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7 June 2018

My research, part II

Part I

In a parallel universe, there was another grammar formalism called Constraint Grammar (CG). While I hardly wrote any such grammars, I was dating someone who wrote them, and at the same time, I was supervised by someone who had an obsession to SAT-solvers. So the two worlds met through me, and thus was a new research question born.

Writing CG

If you are reading this page, you supposedly know English. Thus you would have no trouble determining that the word wish is a noun in the context “a wish”, and a verb in the context “I wish my thesis was finished already”.

Constraint Grammar is the formalisation of this knowledge, in the following format:

SELECT noun IF (0 noun/verb) (-1 article) ;

These rules work on a text that is previously analysed morphologically: that is, some other piece of software has gone through the words one by one, and given them all possible tags they can get. Initially, all instances of wish get both tags, noun and verb. After applying all the rules, hopefully each wish is left with only one tag. If we’re lucky, it is even the correct one.

As an important distinction from other grammar formalisms you might know, CG grammars are inherently heuristic, not absolute. They are born from the notion that all grammars leak, and that we are not supposed to define all and only valid sentences in the language. We just want something that works! Real language that humans write is not perfect–even if you write “✱the boy drink beer”, we would still want to analyse it. The word drink is still a verb, just conjugated incorrectly.

Writing CG rules1 has a more practical and low-level feel to it than writing GF grammars. I’ll be happy to be proven wrong, but I feel that actual knowledge of the language is more crucial for CG. Every rule should reflect some actual sentence that appears in an actual text, and is not given a proper analysis before you wrote the rule. A lot of CG rules include a comment of an example sentence where it works, or alternatively, where it doesn’t work. That’s right–CG grammarians know that their rules fail for some sentences, and include them anyway. What’s the deal with that?

The deal (at least in the current mainstream CG implementation VISL CG-3) is ordering. The rules are applied to the sentences in the order they appear in the file. More reliable and efficient rules are placed in the beginning, and more dubious rules at the end. Thus we’d hope that when the dubious rule gets its turn to act, the falsely triggering sentences are already disambiguated by an appropriate rule, and are thus safe from the dubious rule.

Does that sound complicated? Previous experiments show a difference of a few percent points in precision and recall just by changing the order (Bick, Lager, Nivre) and/or execution strategy (myself but with really shitty grammars so take it with a grain of salt). I’d be pretty annoyed if I wrote CG regularly. I just want to formalise my knowledge about language, not keep track of dependencies of thousands of rules. I only wish someone had done something about this!

Testing CG

So we have a bunch of rules, some of which may conflict in one order, but be fine in another order. Some rules may also be inherently contradictory no matter the order (think “remove all nouns after article” vs. “select all nouns after article”). In addition, some rules can be internally contradictory: “select noun only if it isn’t a noun”.

We could apply all the rules in sequence to a corpus, and see if a rule never triggers. But it may be just that the corpus is incomplete, and doesn’t contain just the sentence needed to trigger that particular rule. Even if we found such a rule that never triggers, we wouldn’t know which rule(s) are those that conflict with it. To solve these problems, we use symbolic evaluation.

We construct an initial sentence of some width, e.g. 10 words. We call this a symbolic sentence. Initially, all symbolic words in the symbolic sentence contain all possible analyses–hundreds or thousands, depending on how granular the morphological analyser is and how complex the language is. (In my experience, that’s the order of importance). We start applying the rules one by one to the symbolic sentence, but instead of rules directly removing or selecting analyses, they form logical formulas. Take the rule we know from before:

SELECT noun IF (0 noun/verb) (-1 article) ;

Now in order for that rule to apply in a particular symbolic word, it has to have 3 things:

We form these requirements for every single word in the symbolic sentence. It is very much expected that not every rule will be able to apply to every single word—say that we apply next a rule that requires verb as a context, and thus there has to be at least one word in the sentence where verb reading is intact. It would be easy to just find a solution where e.g. previous word has no article reading, or the word itself has no noun reading to start with, so by the rule, we are not obligated to remove verb.

But what if two rules are really conflicting (or just in wrong order)? Then it becomes impossible to form a sequence where both rules may apply! Thanks to this analysis, I know exactly if two (or more) rules are in conflict, and I can try to change their order or remove one altogether.

That sounds a bit underwhelmingly simple. More detail is available in my licentiate thesis—but it will be also in a more compact form in my PhD thesis, and with more fancy results. So unless you’re like my supervisor and have a fetish for SAT, why don’t you rather just chill over the summer and read my shiny PhD thesis as soon as I’ve written it.

Doing weird things with CG

I said earlier that CG was not conceived as defining “all and only the grammatical sentences in a language”, but just to do the job and assign some tags even for a grammatically incorrect text. But the mechanisms that we developed in order to test CG have interesting side effects, that let us model CG as a generative formalism.

If we were sculpting a statue, generative formalism would start from an empty place and start splashing some clay around. In contrast, a reductionist formalism would start from a heavy stone block and carve away everything that is not part of the statue. In the end, both have formed an identical statue. An empty generative grammar would output just thin air, and empty reductionist grammar would output an intact stone block.

We can see symbolic sentences as these stone blocks. To give a classic example, say we have the alphabet {a,b} and the language aⁿbⁿ. Then we can construct a CG grammar, apply it to an even-length symbolic sentence consisting of as and bs, and have it output a string where the first half is as, followed by a second half of bs. This language is context-free, and we have only found a method to write such a grammar manually, but we have a method that can translate any regular language into a corresponding CG automatically, and apply it to a symbolic sentence.

If you want to read more, here’s Wen’s blog post. There will also be a section in my thesis on this.

Part III coming up sometime.


1: I once wrote 19 rules to prove a point. I wanted to get to 20, but the 19 already worked better (on a specific corpus) than 300 (probably hand-tailored for another corpus) and I was bored so it became 19. But believe me, I have read a whole lot of CG rules!

tags: research, cg