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A CYK for Any Grammar

A CYK for Any Grammar.

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A CYK for Any Grammar

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  1. A CYK for Any Grammar grammar G, non-terminals A1,...,AK, tokens t1,....tLinput word: w = w(0)w(1) …w(N-1)wp..q = w(p)w(p+1) …w(q-1)Triple (A, p, q) means: A =>*wp..q , Acan be: Ai, tj, or P = {(w(i),i,i+1)| 0 i < N-1} repeat { choose rule (A::=B1...Bm)G if ((A,p0,pm)P && ((m=0 && p0=pm) || (B1,p0,p1), ...,(Bm,pm-1,pm)  P)) P := P U {(A,p0,pm)} } until no more insertions possible What is the maximal number of steps?How long does it take to check step for a rule? for grammar in given normal form

  2. Strategy for Populating Table • Which order to insert (A, p, q)tuples ? • all orders give correct result • efficiency differs • Left-to-right scan of the input: • derive all A,p for given q • then increase q to q+1 • Consider only productive parse attempts • insert (A,p,q) only if we can prove thatS =>* w0..p-1 A Y (Y is any string of symbols)

  3. Dotted Rules Like Nonterminals X ::= Y1 Y2 Y3 Chomsky transformation is (a simplification of) this: X ::= W123 W123 ::= W12 Y3 W12 ::= W1 Y2 W1 ::= WY1 W ::=  Early parser: dotted RHS as names of fresh non-terminals: X ::= (Y1Y2Y3.)(Y1Y2Y3.) ::= (Y1Y2.Y3) Y3 (Y1Y2.Y3) ::= (Y1.Y2Y3) Y2 (Y1.Y2Y3) ::= (.Y1Y2Y3) Y3 (.Y1Y2Y3) ::= 

  4. Earley Parser - group the triples by last element: S(q)={(A,p)|(A,p,q)P} - dotted rules effectively make productions at most binary

  5. S ::= . e EOF | e . EOF | e EOF . e ::= . ID | ID . | . e – e | e . – e | e – . e | e – e . | . e == e | e . == e | e == . e | e == e .

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