1. Basic Parsing Algorithms: Earley Parser and Left Corner Parsing
Alexandr Chernov
Recent Advances in Parsing Technology

2. Chomsky hierarchy
Type-0 grammars (unrestricted grammars) include all formal grammars
Type-1 grammars (context-sensitive grammars) generate the context-sensitive languages
Type-2 grammars (context-free grammars) generate the context-free languages
Type-3 grammars (regular grammars) generate the regular languages

3. Context-free Grammar
A context-free grammar (for short, CFG) is a quadruple G = (V, Σ, P, S), where
V is a finite set of symbols called the vocabulary (or set of grammar symbols);
Σ ⊆ V is the set of terminal symbols (for short, terminals);
S ∈ (V − Σ) is a designated symbol called the start symbol;
P ⊆ (V − Σ) × V∗ is a finite set of productions (or rewrite rules, or rules).
The set N = V −Σ is called the set of nonterminal symbols (for short, nonterminals). Thus, P ⊆ N × V∗, and every production A, α is also denoted as A → α

4. Rewrite Rules
S → NP VP
NP → Det N
Det → the
NP → the N
...

5. Formal Grammar Terminals Nonterminals
Letters, numbers, words (cannot be broken down into "smaller" units)
Nonterminals
Syntactic variable (category), formula, arithmetic expression

6. Parsers
Parsing algorithms for context-free grammar play an important role in the implementation of:
compilers and interpreters for programming languages
programs which "understand" or translate natural languages

7. Two common types of parsers
The main task of parsing is to connect the root node S with the tree leaves, the input
Top-down parsers: starts constructing the parse tree from the root and move down towards the leaves. Easy to implement, but work with restricted grammars. Examples:
Predictive parsers (e.g., LL(k))
Bottom-up parsers: build the nodes on the bottom of the parse tree first. Suitable for automatic parser generation, handle a larger class of grammars. Examples:
Shift-reduce parser (or LR(k) parsers)
Both are general techniques that can be made to work for all languages (but not all grammars!).

8. Basic Parsing Algorithms
Earley parser
Chart parser
CKY (Cocke-Younger-Kasami)
Head Driven / Left Corner Parsing

9. Earley Parser Can parse all context-free languages
Complexity O(n³), where n is the length of the parsed string, O(n²) for unambiguous grammars
Top-down dynamic programming algorithm

10. Special Symbols ┤ - right terminator
. (dot) – position between terminals/nonterminals
E→ .E+T
E→ E.+T
Φ – complete production

11. Earley Parser's Steps
Predictor (applicable to a state when there is a nonterminal to the right of the dot)
Scanner (applicable if there is a terminal to the right of the dot)
Completer (applicable to a state if its dot is at the end of its production)

12. Earley Parser Algorithm
Grammar AE input string = a+a*a
root: E→T | E+T
T→P | T*P
P→a
S0 (x1=a)
Φ→ .E ┤
E→ .E+T
E→ .T
T→ .T*P
T→ .P
P→ .a
S1 (x2=+)
P→ a.
T→ P.
E→ T.
T→ T.*P
Φ→ E. ┤
E→ E.+T
S3 (x4=*)
P→ a.
T→ P.
E→ E+T.
T→ T.*P
S5 (x6= ┤)
P→ a.
T→ T*P.
E→ E+T.
T→ T.*P
Φ→ E.┤
E→ E.+T
S4 (x5=a)
T→ T*.P
P→ .a S6
Φ→ E ┤.
S2 (x3=a)
E→ E+.T
T→ .T*P
T→ .P
P→ .a

13. Left-Corner Parsing
For some grammars top-down prediction can fail to terminate, bottom-up parser is needed
Going Wrong with Top-down Parsing
Input string: John died
S → NP VP
NP → Det N
NP → PN
VP → IV
Det → the
N → robber
PN → John
IV → died

14. Left-Corner Parsing Going Wrong with Bottom-up Parsing
Input string: The plant died
S → NP VP
NP → Det N
VP → IV
VP → TV NP
TV → plant
IV → died
Det → the
N → plant

15. Left-Corner Parsing
The key idea of left-corner parsing is to combine top-down and bottom-up processing
Left corner of a rule
S → NP VP
VP → IV
PN → John

16. Left-Corner Parsing S VP NP IV PN died How does it work? S → NP VP
NP → Det N
NP → PN
VP → IV
Det → the
N → robber
PN → John
IV → died
How does it work? S VP NP PN IV
died

17. Head-Corner Parsing
Head-Corner Parser starts by locating a potential head of the phrase and then proceeds by parsing the daughters to the left and the right of the head
Head-Corner Parser is a generalization of Left- Corner Parser
Left-Corner Parser is 10% faster

18. Head-Corner Parsing
The daughters left of the head are parsed from right to left (starting from the head), the daughters right of the head are parsed from left to right (starting from the head)

20. Head-Corner Parsing
Input string:
Time flies like an arrow

21. Summary
Bottom-up parsing is used for analyzing unknown data relationships in attempt to identify the most fundamental units first, and then to infer higher-order structures from them
Top-down parsing is employed for analyzing unknown data relationships by hypothesizing general parse tree structures and then considering whether the known fundamental structures are compatible with the hypothesis

22. Possible ways of using
Chart parsers can be used for parsing computer languages. Earley Parsers in particular have been used in compiler compilers where their ability to parse using arbitrary CFG eases the task of writing the grammar for a particular language.
Left-Corner Parser can be used for processing of natural languages as long as it recognizes ambiguity

23. Thank you for attention
Questions?

24. Sources
Jay Earley. An efficient context-free parsing algorithm. Communications of the ACM, 13(2):94–102, 1970
Gertjan van Noord. An efficient implementation of the head-corner parser. Computational Linguistics, 23(3):425–456, 1997
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