# LR Parsing LALR Parser Generators

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1 LR Parsing LALR Parser Generators

2 Outline Review of bottom-up parsing Computing the parsing DFA Using parser generators 2

3 Bottom-up Parsing (Review) A bottom-up parser rewrites the input string to the start symbol The state of the parser is described as α I γ α is a stack of terminals and non-terminals γ is the string of terminals not yet examined Initially: I x 1 x 2... x n 3

4 The Shift and Reduce Actions (Review) Recall the CFG: E int E + (E) A bottom-up parser uses two kinds of actions: Shift pushes a terminal from input on the stack E + ( I int ) E + ( int I ) Reduce pops 0 or more symbols off of the stack (production RHS) and pushes a nonterminal on the stack (production LHS) E + (E + ( E ) I ) E + ( E I ) 4

5 Key Issue: When to Shift or Reduce? Idea: use a deterministic finite automaton (DFA) to decide when to shift or reduce The input is the stack The language consists of terminals and non-terminals We run the DFA on the stack and we examine the resulting state X and the token tok after I If X has a transition labeled tok then shift If X is labeled with A βon tok then reduce 5

6 LR(1) Parsing: An Example 0 int 1 E ( 4 accept on \$ 7 E E + (E) on \$, ) E E ( ) 5 11 E int on \$, + int 9 E int on ), + int E E + (E) on ), + I int + (int) + (int)\$ shift int I + (int) + (int)\$ E int E I + (int) + (int)\$ shift (x3) E + (int I ) + (int)\$ E int E + (E I ) + (int)\$ shift E + (E) I + (int)\$ E E+(E) E I + (int)\$ shift (x3) E + (int I )\$ E int E + (E I )\$ shift E + (E) I \$ E E+(E) E I \$ accept

7 Representing the DFA Parsers represent the DFA as a 2D table Recall table-driven lexical analysis Lines correspond to DFA states Columns correspond to terminals and nonterminals Typically columns are split into: Those for terminals: the action table Those for non-terminals: the goto table 7

8 Representing the DFA: Example The table for a fragment of our DFA: ( E ) 7 E E + (E) on \$, + int 5 E int on ), + int + ( ) \$ E 3 s4 4 s5 g6 5 r E int r E int 6 s8 s7 7 r E E+(E) r E E+(E) sk is shift and goto state k r X α is reduce gk is goto state k 8

9 The LR Parsing Algorithm After a shift or reduce action we rerun the DFA on the entire stack This is wasteful, since most of the work is repeated Remember for each stack element on which state it brings the DFA LR parser maintains a stack sym 1, state 1... sym n, state n state k is the final state of the DFA on sym 1 sym k 9

10 The LR Parsing Algorithm let I = w\$ be initial input let j = 0 let DFA state 0 be the start state let stack = dummy, 0 repeat case action[top_state(stack), I[j]] of shift k: push I[j++], k reduce X A: pop A pairs, push X, goto[top_state(stack), X] accept: halt normally error: halt and report error 10

11 Key Issue: How is the DFA Constructed? The stack describes the context of the parse What non-terminal we are looking for What production RHS we are looking for What we have seen so far from the RHS Each DFA state describes several such contexts E.g., when we are looking for non-terminal E, we might be looking either for an int or an E + (E) RHS 11

12 LR(0) Items An LR(0) item is a production with a I somewhere on the RHS The items for T (E) are T I (E) T ( I E) T (E I ) T (E) I The only item for X εis X I 12

13 LR(0) Items: Intuition An item [X α I β] says that the parser is looking for an X it has an α on top of the stack Expects to find a string derived from β next in the input Notes: [X αi aβ] means that a should follow. Then we can shift it and still have a viable prefix [X α I] means that we could reduce X But this is not always a good idea! 13

14 LR(1) Items An LR(1) item is a pair: X α I β, a X αβis a production a is a terminal (the lookahead terminal) LR(1) means 1 lookahead terminal [X α I β, a] describes a context of the parser We are trying to find an X followed by an a, and We have (at least) α already on top of the stack Thus we need to see next a prefix derived from βa 14

15 Note The symbol I was used before to separate the stack from the rest of input α I γ, where α is the stack and γ is the remaining string of terminals In items I is used to mark a prefix of a production RHS: X α I β, a Here β might contain terminals as well In both case the stack is on the left of I 15

16 Convention We add to our grammar a fresh new start symbol S and a production S E Where E is the old start symbol The initial parsing context contains: S I E, \$ Trying to find an S as a string derived from E\$ The stack is empty 16

17 LR(1) Items (Cont.) In context containing E E + I ( E ), + If ( follows then we can perform a shift to context containing E E + ( I E ), + In context containing E E + ( E ) I, + We can perform a reduction with E E + ( E ) But only if a + follows 17

18 LR(1) Items (Cont.) Consider the item E E + ( I E ), + We expect a string derived from E ) + There are two productions for E E int and E E + ( E) We describe this by extending the context with two more items: E I int, ) E I E + ( E ), ) 18

19 The Closure Operation The operation of extending the context with items is called the closure operation Closure(Items) = repeat for each [X α I Yβ, a] in Items for each production Y γ for each b in First(βa) add [Y I γ, b] to Items until Items is unchanged 19

20 Constructing the Parsing DFA (1) Construct the start context: Closure({S I E, \$}) E E + ( E ) int We abbreviate as: S I E, \$ E I E+(E), \$ E I int, \$ E I E+(E), + E I int, + S I E, \$ E I E+(E), \$/+ E I int, \$/+ 20

21 Constructing the Parsing DFA (2) A DFA state is a closed set of LR(1) items The start state contains [S I E, \$] A state that contains [X α I, b] is labelled with reduce with X αon b And now the transitions 21

22 The DFA Transitions A state State that contains [X α I yβ, b] has a transition labeled y to a state that contains the items Transition(State, y) y can be a terminal or a non-terminal Transition(State, y) Items = for each [X α I yβ, b] in State add [X αy I β, b] to Items return Closure(Items) 22

23 Constructing the Parsing DFA: Example 2 S I E, \$ E I E+(E), \$/+ E I int, \$/+ accept on \$ 0 E + S E I, \$ E E I +(E), \$/+ E int 1 E int I, \$/+ E E+ I (E), \$/+ ( E E+(I E), \$/+ E I E+(E), )/+ E I int, )/+ E int on \$, E E+(E I ), \$/+ E E I +(E), )/+ + ) and so on int 5 E int I, )/+ E int on ), + 23

24 LR Parsing Tables: Notes Parsing tables (i.e., the DFA) can be constructed automatically for a CFG But we still need to understand the construction to work with parser generators E.g., they report errors in terms of sets of items What kind of errors can we expect? 24

25 Shift/Reduce Conflicts If a DFA state contains both [X α I aβ, b] and [Y γ I, a] Then on input a we could either Shift into state [X αa I β, b], or Reduce with Y γ This is called a shift-reduce conflict 25

26 Shift/Reduce Conflicts Typically due to ambiguities in the grammar Classic example: the dangling else S if E then S if E then S else S OTHER Will have DFA state containing [S if E then S I, else] [S if E then S I else S, x] If else follows then we can shift or reduce Default (yacc, ML-yacc, etc.) is to shift Default behavior is as needed in this case 26

27 More Shift/Reduce Conflicts Consider the ambiguous grammar E E + E E * E int We will have the states containing [E E * I E, +] [E E * E I, +] [E I E + E, +] E [E E I + E, +] Again we have a shift/reduce on input + We need to reduce (* binds more tightly than +) Recall solution: declare the precedence of * and + 27

28 More Shift/Reduce Conflicts In yacc declare precedence and associativity: %left + %left * Precedence of a rule = that of its last terminal See yacc manual for ways to override this default Resolve shift/reduce conflict with a shift if: no precedence declared for either rule or terminal input terminal has higher precedence than the rule the precedences are the same and right associative 28

29 Using Precedence to Solve S/R Conflicts Back to our example: [E E * I E, +] [E E * E I, +] [E I E + E, +] E [E E I + E, +] Will choose reduce because precedence of rule E E * E is higher than of terminal + 29

30 Using Precedence to Solve S/R Conflicts Same grammar as before E E + E E * E int We will also have the states [E E + I E, +] [E E + E I, +] [E I E + E, +] E [E E I + E, +] Now we also have a shift/reduce on input + We choose reduce because E E + E and + have the same precedence and + is left-associative 30

31 Using Precedence to Solve S/R Conflicts Back to our dangling else example [S if E then S I, else] [S if E then S I else S, x] Can eliminate conflict by declaring else having higher precedence than then But this starts to look like hacking the tables Best to avoid overuse of precedence declarations or we will end with unexpected parse trees 31

32 Precedence Declarations Revisited The term precedence declaration is misleading! These declarations do not define precedence: they define conflict resolutions I.e., they instruct shift-reduce parsers to resolve conflicts in certain ways The two are not quite the same thing! 32

33 Reduce/Reduce Conflicts If a DFA state contains both [X α I, a] and [Y β I, a] Then on input a we don t know which production to reduce This is called a reduce/reduce conflict 33

34 Reduce/Reduce Conflicts Usually due to gross ambiguity in the grammar Example: a sequence of identifiers S ε id id S There are two parse trees for the string id S id S id S id How does this confuse the parser? 34

35 More on Reduce/Reduce Conflicts Consider the states [S id I, \$] [S I S, \$] [S id I S, \$] [S I, \$] id [S I, \$] [S I id, \$] [S I id, \$] [S I id S, \$] [S I id S, \$] Reduce/reduce conflict on input \$ S S id S S id S id Better rewrite the grammar as: S ε id S 35

36 Using Parser Generators Parser generators automatically construct the parsing DFA given a CFG Use precedence declarations and default conventions to resolve conflicts The parser algorithm is the same for all grammars (and is provided as a library function) But most parser generators do not construct the DFA as described before Because the LR(1) parsing DFA has 1000s of states even for a simple language 36

37 LR(1) Parsing Tables are Big But many states are similar, e.g. 1 E int I, \$/+ E int on \$, + and 5 E int I, )/+ E int on ), + Idea: merge the DFA states whose items differ only in the lookahead tokens We say that such states have the same core We obtain 1 E int I, \$/+/) E int on \$, +, ) 37

38 The Core of a Set of LR Items Definition: The core of a set of LR items is the set of first components Without the lookahead terminals Example: the core of {[X α I β, b], [Y γ I δ, d]} is {X α I β, Y γ I δ} 38

39 LALR States Consider for example the LR(1) states {[X α I, a], [Y β I, c]} {[X α I, b], [Y β I, d]} They have the same core and can be merged And the merged state contains: {[X α I, a/b], [Y β I, c/d]} These are called LALR(1) states Stands for LookAhead LR Typically 10 times fewer LALR(1) states than LR(1) 39

40 A LALR(1) DFA Repeat until all states have distinct core Choose two distinct states with same core Merge the states by creating a new one with the union of all the items Point edges from predecessors to new state New state points to all the previous successors A B C A C BE D E F D F 40

41 Conversion LR(1) to LALR(1): Example. ( accept on \$ 7 E E E + (E) on \$, ) 10 int E E ) ( 11 int 5 E int on \$, + int 9 E int on ), + E E + (E) on ), + E accept on \$ E E + (E) on \$, +, ) int 0 1,5 int ( 2 + 3,8 4,9 7,11 ) + 6,10 E E int on \$, +, )

42 The LALR Parser Can Have Conflicts Consider for example the LR(1) states {[X α I, a], [Y β I, b]} {[X α I, b], [Y β I, a]} And the merged LALR(1) state {[X α I, a/b], [Y β I, a/b]} Has a new reduce/reduce conflict In practice such cases are rare 42

43 LALR vs. LR Parsing: Things to keep in mind LALR languages are not natural They are an efficiency hack on LR languages Any reasonable programming language has a LALR(1) grammar LALR(1) parsing has become a standard for programming languages and for parser generators 43

44 A Hierarchy of Grammar Classes From Andrew Appel, Modern Compiler Implementation in ML 44

45 Semantic Actions in LR Parsing We can now illustrate how semantic actions are implemented for LR parsing Keep attributes on the stack On shifting a, push attribute for a on stack On reduce X α pop attributes for α compute attribute for X and push it on the stack 45

46 Performing Semantic Actions: Example Recall the example E T + E 1 { E.val = T.val + E 1.val } T { E.val = T.val } T int * T 1 { T.val = int.val * T 1.val } int { T.val = int.val } Consider the parsing of the string: 4 *

47 Performing Semantic Actions: Example 4 * int * int + int shift int 4 * int + int shift int 4 * int + int shift int 4 * int 9 + int reduce T int int 4 * T 9 + int reduce T int * T T 36 + int shift T 36 + int shift T 36 + int 6 reduce T int T 36 + T 6 reduce E T T 36 + E 6 reduce E T + E E 42 accept 47

48 Notes The previous example shows how synthesized attributes are computed by LR parsers It is also possible to compute inherited attributes in an LR parser 48

49 Notes on Parsing Parsing A solid foundation: context-free grammars A simple parser: LL(1) A more powerful parser: LR(1) An efficiency hack: LALR(1) LALR(1) parser generators Next time we move on to semantic analysis 49

50 Supplement to LR Parsing Strange Reduce/Reduce Conflicts due to LALR Conversion (and how to handle them)

51 Strange Reduce/Reduce Conflicts Consider the grammar S P R, NL N N, NL P T NL : T R T N : T N id T id P - parameters specification R - result specification N - a parameter or result name T -a type name NL - a list of names 51

52 Strange Reduce/Reduce Conflicts In P an id is a N when followed by, or : T when followed by id In R an id is a N when followed by : T when followed by, This is an LR(1) grammar But it is not LALR(1). Why? For obscure reasons 52

53 A Few LR(1) States P I T id P I NL : T id T I id id NL I N : 1 id T id I id N id I : N id I, 3 LALR reduce/reduce conflict on, NL I N, NL : N I id : N I id, LALR merge T id I id/, N id I :/, R I T, R I N : T, 2 id T id I, N id I : 4 T I id, N I id : 53

54 What Happened? Two distinct states were confused because they have the same core Fix: add dummy productions to distinguish the two confused states E.g., add R id bogus bogus is a terminal not used by the lexer This production will never be used during parsing But it distinguishes R from P 54

55 A Few LR(1) States After Fix P I T id P I NL : T id NL I N : NL I N, NL : N I id : N I id, T I id id 1 id T id I id 3 N id I : N id I, Different cores no LALR merging R. T, R. N : T, R.id bogus, T. id, N. id : 2 id T id I, N id I : R id I bogus, 4 55

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