600.465 - Intro to NLP - J. Eisner1 Earley’s Algorithm (1970) Nice combo of our parsing ideas so far:  no restrictions on the form of the grammar:  A.

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Presentation transcript:

Intro to NLP - J. Eisner1 Earley’s Algorithm (1970) Nice combo of our parsing ideas so far:  no restrictions on the form of the grammar:  A  B C spoon D x  incremental parsing (left to right, like humans)  left context constrains parsing of subsequent words  so waste less time building impossible things  makes it faster than O(n 3 ) for many grammars

Intro to NLP - J. Eisner2 … …

3 Overview of Earley’s Algorithm  Finds constituents and partial constituents in input  A  B C. D E is partial: only the first half of the A A BCDE A  B C. D E D + = A BCDE A  B C D. E i j i k jk

Intro to NLP - J. Eisner4 Overview of Earley’s Algorithm  Proceeds incrementally, left-to-right  Before it reads word 5, it has already built all hypotheses that are consistent with first 4 words  Reads word 5 & attaches it to immediately preceding hypotheses. Might yield new constituents that are then attached to hypotheses immediately preceding them …  E.g., attaching D to A  B C. D E gives A  B C D. E  Attaching E to that gives A  B C D E.  Now we have a complete A that we can attach to hypotheses immediately preceding the A, etc.

Intro to NLP - J. Eisner5 Our Usual Example Grammar ROOT  S S  NP VPNP  Papa NP  Det NN  caviar NP  NP PPN  spoon VP  VP PPV  ate VP  V NPP  with PP  P NPDet  the Det  a

Intro to NLP - J. Eisner6 First Try: Recursive Descent  0 ROOT . S 0  0 S . NP VP 0  0 NP . Papa 0  0 NP  Papa. 1  0 S  NP. VP 1  1 VP . VP PP 1 1 VP . VP PP 1 oops, stack overflowed  OK, let’s pretend that didn’t happen.  Let’s suppose we didn’t see VP  VP PP, and used VP  V NP instead. ROOT  SVP  VP PPNP  PapaV  ate S  NP VPVP  V NPN  caviarP  with NP  Det NPP  P NPN  spoonDet  the NP  NP PP Det  a 0 Papa 1 ate 2 the 3 caviar 4 with 5 a 6 spoon 7 “goal stack”

Intro to NLP - J. Eisner7 First Try: Recursive Descent  1 VP . V NP 1 after dot = nonterminal, so recursively look for it (“ predict ”)  1 V . ate 1 after dot = terminal, so look for it in the input (“ scan ”)  1 V  ate. 2 after dot = nothing, so parent’s subgoal is completed (“ attach ”)  1 VP  V. NP 2 predict (next subgoal)  2 NP  do some more parsing and eventually...  2 NP  we complete the parent’s NP subgoal, so attach  1 VP  V NP. 7 attach again  0 S  NP VP. 7 attach again ROOT  SVP  V NPNP  PapaV  ate S  NP VPVP  VP PPN  caviarP  with NP  Det NPP  P NPN  spoonDet  the NP  NP PP Det  a 0 Papa 1 ate 2 the 3 caviar 4 with 5 a 6 spoon 7  0 ROOT . S 0  0 S . NP VP 0  0 NP . Papa 0  0 NP  Papa. 1  0 S  NP. VP 1 after dot = nonterminal, so recursively look for it (“ predict ”)

Intro to NLP - J. Eisner8 First Try: Recursive Descent  0 ROOT . S 0  0 S . NP VP 0  0 NP . Papa 0  0 NP  Papa. 1  0 S  NP. VP 1  1 VP . V NP 1  1 V . ate 1  1 V  ate. 2  1 VP  V. NP 2  2 NP   2 NP   1 VP  V NP. 7  0 S  NP VP. 7 ROOT  SVP  V NPNP  PapaV  ate S  NP VPVP  VP PPN  caviarP  with NP  Det NPP  P NPN  spoonDet  the NP  NP PP Det  a But how about the other parse? must backtrack to try predicting a different VP rule here instead 0 Papa 1 ate 2 the 3 caviar 4 with 5 a 6 spoon 7 implement by function calls: S() calls NP() and VP(), which recurse

Intro to NLP - J. Eisner9 First Try: Recursive Descent  1 VP . V NP 1  1 V . ate 1  1 V  ate. 2  1 VP  V. NP 2  2 NP  do some more parsing and eventually...  2 NP  the correct NP is from 2 to 4 this time (but might we find the one from 2 to 7 instead?) ROOT  SVP  V NPNP  PapaV  ate S  NP VPVP  VP PPN  caviarP  with NP  Det NPP  P NPN  spoonDet  the NP  NP PP Det  a we’d better backtrack here too! (why?) 0 Papa 1 ate 2 the 3 caviar 4 with 5 a 6 spoon 7  0 ROOT . S 0  0 S . NP VP 0  0 NP . Papa 0  0 NP  Papa. 1  0 S  NP. VP 1  1 VP . VP PP 1

Intro to NLP - J. Eisner10 First Try: Recursive Descent  1 VP . VP PP 1 1 VP . VP PP 1 oops, stack overflowed no fix after all – must transform grammar to eliminate left-recursive rules ROOT  SVP  V NPNP  PapaV  ate S  NP VPVP  VP PPN  caviarP  with NP  Det NPP  P NPN  spoonDet  the NP  NP PP Det  a 0 Papa 1 ate 2 the 3 caviar 4 with 5 a 6 spoon 7  0 ROOT . S 0  0 S . NP VP 0  0 NP . Papa 0  0 NP  Papa. 1  0 S  NP. VP 1  1 VP . VP PP 1

Intro to NLP - J. Eisner11 Use a Parse Table (= “Chart”)  Earley’s algorithm resembles recursive descent, but solves the left-recursion problem. No recursive function calls.  Use a parse table as we did in CKY, so we can look up anything we’ve discovered so far. “Dynamic programming.”  Entries in column 5 look like (3, S  NP. VP) (but we’ll omit the  etc. to save space)  Built while processing word 5  Means that the input substring from 3 to 5 matches the initial NP portion of a S  NP VP rule  Dot shows how much we’ve matched as of column 5  Perfectly fine to have entries like (3, S  is it. true that S)

Intro to NLP - J. Eisner12 Use a Parse Table (“Chart”)  Entries in column 5 look like (3, S  NP. VP)  What does it mean if we have this entry?  Unknown right context: Doesn’t mean we’ll necessarily be able to find a VP starting at column 5 to complete the S.  Known left context: Does mean that some dotted rule back in column 3 is looking for an S that starts at 3.  So if we actually do find a VP starting at column 5, allowing us to complete the S, then we’ll be able to attach the S to something.  And when that something is complete, it too will have a customer to its left … just as in recursive descent!  In short, a top-down (i.e., goal-directed) parser: it chooses to start building a constituent not because of the input but because that’s what the left context needs. In the spoon, won’t build spoon as a verb because there’s no way to use a verb there.  So any hypothesis in column 5 could get used in the correct parse, if words 1-5 are continued in just the right way by words 6-n.

Intro to NLP - J. Eisner13 Operation of the Algorithm  Process all hypotheses one at a time in order. (Current hypothesis is shown in blue, with substring.)  This may add to the end of the to-do list, or try to add again. new hypotheses old hypotheses  Process a hypothesis according to what follows the dot – just as in recursive descent:  If a word, scan input and see if it matches  If a nonterminal, predict ways to match it (we’ll predict blindly, but could reduce # of predictions by looking ahead k symbols in the input and only making predictions that are compatible with this limited right context)  If nothing, then we have a complete constituent, so attach it to all its customers (shown in purple).

Intro to NLP - J. Eisner14 BCDE i j A (i, A  B C. D E) column j current hypothesis (incomplete) which action? One entry (hypothesis) … All entries ending at j stored in column j, as in CKY

Intro to NLP - J. Eisner15 (j, D . bludger) new entry to process later Predict BCDE i j A (i, A  B C. D E) column j current hypothesis (incomplete) D jj bludger

Intro to NLP - J. Eisner16 Scan (i, A  B C. D E) column j D jj bludger (j, D  bludger.) column k … new entry to process later k=j+1 in this example Current hypothesis (incomplete) (j, D . bludger) + j k bludger = D j k

Intro to NLP - J. Eisner17 Attach D jk (i, A  B C. D E) column j column k (j, D  bludger.) … current hypothesis (complete)

Intro to NLP - J. Eisner18 Attach D jk BCDE + i j A (i, A  B C. D E) = BCDE i k A (i, A  B C D. E) column j column k (j, D  bludger.) … current hypothesis (complete) customer (incomplete) new entry to process later

Intro to NLP - J. Eisner19 Our Usual Example Grammar ROOT  S S  NP VPNP  Papa NP  Det NN  caviar NP  NP PPN  spoon VP  VP PPV  ate VP  V NPP  with PP  P NPDet  the Det  a 0 Papa 1 ate 2 the 3 caviar 4 with 5 a 6 spoon 7

0 0 ROOT. S initialize Remember this stands for (0, ROOT . S)

0 0 ROOT. S 0 S. NP VP predict the kind of S we are looking for Remember this stands for (0, S . NP VP)

0 0 ROOT. S 0 S. NP VP 0 NP. Det N 0 NP. NP PP 0 NP. Papa predict the kind of NP we are looking for (actually we’ll look for 3 kinds: any of the 3 will do)

0 0 ROOT. S 0 S. NP VP 0 NP. Det N 0 NP. NP PP 0 NP. Papa 0 Det. the 0 Det. a predict the kind of Det we are looking for (2 kinds)

0 0 ROOT. S 0 S. NP VP 0 NP. Det N 0 NP. NP PP 0 NP. Papa 0 Det. the 0 Det. a predict the kind of NP we’re looking for but we were already looking for these so don’t add duplicate goals! Note that this happened when we were processing a left-recursive rule.

0 Papa 1 0 ROOT. S0 NP Papa. 0 S. NP VP 0 NP. Det N 0 NP. NP PP 0 NP. Papa 0 Det. the 0 Det. a scan: the desired word is in the input!

0 Papa 1 0 ROOT. S0 NP Papa. 0 S. NP VP 0 NP. Det N 0 NP. NP PP 0 NP. Papa 0 Det. the 0 Det. a scan: failure

0 Papa 1 0 ROOT. S0 NP Papa. 0 S. NP VP 0 NP. Det N 0 NP. NP PP 0 NP. Papa 0 Det. the 0 Det. a scan: failure

0 Papa 1 0 ROOT. S0 NP Papa. 0 S. NP VP0 S NP. VP 0 NP. Det N0 NP NP. PP 0 NP. NP PP 0 NP. Papa 0 Det. the 0 Det. a attach the newly created NP (which starts at 0) to its customers (incomplete constituents that end at 0 and have NP after the dot)

0 Papa 1 0 ROOT. S0 NP Papa. 0 S. NP VP0 S NP. VP 0 NP. Det N0 NP NP. PP 0 NP. NP PP1 VP. V NP 0 NP. Papa1 VP. VP PP 0 Det. the 0 Det. a predict

0 Papa 1 0 ROOT. S0 NP Papa. 0 S. NP VP0 S NP. VP 0 NP. Det N0 NP NP. PP 0 NP. NP PP1 VP. V NP 0 NP. Papa1 VP. VP PP 0 Det. the1 PP. P NP 0 Det. a predict

0 Papa 1 0 ROOT. S0 NP Papa. 0 S. NP VP0 S NP. VP 0 NP. Det N0 NP NP. PP 0 NP. NP PP1 VP. V NP 0 NP. Papa1 VP. VP PP 0 Det. the1 PP. P NP 0 Det. a1 V. ate predict

0 Papa 1 0 ROOT. S0 NP Papa. 0 S. NP VP0 S NP. VP 0 NP. Det N0 NP NP. PP 0 NP. NP PP1 VP. V NP 0 NP. Papa1 VP. VP PP 0 Det. the1 PP. P NP 0 Det. a1 V. ate predict

0 Papa 1 0 ROOT. S0 NP Papa. 0 S. NP VP0 S NP. VP 0 NP. Det N0 NP NP. PP 0 NP. NP PP1 VP. V NP 0 NP. Papa1 VP. VP PP 0 Det. the1 PP. P NP 0 Det. a1 V. ate 1 P. with predict

0 Papa 1 ate 2 0 ROOT. S0 NP Papa.1 V ate. 0 S. NP VP0 S NP. VP 0 NP. Det N0 NP NP. PP 0 NP. NP PP1 VP. V NP 0 NP. Papa1 VP. VP PP 0 Det. the1 PP. P NP 0 Det. a1 V. ate 1 P. with scan: success!

0 Papa 1 ate 2 0 ROOT. S0 NP Papa.1 V ate. 0 S. NP VP0 S NP. VP 0 NP. Det N0 NP NP. PP 0 NP. NP PP1 VP. V NP 0 NP. Papa1 VP. VP PP 0 Det. the1 PP. P NP 0 Det. a1 V. ate 1 P. with scan: failure

0 Papa 1 ate 2 0 ROOT. S0 NP Papa.1 V ate. 0 S. NP VP0 S NP. VP1 VP V. NP 0 NP. Det N0 NP NP. PP 0 NP. NP PP1 VP. V NP 0 NP. Papa1 VP. VP PP 0 Det. the1 PP. P NP 0 Det. a1 V. ate 1 P. with attach

0 Papa 1 ate 2 0 ROOT. S0 NP Papa.1 V ate. 0 S. NP VP0 S NP. VP1 VP V. NP 0 NP. Det N0 NP NP. PP2 NP. Det N 0 NP. NP PP1 VP. V NP2 NP. NP PP 0 NP. Papa1 VP. VP PP2 NP. Papa 0 Det. the1 PP. P NP 0 Det. a1 V. ate 1 P. with predict

0 Papa 1 ate 2 0 ROOT. S0 NP Papa.1 V ate. 0 S. NP VP0 S NP. VP1 VP V. NP 0 NP. Det N0 NP NP. PP2 NP. Det N 0 NP. NP PP1 VP. V NP2 NP. NP PP 0 NP. Papa1 VP. VP PP2 NP. Papa 0 Det. the1 PP. P NP2 Det. the 0 Det. a1 V. ate2 Det. a 1 P. with predict (these next few steps should look familiar)

0 Papa 1 ate 2 0 ROOT. S0 NP Papa.1 V ate. 0 S. NP VP0 S NP. VP1 VP V. NP 0 NP. Det N0 NP NP. PP2 NP. Det N 0 NP. NP PP1 VP. V NP2 NP. NP PP 0 NP. Papa1 VP. VP PP2 NP. Papa 0 Det. the1 PP. P NP2 Det. the 0 Det. a1 V. ate2 Det. a 1 P. with predict

0 Papa 1 ate 2 0 ROOT. S0 NP Papa.1 V ate. 0 S. NP VP0 S NP. VP1 VP V. NP 0 NP. Det N0 NP NP. PP2 NP. Det N 0 NP. NP PP1 VP. V NP2 NP. NP PP 0 NP. Papa1 VP. VP PP2 NP. Papa 0 Det. the1 PP. P NP2 Det. the 0 Det. a1 V. ate2 Det. a 1 P. with scan (this time we fail since Papa is not the next word)

0 Papa 1 ate 2 the 3 0 ROOT. S0 NP Papa.1 V ate.2 Det the. 0 S. NP VP0 S NP. VP1 VP V. NP 0 NP. Det N0 NP NP. PP2 NP. Det N 0 NP. NP PP1 VP. V NP2 NP. NP PP 0 NP. Papa1 VP. VP PP2 NP. Papa 0 Det. the1 PP. P NP2 Det. the 0 Det. a1 V. ate2 Det. a 1 P. with scan: success!

0 Papa 1 ate 2 the 3 0 ROOT. S0 NP Papa.1 V ate.2 Det the. 0 S. NP VP0 S NP. VP1 VP V. NP 0 NP. Det N0 NP NP. PP2 NP. Det N 0 NP. NP PP1 VP. V NP2 NP. NP PP 0 NP. Papa1 VP. VP PP2 NP. Papa 0 Det. the1 PP. P NP2 Det. the 0 Det. a1 V. ate2 Det. a 1 P. with

0 Papa 1 ate 2 the 3 0 ROOT. S0 NP Papa.1 V ate.2 Det the. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N 0 NP. Det N0 NP NP. PP2 NP. Det N 0 NP. NP PP1 VP. V NP2 NP. NP PP 0 NP. Papa1 VP. VP PP2 NP. Papa 0 Det. the1 PP. P NP2 Det. the 0 Det. a1 V. ate2 Det. a 1 P. with

0 Papa 1 ate 2 the 3 0 ROOT. S0 NP Papa.1 V ate.2 Det the. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon 0 NP. Papa1 VP. VP PP2 NP. Papa 0 Det. the1 PP. P NP2 Det. the 0 Det. a1 V. ate2 Det. a 1 P. with

0 Papa 1 ate 2 the 3 caviar 4 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon 0 NP. Papa1 VP. VP PP2 NP. Papa 0 Det. the1 PP. P NP2 Det. the 0 Det. a1 V. ate2 Det. a 1 P. with

0 Papa 1 ate 2 the 3 caviar 4 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon 0 NP. Papa1 VP. VP PP2 NP. Papa 0 Det. the1 PP. P NP2 Det. the 0 Det. a1 V. ate2 Det. a 1 P. with

0 Papa 1 ate 2 the 3 caviar 4 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon 0 NP. Papa1 VP. VP PP2 NP. Papa 0 Det. the1 PP. P NP2 Det. the 0 Det. a1 V. ate2 Det. a 1 P. with attach

0 Papa 1 ate 2 the 3 caviar 4 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa 0 Det. the1 PP. P NP2 Det. the 0 Det. a1 V. ate2 Det. a 1 P. with attach (again!)

0 Papa 1 ate 2 the 3 caviar 4 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP 0 Det. a1 V. ate2 Det. a 1 P. with attach (again!)

0 Papa 1 ate 2 the 3 caviar 4 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP 0 Det. a1 V. ate2 Det. a4 PP. P NP 1 P. with

0 Papa 1 ate 2 the 3 caviar 4 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP 0 Det. a1 V. ate2 Det. a4 PP. P NP 1 P. with0 ROOT S. attach (again!)

0 Papa 1 ate 2 the 3 caviar 4 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP 0 Det. a1 V. ate2 Det. a4 PP. P NP 1 P. with0 ROOT S.

0 Papa 1 ate 2 the 3 caviar 4 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP 0 Det. a1 V. ate2 Det. a4 PP. P NP 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP 0 Det. a1 V. ate2 Det. a4 PP. P NP 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with 5 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar.4 P with. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP 0 Det. a1 V. ate2 Det. a4 PP. P NP 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with 5 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar.4 P with. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.4 PP P. NP 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP 0 Det. a1 V. ate2 Det. a4 PP. P NP 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with 5 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar.4 P with. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.4 PP P. NP 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.5 NP. Det N 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP. NP PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.5 NP. Papa 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP 0 Det. a1 V. ate2 Det. a4 PP. P NP 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with 5 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar.4 P with. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.4 PP P. NP 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.5 NP. Det N 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP. NP PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.5 NP. Papa 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP5 Det. the 0 Det. a1 V. ate2 Det. a4 PP. P NP5 Det. a 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with 5 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar.4 P with. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.4 PP P. NP 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.5 NP. Det N 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP. NP PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.5 NP. Papa 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP5 Det. the 0 Det. a1 V. ate2 Det. a4 PP. P NP5 Det. a 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with 5 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar.4 P with. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.4 PP P. NP 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.5 NP. Det N 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP. NP PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.5 NP. Papa 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP5 Det. the 0 Det. a1 V. ate2 Det. a4 PP. P NP5 Det. a 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with 5 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar.4 P with. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.4 PP P. NP 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.5 NP. Det N 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP. NP PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.5 NP. Papa 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP5 Det. the 0 Det. a1 V. ate2 Det. a4 PP. P NP5 Det. a 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with 5 a 6 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar.4 P with.5 Det a. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.4 PP P. NP 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.5 NP. Det N 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP. NP PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.5 NP. Papa 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP5 Det. the 0 Det. a1 V. ate2 Det. a4 PP. P NP5 Det. a 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with 5 a 6 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar.4 P with.5 Det a. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.4 PP P. NP5 NP Det. N 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.5 NP. Det N 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP. NP PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.5 NP. Papa 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP5 Det. the 0 Det. a1 V. ate2 Det. a4 PP. P NP5 Det. a 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with 5 a 6 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar.4 P with.5 Det a. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.4 PP P. NP5 NP Det. N 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.5 NP. Det N6 N. caviar 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP. NP PP6 N. spoon 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.5 NP. Papa 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP5 Det. the 0 Det. a1 V. ate2 Det. a4 PP. P NP5 Det. a 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with 5 a 6 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar.4 P with.5 Det a. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.4 PP P. NP5 NP Det. N 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.5 NP. Det N6 N. caviar 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP. NP PP6 N. spoon 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.5 NP. Papa 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP5 Det. the 0 Det. a1 V. ate2 Det. a4 PP. P NP5 Det. a 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with 5 a 6 spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar.4 P with.5 Det a.6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.4 PP P. NP5 NP Det. N 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.5 NP. Det N6 N. caviar 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP. NP PP6 N. spoon 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.5 NP. Papa 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP5 Det. the 0 Det. a1 V. ate2 Det. a4 PP. P NP5 Det. a 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with 5 a 6 spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar.4 P with.5 Det a.6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.4 PP P. NP5 NP Det. N5 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.5 NP. Det N6 N. caviar 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP. NP PP6 N. spoon 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.5 NP. Papa 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP5 Det. the 0 Det. a1 V. ate2 Det. a4 PP. P NP5 Det. a 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with 5 a 6 spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar.4 P with.5 Det a.6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.4 PP P. NP5 NP Det. N5 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.5 NP. Det N6 N. caviar4 PP P NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP. NP PP6 N. spoon5 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.5 NP. Papa 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP5 Det. the 0 Det. a1 V. ate2 Det. a4 PP. P NP5 Det. a 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with a spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. … 6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.5 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.4 PP P NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.2 NP NP PP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP1 VP VP PP. 0 Det. a1 V. ate2 Det. a4 PP. P NP 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with a spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. … 6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.5 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.4 PP P NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.2 NP NP PP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP1 VP VP PP. 0 Det. a1 V. ate2 Det. a4 PP. P NP7 PP. P NP 1 P. with0 ROOT S. 4 P. with

0 Papa 1 ate 2 the 3 caviar 4 with a spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. … 6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.5 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.4 PP P NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.2 NP NP PP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP1 VP VP PP. 0 Det. a1 V. ate2 Det. a4 PP. P NP7 PP. P NP 1 P. with0 ROOT S.1 VP V NP. 4 P. with2 NP NP. PP

0 Papa 1 ate 2 the 3 caviar 4 with a spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. … 6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.5 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.4 PP P NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.2 NP NP PP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP1 VP VP PP. 0 Det. a1 V. ate2 Det. a4 PP. P NP7 PP. P NP 1 P. with0 ROOT S.1 VP V NP. 4 P. with2 NP NP. PP 0 S NP VP. 1 VP VP. PP

0 Papa 1 ate 2 the 3 caviar 4 with a spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. … 6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.5 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.4 PP P NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.2 NP NP PP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP1 VP VP PP. 0 Det. a1 V. ate2 Det. a4 PP. P NP7 PP. P NP 1 P. with0 ROOT S.1 VP V NP. 4 P. with2 NP NP. PP 0 S NP VP. 1 VP VP. PP 7 P. with

0 Papa 1 ate 2 the 3 caviar 4 with a spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. … 6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.5 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.4 PP P NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.2 NP NP PP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP1 VP VP PP. 0 Det. a1 V. ate2 Det. a4 PP. P NP7 PP. P NP 1 P. with0 ROOT S.1 VP V NP. 4 P. with2 NP NP. PP 0 S NP VP. 1 VP VP. PP 7 P. with

0 Papa 1 ate 2 the 3 caviar 4 with a spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. … 6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.5 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.4 PP P NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.2 NP NP PP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP1 VP VP PP. 0 Det. a1 V. ate2 Det. a4 PP. P NP7 PP. P NP 1 P. with0 ROOT S.1 VP V NP. 4 P. with2 NP NP. PP 0 S NP VP. 1 VP VP. PP 7 P. with

0 Papa 1 ate 2 the 3 caviar 4 with a spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. … 6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.5 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.4 PP P NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.2 NP NP PP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP1 VP VP PP. 0 Det. a1 V. ate2 Det. a4 PP. P NP7 PP. P NP 1 P. with0 ROOT S.1 VP V NP. 4 P. with2 NP NP. PP 0 S NP VP. 1 VP VP. PP 7 P. with 0 ROOT S.

0 Papa 1 ate 2 the 3 caviar 4 with a spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. … 6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.5 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.4 PP P NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.2 NP NP PP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP1 VP VP PP. 0 Det. a1 V. ate2 Det. a4 PP. P NP7 PP. P NP 1 P. with0 ROOT S.1 VP V NP. 4 P. with2 NP NP. PP 0 S NP VP. 1 VP VP. PP 7 P. with 0 ROOT S.

0 Papa 1 ate 2 the 3 caviar 4 with a spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. … 6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.5 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.4 PP P NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.2 NP NP PP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP1 VP VP PP. 0 Det. a1 V. ate2 Det. a4 PP. P NP7 PP. P NP 1 P. with0 ROOT S.1 VP V NP. 4 P. with2 NP NP. PP 0 S NP VP. 1 VP VP. PP 7 P. with 0 ROOT S.

0 Papa 1 ate 2 the 3 caviar 4 with a spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. … 6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.5 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.4 PP P NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.2 NP NP PP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP1 VP VP PP. 0 Det. a1 V. ate2 Det. a4 PP. P NP7 PP. P NP 1 P. with0 ROOT S.1 VP V NP. 4 P. with2 NP NP. PP 0 S NP VP. 1 VP VP. PP 7 P. with 0 ROOT S.

0 Papa 1 ate 2 the 3 caviar 4 with a spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. … 6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.5 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.4 PP P NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP NP. PP 0 NP. Papa1 VP. VP PP2 NP. Papa0 S NP VP.2 NP NP PP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP1 VP VP PP. 0 Det. a1 V. ate2 Det. a4 PP. P NP7 PP. P NP 1 P. with0 ROOT S.1 VP V NP. 4 P. with2 NP NP. PP 0 S NP VP. 1 VP VP. PP 7 P. with 0 ROOT S.

Intro to NLP - J. Eisner81 Left Recursion Kills Pure Top-Down Parsing … VP

Intro to NLP - J. Eisner82 Left Recursion Kills Pure Top-Down Parsing … VP PP

Intro to NLP - J. Eisner83 Left Recursion Kills Pure Top-Down Parsing … VP PP VP PP

Intro to NLP - J. Eisner84 Left Recursion Kills Pure Top-Down Parsing … VP PP VP PP VP PP makes new hypotheses ad infinitum before we’ve seen the PPs at all hypotheses try to predict in advance how many PP’s will arrive in input

Intro to NLP - J. Eisner85 … but Earley’s Alg is Okay! VP PP VP 1 VP . VP PP (in column 1)

Intro to NLP - J. Eisner86 … but Earley’s Alg is Okay! VP V NP 1 VP  V NP. ate the caviar VP PP VP 1 VP . VP PP (in column 1) (in column 4)

Intro to NLP - J. Eisner87 … but Earley’s Alg is Okay! VP V NP VP PP VP V NP attach ate the caviar 1 VP  VP. PP VP PP VP 1 VP . VP PP (in column 1) (in column 4)

Intro to NLP - J. Eisner88 … but Earley’s Alg is Okay! VP V NP VP PP VP V NP ate the caviar with a spoon 1 VP  VP PP. VP PP VP 1 VP . VP PP (in column 1) (in column 7)

Intro to NLP - J. Eisner89 … but Earley’s Alg is Okay! VP V NP VP PP VP V NP ate the caviar with a spoon 1 VP  VP PP. VP PP VP 1 VP . VP PP can be reused (in column 1) (in column 7)

Intro to NLP - J. Eisner90 … but Earley’s Alg is Okay! VP V NP VP PP VP V NP VP PP 1 VP  VP. PP ate the caviar with a spoon VP PP VP 1 VP . VP PP can be reused (in column 1) (in column 7) attach

Intro to NLP - J. Eisner91 … but Earley’s Alg is Okay! VP V NP VP PP VP V NP VP PP ate the caviar with a spoon in his bed 1 VP  VP PP. VP PP VP 1 VP . VP PP can be reused (in column 1) (in column 10)

Intro to NLP - J. Eisner92 … but Earley’s Alg is Okay! VP V NP VP PP VP V NP VP PP ate the caviar with a spoon in his bed 1 VP  VP PP. VP PP VP 1 VP . VP PP can be reused again (in column 1) (in column 10)

Intro to NLP - J. Eisner93 … but Earley’s Alg is Okay! VP V NP VP PP VP V NP VP PP VP 1 VP . VP PP can be reused again VP PP VP PP 1 VP  VP. PP ate the caviar with a spoon in his bed (in column 1) (in column 10) attach

0 Papa 1 ate 2 the 3 caviar 4 with a spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. … 6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.5 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.4 PP P NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP NP. PP 0 NP. Papa 1 VP. VP PP 2 NP. Papa0 S NP VP.2 NP NP PP. 0 Det. the1 PP. P NP2 Det. the 1 VP VP. PP 1 VP VP PP. 0 Det. a1 V. ate2 Det. a4 PP. P NP7 PP. P NP 1 P. with0 ROOT S.1 VP V NP. 4 P. with2 NP NP. PP 0 S NP VP. 1 VP VP. PP 7 P. with 0 ROOT S. completed a VP in col 4 col 1 lets us use it in a VP PP structure

0 Papa 1 ate 2 the 3 caviar 4 with a spoon 7 0 ROOT. S0 NP Papa.1 V ate.2 Det the.3 N caviar. … 6 N spoon. 0 S. NP VP0 S NP. VP1 VP V. NP2 NP Det. N2 NP Det N.5 NP Det N. 0 NP. Det N0 NP NP. PP2 NP. Det N3 N. caviar1 VP V NP.4 PP P NP. 0 NP. NP PP1 VP. V NP2 NP. NP PP3 N. spoon2 NP NP. PP5 NP NP. PP 0 NP. Papa 1 VP. VP PP 2 NP. Papa0 S NP VP.2 NP NP PP. 0 Det. the1 PP. P NP2 Det. the1 VP VP. PP1 VP VP PP. 0 Det. a1 V. ate2 Det. a4 PP. P NP7 PP. P NP 1 P. with0 ROOT S.1 VP V NP. 4 P. with2 NP NP. PP 0 S NP VP. 1 VP VP. PP 7 P. with 0 ROOT S. completed that VP = VP PP in col 7 col 1 would let us use it in a VP PP structure can reuse col 1 as often as we need

Intro to NLP - J. Eisner96 What’s the Complexity?