Just like Chomsky, we'll introduce exactly one transformation. From Lambda calculus it's known like Lambda abstraction. Let's say (λx E1 E2) is an application of function (λx E1) to E2, and the result of this application will be E3 which is E1 with all occurrences of x replaced with E2. Abstraction is a reverse operation. Having a complex expression E3 where E2 occurs, we'll replace the occurrences of E2 with x and append λx to the beginning. Of course, any variable name could be in place of x.

This abstraction may be used when there are several functions applied on one argument. The most frequent example here is predicate coordination:

They stopped and looked across the river.

Here we have both 'stopped' and 'looked' applied to 'they'. Plain functional composition can't apply two functions to one argument simultaneously, so we'll need something that can. Let's suppose that 'and' function does it. It receives several component functions, merges them somehow and produces a single function as a result. This function will be a Composite, meaning that its action on an argument basically would be to apply all components to that argument. Here the merged functions will be 'stopped' and '(λx looked x across (the river))'. The second one will receive the argument and put it into correct position in its argument structure. So the result will be

((and stopped (λx looked x across (the river))) they)

## Sunday, March 22, 2009

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