# CS 314 Principles of Programming Languages

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1 CS 314 Principles of Programming Languages Lecture 5: Syntax Analysis (Parsing) Zheng (Eddy) Zhang Rutgers University January 31, 2018

2 Class Information Homework 1 is being graded now. The sample solution will be posted soon. Homework 2 posted. Due in one week. 2

3 Review: Context Free Grammars (CFGs) A formalism to for describing languages A CFG G = < T, N, P, S >: 1. A set T of terminal symbols (tokens). 2. A set N of nonterminal symbols. 3. A set P production (rewrite) rules. 4. A special start symbol S. The language L(G) is the set of sentences of terminal symbols in T* that can be derived from the start symbol S: L(G) = {w T* S * w} 3

4 Review: Context Free Grammar <if-stmt> ::= if <expr> then <stmt> <expr> ::= id <= id <stmt> ::= id := num Rule 1 \$1 1& Rule 2 \$0 0\$ Rule 3 &1 1\$ Rule 4 &0 0& Rule 5 \$# A Rule 6 &# B Context free grammar Not a context free grammar CFGs are rewrite systems with restrictions on the form of rewrite (production) rules that can be used. The left hand side of a production rule can only be one non-terminal symbol. 4

5 A Language May Have Many Grammars Consider G : The Original Grammar G: 1. < letter > ::= A B C Z 2. < digit > ::= < ident > ::= < letter > 1. < letter > ::= A B C Z 2. < digit > ::= < identifier > ::= < letter > 4. < ident > < letterordigit > 4. < identifier > < letter > 5. < stmt > ::= < ident > := 0 5. < identifier > < digit > 6. < letterordigit > ::= < letter > < digit > 6. < stmt > ::= < identifier > := 0 <stmt> <ident> := 0 X2 := 0 <stmt> <identifier> := 0 <ident> <letterordigit> <identifier> <digit> <letter> <digit> <letter> 2 X 2 X 5

6 Review: Grammars and Programming Languages Many grammars may correspond to one programming language. Good grammars: Captures the logic structure of the language structure carries some semantic information (example: expression grammar) Use meaningful names Are easy to read Are unambiguous 6

7 Review: Ambiguous Grammars Time flies like an arrow; fruit flies like a banana. A grammar G is ambiguous iff there exists a w L(G) such that there are: two distinct parse trees for w, or two distinct leftmost derivations for w, or two distinct rightmost derivations for w. We want a unique semantics of our programs, which typically requires a unique syntactic structure. 7

8 Review: Arithmetic Expression Grammar Parse 8-3 * 2 : < expr > < start > ::= < expr > < expr > :: = < expr > - < expr > < expr > * < expr > < d > < l > < d > :: = < l > :: = a b c z Two parse trees! < expr > < expr > - < expr > < expr > * < expr > < d > 8 < expr > * < expr > < d > < d > < expr > - < expr > < d > < d > < d >

9 Review: Arithmetic Expression Grammar Parse 8-3 * 2 : Two leftmost derivations! < expr > < expr > - < expr > < expr > < expr > * < expr > < d > < expr > * < expr > < d > < d > < expr > - < expr > < d > < d > < d > leftmost derivation <expr> <expr> - <expr> <d> - <expr> <d> - <expr> * <expr> <d> - <d> * <expr> <d> - <d> * <d> L L L L L leftmost derivation <expr> <expr> * <expr> L L <expr> - <expr> * <expr> L <d> - <expr> * <expr> <d> - <d> * <expr> <d> - <d> * <d> L L 9

10 Review: Ambiguity How to deal with ambiguity? Change the language Example: Adding new terminal symbols as delimiters. Fix the dangling else, expression grammars. Change the grammar Example: Impose associativity and precedence in an arithmetic expression grammar. 10

11 Changing the Grammar to Impose Precedence < start > ::= < expr > < expr > ::= < expr > + < expr > < expr > - < expr > < expr > * < expr > < expr > / < expr > < d > < l > < d > ::= < l > ::= a b c z < start > ::= < expr > < expr > ::= < expr > + < expr > < expr > - < expr > < term > < term > ::= < term > * < term > < term > / < term > < d > < l > < d > ::= < l > ::= a b c z Original Grammar G Modified Grammar G 11

12 Grouping in Parse Tree Now Reflects Precedence Parse 8-3 * 2 : < start > ::= < expr > < expr > < expr > :: = < expr > + < expr > < expr > - < expr > < expr > - < expr > < term > < term > < term > * < term > < term > ::= < term > * < term > < term > / < term > < d > < d > < d > < d > < l > < d > :: = < l > :: = a b c z Modified Grammar G Only One Possible Parse Tree 12

13 Precedence Low Precedence: Addition + and Subtraction - Medium Precedence: Multiplication * and Division / Highest Precedence: Exponentiation ^ < start > ::= < expr > < expr > :: = < expr > + < expr > < expr > - < expr > < term > < term > ::= < term > * < term > < term > / < term > < d > < l > < d > :: = < l > :: = a b c z 13

14 Still Have Ambiguity How about ? < start > OR? < expr > - < expr > < expr > 1 3 < expr > - < expr > 3 2 < start > < expr > < expr > - < expr > 2 1 Eval: (3-2) - 1 = 0 Eval: 3 - (2-1) = 2 < start > ::= < expr > < expr > :: = < expr > + < expr > < expr > - < expr > < term > < term > ::= < term > * < term > < term > / < term > < d > < l > < d > :: = < l > :: = a b c z 14

15 Still Have Ambiguity How about ? OR? < start > < expr > - < expr > < expr > 1 3 < expr > - < expr > < start > - < expr > < expr > - < expr > 2 1 < start > ::= < expr > < expr > :: = < expr > + < expr > < expr > - < expr > < term > < term > ::= < term > * < term > < term > / < term > < d > < l > < d > :: = < l > :: = a b c z 15

16 Still Have Ambiguity Grouping of operators of same precedence not disambiguated. Non-commutative operators: only one parse tree correct. < start > ::= < expr > < expr > :: = < expr > + < expr > < expr > - < expr > < term > < term > ::= < term > * < term > < term > / < term > < d > < l > < d > :: = < l > :: = a b c z 16

17 Imposing Associativity Same grammar with left / right recursion for - : Our choices: < expr > :: = < d > - < expr > < d > < d > :: = Or: < expr > :: = < expr > - < d > < d > < d > :: = <expr> < d > - <expr> < d > - < d > - <expr> < d > - < d > - < d > - <expr> <expr> <expr> - < d > <expr> - < d > - < d > <expr> - < d > - < d > - <d> Which one do we want for - in the calculator language? 17

18 Associativity Deals with operators of same precedence Implicit grouping or parenthesizing Left to right: *, /, +, - Right to left: ^ 18

19 Complete, Unambiguous Arithmetic Expression Grammar < start > ::= < expr > < expr > :: = < expr > + < expr > < expr > - < expr > < expr > * < expr > < expr > / < expr > < expr > ^ < expr > < d > < l > < d > :: = < l > :: = a b c z Original Ambiguous Grammar G < start > ::= < expr > < expr > ::= < expr > + < term > < expr > < term > < term > < term > ::= < term > * < factor > < term > / < factor > < factor > < factor >::= < g > ^ < factor > < g > < g > ::= ( < expr > ) < d > < l > < d > ::= < l > ::= a b c z Unambiguous Grammar G 19

20 Abstract versus Concrete Syntax Concrete Syntax: Representation of a construct in a particular language, including placement of keywords and delimiters. Abstract Syntax: Structure of meaningful components of each language construct. 20

21 Example: Consider A * B C: Concrete Syntax Tree < S > < E > < E > < T > < S > ::= < E > < E > ::= < E > < T > < T > < T > ::= < T > * id id Abstract Syntax Tree (AST) < T > < T > * id id id id * id id 21

22 Abstract versus Concrete Syntax Same abstract syntax, different concrete syntax: Pascal while x <> A[i], do i := i + 1 end C while ( x!= A[i]) i = i + 1; 22

23 Regular vs. Context Free All Regular languages are context free languages Not all context free languages are regular languages Example: N ::= X Y X ::= a X b Y ::= c Y c is equivalent to: a b* c + Question: Is {a n b n n 0} a context free language? Is {a n b n n 0} a regular language? 23

24 Regular Grammars CFGs with restrictions on the shape of production rules. Left-linear: X ::= a X b N ::= X a b Right-linear: Y ::= a b a b Y N ::= b b Y 24

25 Complexity of Parsing Classification of languages that can be recognized by specific grammars. Complexity: Regular grammars DFAs O(n) LR grammars Kunth s algorithm O(n) Arbitrary CFGs Earley s algorithm O(n 3 ) CFG CSG RG LR Reading: Arbitrary CSGs LBAs P-SPACE COMPLETE Scott Chapter (for LR parser) and Chapter for language class. Earley, Jay (1970), An efficient context-free parsing algorithm, Communications of the ACM. 25

26 Top - Down Parsing - LL(1) <S> <A> α β Basic Idea: x The parse tree is constructed from the root, expanding non-terminal nodes on the tree s frontier following a leftmost derivation. The input program is read from left to right, and input tokens are read (consumed) as the program is parsed. The next non-terminal symbol is replaced using one of its rules. The particular choice has to be unique and uses parts of the input (partially parsed program), for instance the first token of the remaining input. y 26

27 Top - Down Parsing - LL(1) (cont.) Example: S ::= a S b ε How can we parse (automatically construct a leftmost derivation) the input string a a a b b b using a PDA (push-down automaton) and only the first symbol of the remaining input? INPUT: a a a b b b eof 27

28 Predictive Parsing Basic idea: For any two productions A ::= α β, we would like a distinct way of choosing the correct production to expand. For some rhs α G, define FIRST(α) as the set of tokens that appear as the first symbol in some string derived from α. That is x FIRST(α) iff α xγ for some γ, and ε FIRST(α) iff α ε 28

29 Predictive Parsing For a non-terminal A, define FOLLOW(A) as the set of terminals that can appear immediately to the right of A in some sentential form. Thus, a non-terminal s FOLLOW set specifies the tokens that can legally appear after it. A terminal symbol has no FOLLOW set. FIRST and FOLLOW sets can be constructed automatically 29

30 Predictive Parsing (cont.) Key Property: Whenever two productions A ::= α and A ::= β both appear in the grammar, we would like FIRST (α) FIRST (β) =, and if α * ε, then FIRST (β) FOLLOW (A) = Analogue case for β * ε. Note: due to first condition, at most one of α and β can derive ε. This would allow the parser to make a correct choice with a lookahead of only one symbol! 30

31 LL(1) Grammar Define FIRST+(δ) for rule A ::= δ FIRST (δ) - { ε } U Follow (A), if ε FIRST(δ) FIRST (δ) otherwise A Grammar is LL(1) iff (A ::= α and A ::= β) implies FIRST + (α) FIRST + (β) = 31

32 Next Lecture Next Time: Review and more on LL(1) parsing Read Scott, Chapter (and some chapter on companion site) 32

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