Languages & Grammars

Grammars  A set of rules which govern the structure of a language Fritz Fritz The dog The dog ate ate left left

Formal Grammar Notation  G = {V,T,S,P} –V are variables ( ) –T are terminals (Fritz) –S is the start variable ( ) –P are the production rules  Let W be a string of variables and terminals  W Y means that W can be transformed into Y using the production rules

Languages + Grammars  A grammar defines a language  Many grammars can define the same language  Grammars that generate the same language are equivalent.

Formal languages  Forget words; letters only  S is usually start symbol  What language?  S -> aA |  S -> aA |  A -> bS

Exercises... or Puzzles?  Write a grammar that generates: L={a n b m, n<2, m<=2}  Write a grammar that generates: L={a n b m, n>0, m>=0}  Write a grammar that generates all strings on {a,b}* with exactly 2 a ’ s  Write a grammar that generates: L={a n b n, n>0}  Write a grammar that generates:  L = {w  {a}*: |w| mod 4 = 0}

Mathy Questions  Suppose a grammar for L1 has start symbol S1 and a grammar for L2 has start symbol S2. What grammar describes –L1 U L2? –L1L2? –L1*?  Can you prove that your answer is correct?  Can you prove your neighbor ’ s answer is wrong?

Recursion  Generate a+ –S -> aS | a  Generate a* –S -> aS | –S -> aS |

Impose Order  Language: a + b + –S -> AB –A -> aA | a –B -> bB | b

Relationship between symbols  Language: a n b n n>0 –S -> aSb | ab

Context-Free Grammars  A grammar is context-free if all production rules have only one non- terminal on the left-hand side A -> aSa A -> AB A -> a  Not context-free: ABB -> aaSB

Members of CFLs  To see if a string is a member of a CFL, replace every non-terminal with the right side of one of its production rules.  S -> AB A -> aaA | λ B -> Bb | λ  Derivation of string aab: S -> AB -> aaAB -> aaB -> aaBb -> aab  So, aab is a sentence in the language, and aaAB is a sentential form.

Theory  Not all Context-free languages are regular.  Example: a n b n  Can you write a CF grammar?  Can you write an automaton?

Theory  BUT: All regular languages are also context-free.  How could we prove this? CF Reg

Automata to Grammar  States are non-terminals  Alphabet letters are the terminals  Start state corresponds to Start Symbol  Final States go to lambda

 S -> aA  A -> aA | aS | bB | C  B -> bC  C -> λ

Derivations  Leftmost derivation always replaces the leftmost variable in the sentential form next.  Rightmost derivation always replaces the one on the right.  Derivations can be shown using a derivation tree.

Example S -> aAB A -> bBb B -> A | λ Derive the string abbbb Leftmost: S-> a A B -> a bBb B -> a bAb B -> a b bBb b B -> a b b b b B -> a b b b b Rightmost:

Derivation Trees  Ordered tree in which: –Interior nodes are left-hand sides of rules (variables) –Children of a node are right-hand sides –Root is start symbol –Leaves are terminals  Reading the leaves from left to right is the yield of the tree (a sentence in the language)

Example:  S -> aAB A -> bBb B -> A | λ  Show derivation tree for derivation of abbbb S a A B b B b λ A b B b λ

Ambiguity  For any string in a context-free language, there may be more than one derivation tree that produces it. This is ambiguity.  Programming Languages cannot have ambiguity.  Try to find equivalent, unambiguous grammars, but it can ’ t always be done.  If there is any unambiguous grammar for a language, it is unambiguous.  If the language has no unambiguous grammar, the language is inherently ambiguous.

Exercises  Grammar: –S -> Sa | SSb | b | –S -> Sa | SSb | b |  Give rightmost derivation of string bbaaba  Draw derivation tree  Show that the grammar is ambiguous