CS 321 Programming Languages and Compilers

1
CS 321Programming Languages and Compilers

• Bottom Up Parsing

2
Bottom-up Parsing Shift-reduce parsing

• Grammar H L ? EL E
• E ? a b
• Input aab
• has parse tree

L
L

E
L

E
a
a
b
3
Data for Shift-reduce Parser

• Input string sequence of tokens being checked
  for grammatical correctness
• Stack sentential form representing the input
  seen so far
• Trees Constructed the parse trees that have been
  constructed so far at some point in the parsing

4
Operations of shift-reduce parser

• Shift move input token to the set of trees as a
  singleton tree.
• Reduce coalesce one or more trees into single
  tree (according to some production).
• Accept terminate and accept the token stream as
  grammatically correct.
• Reject Terminate and reject the token stream.

5
A parse for grammar H

• Input Stack Action Trees
• aab Shift
• ab a Reduce E? a
• ab E Shift
• ab E Shift

a

6
A parse for grammar H

• Input Stack Action Trees
• b Ea Reduce E? a
• b EE Shift
• b EE Shift

7
A parse for grammar H

• Input Stack Action
  Trees
• EEb Reduce E?b
• EEE Reduce L?E
• EEL Reduce L?EL

b


L


8
A parse for grammar H

• Input Stack Action Trees
• EL Reduce L?EL
• L Acc

9
Bottom-up Parsing

• Characteristic automata for an LR grammar
  tells when to shift/reduce/accept or reject
• handles
• viable prefixes

10
A parse for grammar H

• Input Stack Action StackInput
• aab Shift aab
• ab a Reduce E? a aab
• ab E Shift Eab
• ab E Shift
  Eab
• b Ea Reduce E? a Eab
• b EE Shift EEb
• b EE Shift EEb
• EEb Reduce E?b EEb
• EEE Reduce L?E EEE
• EEL Reduce L?EL EEL
• EL Reduce L?EL EL
• L Accept L

11
Bottom-up parsing?

• StackInput
• L
• EL
• EEL
• EEE
• EEb
• EEb
• EEb
• Eab
• Eab
• Eab
• aab
• aab

12
Handles and viable prefixes

• Stackremaining input sentential form
• Handle the part of the sentential form that
  is reduced in each step
• Viable prefix the prefix of the sentential
  form in a right-most derivation that do not
  extend beyond the end of the handle
• E.g. viable prefixes for H (E)(E L a
  b)
• Viable prefixes form a regular set.

13
Characteristic Finite State Machine (CFSM)

• Viable prefixes of H are recognized by this CFSM

1
L

L
2
5
6
E
E
0
a
a
3
b
b
4
14
How a Bottom-Up Parser Works

• Run the CFSM on symbols in the stack
• If a transition possible on the incoming input
  symbol, then shift, else reduce.
• Still need to decide which rule to use for the
  reduction.

15
Characteristic automaton
Viable Prefixes
start
aab leads to state 3 after a Eab leads to
state 3 after Ea EEb leads to state 4 after
EEb EEE leads to state 2 after EEE EEL
leads to state 6 after EEL EL leads to state 6
after EL
16
Characteristic automaton
start
State Action 0,5 shift (if possible) 1 accept 2 r
educe L?E, if EOF shift otherwise 3 reduce
E?a 4 reduce E?b 6 reduce L?EL
17
Example expression grammar

• E ? ET T
• T ? TP P
• P ? id
• ididididhas parse tree

E
T

E
T

E
P
T

E
P
id
T
P
id
P
id
id
18
A parse in this grammar

• idididid Shift
• ididid id Reduce
• ididid P Reduce
• ididid T Reduce
• ididid E Shift
• ididid E Shift
• idid Eid Reduce
• idid EP Reduce
• idid ET Reduce
• idid E Shift
• idid E Shift

19
A parse in this grammar (cont.)

• id E id Reduce
• id EP Reduce
• id ET Reduce
• id E Shift
• id E Shift
• Eid Reduce
• EP Reduce
• ET Reduce
• E Reduce
• E Accept

20
Characteristic Finite State Machine

• The CFSM recognizes viable prefixes (strings of
  grammar symbols that can appear on the stack)

T

1
5
8
E
id
4

id
P
0
id
T
P
2

7
9
P
3
21
Definitions

• Rightmost derivation
• Right-sentential form
• Handle
• Viable prefix
• Characteristic automaton

22
Rightmost Derivation
Definition A rightmost derivation in G is a
derivation
such that for each step i, the rightmost
non-terminal in
is replaced to obtain
1.
is not right-most
2.
is right-most
23
Right-sentential Forms
Definition A right-sentential form is any
sentential form that occurs in a right-most
derivation.
E.g., any of these
24
Handles
Definition Assume the i-th step of a rightmost
derivation is wiuiAvi ? ui?viwi1 Then,
(?, ui?) is the handle of wi1
In an unambiguous grammar, any sentence has a
unique rightmost derivation, and so we can talk
about the handle rather than a handle.
25
The Plan

• Construct a parser by first constructing the
  CFSM, then constructing GOTO and ACTION tables
  from it.
• Construction has two parts
• LR(0) construction of the CFSM
• SLR(1) construction of tables

26
Constructing the CFSM States
The states in the CFSM are created by taking the
closure of LR(0) items
L ? E L L ? E L L ? E L L ? E L
Given a production L ? E L, these are all
induced LR(0) items
27
What is ?
The in L ? E L represents the state of
the parse.
L
L
E
Only this part of the tree is fully developed
28
A State in the CFSM Closure of LR(0) Item
For set I of LR(0) items, calculate
closure(I) 1. if A ? ? B ? is in
closure(I), then for every production B
? ?, B ? ? is in closure(I) 2.
closure(I) is the smallest set with property (1)
29
Closure of LR(0) Item Example
H L ? EL E E ? a b
closure(L ? E L) L ? E L , L ?
E L , L ? E, E ? a , E ? b
30
LR(0) Machine

• Given grammar G with goal symbol S, augment the
  grammar by adding new goal symbol S and
  production S ? S.
• States sets of LR(0) items
• Start state closure(S ? S)
• All states are considered to be final (set of
  viable prefixes closed under prefix)
• transition(I, X) closure(A ? ?X? A
  ? ?X????I).

31
Example LR(0) CFSM Construction
H L? L L ? EL E E ?
a b
Augment the grammar
Initial State is closure of this augmenting
rule
L? L L ? EL L ? E E ? a E ? b
closure(L? L)
32
Example Transitions from I0
transition( , L) L ? L
transition( , E) L ? E L , L ? E
transition( , a ) E ? a
transition( , b) E ? b
There are no other transitions from I0. There
are no transitions possible from I1. Now consider
the transitions from I2.
33
Transitions from I2
L ? E L L ? EL L ? E E ? a E ? b
transition( , )
transition( , L) L ? EL
New state
transition( , E)
transition( , a)
transition( , b)
34
The CFSM Transition Diagram for H
L ? L
L
L ? E L L ? EL L ? E E ? a E ? b
L? L L ? EL L ? E E ? a E ? b

L ? E L L ? E
E
E
L
L ? EL
a
a
E ? a
b
b
E ? b
35
Characteristic Finite State Machine for H
1
L

L
2
5
6
E
E
0
a
a
3
b
b
4
36
How LR(1) parsers work

• GOTO table transition function of characteristic
  automaton in tabular form
• ACTION table State? ? ?Action
• Procedure
• Use the GOTO table to run the CFSM over the
  stack. Suppose state reached at top of stack is
  ?.
• Take action given by ACTION(?,a), where a is
  incoming input symbol.

37
Action Table for H
38
SLR(1) Parser Construction

• GOTO table is the move function from the LR(0)
  CFSM.
• ACTION table is constructed from the CFSM as
  follows
• If state i contains A ? ?a?, then ACTION(i,a)
  Shift.
• If state i contains A ? ?, then ACTION(i,a)
  Reduce A????(But, if A is L, action is Accept.)
• Otherwise, reject.
• But,...

39
SLR(1) Parser Construction

• Rules for the ACTION table can involve
  shift/reduce conflicts.
• So the actual rule for reduce actions is
• If state i contains A ? ?, then ACTION(i,a)
  Reduce A ? ?, for all a ? FOLLOW(A).
• E.g. state 2 for grammar H yields a shift/reduce
  conflict. Namely, should you shift the or
  reduce by L?E. This is resolved by looking at
  the follow set for L.
• Follow(L)

40
FIRST and FOLLOW sets

• FIRST(?) a ? ? ? ?? a? ? ? a ?? ?
• FOLLOW(A) a ? ? S ?? ?Aa?

41
Calculating FIRST sets

• Create table Fi mapping N to ? ? ? initially,
  Fi(A) ? for all A.
• Repeat until Fi does not change
• For each A ? ??? P, Fi(A) Fi(A) ?
  FIRST(?, Fi)
• where FIRST(?, Fi) is defined as follows
• FIRST(?, Fi) ?
• FIRST(a?, Fi) a
• FIRST(B?, Fi) Fi(B) ??FIRST(b,Fi), if ??Fi(B)
  Fi(B), o.w.

42
Calculating FOLLOW sets

• Calculate FOLLOW sets
• Create table Fo N ? ? ? initially, Fo(A)
  ? for all A, except Fo(S)
• Repeat until Fo does not change
• For each production A ? ?B?, Fo(B) Fo(B)
  ? FIRST(?) - ?
• For each production A ? ?B? Fo(B) Fo(B) ?
  Fo(A)
• For each production A ? ?B???if ?? FIRST(?),
  Fo(B) Fo(B) ? Fo(A)