# CSE P 501 Compilers. LR Parsing Hal Perkins Spring UW CSE P 501 Spring 2018 D-1

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1 CSE P 501 Compilers LR Parsing Hal Perkins Spring 2018 UW CSE P 501 Spring 2018 D-1

2 Agenda LR Parsing Table-driven Parsers Parser States Shift-Reduce and Reduce-Reduce conflicts UW CSE P 501 Spring 2018 D-2

3 Bottom-Up Parsing Idea: Read the input left to right Whenever we ve matched the right hand side of a production, reduce it to the appropriate non-terminal and add that non-terminal to the parse tree The upper edge of this partial parse tree is known as the frontier UW CSE P 501 Spring 2018 D-3

4 Example Grammar Bottom-up Parse S ::= aab e A ::= Abc b B ::= d a b b c d e UW CSE P 501 Spring 2018 D-4

5 LR(1) Parsing We ll look at LR(1) parsers Left to right scan, Rightmost derivation, 1 symbol lookahead Almost all practical programming languages have a LR(1) grammar LALR(1), SLR(1), etc. subsets of LR(1) LALR(1) can parse most real languages, tables are more compact, and is used by YACC/Bison/CUP/etc. UW CSE P 501 Spring 2018 D-5

6 LR Parsing in Greek The bottom-up parser reconstructs a reverse rightmost derivation Given the rightmost derivation S =>b 1 =>b 2 => =>b n-2 =>b n-1 =>b n = w the parser will first discover b n-1 =>b n, then b n-2 =>b n-1, etc. Parsing terminates when b 1 reduced to S (start symbol, success), or No match can be found (syntax error) UW CSE P 501 Spring 2018 D-6

7 How Do We Parse with This? Key: given what we ve already seen and the next input symbol (the lookahead), decide what to do. Choices: Perform a reduction Look ahead further Can reduce A=>b if both of these hold: A=>b is a valid production, and A=>b is a step in this rightmost derivation This is known as a shift-reduce parser UW CSE P 501 Spring 2018 D-7

8 Sentential Forms If S =>* a, the string a is called a sentential form of the grammar In the derivation S =>b 1 =>b 2 => =>b n-2 =>b n-1 =>b n = w each of the b i are sentential forms A sentential form in a rightmost derivation is called a right-sentential form (similarly for leftmost and leftsentential) UW CSE P 501 Spring 2018 D-8

9 Handles Informally, a production whose right hand side matches a substring of the tree frontier that is part of the rightmost derivation of the current input string (i.e., the correct production) Even if A ::= b is a production, it is a handle only if b matches the frontier at a point where A ::= b was used in this specific derivation b may appear in many other places in the frontier without designating a handle Bottom-up parsing is all about finding handles UW CSE P 501 Spring 2018 D-9

10 Handle Examples In the derivation S => aabe => aade => aabcde => abbcde abbcde is a right sentential form whose handle is A::=b at position 2 aabcde is a right sentential form whose handle is A::=Abc at position 4 Note: some books take the left end of the match as the position UW CSE P 501 Spring 2018 D-10

11 Handles The Dragon Book Defn. Formally, a handle of a right-sentential form g is a production A ::= b and a position in g where b may be replaced by A to produce the previous right-sentential form in the rightmost derivation of g UW CSE P 501 Spring 2018 D-11

12 Implementing Shift-Reduce Parsers Key Data structures A stack holding the frontier of the tree A string with the remaining input (tokens) We also need something to encode the rules that tell us what action to take next, given the state of the stack and the lookahead symbol Typically a table that encodes a finite automata UW CSE P 501 Spring 2018 D-12

13 Shift-Reduce Parser Operations Reduce if the top of the stack is the right side of a handle A::=b, pop the right side b and push the left side A Shift push the next input symbol onto the stack Accept announce success Error syntax error discovered UW CSE P 501 Spring 2018 D-13

14 Shift-Reduce Example S ::= aabe A ::= Abc b B ::= d Stack Input Action \$ abbcde\$ shift UW CSE P 501 Spring 2018 D-14

15 How Do We Automate This? Cannot use clairvoyance in a real parser (alas ) Defn. Viable prefix a prefix of a right-sentential form that can appear on the stack of the shift-reduce parser Equivalent: a prefix of a right-sentential form that does not continue past the rightmost handle of that sentential form In Greek: g is a viable prefix of G if there is some derivation S =>* rm aaw =>* rm abw and g is a prefix of ab. The occurrence of b in abw is a handle of abw UW CSE P 501 Spring 2018 D-15

16 How Do We Automate This? Fact: the set of viable prefixes of a CFG is a regular language(!) Idea: Construct a DFA to recognize viable prefixes given the stack and remaining input Perform reductions when we recognize them UW CSE P 501 Spring 2018 D-16

17 DFA for prefixes of S ::= aabe A ::= Abc b B ::= d e accept 8 9 \$ B start 1 a 2 A 3 b 6 c 7 S ::= aabe A ::= Abc b d 4 5 A ::= b B ::= d UW CSE P 501 Spring 2018 D-17

18 Trace S ::= aabe A ::= Abc b B ::= d Stack Input \$ abbcde\$ accept e 8 9 B \$ start a A b c S ::= aabe A ::= Abc b d 4 5 A ::= b B ::= d UW CSE P 501 Spring 2018 D-18

19 Observations Way too much backtracking We want the parser to run in time proportional to the length of the input Where the heck did this DFA come from anyway? From the underlying grammar Defer construction details for now UW CSE P 501 Spring 2018 D-19

20 Avoiding DFA Rescanning Observation: no need to restart DFA after a shift. Stay in the same state and process next token. Observation: after a reduction, the contents of the stack are the same as before except for the new nonterminal on top \ Scanning the stack will take us through the same transitions as before until the last one \ If we record state numbers on the stack, we can go directly to the appropriate state when we pop the right hand side of a production from the stack UW CSE P 501 Spring 2018 D-20

21 Stack Change the stack to contain pairs of states and symbols from the grammar \$s 0 X 1 s 1 X 2 s 2 X n s n State s 0 represents the accept (start) state (Not always explicitly on stack depends on particular presentation) When we push a symbol on the stack, push the symbol plus the FA state When we reduce, popping the handle will reveal the state of the FA just prior to reading the handle Observation: in an actual parser, only the state numbers are needed since they implicitly contain the symbol information. But for explanations / examples it can help to show both. UW CSE P 501 Spring 2018 D-21

22 Encoding the DFA in a Table A shift-reduce parser s DFA can be encoded in two tables One row for each state action table encodes what to do given the current state and the next input symbol goto table encodes the transitions to take after a reduction UW CSE P 501 Spring 2018 D-22

23 Actions (1) Given the current state and input symbol, the main possible actions are si shift the input symbol and state i onto the stack (i.e., shift and move to state i ) rj reduce using grammar production j The production number tells us how many <symbol, state> pairs to pop off the stack (= number of symbols on rhs of production) UW CSE P 501 Spring 2018 D-23

24 Actions (2) Other possible action table entries accept blank no transition syntax error A LR parser will detect an error as soon as possible on a left-to-right scan A real compiler needs to produce an error message, recover, and continue parsing when this happens UW CSE P 501 Spring 2018 D-24

25 Goto When a reduction is performed using A ::= β, we pop β <symbol, state> pairs from the stack revealing a state uncovered_s on the top of the stack goto[uncovered_s, A] is the new state to push on the stack when reducing production A ::= b (after popping handle β and pushing A) UW CSE P 501 Spring 2018 D-25

26 Reminder: DFA for S ::= aabe A ::= Abc b B ::= d accept 8 e 9 \$ B start 1 a 2 A 3 b 6 c 7 S ::= aabe A ::= Abc b d 4 5 A ::= b B ::= d UW CSE P 501 Spring 2018 D-26

27 LR Parse Table for 1. S ::= aabe 2. A ::= Abc 3. A ::= b 4. B ::= d State action goto a b c d e \$ A B S 0 acc 1 s2 g0 2 s4 g3 3 s6 s5 g8 4 r3 r3 r3 r3 r3 r3 5 r4 r4 r4 r4 r4 r4 6 s7 7 r2 r2 r2 r2 r2 r2 8 s9 9 r1 r1 r1 r1 r1 r1 UW CSE P 501 Spring 2018 D-27

28 LR Parsing Algorithm tok = scanner.gettoken(); while (true) { s = top of stack; if (action[s, tok] = si ) { push tok; push i (state); tok = scanner.gettoken(); } else if (action[s, tok] = rj ) { pop 2 * length of right side of production j (2* b ); uncovered_s = top of stack; push left side A of production j ; push state goto[uncovered_s, A]; } } else if (action[s, tok] = accept ) { return; } else { // no entry in action table report syntax error; halt or attempt recovery; } UW CSE P 501 Spring 2018 D-28

29 Example Stack Input \$ abbcde\$ S 1. S ::= aabe 2. A ::= Abc 3. A ::= b 4. B ::= d action goto a b c d e \$ A B S 0 s2 ac 1 s2 g0 2 s4 g3 3 s6 s5 g8 4 r3 r3 r3 r3 r3 r3 5 r4 r4 r4 r4 r4 r4 6 s7 7 r2 r2 r2 r2 r2 r2 8 s9 9 r1 r1 r1 r1 r1 r1 UW CSE P 501 Spring 2018 D-29

30 LR States Idea is that each state encodes The set of all possible productions that we could be looking at, given the current state of the parse, and Where we are in the right hand side of each of those productions UW CSE P 501 Spring 2018 D-30

31 Items An item is a production with a dot in the right hand side Example: Items for production A ::= X Y A ::=. X Y A ::= X. Y A ::= X Y. Idea: The dot represents a position in the production UW CSE P 501 Spring 2018 D-31

32 DFA for S ::= aabe A ::= Abc b B ::= d S ::=.aabe a S ::= a.abe A ::=.Abc A ::=.b b A ::= b. \$ A 3 5 accept S ::= aa.be A ::= A.bc B ::=.d d B ::= d. 8 9 B 6 b 7 S ::= aab.e e S ::= aabe. A ::= Ab.c c A ::= Abc. UW CSE P 501 Spring 2018 D-32

33 Problems with Grammars Grammars can cause problems when constructing a LR parser Shift-reduce conflicts Reduce-reduce conflicts UW CSE P 501 Spring 2018 D-33

34 Shift-Reduce Conflicts Situation: both a shift and a reduce are possible at a given point in the parse (equivalently: in a particular state of the DFA) Classic example: if-else statement S ::= ifthen S ifthen S else S UW CSE P 501 Spring 2018 D-34

35 Parser States for 1. S ::= ifthen S 2. S ::= ifthen S else S S ::=. ifthen S S ::=. ifthen S else S ifthen S ::= ifthen. S S ::= ifthen. S else S S S ::= ifthen S. S ::= ifthen S. else S else S ::= ifthen S else. S State 3 has a shiftreduce conflict Can shift past else into state 4 (s4) Can reduce (r1) S ::= ifthen S (Note: other S ::=. ifthen items not included in states 2-4 to save space) UW CSE P 501 Spring 2018 D-35

36 Solving Shift-Reduce Conflicts Fix the grammar Done in Java reference grammar, others Use a parse tool with a longest match rule i.e., if there is a conflict, choose to shift instead of reduce Does exactly what we want for if-else case Guideline: a few shift-reduce conflicts are fine, but be sure they do what you want (and that this behavior is guaranteed by the tool specification) UW CSE P 501 Spring 2018 D-36

37 Reduce-Reduce Conflicts Situation: two different reductions are possible in a given state Contrived example S ::= A S ::= B A ::= x B ::= x UW CSE P 501 Spring 2018 D-37

38 Parser States for 1. S ::= A 2. S ::= B 3. A ::= x 4. B ::= x 1 2 S ::=.A S ::=.B A ::=.x B ::=.x x A ::= x. B ::= x. State 2 has a reducereduce conflict (r3, r4) UW CSE P 501 Spring 2018 D-38

39 Handling Reduce-Reduce Conflicts These normally indicate a serious problem with the grammar. Fixes Use a different kind of parser generator that takes lookahead information into account when constructing the states Most practical tools use this information Fix the grammar UW CSE P 501 Spring 2018 D-39

40 Another Reduce-Reduce Conflict Suppose the grammar tries to separate arithmetic and boolean expressions expr ::= aexp bexp aexp ::= aexp * aident aident bexp ::= bexp && bident bident aident ::= id bident ::= id This will create a reduce-reduce conflict after recognizing id UW CSE P 501 Spring 2018 D-40

41 Covering Grammars A solution is to merge aident and bident into a single non-terminal like ident (or just use id in place of aident and bident everywhere they appear) This is a covering grammar Will generate some programs (sentences) that are not generated by the original grammar Use the type checker or other static semantic analysis to weed out illegal programs later UW CSE P 501 Spring 2018 D-41

42 Coming Attractions Constructing LR tables We ll present a simple version (SLR(0)) in lecture, then talk about adding lookahead and then a little bit about how this relates to LALR(1) used in most parser generators LL parsers and recursive descent Continue reading ch. 3 UW CSE P 501 Spring 2018 D-42

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