Algorithms for NLP. LR Parsing. Reading: Hopcroft and Ullman, Intro. to Automata Theory, Lang. and Comp. Section , pp.

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1 Algorithms for NL L arsing eading: Hopcroft and Ullman, Intro. to Automata Theory, Lang. and Comp. Section , pp

2 Shift-educe arsing A class of parsers with the following principles: arsing is done Bottom-Up, reducing the input into the grammar start symbol The parser builds a right-most derivation of the input in reverse arsing algorithm simulates the operation of a DA refix of the sentential form is kept on the stack Two types of operation: Shift the next input symbol onto the stack educe the stack by popping the HS of a grammar rule, and pushing the corresponding LHS non-terminal symbol arser is usually deterministic and with no back-tracking Extremely efficient, operating in linear time - But - possible to construct for only a limited class of CFGs Algorithms for NL

3 L arsing General rinciples: Use sets of dotted grammar rules to reflect the state of the parser: What constituents have we constructed so far What constituents are we predicting next re-compile the grammar into a collection of finite sets of dotted rules Use these sets to capture the state of the parser during parsing The arser is a deterministic shift-reduce parser. Developed by Knuth in the late 1960s - as a framework for compiling programming languages Algorithms for NL

4 L arsing Algorithm erforms shift and reduce parsing actions on the stack, and changes state with each operation Is driven by a pre-compiled parsing table that has two parts The action table specifies the next shift or reduce parsing operation The goto table specifies which state to transfer to after a reduction The stack stores a string of the form are parser states and the are grammar symbols where the At each step the parser does one of the following types of operations: Shifts): ush the current input symbol the new state on the stack followed by educei): educe the stack according to rule of the grammar eject: eject the input as ungrammatical and signal an error Accept: Accept the input as grammatical and halt Algorithms for NL

5

6 L arsing - Example The Grammar: The original input: The large can can hold the water OS assigned input: art adj n aux v art n arser input: art adj n aux v art n $ Algorithms for NL

7 L arsing - Example Constructed arsing Table for the Grammar: Algorithms for NL

8 educe Shift Goto State art adj n aux v $ N V S 0 sh3 sh acc 2 sh6 sh7 5 3 sh8 sh9 4 sh10 5 r1 6 sh6 sh sh3 sh sh13 9 r3 10 r4 11 r5 12 r6 13 r Algorithms for NL

9 L arsing - Example The input: art adj n aux v art n $ Algorithms for NL

10 * % Step Action Stack after action 0 1 sh3 2 sh8 3 sh13 4 r2 6 sh7 7 sh3 8 sh9 9 r3 10 r6 11 r5 12 r1 13 accept 0 2!) * 6+ 7!" 0 2!) 05 sh6 2!) 0 0! " # 3!$ 0! " # 3!$ 0! " 0 # , 0 2!) * 6, 0 2!) * !) * 6+ 7!" 2 %8 8&13 * #3 &# Algorithms for NL

11 . / A 6 5 <;: = > <;: = > 4 A Constructing an SL arsing Table An item is a dotted grammar rule 021. / 0 >=<;:is also in We construct a deterministic FSA that recognizes prefixes of rightmost sentential forms of the grammar 7. The states of the FSA are sets of L0) items We augment the grammar with a new start rule 98 We define the closure operation on a set of items: 1. Every item in 2. If 0 1 and is a rule in 7, then add 0 4 to The closure operation adds predicted new items to the set similar to Earley s redictor operation) Algorithms for NL

12 I L G G G G H K C <7 < G G G G F J D E6 5 3 B 5 6 C B <7 < 3 B Constructing an SL arsing Table Wedefine the Goto operation for an item set and a grammar Bsymbol : is the closure of the set of all items such that 021 6? 0 1 Example: Similar to Earley s Scanner and Completer operations Algorithms for NL

13 Constructing an SL arsing Table l We construct the collection of sets of L0) items for an augmented grammar G l We start with the item set So = {closure{[s1 + as]))) l The algorithm: procedure iterns G ); begin C : = {~.losure{ls1 -..SI})}; repeat for each set of items I in C and each grammar symbol X such that goto I. X) is not empty and not in C do add goto/, X) to C until no more sets of items can be added to C end Algorithms for NL

14 % Q &# Q % $!#, + Q,, &# $!# : --5:,-6: --1:,-7: --2: &# $!# O, + Q $!# Q O "! # Q & )! Q *, $! Q % & Q, "! Q "! Q % & - Q + Q $! Q % & "! Q )! Q *, "! Q % & )! * Q, Q, -Q, Q Constructing an SL arsing Table - Example Building the collection of sets of M N items for our simple NL Grammar:,-11:,-12: -13: -3: "! # Q $! % & -8: -9: -4: $! Q & -10: Algorithms for NL $! % & "! "! Q & "! % & + Q )! *, Q

15 <7 < <7 < = : T < C C 4 C 3 S S > > : 1 < : T < : T <. / C 4 C 0 C $ 3 5 = S 1 : > : 3 4 W 5 U= V T S = Constructing an SL arsing Table Building the FSA and arsing Table: 1. Construct the collection of sets of grammar items from the 2. State corresponds to item set a) For any terminal symbol, if then set b) If 0 1 c) If 0 8 = 6? 6? and then set 0 1 d) For any non-terminal symbol then set X < = = = T3is rule 6? then set and T for all terminal symbols 0214, if and e) All table entries not defined in a)-d) are set as error Algorithms for NL

16 Constructing an SL arsing Table - Example The constructed FSA for our example grammar: S0 S S1 N adj art S2 V S5 S4 S3 art v aux S6 aux n n adj v V S7 S10 S9 S8 N S11 n S13 S12 adj Algorithms for NL

17 ] Y [ZY \ Lk) arsing How to handle conflicts in the SL table: A table conflict: more than one action is specified in Conflicts can be either shift-reduce or reduce-reduce arser will not be able to parse deterministically A Grammar for which this happens is not SL More powerful techniques for building item sets can sometimes resolve the problem, by making use of lookaheads into the input Known techniques: Canonical Lk), LALk) A lookahead of one is sufficient and optimal) in many cases Another option - extending the L arsing algorithm: GL arsing Algorithms for NL

18 arsing with an L arser The pointers that form the parse tree can be created while performing reduce actions A parse node is created for each constituent that is pushed onto the stack When we do a reduce - we create a new parse node for the LHS non-terminal and link it to the parse-nodes of the popped HS constituents At the end - the the parse tree constituent on the stack points to the root of Algorithms for NL

19 L arsing - Example The input: art adj n aux v art n $ Algorithms for NL

20 Step Action Stack after action arse Node 0 1 sh3 2 sh8 3 sh13 4 r2 6 sh7 7 sh3 8 sh9 9 r3 10 r6 11 r5 12 r1 13 accept 0 0! " 0! " 0! " sh6 #1 #1 3! $ #1 3! $ ! ) 4 2! ) 4 2! ) 4 2! ) 4 2! ) 4 2! ) 2, 3 1 art %2 %2 82 adj & n 2 4 N 1 2 3) *5 *5 11 6, 6 5 aux 6+ *5 6+ *5 6+ * ! " 6 7! " 76 v #7 3 7 art 3& #7 6+ * n 12 9 N 7 8) V 6 9) 5 11 V 5 10) 112 S 4 11) Algorithms for NL

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