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Chart Parsing

Allen’s Chapter 3

J & M’s Chapter 10

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Chart Parsing

General Principles: A Bottom-Up parsing method

– Construct a parse starting from the input symbols– Build constituents from sub-constituents– When all constituents on the RHS of a rule are matched, create a

constituent for the LHS of the rule

The Chart allows storing partial analyses, so that they can be shared.

Data structures used by the algorithm:– The Key: the current constituent we are attempting to “match”– An Active Arc: a grammar rule that has a partially matched RHS– The Agenda: Keeps track of newly found unprocessed constituents– The Chart: Records processed constituents (non-terminals) that span

substrings of the input

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Chart Parsing

Steps in the Process: Input is processed left-to-right, one word at a time

1. Find all POS of the current word (terminal-level)2. Initialize Agenda with all POS of the word3. Pick a Key from the Agenda4. Add all grammar rules that start with the Key as active

arcs5. Extend any existing active arcs with the Key6. Add LHS constituents of newly completed rules to the

Agenda7. Add the Key to the Chart8. If Agenda not empty – go to (3), else go to (1)

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The Chart Parsing Algorithm

Extending Active Arcs with a Key:- Each Active Arc has the form:

<pi> [A X1…C…Xm] <pj>

- A Key constituent has the form: <pj >C<pk>- When processing the Key <p1>C<p2>, we search the active

arc list for an arc <p0>[A X1…C…Xm]<p1>, and then create a new active arc <p0>[A X1…C… Xm]<p2>

- If the new active arc is a completed rule: <p0>[A X1…C]<p2>, then we add <p0>A<p2> to the Agenda

- After “using” the key to extend all relevant arcs, it is entered into the Chart

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The Chart Parsing Algorithm

The Main Algorithm: parsing input x = x1…xn

1. i = 02. If Agenda is empty and i < n then set i = i + 1, find all

POS of xi and add them as constituents <pi>C<pi+1> to the Agenda

3. Pick a Key constituent <pj>C< pk > from the Agenda4. For each grammar rule of form A CX1…Xm, add

<pj>[A CX1…Xm]<pj> to the list of active arcs5. Use Key to extend all relevant active arcs6. Add LHS of any completed active arcs into the Agenda7. Insert the Key into the Chart8. If Key is <1>S<n> then Accept the input Else goto (2)

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Chart Parsing - Example

The Grammar:(1) S NP VP

(2) NP ART ADJ N

(3) NP ART N

(4) NP ADJ N

(5) VP AUX VP

(6) VP V NP

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Chart Parsing - Example

The input: “x = The large can can hold the water”

POS of Input Words:– the: ART– large: ADJ– can: N, AUX, V– hold: N, V– water: N, V

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The large can can hold the water

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The large can can hold the water

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The large can can hold the water

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The large can can hold the water

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The final Chart

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Time Complexity of Chart Parsing

• Algorithm iterates for each word of input (i.e. n iterations)• How many keys can be created in a single iteration i?

– Each key in Agenda has the form <pj>C< pi+1>, 1 j i

– Thus O(n) keys

• Each key <pj>C<pi+1> can extend active arcs of the form• <pk>[A X1…C…Xm]<pj>, for 1 k j• Each key can thus extend at most O(n) active arcs• Time required for each iteration is thus O(n2)• Time bound on entire algorithm is therefore O(n3)

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Improved Chart Parser

Increasing the Efficiency of the Algorithm• Observation: The algorithm creates constituents that cannot “fit” into

a global parse structure. Example: <2>[NP ADJ N]<3>

• These can be eliminated using predictions• Using Top-down predictions:

– For each non-terminal A, define First(A) = the set of all leftmost non-terminals derivable from A

– Compute in advance First(A) for all non-terminals A– When processing the key <pj>C<pi+1> we look for grammar rules that form

an active arc <pj>[A C X2…Xm]<pj>– Add the active arc only if A is predicted: there already exists an active arc

<pk>[B X1…D…Xm]<pj> such that A First(D)– For pk = 1 (arcs of the form <1>[A C X2…Xm]<1>), add the active arc

only if A First(S)

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Improved Chart Parser - Example

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The large can can hold the water

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The large can can hold the water

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