1. LR-Grammars
LR(0), LR(1), and LR(K)

2. Deterministic Context-Free Languages
DCFL
A family of languages that are accepted by a Deterministic Pushdown Automaton (DPDA)
Many programming languages can be described by means of DCFLs

3. Prefix and Proper Prefix
Prefix (of a string)
Any number of leading symbols of that string
Example: abc
Prefixes: , a, ab, abc
Proper Prefix (of a string)
A prefix of a string, but not the string itself
Proper prefixes: , a, ab

4. Prefix Property
Context-Free Language (CFL) L is said to have the prefix property whenever w is in L and no proper prefix of w is in L
Not considered a serve restriction
Why?
Because we can easily convert a DCFL to a DCFL with the prefix property by introducing an endmarker

5. Suffix and Proper Suffix
Suffix (of a string)
Any number of trailing symbols
Proper Suffix
A suffix of a string, but not the string itself

6. Example Grammar
This is the grammar that will be used in many of the examples:
S’  Sc
S  SA | A
A  aSb | ab

7. LR-Grammar
Left-to-right scan of the input producing a rightmost derivation
Simply:
L stands for Left-to-right
R stands for rightmost derivation

8. LR-Items An item (for a given CFG)
A production with a dot anywhere in the right side (including the beginning and end)
In the event of an -production: B  
B  · is an item

9. Example: Items Given our example grammar:
S’  Sc, S  SA|A, A  aSb|ab
The items for the grammar are:
S’·Sc, S’S·c, S’Sc·
S·SA, SS·A, SSA·, S·A, SA·
A·aSb, Aa·Sb, AaS·b, AaSb·, A·ab, Aa·b, Aab·

10. Some Notation * = 1 or more steps in a derivation
*rm = rightmost derivation
rm = single step in rightmost derivation

11. Right-Sentential Form
A sentential form that can be derived by a rightmost derivation
A string of terminals and variables  is called a sentential form if S* 

12. More terms Handle If the grammar is unambiguous:
A substring which matches the right-hand side of a production and represents 1 step in the derivation
Or more formally:
(of a right-sentential form  for CFG G)
Is a substring  such that:
S *rm w
w = 
If the grammar is unambiguous:
There are no useless symbols
The rightmost derivation (in right-sentential form) and the handle are unique

13. Example Given our example grammar: An example right-most derivation:
S’  Sc, S  SA|A, A  aSb|ab
An example right-most derivation:
S’  Sc  SAc  SaSbc
Therefore we can say that: SaSbc is in right-sentential form
The handle is aSb

14. More terms Viable Prefix Complete item
(of a right-sentential form for )
Is any prefix of  ending no farther right than the right end of a handle of .
Complete item
An item where the dot is the rightmost symbol

15. Example Given our example grammar: The right-sentential form abc:
S’  Sc, S  SA|A, A  aSb|ab
The right-sentential form abc:
S’ *rm Ac  abc
Valid prefixes:
A  ab for prefix ab
A  ab for prefix a
A  ab for prefix 
Aab is a complete item,  Ac is the right-sentential form for abc

16. LR(0)
Left-to-right scan of the input producing a rightmost derivation with a look-ahead (on the input) of 0 symbols
It is a restricted type of CFG
1st in the family of LR-grammars
LR(0) grammars define exactly the DCFLs having the prefix property

17. Computing Sets of Valid Items
The definition of LR(0) and the method of accepting L(G) for LR(0) grammar G by a DPDA depends on:
Knowing the set of valid items for each prefix 
For every CFG G, the set of viable prefixes is a regular set
This regular set is accepted by an NFA whose states are the items for G

18. Continued
Given an NFA (whose states are the items for G) that accepts the regular set
We can apply the subset construction to this NFA and yield a DFA
The DFA whose state is the set of valid items for 

19. NFA M Three Rules NFA M recognizes the viable prefixes for CFG
M = (Q, V  T, , q0, Q)
Q = set of items for G plus state q0
G = (V, T, P, S)
Three Rules
(q0,) = {S| S is a production}
(AB,) = {B| B is a production}
Allows expansion of a variable B appearing immediately to the right of the dot
(AX, X) = {AX}
Permits moving the dot over any grammar symbol X if X is the next input symbol

20. Theorem 10.9
The NFA M has property that (q0, ) contains A iff A is valid for 
This theorem gives a method for computing the sets of valid items for any viable prefix
Note: It is an NFA. It can be converted to a DFA. Then by inspecting each state it can be determine if it is a valid LR(0) grammar

21. Definition of LR(0) Grammar
G is an LR(0) grammar if
The start symbol does not appear on the right side of any productions
 prefixes  of G where A is a complete item, then it is unique
i.e., there are no other complete items (and there are no items with a terminal to the right of the dot) that are valid for 

22. Facts we now know: Every LR(0) grammar generates a DCFL
Every DCFL with the prefix property has a LR(0) grammar
Every language with LR(0) grammar have the prefix property
L is DCFL iff L has a LR(0) grammar

23. DPDA’s from LR(0) Grammars
We trace out the rightmost derivation in reverse
The stack holds a viable prefix (in right-sentential form) and the current state (of the DFA)
Viable prefixes: X1X2…Xk
States: s1, s2,…,sk
Stack: s0X1s1…Xksk

24. Reduction If sk contains A Let Then A is valid for X1X2…Xk
 = suffix of X1X2…Xk
Let
 = Xi+1…Xk
w such that X1…Xkw is a right-sentential form.

25. Reduction Continued There is a derivation:
S *rm X1…XiAw rm X1…Xkw
To obtain the right-sentential form (X1…Xkw) in a right derivation we reduce  to A
Therefore, we pop Xi+1…Xk from the stack and push A onto the stack

26. Shift If sk contains only incomplete items
Then the right-sentential form (X1…Xkw) cannot be formed using a reduction
Instead we simply “shift” the next input symbol onto the stack

27. Theorem 10.10
If L is L(G) for an LR(0) grammar G, then L is N(M) for a DPDA M
N(M) = the language accepted by empty stack or null stack

28. Proof Construct from G the DFA D Stack Symbols of M are
Transition function: recognizes G’s prefixes
Stack Symbols of M are
Grammar Symbols of G
States of D
M has start state q and other states used to perform reduction

29. We know that: If G is LR(0) then
Reductions are the only way to get the right-sentential form when the state of the DFA (on the top of the stack) contains a complete item
When M starts on input w it will construct a right-most derivation for w in reverse order

30. What we need to prove:
When a shift is called for and the top DFA state on the stack has only incomplete items then there are no handles
(Note: if there was a handle, then some DFA state on the stack would have a complete item)

31. Suppose  state A (complete item)
Each state is put onto the top of the stack
It would then immediately be reduced to A
Therefore, a complete item cannot possibly become buried on the stack

32. Proof continued
The acceptance of G occurs when the top of the stack contains the start symbol
The start symbol by definition of LR(0) grammars cannot appear on the right side of a production
L(G) always has a prefix property if G is LR(0)

33. Conclusion of Proof
Thus, if w is in L(G), M finds the rightmost derivation of w, reduces w to S, and accepts
If M accepts w, then the sequence of right-sentential forms provides a derivation of w from S
N(M) = L(G)

34. Corollary of Theorem 10.10 Every LR(0) grammar is unambiguous Why?
The rightmost derivation of w is unique
(Given the construction we provided)

35. LR(1) Grammars LR grammar with 1 look-ahead
All and only deterministic CFL’s have LR(1) grammars
Are greatly important to compiler design
Why?
Because they are broad enough to include the syntax of almost all programming languages
Restrictive enough to have efficient parsers (that are essentially DPDAs)

36. LR(1) Item
Consists of an LR(0) item followed by a look-ahead set consisting of terminals and/or the special symbol $
$ = the right end of the string
General Form:
A  , {a1, a2, …, an}
The set of LR(1) items forms the states of a viable prefix by converting the NFA to a DFA

37. A grammar is LR(1) if
The start symbol does not appear on the right side of any productions
The set of items, I, valid for some viable prefix includes some complete item A, {a1,…,an} then
No ai appears immediately to the right of the dot in any item of I
If B, {b1,…,bk} is another complete item in I, then ai  bj for any 1  i  n and 1  j  k

38. Accepting LR(1) language:
Similar to the DPDA used with LR(0) grammars
However, it is allowed to use the next input symbol during it’s decision making
This is accomplished by appending a $ to the end of the input and the DPDA keeps the next input symbol as part of the state

39. LR(1) Rules for Reduce/Shift
If the top set of items has a complete item A, {a1, a2, …, an}, where A  S, reduce by A if the current input symbol is in {a1, a2, …, an}
If the top set of items has an item S, {$}, then reduce by S and accept if the current symbol is $ (i.e., the end of the input is reached)
If the top set of items has an item AaB, T, and a is the current input symbol, then shift

40. Regarding the Rules
Guarantees that at most one of the rules will be applied for any input symbol or $
Often for practicality the information is summarized into a table
Rows: sets of items
Columns: terminals and $