“Meaning” is what you want in your mind when working with this stuff, “representation” is the irrelevant nuts-and-bolts that get in the way.

Instead of the meaning of the number “3″, you could get sidetracked for e.g. with the fontography of the symbol, its various representational history and evolution, etc.

Conal is building a new set of tools (i.e. a library). Every code library is essentially an abstraction layer. How to go about reasoning with the tools, using them effectively, is what he’s trying to get at with “meaning”, as opposed to the innards of the abstraction layer, the “representation.”

(I’m a complete hick when it comes to most mathematics, let alone this stuff, so apologies if I’m being really painfully clueless.)

Conal, I interpret what you say at the start of your post to mean that Behaviour is used in an actual program e.g. Haskell code that can be compiled and run, whereas B is not about Haskell but about meaning/semantics/’purely’ math, but you happen to use Haskell notation to talk about it.

??

In FRP, Behavior is a parameterized abstract data type

type Behavior a

This type can be represented in various ways (e.g., see Functional Implementations of Continuous Modeled Animation). In this post, however, I don’t address the representation at all, leaving Behavior abstract. Instead, I talk about the meaning of behaviors, in the sense of denotational semantics. (In case this notion of meaning is not clear yet, consider the relationship between the numeral “3″ and the number 3. We conventionally interpret numerals as numbers, but numerals are symbols, while numbers are quantities.) What’s perhaps unusual here is that I’m giving the meaning of a data type, not a language. (Though languages can be viewed of data types in a meta-language.)

A behavior represents (can be interpreted as) a value that varies in time. The semantic domain B and semantic function, at :: Behavior a -> B a, make that interpretation formal. For instance, in the meaning of the behavior time

at time == id

note that time :: Behavior Time, while id :: B Time, i.e., id :: Time -> Time.

Thanks,

Paul.