Compilers Derivations Alex Aiken Derivations A derivation is a - PowerPoint PPT Presentation

Compilers Derivations Alex Aiken

Derivations A derivation is a sequence of productions S  …  …  …  …  … A derivation can be drawn as a tree – Start symbol is the tree’s root – For a production X  Y 1 … Y n add children Y 1 … Y n to node X Alex Aiken

Derivations • Grammar E  E + E | E  E | (E) | id • String id  id + id Alex Aiken

Derivations E E  E+E E + E   E E+E   E * E id id E + E   id id + E id id   id id + id Alex Aiken

Derivations E E Alex Aiken

Derivations E E + E E  E+E Alex Aiken

Derivations E E + E E  E+E E * E   E E E + Alex Aiken

Derivations E E E + E  E+E   E * E E E+E   id E + E id Alex Aiken

Derivations E E  E + E E+E   E E+E E * E   id E + E   id id id id + E Alex Aiken

Derivations E E  E+E E + E   E E+E   E * E id id E + E   id id + E id id   id id + id Alex Aiken

Derivations • A parse tree has – Terminals at the leaves – Non-terminals at the interior nodes • An in-order traversal of the leaves is the original input • The parse tree shows the association of operations, the input string does not Alex Aiken

Derivations • The example is a left-most E derivation  E+E – At each step, replace the left- most non-terminal  E+id   E E + id • There is an equivalent   E id + id notion of a right-most derivation   id id + id Alex Aiken

Derivations E E Alex Aiken

Derivations E E + E E  E+E Alex Aiken

Derivations E E + E E  E+E id  E+ id Alex Aiken

Derivations E E E + E  E+E  E * E id E+id   E E + id Alex Aiken

Derivations E E  E + E E+E  E+id E * E id   E E + id   id E id + id Alex Aiken

Derivations E E  E+E E + E  E+id   E * E id E E + id   E id + id id id   id id + id Alex Aiken

Derivations Note that right-most and left-most derivations have the same parse tree Alex Aiken

Derivations Which of the following is a valid S  aXa derivation of the given grammar? X   | bY S Y   | cXc | d aXa S abYa aa acXca acca S S aXa aXa abYa abYa abcXca abcXcda abcbYca abccda abcbdca

Derivations Which of the following is a valid S  aXa parse tree for the given grammar? X   | bY S a X a Y   | cXc | d S b Y a X a c X c d b Y  S S a X a a X a b Y  b Y c X c  c X c d 

Derivations • We are not just interested in whether s  L(G) – We need a parse tree for s • A derivation defines a parse tree – But one parse tree may have many derivations • Left-most and right-most derivations are important in parser implementation Alex Aiken