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LR parser

An LR parser is a type of parser for context-free grammars that is very commonly used by computer programming language compilers (and other associated tools). LR parsers read their input from Left to right and produce a Rightmost derivation (hence LR, compare with LL parser). The term "LR(k) parser" is also used; here the k refers to the number of unconsumed "look ahead" input symbols that are used in making parsing decisions. Usually k is 1 and is often omitted. A context-free grammar is called LR(k) if there exists a LR(k) parser for it.

In typical use when we refer to an LR parser we mean a particular parser capable of recognising a particular language specified by a context free grammar. It is not uncommon, however, to refer to an LR parser meaning a driver program that can be supplied with a suitable table to produce a wide number of different particular LR parsers.

LR parsing has many benefits:

LR parsers are difficult to produce by hand; they are usually constructed by a parser generator or a compiler-compiler. Depending on how the parsing table is generated these parsers are called Simple LR parser (SLR for short), Look-ahead LR parser (LALR), and Canonical LR parser. These types of parsers can deal with increasingly large sets of grammars; LALR parsers can deal with more grammars than SLR, Canonical LR parsers works on more grammars than LALR parsers. The program Yacc produces LALR parsers and is fairly popular.

Table of contents
1 Architecture of LR Parsers
2 Constructing LR(0) parsing tables

Architecture of LR Parsers

A table-based bottom-up parser can be schematically presented as in Figure 1. 


        +---+---+---+---+
 Input: | 1 | + | 1 | $ |
        +---+---+---+---+
                  ^
                  |
 Stack:           |
             +----------+
 +---+       |          |
 | 6 |<------|  Parser  |----->  Output
 +---+       |          |
 | 3 |       +----------+
 +---+            |                  
 | 0 |         +--+--------+             
 +---+         |           |            
         +----------+ +----------+ 
         | Action   | | Goto     | 
         | table    | | table    | 
         +----------+ +----------+ 

 Figure 1. Architecture of a table-based bottom-up parser
The parser has an input buffer, a stack on which it keeps a list of states it has been in, an action table and a goto table that tell it to what new state it should move or which grammar rule it should use given the state it is currently in and the terminal or nonterminal it has just read on the input stream. To explain its workings we will use the following small grammar:

(1) E → E * B
(2) E → E + B
(3) E → B
(4) B → 0
(5) B → 1

The Action and Goto Table

The two LR(0) parsing tables for this grammar look as follows:

action
goto
state * + 0 1 $ E B
0 s1 s2 3 4
1 r4 r4 r4 r4 r4
2 r5 r5 r5 r5 r5
3 s5 s6 acc
4 r3 r3 r3 r3 r3
5 s1 s2 7
6 s1 s2 8
7 r1 r1 r1 r1 r1
8 r2 r2 r2 r2 r2

The action table is indexed by a state of the parser and a terminal (including a special terminal $ that indicates the end of the input stream) and contains three types of actions: a shift that is written as 'sn ' and indicates that the next state is n, a reduce that is written as 'rm ' and indicates that a reduction with grammar rule m should be performed and an accept that is written as 'acc' and inidcates that the parser accepts the string in the input stream.

The goto table is indexed by a state of the parser and a nonterminal and simply indicates what the next state of the parser will be if it has recognized a certain nonterminal.

The Parsing Algorithm

The LR parsing algorithm now works as follows:

  1. The stack is initialized with [ 0 ]. The current state will always be the state that is on top of the stack.
  2. Given the current state and the current terminal on the input stream an action is looked up in the action table. There are four cases:
    • a shift sn :
      • the current terminal is removed from the input stream, and
      • the state n is pushed onto the stack and becomes the current state,
    • a reduce rm:
      • the number m is written to the output stream,
      • for every symbol in the right-hand side of rule m a state is removed from the stack and
      • given the state that is then on top of the stack and the left-hand side of rule m a new state is looked up in the goto table and made the new current state by pushing it onto the stack.
    • an accept : the string is accepted
    • no action : a syntax error is reported
  3. The previous step is repeated until the string is accepted or a syntax error is reported.

In practice the parser will usually try to recover from a syntax error by ignoring unexpected symbols and/or inserting missing symbols, and continue parsing, but this is outside the scope of this article.

An Example

To explain why this algorithm works we now proceed with showing how a string like "1 + 1" would be parsed by such a parser. When the parser starts it always starts with the initial state 0 and the following stack:

[ 0 ]

The first terminal that the parser sees is the '1' and according to the action table it should then go to state 2 resulting in the following stack:

[ 0 '1' 2 ]

The top of the stack is shown on the right-hand side. For the sake of our explanation we also show the symbol that caused the transition to the next state although strictly speaking it is not part of the stack.

In state 2 the action table says that whatever terminal we see on the input stream we should do a reduction with grammar rule 5. If the table is correct then this means that the parser has just recognized the right-hand side of rule 5, which is indeed the case. So in this case we write 5 to the output stream, pop one state from the stack, and push on the stack the state from the cell in the goto table for state 0 and B, i.e., state 4, onto the stack. The resulting stack is:

[ 0 B 4 ]

However, also in state 4 the action table says we also do a reduction with rule 3. So we write 3 to the output stream, pop one state from the stack, and find the new state in the goto table for state 0 and E, which is state 3. The resulting stack:

[ 0 E 3 ]

The next terminal that the parser sees is a '+' and according to the action table it should then go to state 6:

[ 0 E 3 '+' 6 ]

Note that the resulting stack can be interpreted as the history of a finite state automaton that has just read a nonterminal E followed by a terminal '+'. The transition table of this automaton is defined by the shift actions in the action table and the goto actions in the goto table.

The next terminal is now '1' and this means that we perform a shift and go to state 2:

[ 0 E 3 '+' 6 '1' 2 ]

Just as the previous '1' this one is reduced to B giving the following stack:

[ 0 E 3 '+' 6 B 8 ]

Again note that the stack corresponds with a list of states of a finite automaton that has read a nonterminal E, followed by a '+' and then a nonterminal B. In state 8 we always perform a reduce with rule 2. Note that the top 3 states on the stack correspond with the 3 symbols in the right-hand side of rule 2.

[ 0 E 3 ]

Finally, we read a '$' from the input stream which means that according to the action table (the current state is 3) the parser accepts the input string. The rule numbers that will then have been written to the output stream will be [ 5, 3, 5, 2 ] which is indeed a rightmost derivation of the string "1 + 1" in reverse.

Constructing LR(0) parsing tables

Items

The construction of these parsing tables is based on the notion of LR(0) item (simply called item here) which are grammar rules with a special dot added somewhere in the right-hand side. For example the rule E → E + B has the following four corresponding items:

E → · E + B
E → E · + B
E → E + · B
E → E + B ·
An exception are rules of the form A → ε with which only the item A → · corresponds. These rules will be used to denote that state of the parser. The item E → E · + B, for example, indicates that the parser has recognized a string corresponding with E on the input stream and now expects to read a '+' followed another string corresponding with E.

Item sets

It is usually not possible to characterize the state of the parser with a single item because it may not know in advance which rule it is going to use for reduction. For example if there is also a rule E → E * B then the items E → E · + B and E → E · * B will both apply after a string corresponding with E has been read. Therefore we will characterize the state of the parser by a set of items, in this case the set { E → E · + B, E → E · * B }.

Closure of item sets

An item with a dot in front of a nonterminal, such as E → E + · B, indicates that the parser expects to parse the nonterminal B next. To ensure the item set contains all possible rules the parser may be in the midst of parsing, it must include all items describing how B itself will be parsed. This means that if there are rules such as B → 1 and B → 0 then the item set must also include the items B → · 1 and B → · 0. In general this can be formulated as follows:

If there is an item of the form A → v·Bw in an item set and in the grammar there is a rule of the form B → w' then the item B → · w' should also be in the item set.

Any set of items can be extended such that it satisfies this rule: simply continue to add the appropriate items until all nonterminals preceeded by dots are accounted for. The minimal extension is called the closure of an item set and written as clos(I) where I is an item set. It are these closed item sets that we will take as the states of the parser, although only the ones that are actually reachable from the begin state will be included in the tables.

The augmented grammar

Before we start determining the transitions between the different states, the grammar is always augmented with an extra rule

(0) S → E

where S is a new start symbol and E the old start symbol. The parser will use this rule for reduction exactly when it has accepted the input string.

For our example we will take the same grammar as before and augment it:

(0) S → E
(1) E → E * B
(2) E → E + B
(3) E → B
(4) B → 0
(5) B → 1

It is for this augmented grammar that we will determine the item sets and the transitions between them.

Finding the reachable item sets and the transitions between them

The first step of constructing the tables consists of determining the transitions between the closed item sets. These transitions will be determined as if we are considering an finite automaton that can read terminals as well as nonterminals. The begin state of this automaton is always the closure of the first item of the added rule: S → · E:

Item set 0
S → · E
+ E → · E * B
+ E → · E + B
+ E → · B
+ B → · 0
+ B → · 1

The '+' in front of an item indicates the items that were added for the closure. The original items without a '+' are called the kernel of the item set.

Beginning from the begin state we will now determine all the states that can be reached from this state. The possible transitions for an item set can be found by looking at the symbols (terminals and nonterminals) we find right behind the dots; in the case of item set 0 these are the terminals '0' and '1' and the nonterminal E and B. To find the item set that a symbol x leads to we follow the following procedure:

  1. Take the set of all items in the current item set where there is a dot in front of x.
  2. Move in all these items the dot behind the x.
  3. Close the resulting set of items.

For the terminal '0' this results in:

Item set 1
B → 0 ·

and for the terminal '1' in:

Item set 2
B → 1 ·

and for the nonterminal E in:

Item set 3
S → E ·
E → E · * B
E → E · + B

and for the nonterminal B in:

Item set 4
E → B ·

Note that in all cases the closure does not add any new items. We continue this process until no more new item sets are found. For the item sets 1 and 2 there will be no transitions since the dot is not in front of any symbol. For item set 3 we see that the dot is in front of the terminals '*' and '+'. For '*' the transition goes to:

Item set 5
E → E * · B
+ B → · 0
+ B → · 1

and for '+' the transition goes to:

Item set 6
E → E + · B
+ B → · 0
+ B → · 1

For item set 5 we have to consider the terminals '0' and '1' and the nonterminal B. For the terminals we see that the resulting closed item sets are equal to the already found item sets 1 and 2, respectively. For the nonterminal B the transition goes to:

Item set 7
E → E * B ·

For item set 6 we also have to consider the terminal '0' and '1' and the nonterminal B. Als here the resulting item sets for the terminals are equal to the already found item sets 1 and 2. For the nontermal B the transition goes to:

Item set 8
E → E + B ·

These final item sets have no symbols behind their dots so no more new item sets are added and we are finished. The transition table for the automaton now looks as follows:

item set*+01 EB
0 1 2 3 4
1
2
3 5 6
4
5 1 2 7
6 1 2 8
7
8

Constructing the action and goto table

From this table and the found item sets we construct the action and goto table as follows:

  1. the columns for nonterminals are copied to the goto table
  2. the columns for the terminals are copied to the action table as shift actions
  3. an extra column for '$' (end of input) is added to the action table that contains acc for every item set that contains S → E ·.
  4. if an item set i contains an item of the form A → w · and A → w is rule m with m > 0 then the row for state i in the action table is completely filled with the reduce action rm.
The reader may verify that this results indeed in the action and goto table that were presented earlier on.

A note about LR(0) versus SLR and LALR parsing

Note that the reduce actions produced by the above procedure always occupy an entire table row, causing the reduction to occur regardless of the next symbol in the input stream. This is why these are LR(0) parse tables: they are only suitable for grammars that do not require lookahead. A grammar that needs lookahead to disambiguate reductions would need a table row containing different reduce actions in different columns, and the above procedure is not capable of creating such rows.

Refinements to the LR(0) table construction procedure (namely SLR, LALR) are capable of constructing reduce actions that do not occupy entire rows. Therefore, they are capable performing different reductions (or even shifts) from the same parser state for different input symbols, allowing them to parse grammars not parseable by an LR(0) table.

Conflicts in the constructed tables

The automaton that is constructed is by the way it is constructed always a deterministic automaton. However, when reduce actions are added to the action table it can happen that the same cell is filled with a reduce action and a shift action (a shift-reduce conflict) or with two different reduce actions (a reduce-reduce conflict). However, it can be shown that when this happens the grammar is not an LL(0) grammar.

A small example of a non-LL(0) grammar with a shift-reduce conflict is:

(1) E → 1 E
(2) E → 1

One of the item sets we then find is:

Item set 1
E → 1 · E
E → 1 ·
+ E → · 1 E
+ E → · 1

There is a shift-reduce conflict in this item set because in the cell in the action table for this item set and the terminal '1' there will be both a shift action to state 1 and a reduce action with rule 2.

A small example of a non-LL(0) grammar with a reduce-reduce conflict is:

(1) E → A 1
(2) E → B 2
(3) A → 1
(4) B → 1

In this case we obtain the following item set:

Item set 1
A → 1 ·
B → 1 ·

There is a reduce-reduce conflict in this item set because in the cells in the action table for this item set there will be both a reduce action for rule 3 and one for rule 4.

Both examples above can be solved by letting the parser use the follow set (see LL parser) of a nonterminal A to decide if it is going to use one of As rules for a reduction; it will only use the rule A → w for a reduction if the next symbol on the input stream is in the follow set of A. This solution results in so-called Simple LR parsers.