# If-Then-Else Problem (a motivating example for LR grammars)

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1 If-Then-Else Problem (a motivating example for LR grammars) If x then y else z If a then if b then c else d this is analogous to a bracket notation when left brackets >= right brackets: [ [ ] ([ i ] j, i>=j) Grammar: S [ S C S λ C ] [ [ ] SSλC or SSCλ ambiguous C λ 24

2 Solving the If-Then-Else Problem The ambiguity exists at the language level as well. The semantics need to be defined properly: e.g., the then part belongs to the closest matching if S [ S S S1 S1 [ S1 ] S1 λ This grammer is still not LL(1), nor is it LL(k) Show that this is so. 25

3 Parsing the If-Then-Else Construct LL(k) parsers can look ahead k tokens. (LL(k) will not be discussed in this class, but be able to explain in English what LL(k) means and recognize a simple LL(k) grammar.) For the If-Then-Else construct, a parsing strategy is needed that can look ahead at the entire RHS of a production (not just k tokens) before deciding what production to take. LR parsers can do that. 26

4 LR Parsers A Shift-Reduce Parser: Basic idea: put tokens on a stack until an entire production is found. Issues: recognize the end point of a production find the length of the production (RHS) find the corresponding nonterminal (i.e., the LHS of the production) 27

5 Data Structures for Shift-Reduce Parsers At each state, given the next token, a goto table defines the successor state an action table defines whether to shift (put the next state and token on the stack) reduce (a RHS is found, process the production) terminate (parsing is complete) 28

6 Example of Shift-Reduce Parsing Consider the simple Grammar: <program> begin <stmts> end \$ <stmts> SimpleStmt ; <stmts> <stmts> begin <stmts> end ; <stmts> <stmts> λ Shift Reduce Driver Algorithm on page 142, Fig

7 LR Parser Generators (OR: HOW TO COME UP WITH GOTO AND ACTION TABLES) Basic idea: shift tokens onto the stack; at any step keep the set of productions that match the read-in tokens. Reduce the RHS of recognized productions to the corresponding nonterminal on the LHS of the production by replacing the RHS tokens on the stack wit the LHS non-terminal. 30

8 LR(k) Parsers LR(0) parsers: no lookahead predict which production to use by looking only at the symbols already read. LR(k) parsers: k symbol lookahead most powerful class of deterministic bottom-up parsers (in terms of grammars read) 31

9 Terminology for LR Parsing Configuration: A X 1... X i X i+1... X j Configuration set: marks the point to which the production has been recognized all the configurations that apply at a given point in the parse. For example: A B CD A B GH T B Z 32

10 Configuration Closure Set Include all configurations necessary to recognize the next symbol after the. For example: S E\$ E E + T T T ID (E) closure0({s E\$})={ S E \$ E E+T E T T ID T (E) } 33

11 Successor Configuration Set Starting with the initial configuration set s0 = closure0({s α \$}), an LR(0) parser will find the successor, given a (next) symbol X. X can be either a terminal (a token from the scanner) or a nonterminal (the result of a reduction) Determining the successor s = go_to(s,x) : 1. pick all configurations in s of the form A β X γ 2. take closure0 of this set 34

12 Building the Characteristic Finite State Machine (CFSM) Nodes are configuration sets Arcs are go_to relationships Example: S S\$ S ID State 0: State 1: ID S S\$ S ID S ID State 2: State 3: S \$ S S \$ S S \$ 35

13 Building the go_to Table Building the go_to table is straightforward from the CFSM: For the previous example the table looks like this: State Symbol ID \$ S

14 Building the Action Table Given the configuration set s: We shift if the next token matches a terminal after the in some configuration in A α a β s and a V t, else error We reduce production P if the is at the end of a production B α s and production P is B α 37

15 Exercise Create CFSM, go_to table, and action table for S E\$ E E + T T T ID (E) 1: S E\$ 2: E E + T 3: E T 4: T ID 5: T (E) 38

16 LR(0) and LR(k) Grammars For LR(0) grammars the action table entries just described are unique. For most useful grammars we cannot decide on shift or reduce based on the symbols read. Instead, we have to look ahead k tokens. This leads to LR(k). However, it is possible to create an LR(0) grammar that is equivalent to any given LR(k) grammar (provided there is an end marker). This is only of theoretical interest because this grammar may be very complex and unreadable. 39

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