LALR(1)
Expanding on the topic of LALR(1) parsing and LALR(1) grammars, we can discuss some advanced features and considerations:
Precedence Declarations
In some grammars, operators may have different precedences, such as multiplication having higher precedence than addition. This can be represented in a grammar using precedence declarations. For instance:
These declarations would cause the parser to reduce the left state on token + and to shift the right state on token *, achieving the desired precedence. LALR parser generators also allow a precedence declaration to be attached directly to a production, specifying its priority with respect to other productions of the same nonterminal .
LR(1) Construction
The LR(1) construction extends the LR(0) automaton construction to an LR(1) automaton. Just as in the LR(0) automaton, the states are a set of items that is closed under prediction. However, the items now contain a set of lookahead tokens. Thus, an LR(1) item has the form X→α.,β~~~~λ, where the symbols α represent the top of the automaton stack, the dot represents the current input position, the symbols β derive possible future input, and the set of tokens λ describes tokens that could appear in the input stream after the derivation of β .
LALR Grammars
The number of LR(1) states is often unnecessarily large, because the LR(1) automaton ends up with many states that are identical other than lookahead tokens. This insight leads to LALR automata. An LALR automaton is exactly the same as an LR(1) automaton except that it merges together all states that are identical other than lookaheads. In the merge, the lookahead sets are combined for each item. Many parser generators are based on LALR, including commonly used software like yacc, bison, CUP, mlyacc, ocamlyacc, and Menhir.
While LALR is in practice just about as good as full LR, it does occasionally lose some expressive power. To see why, consider what happens when the two LR(1) states in the following diagram are merged to form the state marked M: The two states on the top are unambiguous LR(1) states in which the lookahead character indicates which of the two productions to reduce. But when merged, the resulting state has reduce–reduce conflicts on both + and $. When merging LR(1) states creates a new conflict, the grammar must be LR(1) but not LALR(1) .
LALR(1) Parser Behavior
The LALR(1) parser behaves similarly to the LR(1) parser for correct inputs, producing the same sequence of reduce and shift actions. On an incorrect input, the LALR parser produces the same sequence of actions up to the last shift action, although it might then do a few more reduce actions before reporting the error. So although the LALR parser has fewer states, its behavior is identical for correct inputs, and extremely similar for incorrect inputs .
Let's illustrate LALR(1) parsing with an example. Consider the following context-free grammar:
And suppose we want to parse the input string xxzxx.
First, we need to construct the LALR(1) parsing table. This table guides the parsing process by mapping pairs of the form (state, input symbol) to actions (shift, reduce, accept, or error). The construction of the LALR(1) parsing table involves several steps, including the calculation of the FIRST and FOLLOW sets, the creation of the canonical collection of LR(1) items, and the merging of states to eliminate reduce-reduce conflicts.
Once the LALR(1) parsing table is constructed, we can start parsing the input string. At each step, we examine the top symbol of the stack and the next input symbol, and follow the corresponding entry in the parsing table to decide whether to shift, reduce, or accept.
Here's a simplified version of the parsing process for the input string xxzxx:
State Stack Input Action
0 A xxzxx Shift
1 Ax zxx Reduce A -> CxA
2 CxA zxx Shift
3 zxA xx Shift
4 xxzxA xx Reduce B -> xC y
5 xy xx Reduce C -> xB x
6 xx zx Reduce C -> z
7 z xx Reduce A -> ε
8 xx xx Accept
In this example, the LALR(1) parser successfully parses the input string xxzxx according to the given grammar. Note that the actual parsing process would involve more steps and more complex entries in the parsing table.
Keep in mind that this is a simplified example. In practice, the construction of the LALR(1) parsing table can be complex and time-consuming, especially for larger grammars. Also, the LALR(1) parser may produce different results for incorrect inputs, depending on how the parsing table is constructed and how conflicts are resolved.
Let's illustrate LALR(1) parsing with an example. Consider the following context-free grammar:
We want to parse the input string xqy.
First, let's construct the LALR(1) parsing table. The construction of the LALR(1) parsing table involves several steps, including the calculation of the FIRST and FOLLOW sets, the creation of the canonical collection of LR(1) items, and the merging of states to eliminate reduce-reduce conflicts.
Once the LALR(1) parsing table is constructed, we can start parsing the input string. At each step, we examine the top symbol of the stack and the next input symbol, and follow the corresponding entry in the parsing table to decide whether to shift, reduce, or accept.
Here's a simplified version of the parsing process for the input string xqy:
State Stack Input Action
0 S xqy Shift
1 Sx qy Reduce A -> q
2 Sxq y Shift
3 Sxy q Reduce S -> xBy
4 By q Reduce B -> q
5 S q Accept
In this example, the LALR(1) parser successfully parses the input string xqy according to the given grammar. Note that the actual parsing process would involve more steps and more complex entries in the parsing table.
Keep in mind that this is a simplified example. In practice, the construction of the LALR(1) parsing table can be complex and time-consuming, especially for larger grammars. Also, the LALR(1) parser may produce different results for incorrect inputs, depending on how the parsing table is constructed and how conflicts are resolved.