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42-519: An LALR parser is a simplified version of a canonical LR parser . The LALR parser was invented by Frank DeRemer in his 1969 PhD dissertation, Practical Translators for LR(k) languages , in his treatment of the practical difficulties at that time of implementing LR(1) parsers. He showed that the LALR parser has more language recognition power than the LR(0) parser, while requiring the same number of states as

84-610: A LR(1) parser ) is a type of bottom-up parsing algorithm used in computer science to analyze and process programming languages . It is based on the LR parsing technique, which stands for "left-to-right, rightmost derivation in reverse." Formally, a canonical LR parser is an LR(k) parser for k=1 , i.e. with a single lookahead terminal . The special attribute of this parser is that any LR(k) grammar with k>1 can be transformed into an LR(1) grammar. However, back-substitutions are required to reduce k and as back-substitutions increase,

126-463: A '•' have to be recursively included into the item set until all of those nonterminals are dealt with. The resulting item set is called the closure of the item set we began with. For LR(1) for each production rule an item has to be included for each possible lookahead terminal following the rule. For more complex languages this usually results in very large item sets, which is the reason for the large memory requirements of LR(1) parsers. In our example,

168-617: A LALR(1) parser to an unambiguous LR(1) grammar are reduce/reduce conflicts. The SLR(1) parser performs further merging, which introduces additional conflicts. The standard example of an LR(1) grammar that cannot be parsed with the LALR(1) parser, exhibiting such a reduce/reduce conflict, is: In the LALR table construction, two states will be merged into one state and later the lookaheads will be found to be ambiguous. The one state with lookaheads is: An LR(1) parser will create two different states (with non-conflicting lookaheads), neither of which

210-502: A LALR(2) grammar is usually adequate. If the parser generator allows only LALR(1) grammars, the parser typically calls some hand-written code whenever it encounters constructs needing extended lookahead. Similar to an SLR parser and Canonical LR parser generator, an LALR parser generator constructs the LR(0) state machine first and then computes the lookahead sets for all rules in the grammar, checking for ambiguity. The Canonical LR constructs full lookahead sets. LALR uses merge sets, that

252-453: A marker, to note at which point the currently processed symbol appears in the rule the item represents. For LR(1) parsers, each item is specific to a lookahead terminal, thus the lookahead terminal has also been noted inside each item. For example, assume a language consisting of the terminal symbols 'n', '+', '(', ')', the nonterminals 'E', 'T', the starting rule 'S' and the following production rules: Items sets will be generated by analog to

294-416: A parser given a formal grammar. The first algorithms for LALR parser generation were published in 1973. In 1982, DeRemer and Tom Pennello published an algorithm that generated highly memory-efficient LALR parsers. LALR parsers can be automatically generated from a grammar by an LALR parser generator such as Yacc or GNU Bison . The automatically generated code may be augmented by hand-written code to augment

336-437: A rule. The lookahead can help avoid reducing a specific rule if the lookahead is not valid, which would probably mean that the current state should be combined with the following instead of the previous state. That means in the following example the sequence can be reduced to instead of if the lookahead after the parser went to state B wasn't acceptable, i.e. no transition rule existed. Reductions can be produced directly from

378-606: A series of optimizations for the LALR parser that would further improve its memory efficiency. Their work was published in 1982. Generally, the LALR parser refers to the LALR(1) parser, just as the LR parser generally refers to the LR(1) parser. The "(1)" denotes one-token lookahead, to resolve differences between rule patterns during parsing. Similarly, there is an LALR(2) parser with two-token lookahead, and LALR( k ) parsers with k -token lookup, but these are rare in actual use. The LALR parser

420-491: A terminal as in which allows for multiple sequences to appear. LR(1) parsers have the requirement that each rule should be expressed in a complete LR(1) manner, i.e. a sequence of two states with a specific lookahead. That makes simple rules such as requiring a great many artificial rules that essentially enumerate the combinations of all the possible states and lookahead terminals that can follow. A similar problem appears for implementing non-lookahead rules such as where all

462-550: Is LALR( k ) for any k > 0 {\displaystyle k>0} . Depending on the presence of empty derivations, a LL(1) grammar can be equal to a SLR(1) or a LALR(1) grammar. If the LL(1) grammar has no empty derivations it is SLR(1) and if all symbols with empty derivations have non-empty derivations it is LALR(1). If symbols having only an empty derivation exist, the grammar may or may not be LALR(1). Canonical LR parser A canonical LR parser (also called

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504-407: Is ambiguous. In an LALR parser this one state has conflicting actions (given lookahead c or d, reduce to E or F), a "reduce/reduce conflict"; the above grammar will be declared ambiguous by a LALR parser generator and conflicts will be reported. To recover, this ambiguity is resolved by choosing E, because it occurs before F in the grammar. However, the resultant parser will not be able to recognize

546-502: Is based on the LR(0) parser, so it can also be denoted LALR(1) = LA(1)LR(0) (1 token of lookahead, LR(0)) or more generally LALR( k ) = LA( k )LR(0) (k tokens of lookahead, LR(0)). There is in fact a two-parameter family of LA( k )LR( j ) parsers for all combinations of j and k , which can be derived from the LR( j  +  k ) parser, but these do not see practical use. As with other types of LR parsers, an LALR parser

588-592: Is being offered by several parser generators. Like most parsers, the LR(1) parser is automatically generated by compiler-compilers like GNU Bison , MSTA, Menhir, HYACC, and LRSTAR. In 1965 Donald Knuth invented the LR(k) parser ( L eft to right, R ightmost derivation parser) a type of shift-reduce parser , as a generalization of existing precedence parsers . This parser has the potential of recognizing all deterministic context-free languages and can produce both left and right derivations of statements encountered in

630-550: Is followed by B. The FIRST sets can be determined directly from the closures of all nonterminals in the language, while the FOLLOW sets are determined from the items under usage of the FIRST sets. In our example, as one can verify from the full list of item sets below, the first sets are: Within item set 0 the follow sets can be found to be: From this the full item set 0 for an LR(1) parser can be created, by creating for each item in

672-495: Is followed by a different terminal. Starting from the production rules of a language, at first the item sets for this language have to be determined. In plain words, an item set is the list of production rules, which the currently processed symbol might be part of. An item set has a one-to-one correspondence to a parser state, while the items within the set, together with the next symbol, are used to decide which state transitions and parser action are to be applied. Each item contains

714-743: Is in state I k and I k moves to state I m with label b, then we add the action If [A→α •, a] is in state I k , then we add the action LALR parser generator A lookahead LR parser (LALR) generator is a software tool that reads a context-free grammar (CFG) and creates an LALR parser which is capable of parsing files written in the context-free language defined by the CFG. LALR parsers are desirable because they are very fast and small in comparison to other types of parsers. There are other types of parser generators , such as Simple LR parser , LR parser , GLR parser , LL parser and GLL parser generators. What differentiates one from another

756-399: Is in the mathematical grammar analysis algorithm used by the parser generation tool. LALR generators accept more grammars than do SLR generators, but fewer grammars than full LR(1). Full LR involves much larger parse tables and is avoided unless clearly needed for some particular computer language. Real computer languages can often be expressed as LALR(1) grammars. In cases where they can't,

798-399: Is it merges lookahead sets where the LR(0) core is the same. The SLR uses FOLLOW sets as lookahead sets which associate the right hand side of a LR(0) core to a lookahead terminal. This is a greater simplification than that in the case of LALR because many conflicts may arise from LR(0) cores sharing the same right hand side and lookahead terminal, conflicts which are not present in LALR. This

840-454: Is quite efficient at finding the single correct bottom-up parse in a single left-to-right scan over the input stream, because it does not need to use backtracking . Being a lookahead parser by definition, it always uses a lookahead, with LALR(1) being the most-common case. The LALR(1) parser is less powerful than the LR(1) parser, and more powerful than the SLR(1) parser, though they all use

882-408: Is quite long and thus will not be stated here. Detailed LR(k) treatment of this grammar can e.g. be found in [1] . The lookahead of an LR(1) item is used directly only when considering reduce actions (i.e., when the • marker is at the right end). The core of an LR(1) item [S → a A • B e, c] is the LR(0) item S → a A • B e. Different LR(1) items may share the same core. For example, in item set 2

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924-416: Is the set of terminals which can appear as the first element of any chain of rules matching nonterminal A. FOLLOW(I) of an Item I [A → α • B β, x] is the set of terminals that can appear immediately after nonterminal B, where α, β are arbitrary symbol strings, and x is an arbitrary lookahead terminal. FOLLOW(k,B) of an item set k and a nonterminal B is the union of the follow sets of all items in k where '•'

966-778: Is the type of CFG which they are capable of accepting and the type of parsing algorithm which is used in the generated parser. An LALR parser generator accepts an LALR grammar as input and generates a parser that uses an LALR parsing algorithm (which is driven by LALR parser tables). In practice, LALR offers a good solution, because LALR(1) grammars are more powerful than SLR(1), and can parse most practical LL(1) grammars. LR(1) grammars are more powerful than LALR(1), but ("canonical") LR(1) parsers can be extremely large in size and are considered not practical. Minimal LR(1) parsers are small in size and comparable to LALR(1) parsers. Frank DeRemer invented LALR parsers with his PhD dissertation, called "Practical LR(k) Translators", in 1969, at MIT. This

1008-546: The LR parser, namely the Look-Ahead LR (LALR) and the Simple LR parser (SLR) that had much lower memory requirements at the cost of less language-recognition power, with the LALR parser being the most-powerful alternative. In 1977, memory optimizations for the LR parser were invented but still the LR parser was less memory-efficient than the simplified alternatives. In 1979, Frank DeRemer and Tom Pennello announced

1050-645: The LR(0) item set one copy for each terminal in the follow set of the LHS nonterminal. Each element of the follow set may be a valid lookahead terminal: The rest of the item sets can be created by the following algorithm In the example we get 5 more sets from item set 0, item set 1 for nonterminal E, item set 2 for nonterminal T, item set 3 for terminal n, item set 4 for terminal '+' and item set 5 for '('. Item set 1 (E): Item set 2 (T): Item set 3 (n): Item set 4 ('+'): Item set 5 ('('): From item sets 2, 4 and 5 several more item sets will be produced. The complete list

1092-532: The LR(0) parser for a language that can be recognized by both parsers. This makes the LALR parser a memory-efficient alternative to the LR(1) parser for languages that are LALR. It was also proven that there exist LR(1) languages that are not LALR. Despite this weakness, the power of the LALR parser is sufficient for many mainstream computer languages, including Java , though the reference grammars for many languages fail to be LALR due to being ambiguous . The original dissertation gave no algorithm for constructing such

1134-454: The LR(1) parser are parameterized with a lookahead terminal. Simple parsing tables, like those used by the LR(0) parser represent grammar rules of the form which means that if we go have input A followed by B then we will reduce the pair to A1 regardless of what follows. After parameterizing such a rule with a lookahead we have: which means that the reduction will now be performed only if

1176-503: The grammar can quickly become large, repetitive and hard to understand. LR(k) can handle all deterministic context-free languages . In the past this LR(k) parser has been avoided because of its huge memory requirements in favor of less powerful alternatives such as the LALR and the LL(1) parser. Recently, however, a "minimal LR(1) parser" whose space requirements are close to LALR parsers ,

1218-568: The input file. Knuth proved that it reaches its maximum language recognition power for k=1 and provided a method for transforming LR(k), k > 1 grammars into LR(1) grammars. Canonical LR(1) parsers have the practical disadvantage of having enormous memory requirements for their internal parser-table representation. In 1969, Frank DeRemer suggested two simplified versions of the LR parser called LALR and SLR . These parsers require much less memory than Canonical LR(1) parsers, but have slightly less language-recognition power. LALR(1) parsers have been

1260-417: The lookahead terminal is a . This allows for richer languages where a simple rule can have different meanings depending on the lookahead context. For example, in a LR(1) grammar, all of the following rules perform a different reduction in spite of being based on the same state sequence. The same would not be true if a lookahead terminal was not being taken into account. Parsing errors can be identified without

1302-461: The most common implementations of the LR Parser. However, a new type of LR(1) parser, some people call a "Minimal LR(1) parser" was introduced in 1977 by David Pager who showed that LR(1) parsers can be created whose memory requirements rival those of LALR(1) parsers. Recently , some parser generators are offering Minimal LR(1) parsers, which not only solve the memory requirement problem, but also

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1344-434: The mysterious-conflict-problem inherent in LALR(1) parser generators. In addition, Minimal LR(1) parsers can use shift-reduce actions, which makes them faster than Canonical LR(1) parsers. The LR(1) parser is a deterministic automaton and as such its operation is based on static state transition tables . These codify the grammar of the language it recognizes and are typically called "parsing tables". The parsing tables of

1386-441: The parser having to read the whole input by declaring some rules as errors. For example, can be declared an error, causing the parser to stop. This means that the lookahead information can also be used to catch errors, as in the following example: In this case A B will be reduced to A1 when the lookahead is a, b or c and an error will be reported when the lookahead is d. The lookahead can also be helpful in deciding when to reduce

1428-456: The parser is required to perform the reduction [E → T] if the next symbol is '$ ', but to do a shift if the next symbol is '+'. Note that a LR(0) parser would not be able to make this decision, as it only considers the core of the items, and would thus report a shift/reduce conflict. A state containing [A → α • X β, a] will move to a state containing [A → α X • β, a] with label X. Every state has transitions according to Goto. If [A → α • b β, a]

1470-408: The possible lookaheads must be enumerated. That is the reason why LR(1) parsers cannot be practically implemented without significant memory optimizations. LR(1) parsing tables are constructed in the same way as LR(0) parsing tables with the modification that each Item contains a lookahead terminal . This means, contrary to LR(0) parsers, a different action may be executed, if the item to process

1512-469: The power of the resulting parser. In 1965, Donald Knuth invented the LR parser ( L eft to Right, R ightmost derivation ). The LR parser can recognize any deterministic context-free language in linear-bounded time. Rightmost derivation has very large memory requirements and implementing an LR parser was impractical due to the limited memory of computers at that time. To address this shortcoming, in 1969, Frank DeRemer proposed two simplified versions of

1554-522: The procedure for LR(0) parsers. The item set 0 which represents the initial state will be created from the starting rule: The dot '•' denotes the marker of the current parsing position within this rule. The expected lookahead terminal to apply this rule is noted after the comma. The '$ ' sign is used to denote 'end of input' is expected, as is the case for the starting rule. This is not the complete item set 0, though. Each item set must be 'closed', which means all production rules for each nonterminal following

1596-445: The same production rules . The simplification that the LALR parser introduces consists in merging rules that have identical kernel item sets , because during the LR(0) state-construction process the lookaheads are not known. This reduces the power of the parser because not knowing the lookahead symbols can confuse the parser as to which grammar rule to pick next, resulting in reduce/reduce conflicts . All conflicts that arise in applying

1638-421: The starting symbol requires the nonterminal 'E' which in turn requires 'T', thus all production rules will appear in item set 0. At first, we ignore the problem of finding the lookaheads and just look at the case of an LR(0), whose items do not contain lookahead terminals. So the item set 0 (without lookaheads) will look like this: To determine lookahead terminals, so-called FIRST and FOLLOW sets are used. FIRST(A)

1680-417: The valid input sequence b e c , since the ambiguous sequence e c is reduced to (E → e) c , rather than the correct (F → e) c , but b E c is not in the grammar. The LALR( j ) parsers are incomparable with LL( k ) parsers : for any j and k both greater than 0, there are LALR( j ) grammars that are not LL( k ) grammars and vice versa. In fact, it is undecidable whether a given LL(1) grammar

1722-502: Was an important breakthrough, because LR(k) translators, as defined by Donald Knuth in his 1965 paper, "On the Translation of Languages from Left to Right", were much too large for implementation on computer systems in the 1960s and 70's. An early LALR parser generator and probably the most popular one for many years was " yacc " (Yet Another Compiler Compiler), created by Stephen Johnson in 1975 at AT&T Labs. Another, "TWS",

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1764-583: Was created by Frank DeRemer and Tom Pennello. Today, there are many LALR parser generators available, many inspired by and largely compatible with the original Yacc, for example GNU bison , a pun on the original Yacc/ Yak . See Comparison of deterministic context-free language parser generators for a more detailed list. The LALR parser and its alternatives, the SLR parser and the Canonical LR parser , have similar methods and parsing tables; their main difference

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