This seems to be a pretty fundamental flaw of greedy parsers that you cannot parse left associative operators because infinite recursion will always occur. Is there a way to prevent this from happening or is this just reality?
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I'm working on a project for my OOP class. Part of the task is developing a parser for a very simple grammar. As far as I understood, by far the simplest parser to implement by hand is recursive-descent parser.
However all operators for the language that I'm parsing are left-associative by nature. As far as I know best way to deal with left recursion enforced by left associativity is to use LR parser instead.
My idea is to parse tokens right-to-left instead, which I believe should enable me to rewrite left associative rules to right associative ones instead.
Will this work, and if not, why not?
You can do this if you'd like, though this won't necessarily solve all your problems. If you're familiar with the LL or LR parsers, there are corresponding versions that work right-to-left called RR and RL parsers that pretty much work like LL or LR parsers scanning in the opposite direction. As a result, they have similar weaknesses to the original LL or LR parsing algorithms, so while this might help you, it might not actually solve anything.
As alternatives, you can try rewriting the grammar to see if you can encode precedence and associativity directly. You could also, depending on the grammar, consider using a precedence parsing algorithm like Dijkstra's shunting-yard algorithm. You could also consider using recursive descent parser with backtracking. Finally, you could use something like an Earley parser, which can handle any grammar and isn't too hard to code up.
Good insight -- it turns out this works really well to solve the left recursion problem, but you also have to parse bottom-up, not just right to left. I published a preprint about this.
Pika parsing: parsing in reverse solves the left recursion and error recovery problems
https://arxiv.org/abs/2005.06444
Given an ambiguous grammar, to remove operator precedence problems we would convert the grammar to follow the operator precedence rules. To solve the operator associativity problem, we convert the grammar into left recursive or right recursive by considering the operator it is associated with.
Now when the computer has to do the parsing, suppose if it uses the recursive descent algorithm, should the grammar be unambiguous in the first place? Or the grammar should have different requirements according to the algorithm?
If the grammar is left recursive, the recursive descent algorithm doesn't terminate. Now how do I give an unambiguous grammar(with associativity problems solved) to the algorithm as the input?
The grammar must be LL(k) to use the standard efficient recursive descent algorithm with no backtracking. There are standard transformations useful for taking a general LR grammar (basically any grammar parsable by a deterministic stack-based algorithm) to LL(k) form. They include left recursion elimination and left factoring. These are extensive topics I won't attempt to cover here. But they are covered well in most any good compiler text and reasonably well in online notes available thru search. Aho Sethi and Ullman Compiler Design is a great reference for this and most other compiler basics.
I am wondered why there is no generalized parser combinators for Bottom-up parsing in Haskell like a Parsec combinators for top down parsing.
( I could find some research work went during 2004 but nothing after
https://haskell-functional-parsing.googlecode.com/files/Ljunglof-2002a.pdf
http://www.di.ubi.pt/~jpf/Site/Publications_files/technicalReport.pdf )
Is there any specific reason for not achieving it?
This is because of referential transparency. Just as no function can tell the difference between
let x = 1:x
let x = 1:1:1:x
let x = 1:1:1:1:1:1:1:1:1:... -- if this were writeable
no function can tell the difference between a grammar which is a finite graph and a grammar which is an infinite tree. Bottom-up parsing algorithms need to be able to see the grammar as a graph, in order to enumerate all the possible parsing states.
The fact that top-down parsers see their input as infinite trees allows them to be more powerful, since the tree could be computationally more complex than any graph could be; for example,
numSequence n = string (show n) *> option () (numSequence (n+1))
accepts any finite ascending sequence of numbers starting at n. This has infinitely many different parsing states. (It might be possible to represent this in a context-free way, but it would be tricky and require more understanding of the code than a parsing library is capable of, I think)
A bottom up combinator library could be written, though it is a bit ugly, by requiring all parsers to be "labelled" in such a way that
the same label always refers to the same parser, and
there is only a finite set of labels
at which point it begins to look a lot more like a traditional specification of a grammar than a combinatory specification. However, it could still be nice; you would only have to label recursive productions, which would rule out any infinitely-large rules such as numSequence.
As luqui's answer indicates a bottom-up parser combinator library is not a realistic. On the chance that someone gets to this page just looking for haskell's bottom up parser generator, what you are looking for is called the Happy parser generator. It is like yacc for haskell.
As luqui said above: Haskell's treatment of recursive parser definitions does not permit the definition of bottom-up parsing libraries. Bottom-up parsing libraries are possible though if you represent recursive grammars differently. With apologies for the self-promotion, one (research) parser library that uses such an approach is grammar-combinators. It implements a grammar transformation called the uniform Paull transformation that can be combined with the top-down parser algorithm to obtain a bottom-up parser for the original grammar.
#luqui essentially says, that there are cases in which sharing is unobservable. However, it's not the case in general: many approaches to observable sharing exist. E.g. http://www.ittc.ku.edu/~andygill/papers/reifyGraph.pdf mentions a few different methods to achieve observable sharing and proposes its own new method:
This looping structure can be used for interpretation, but not for
further analysis, pretty printing, or general processing. The
challenge here, and the subject of this paper, is how to allow trees
extracted from Haskell hosted deep DSLs to have observable back-edges,
or more generally, observable sharing. This a well-understood problem,
with a number of standard solutions.
Note that the "ugly" solution of #liqui is mentioned by the paper under the name of explicit labels. The solution proposed by the paper is still "ugly" as it uses so called "stable names", but other solutions such as http://www.cs.utexas.edu/~wcook/Drafts/2012/graphs.pdf (which relies on PHOAS) may work.
I am confused by the following quote from Wikipedia:
In other words, if a language was reasonable enough to allow an
efficient one-pass parser, it could be described by an LR(k) grammar.
And that grammar could always be mechanically transformed into an
equivalent (but larger) LR(1) grammar. So an LR(1) parsing method was,
in theory, powerful enough to handle any reasonable language. In
practice, the natural grammars for many programming languages are
close to being LR(1).[citation needed]
This means that a parser generator, like bison, is very powerful (since it can handle LR(k) grammars), if one is able to convert a LR(k) grammar to a LR(1) grammar. Do some examples of this exist, or a recipe on how to do this? I'd like to know this since I have a shift/reduce conflict in my grammar, but I think this is because it is a LR(2) grammar and would like to convert it to a LR(1) grammar. Side question: is C++ an unreasonable language, since I've read, that bison-generated parsers cannot parse it.
For references on the general purpose algorithm to find a covering LR(1) grammar for an LR(k) grammar, see Real-world LR(k > 1) grammars?
The general purpose algorithm produces quite large grammars; in fact, I'm pretty sure that the resulting PDA is the same size as the LR(k) PDA would be. However, in particular cases it's possible to come up with simpler solutions. The general principle applies, though: you need to defer the shift/reduce decision by unconditionally shifting until the decision can be made with a single lookahead token.
One example: Is C#'s lambda expression grammar LALR(1)?
Without knowing more details about your grammar, I can't really help more than that.
With regard to C++, the things that make it tricky to parse are the preprocessor and some corner cases in parsing (and lexing) template instantiations. The fact that the parse of an expression depends on the "kind" (not type) of a symbol (in the context in which the symbol occurs) makes precise parsing with bison complicated. [1] "Unreasonable" is a value judgement which I'm not comfortable making; certainly, tool support (like accurate syntax colourizers and tab-completers) would have been simple with a different grammar, but the evidence is that it is not that hard to write (or even read) good C++ code.
Notes:
[1] The classic tricky parse, which also applies to C, is (a)*b, which is a cast of a dereference if a represents a type, and otherwise a multiplication. If you were to write it in the context: c/(a)*b, it would be clear that an AST cannot be constructed without knowing whether it's a cast or a product, since that affects the shape of the AST,
A more C++-specific issue is: x<y>(z) (or x<y<z>>(3)) which parse (and arguably tokenise) differently depending on whether x names a template or not.
I've recently being trying to teach myself how parsers (for languages/context-free grammars) work, and most of it seems to be making sense, except for one thing. I'm focusing my attention in particular on LL(k) grammars, for which the two main algorithms seem to be the LL parser (using stack/parse table) and the Recursive Descent parser (simply using recursion).
As far as I can see, the recursive descent algorithm works on all LL(k) grammars and possibly more, whereas an LL parser works on all LL(k) grammars. A recursive descent parser is clearly much simpler than an LL parser to implement, however (just as an LL one is simpler than an LR one).
So my question is, what are the advantages/problems one might encounter when using either of the algorithms? Why might one ever pick LL over recursive descent, given that it works on the same set of grammars and is trickier to implement?
LL is usually a more efficient parsing technique than recursive-descent. In fact, a naive recursive-descent parser will actually be O(k^n) (where n is the input size) in the worst case. Some techniques such as memoization (which yields a Packrat parser) can improve this as well as extend the class of grammars accepted by the parser, but there is always a space tradeoff. LL parsers are (to my knowledge) always linear time.
On the flip side, you are correct in your intuition that recursive-descent parsers can handle a greater class of grammars than LL. Recursive-descent can handle any grammar which is LL(*) (that is, unlimited lookahead) as well as a small set of ambiguous grammars. This is because recursive-descent is actually a directly-encoded implementation of PEGs, or Parser Expression Grammar(s). Specifically, the disjunctive operator (a | b) is not commutative, meaning that a | b does not equal b | a. A recursive-descent parser will try each alternative in order. So if a matches the input, it will succeed even if b would have matched the input. This allows classic "longest match" ambiguities like the dangling else problem to be handled simply by ordering disjunctions correctly.
With all of that said, it is possible to implement an LL(k) parser using recursive-descent so that it runs in linear time. This is done by essentially inlining the predict sets so that each parse routine determines the appropriate production for a given input in constant time. Unfortunately, such a technique eliminates an entire class of grammars from being handled. Once we get into predictive parsing, problems like dangling else are no longer solvable with such ease.
As for why LL would be chosen over recursive-descent, it's mainly a question of efficiency and maintainability. Recursive-descent parsers are markedly easier to implement, but they're usually harder to maintain since the grammar they represent does not exist in any declarative form. Most non-trivial parser use-cases employ a parser generator such as ANTLR or Bison. With such tools, it really doesn't matter if the algorithm is directly-encoded recursive-descent or table-driven LL(k).
As a matter of interest, it is also worth looking into recursive-ascent, which is a parsing algorithm directly encoded after the fashion of recursive-descent, but capable of handling any LALR grammar. I would also dig into parser combinators, which are a functional way of composing recursive-descent parsers together.