Context-Free Grammars

1. Context-Free Grammars Sipser 2.1 (pages 99 – 109)

2. What are we missing? • So far: • We know how to recognize languages • With finite state automata • As people… • We know how to generate languages • With regular expressions • As people… • Finite state automata and regular expressions are limited, though!

3. Bring back memories? • In English, a grammar tells us whether a particular sentence is well formed or not • For instance, “a sentence can consist of a noun phrase followed by a verb phrase” • More concisely, we could write <sentence> →G <noun_phrase><verb_phrase>

4. Great, but what’s a noun phrase? • A sentence is • <sentence> →G <noun_phrase><verb_phrase> • We need to provide definitions for the newly introduced constructs <noun_phrase> and <verb_phrase> • <noun_phrase> →G <article><noun> • <verb_phrase> →G <verb>

5. Generating well-formed sentences • Grammar rules so far: • <sentence> →G <noun_phrase><verb_phrase> • <noun_phrase> →G <article><noun> • <verb_phrase> →G <verb> • To complete our simple grammar, we associate actual words with the terms <article>, <noun>, and <verb> • <article> →G a • <article> →G the • <noun> →G student • <verb> →G relaxes • <verb> →G studies.

6. Context-free grammars • A context-free grammar G is a quadruple (V, Σ, R, S), where • V is a finite set called the variables • Σis a finite set, disjoint fromV, called the terminals • Ris a finite subset ofV × (V∪Σ)* called the rules • S∈Vis called the start symbol • For any A∈V and u∈(V∪Σ)*, we write A→Gu whenever (A, u)∈R

7. The language of a grammar • If • u, v, w∈ (V∪Σ)* • A →Gw is a rule then • We say uAvyields uwv • Write uAv⇒Guwv • If • u⇒G u1 ⇒G u2 ⇒G ... ⇒G uk⇒G v then • We write u ⇒*Gv • The language of the grammar G is L(G) = {w∈Σ* | S ⇒*G w}

8. For example… • ConsiderG = (V, Σ, R, S), where • V={S} • Σ={a,b} • R ={ S →GaSa | bSb | aSb | bSa | ε} • Is there a grammar whose language is PAL = {w∈ Σ* | w = reverse(w)}?

9. Arithmetic expressionsand parse trees • ConsiderG = (V, Σ, R, S), where • V ={<EXPR>, <TERM>, <FACTOR>} • Σ ={a, +, ×, (, )} • R ={ <EXPR> →G <EXPR>+<TERM> | <TERM>, <TERM> →G <TERM>×<FACTOR> | <FACTOR>, <FACTOR> →G (<EXPR>) | a } • S = <EXPR> • What about a × a +a?