Babel Development Log

In fuzzy logic, a ‘hedge’ is a type of unary operator which acts on the membership function of a fuzzy set. Hedges typically correspond in meaning to adverbs such as ‘very’ and ’slightly’. Effects may include displacing the peak of the membership function, or causing it to become more concentrated or spread out.

Many predicates need to have fuzzy semantics if they are to accurately represent their linguistic counterparts. For example, colour:bright is fully true for colours with a saturation and value of 100%, but would still be mostly true if the value were lowered to 90%. The way I envisage the membership function to be defined, it would become completely false only on reaching the grey line. Similar considerations apply to colour:pale and colour:dark.

Because there are degrees of truth to these predicates it is possible to talk of colours which are (for example) ‘very bright’ or ’slightly dark’. All that is needed are predicates to represent the required hedges. Here are the ones I intend to make available in the first instance:

• not[1] inverts the membership function, so that members become non-members and vice versa. It is normally implemented by the function f(x) = 1−x.
• very concentrates the membership function, peaking at the highest possible values. It is often implemented by the function f(x) = x² but there are other ways to define it.
• slightly concentrates the membership function, peaking at what would otherwise be the lowest possible values. A possible definition is (very not).
• moderately concentrates the membership function, peaking at what would otherwise be middling values. A possible definition is the union of (not slightly) and (not very).

Hedges are sufficiently fundamental that I see no objection to their being placed in the global namespace.

Given that slightly and moderately could be defined in terms of other predicates it may be asked whether they are necessary. I’ve included them for three reasons: in the interests of brevity, to avoid committing to any particular definition of the membership function, and because I’ve not seen any evidence that decomposing these hedges would be of value linguistically.

On the other hand, I do intend to make use of constructs such as (not very) and (very slightly) because equivalent forms are used in many languages.

I’ve not (yet) defined a somewhat hedge because I’m not convinced that the way this term is used in fuzzy logic (typically implemented by means of the square root function) corresponds to ’somewhat’ in English, or indeed to any word that I have been able to identify. To my mind, if there were to be a somewhat hedge then it should peak between slightly and moderately.

A potentially useful benefit of defining very to mean f(x) = x² is that it allows ((very dark) orange) to be re-expressed as (dark dark orange), which could then be translated as ‘dark brown’. I’m not going to commit to saying that very is defined this way, because I want to avoid that level of detail. I will say that it is an acceptable approximation, which is all the translation system needs to know in order to make use of it (since translation is not an exact process).

There will probably need to be restrictions on when and where these predicates can be used, and in what combinations. For example, whereas (very dark) makes sense and ought, (very (cardinal 5)) does not (because being fivefold is not an attribute of degree). I’ll return to this subject when I have more experience of implementation.

[1] Some sources class not as a hedge, others as part of a larger class of functions called modifiers.

This entry was posted on Sunday, August 9th, 2009 at 10:17 am and is filed under Predicates. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.