the –t flag, e.g. “calc –t”.

calc> 2*2^3/3 = 5.333333 3.000 /(/) \ 3.000 / (^) \ 2.000 / (*) \ 2.000

A calculator example. Type “2*2^3/3 and press enter:

A Lex/Yacc Infix Calculator

Review

• What are required:• Able to write simple Context Free

Grammars, similar to those used in implementing FIZ

• Able to determine whether a string of tokens is accepted by a grammar

• Able to show how a string of tokens is parsed into some non-terminal (i.e., draw the parsing tree)