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Is there an (efficient?) algorithm which given a context-free language $L$ (given as a grammar) and a string $x$ with $x \not \in L$ computes a $y$ with $y \in L$ and $\forall y': y' \in L \implies d(x, y) \le d(x, y')$, where $d$ is the Levenshtein distance? (Secondarily, can one enumerate all such $y$?)

The motivation is to give more useful parse error messages: if you give me an almost-valid C program, I would like to tell you "if you delete these parentheses and insert a semicolon here, it'll parse".

A family of related questions:

  • What happens when you vary the distance function? (Can you think of other candidates? Do non-metric distance functions make sense? Do they change the answer?)
  • Does the answer change if you restrict $y$ to e.g.
    • $x$ is a prefix of $y$ (i.e. you can do append-only error-correction)
    • $y$ is a prefix of $x$ (i.e. you can do pop-last-only error-correction)
    • all the characters of $x$ occur in $y$ and in the same relative order, but maybe not adjacently (i.e. you can error-correct by inserting anywhere)
    • [...] $y$ occur in $x$ [...] by deleting anywhere.
  • Does the answer change if you change the optimization criterion? E.g. "find a $y$ in $L$ which maximizes the length of common_prefix(x, y)", or something else?
  • Does the answer change for other language classes, e.g. regular?
  • Does this change if the language is given as $LALR(k)$, $LL(k)$, deterministic or other sub-CF grammar?
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  • $\begingroup$ 1) How is L given? 2) Please post only one question per post; follow-up questions can be posted separatedly later. $\endgroup$ – Raphael Jul 3 '16 at 15:17
  • $\begingroup$ Noted. Given that my question is here, in its current form, should I split it? $\endgroup$ – Jonas Kölker Jul 3 '16 at 15:44
  • $\begingroup$ Yes, definitely. Your initial question is quite enough for the time being. I recommend you store your related questions elsewhere, and post them one by one. (Chances are that answers to this or some related questions clear up or change others!) $\endgroup$ – Raphael Jul 3 '16 at 16:24
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Yes. This can be done, using Levenshtein automata.

Let $S_k = \{y \in \{0,1\}^* : d(x,y) \le k)\}$. Then the set $S_k$ is regular, and one can construct a finite-state automaton for it, called a Levenshtein automaton. Now the intersection of a CFL and a regular language is another CFL. Also, given a CFL, you can efficiently determine whether it is non-empty or not, and if not, you can efficiently find an example of an element of it.

So, we obtain the following simple algorithm:

  • For $k:= 1,2,3,\dots$, do:
    • If $L \cap S_k$ is non-empty, find $y \in L \cap S_k$ and output it.

This can be sped up by using binary search to find the smallest $k$ such that $L \cap S_k$ is non-empty.

The same approach can handle your proposed restrictions as well, as those simply amount to intersecting with another regular language. It also works if $L$ is regular, as any regular language is necessarily context-free.

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    $\begingroup$ I assume you mean $S_k = \{y \in L: d(x,y) \le k)\}$? It's not immediately obviously to me why this is regular if $L$ is non-regular. Can you expand on that? This does $O(\log(\delta))$ product constructions (of fairly big automata for big $k$, I assume) one find-member on each automaton ($\delta$ is the edit distance). Do you know if anyone has done this in practice? Is there some simple dynamic programming based on the PDA state when it crashed or something which is "only" $O(n^3 \cdot \delta)$? $\endgroup$ – Jonas Kölker Jul 3 '16 at 15:40
  • $\begingroup$ @JonasKölker $S_k$ is finite, hence regular. $\endgroup$ – Bakuriu Jul 3 '16 at 19:12
  • $\begingroup$ @JonasKölker, my apologies, I had two errors in the definition of $S_k$. See the corrected answer. With these corrections, the fact that it is regular follows from the factoid on the Wikipedia page I link to. I don't know what the size of the Levenshtein automaton is, but at a guess, it's probably in the vicinity of $O(nk)$ where $n$ is the length of $x$, if expressed as a NFA. If that's right, the total running time should be something like $O(n^3 \sigma^3 \lg \sigma)$. $\endgroup$ – D.W. Jul 3 '16 at 21:11
  • $\begingroup$ @Bakuriu: that dawned on me just five seconds before reading your comment. $\endgroup$ – Jonas Kölker Jul 3 '16 at 22:23
  • $\begingroup$ @D.W.: $O(nk)$ sounds right: let a "track" be an automaton accepting $x$ (it has the shape of a simple path), i.e. there's a transition from state $p$ to state $p+1$ on letter $x_p$. Then, have $k+1$ parallel tracks, and a transition from state $p$ on track $i$ to state $p$ on track $i+1$ on some letter (which was inserted) and to state $p+1$ on track $i+1$ on lambda ($x_p$ was deleted) and to state $p+1$ on track $i+1$ on each letter other than $x_p$ (it was modified). Maybe with a few extra states on the end of each track, and a formal proof would be nice, but you get the idea :-) $\endgroup$ – Jonas Kölker Jul 3 '16 at 22:28
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If we limited ourselves to those $y$ with $|y| \le |x|$ (or any finite set) and a computable strict partial order, we can construct the (finite) DAG of the order on our $y$s and find the set of all nodes of in-degree 0, those being the optimal ones. Given this set, we can answer is-empty, enumerate-all and arbitrary-member queries.

This addresses the question where $y$ is a prefix of $x$ and where $y$ is a delete-anywhere substring of $x$, for any decidable language class (e.g. context-free and regular).

The downside is that for delete-anywhere, we iterate a $y$-set of size $2^{|x|}-1$ (we can omit $x$ itself since it's not a member). For prefix-only $y$s it's only $|x|$ strings which is reasonable.

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The language $E = \{x\} \cdot \Sigma^*$ is regular, as is $I = \Sigma^* \{x_1\} \Sigma^* \{x_2\} \Sigma^* \cdots \Sigma^* \{x_n\} \Sigma^*$. For any language class of $L$ for which we can compute the shortest member of $L \cap R$ for regular $R$, we can answer the insert-at-end and insert-anywhere optimization problem (note that shorter implies smaller edit distance when we have only done inserts).

This is the case for regular and context-free languages:

If we have a DFA for $L$ we can use the product construction to get the language of valid candidates $C \in \{L \cap I, L \cap E\}$ as appropriate. Given this we can test $C$ for emptiness and (if not) find a shortest path from the starting state to an accept state, giving us the desired $y$.

The intersection of a context-free and a regular language is context free. We can compute the shortest string in a context-free language, see https://math.stackexchange.com/questions/606518/the-shortest-word-in-context-free-language. (My slight modification: for $k$ in 1..*: If $A → w$ with $|w| \le k$ replace $A$ with $w$ in all right-hand-sides. Once $S$ produces a string, that's your result.)

By computing $C_i = L \cap (\{x_1 \cdots x_i\} \cdot \Sigma^*)$ for $i = 0, \ldots, |x| - 1$ and testing for non-emptiness, we can find the $y$ in $L$ which shares the longest prefix with $x$. Among them, we can also find the shortest by the above method.

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