Context-Free Grammars

1. Context-Free Grammars

2. Strings A string is a finite sequence, possibly empty, of symbols drawn from some alphabet . •  or  or “” is the empty string. • * is the set of all possible strings over an alphabet .

3. Languages A language is a (possibly infinite) set of finite length strings over a finite alphabet.

4. Languages and Machines

5. Regular Languages A Finite State Machine to accept a*b*: An FSM to accept AnBn = {anbn : n 0}

6. Regular Expressions (a | b)* abba (a | b)* \b[A-Za-z0-9_%-]+@[A-Za-z0-9_%-]+ (\.[A-Za-z]+){1,4}\b

7. Floating Point Numbers Example strings: +3.0, 3.0, 0.3E1, 0.3E+1, -0.3E+1, -3E8 The language is accepted by the FSM: A regular expression:

8. When We Need More Power (())()  ()( 

9. Rewrite Systems and Grammars A rewrite system (or production system or rule-based system) is: ● a list of rules, and ● an algorithm for applying them. Each rule has a left-hand side (lhs) and a right hand side (rhs).

10. Grammars A grammar is a production system designed to describe a language. So it has: ● a list of rules, and ● an algorithm for applying them. Each rule has a left-hand side (lhs) and a right hand side (rhs). Example rules: SaSb aS aSbbSabSa

11. A Grammar Interpreter generate(G: grammar, w: initial string) = 1. Set working-string to w. 2. Until no working symbols remain do: Match the lhs of some rule against some part of working-string. Replace the matched part of working-string with the rhs of the rule that was matched. 3. Return working-string.

12. Context-Free Grammars No restrictions on the form of the right hand sides. SabDeFGab But require single non-terminal on left hand side. S not ASB Generally a unique starting symbol is included as part of the grammar definition. Also, generally, some symbols are marked as terminal symbols.

13. AnBn

14. AnBn SaSb S

15. Balanced Parentheses

16. Balanced Parentheses S (S) S SSS

17. Context-Free Grammars A context-free grammar G is a quadruple, (V, , R, S), where: ● V is the rule alphabet, which contains nonterminals and terminals. ●  (the set of terminals) is a subset of V, ● R (the set of rules) is a finite subset of (V - ) V*, ● S (the start symbol) is an element of V - . Example: ({S, a, b}, {a, b}, {SaSb, S}, S)

18. Definition of a Context-Free Language A language L is context-free iff it is generated by some context-free grammar G.

19. Where Context-Free Grammars Get Their Power Self-embedding rules: SaSb is self-embedding S  SaS is recursive but not self-embedding S  SaT is self-embedding TSa

20. PalEven = {wwR : w {a, b}*}

21. PalEven = {wwR : w {a, b}*} G = {{S, a, b}, {a, b}, R, S}, where: R =

22. PalEven = {wwR : w {a, b}*} G = {{S, a, b}, {a, b}, R, S}, where: R = { SaSa SbSb S }.

23. WcW = {wcw : w {a, b}*}

24. WcW = {wcw : w {a, b}*} Why might we care?

25. WcW = {wcw : w {a, b}*} Why might we care? string winniethepooh; winniethepooh = “bearofverylittlebrain”; Type Checking

26. Arithmetic Expressions E E + E E E E E id E  (E)

27. BNF A notation for writing practical context-free grammars • The symbol | should be read as “or”. Example: SaSb | bSa | SS |  • Allow a nonterminal symbol to be any sequence of characters surrounded by angle brackets. Examples of nonterminals: <program> <variable>

28. BNF for a Java Fragment <block> ::= {<stmt-list>} | {} <stmt-list> ::= <stmt> | <stmt-list> <stmt> <stmt> ::= <block> | while (<cond>) <stmt> | if (<cond>) <stmt> | do <stmt> while (<cond>); | <assignment-stmt>; | return | return <expression> | <method-invocation>;

29. Grammar vs Prose Why use a grammar? Why not just write prose? http://www.w3.org/Protocols/rfc2616/rfc2616-sec4.html#sec4.2

30. Spam Generation These production rules yield 1,843,200 possible spellings. How Many Ways Can You Spell V1@gra? By Brian Hayes American Scientist, July-August 2007 http://www.americanscientist.org/issues/pub/how-many-ways-can-you-spell-v1gra

31. HTML <ul> <li>Item 1, which will include a sublist</li> <ul> <li>First item in sublist</li> <li>Second item in sublist</li> </ul> <li>Item 2</li> </ul> A grammar: /* Text is a sequence of elements. HTMLtextElementHTMLtext |  Element UL | LI| … (and other kinds of elements that are allowed in the body of an HTML document) /* The <ul> and </ul> tags must match. UL <ul>HTMLtext</ul> /* The <li> and </li> tags must match. LI <li>HTMLtext</li>

32. English S  NPVP NP  theNominal | aNominal | Nominal | ProperNoun | NPPP Nominal  N | Adjs N N  cat | dogs | bear | girl | chocolate | rifle ProperNoun  Chris | Fluffy Adjs  Adj Adjs | Adj Adj  young | older | smart VP  V | V NP | VPPP V  like | likes | thinks | shots | smells PP  Prep NP Prep  with

33. A Bigger English Grammar Generating fake conference papers: SCIgen

34. Outside-In Structure and RNA Folding

35. A Grammar for RNA Folding <family>  <tail> <stemloop> [1] <tail>  <base> <base> <base> [1] <stemloop>  C <stemloop-5> G [.23] <stemloop>  G <stemloop-5> C [.23] <stemloop>  A <stemloop-5> U [.23] <stemloop>  U <stemloop-5> A [.23] <stemloop> G <stemloop-5> U [.03] <stemloop> U <stemloop-5> G [.03] <stemloop-5> …

36. Proving the Correctness of a Grammar AnBn = {anbn : n 0} G = ({S, a, b}, {a, b}, R, S), R = { S aSb S  } ● Prove that G generates only strings in L. ● Prove that G generates all the strings in L.

37. Proving the Correctness of a Grammar To prove that G generates only strings in L: Imagine the process by which G generates a string as the following loop: • st := S. • Until no nonterminals are left in st do: 2.1. Apply some rule in R to st. 3. Output st. Then we construct a loop invariant I and show that: ●I is true when the loop begins, ●I is maintained at each step through the loop, and ●I  (st contains only terminal symbols) st L.

38. Proving the Correctness of a Grammar AnBn = {anbn : n 0}.G = ({S, a, b}, {a, b}, R, S), R = { S aSb S  }. ● Prove that G generates only strings in L: Let I = (#a(st) = #b(st))  (st a*(S ) b*). ● Prove that G generates all strings in L: By induction on the length of st.

39. Structure (Parse Trees) Context free languages: We care about structure. E E + E idE * E 3 idid 5 7 3 + 5 * 7

40. Structure in English S NP VP Nominal VNP AdjsNNominal AdjN the smart cat smells chocolate

41. Parsing • Top down • Bottom up

42. Generative Capacity Because parse trees matter, it makes sense, given a grammar G, to distinguish between: ● G’s weak generative capacity, defined to be the set of strings, L(G), that G generates, and ● G’s strong generative capacity, defined to be the set of parse trees that G generates.

43. Ambiguity A grammar is ambiguous iff there is at least one string in L(G) for which G produces more than one parse tree. For most applications of context-free grammars, this is a problem.

44. An Arithmetic Expression Grammar EE + E (1) EEE (2) E (E) (3) Eid (4) 2 + 3 * 5

45. An Arithmetic Expression Grammar EE + E (1) EEE (2) E (E) (3) Eid (4)

46. Inherent Ambiguity Both of the following problems are undecidable: • Given an arbitrary context-free grammar G, is G ambiguous? • Given an arbitrary context-free language L, is L inherently ambiguous?

47. Arithmetic Expressions EE + E EEE E (E) Eid } Problem 1: Associativity E E EE EE EE EE id  id  id id  id  id

48. Arithmetic Expressions EE + E EEE E (E) Eid } Problem 2: Precedence E E EE EE EE EE id  id + id id  id + id

49. Arithmetic Expressions - A Better Way EE + T ET TT * F TF F (E) F id Examples: id + id * id id * id * id

50. Arithmetic Expressions - A Better Way EE + T ET TT * F TF F (E) F id