LL grammar
In formal language theory, an LL grammar is a context-free grammar that can be parsed by an LL parser, which parses the input from Left to right, and constructs a Leftmost derivation of the sentence (hence LL, compared with LR parser that constructs a rightmost derivation). A language that has an LL grammar is known as an LL language. These form subsets of deterministic context-free grammars (DCFGs) and deterministic context-free languages (DCFLs), respectively. One says that a given grammar or language "is an LL grammar/language" or simply "is LL" to indicate that it is in this class. LL parsers are table-based parsers, similar to LR parsers. LL grammars can alternatively be characterized as precisely those that can be parsed by a predictive parser – a recursive descent parser without backtracking – and these can be readily written by hand. This article is about the formal properties of LL grammars; for parsing, see LL parser or recursive descent parser. Formal definition

Finite case
Given a natural number ,
a context-free grammar 
is an LL(k) grammar if
 for each terminal symbol string  of length up to  symbols,
 for each nonterminal symbol , and
 for each terminal symbol string ,
there is at most one production rule  such that for some terminal symbol strings ,
 the string  can be derived from the start symbol ,
  can be derived from  after first applying rule , and
 the first  symbols of  and of  agree. An alternative, but equivalent, formal definition is the following:
 is an LL(k) grammar if, for arbitrary derivations
 
when the first  symbols of  agree with those of , then . Informally, when a parser has derived , with  its leftmost nonterminal and  already consumed from the input, then by looking at that  and peeking at the next  symbols  of the current input, the parser can identify with certainty the production rule  for . When rule identification is possible even without considering the past input , then the grammar is called a strong LL(k) grammar. In the formal definition of a strong LL(k) grammar, the universal quantifier for  is omitted, and  is added to the "for some" quantifier for . For every LL(k) grammar, a structurally equivalent strong LL(k) grammar can be constructed. The class of LL(k) languages forms a strictly increasing sequence of sets: LL(0) ⊊ LL(1) ⊊ LL(2) ⊊ …. It is decidable whether a given grammar G is LL(k), but it is not decidable whether an arbitrary grammar is LL(k) for some k. It is also decidable if a given LR(k) grammar is also an LL(m) grammar for some m.

Every LL(k) grammar is also an LR(k) grammar. An ε-free LL(1) grammar is also an SLR(1) grammar. An LL(1) grammar with symbols that have both empty and non-empty derivations is also an LALR(1) grammar. An LL(1) grammar with symbols that have only the empty derivation may or may not be LALR(1). LL grammars cannot have rules containing left recursion. Each LL(k) grammar that is ε-free can be transformed into an equivalent LL(k) grammar in Greibach normal form (which by definition does not have rules with left recursion). Regular case
Let  be a terminal alphabet. A partition  of  is called a regular partition if for every  the language  is regular. Let  be a context free grammar and let  be a regular partition of . We say that  is an LL() grammar if, for arbitrary derivations
 
such that  it follows that . A grammar G is said to be LL-regular (LLR) if there exists a regular partition of  such that G is LL(). A language is LL-regular if it is generated by an LL-regular grammar. LLR grammars are unambiguous and cannot be left-recursive. Every LL(k) grammar is LLR. Every LL(k) grammar is deterministic, but there exists a LLR grammar that is not deterministic. Hence the class of LLR grammars is strictly larger than the union of LL(k) for each k.

It is decidable whether, given a regular partition , a given grammar is LL(). It is, however, not decidable whether an arbitrary grammar G is LLR. This is due to the fact that deciding whether a grammar G generates a regular language, which would be necessary to find a regular partition for G, can be reduced to the Post correspondence problem. Every LLR grammar is LR-regular (LRR, the  equivalent for LR(k) grammars), but there exists an LR(1) grammar that is not LLR. Historically, LLR grammars followed the invention of the LRR grammars. Given a regular partition a Moore machine can be constructed to transduce the parsing from right to left, identifying instances of regular productions. Once that has been done, an LL(1) parser is sufficient to handle the transduced input in linear time. Thus, LLR parsers can handle a class of grammars strictly larger than LL(k) parsers while being equally efficient. Despite that the theory of LLR does not have any major applications. One possible and very plausible reason is that while there are generative algorithms for LL(k) and LR(k) parsers, the problem of generating an LLR/LRR parser is undecidable unless one has constructed a regular partition upfront. But even the problem of constructing a suitable regular partition given grammar is undecidable. Simple deterministic languages

A context-free grammar is called simple deterministic, or just simple, if
 it is in Greibach normal form (i.e. each rule has the form ), and
 different right hand sides for the same nonterminal  always start with different terminals . A set of strings is called a simple deterministic, or just simple, language, if it has a simple deterministic grammar. The class of languages having an ε-free LL(1) grammar in Greibach normal form equals the class of simple deterministic languages. This language class includes the regular sets not containing ε. Equivalence is decidable for it, while inclusion is not. Applications
LL grammars, particularly LL(1) grammars, are of great practical interest, as they are easy to parse, either by LL parsers or by recursive descent parsers, and  are designed to be LL(1) for this reason. Languages based on grammars with a high value of k have traditionally been considered  to be difficult to parse, although this is less true now given the availability and widespread use  of parser generators supporting LL(k) grammars for arbitrary k.

See also
 Comparison of parser generators for a list of LL(k) and LL(*) parsers

Notes

Sources

Further reading 

Formal languages