[Haskell-cafe] Lambda Calculus: Bound and Free formal definitions

Mark Spezzano valheru at chariot.net.au
Thu Dec 30 04:50:34 CET 2010

```Hi all,

Maybe I should clarify...

For example,

5.2 FREE:
========
If E1 = \y.xy then x is free
If E2 = \z.z then x is not even mentioned

So E = E1 E2 = x (\z.z) and x is free as expected
So E = E2 E1 = \y.xy and x is free as expected

5.3 BOUND:
=========
If E1 = \x.xy then x is bound
If E2 = \z.z then is not even mentioned

So E = E1 E2 = (\x.xy)(\z.z) = (\z.z)y -- Error: x is not bound but should be by the rule of 5.3
So E = E2 E1 = (\z.z)(\x.xy) = (\x.xy) then x is bound.

Where's my mistake in the second-to-last example? Shouldn't x be bound (somewhere/somehow)?

Thanks,

Mark

On 30/12/2010, at 1:52 PM, Mark Spezzano wrote:

> Duh, Sorry. Yes, there was a typo
>
> the second one should read
>
> If E is a combination E1 E2 then X is bound in E if and only if X is bound in E1 or is bound in E2.
>
> Apologies for that oversight!
>
> Mark
>
>
> On 30/12/2010, at 1:21 PM, Antoine Latter wrote:
>
>> Was there a typo in your email? Because those two definitions appear
>> identical. I could be missing something - I haven't read that book.
>>
>> Antoine
>>
>> On Wed, Dec 29, 2010 at 9:05 PM, Mark Spezzano
>> <mark.spezzano at chariot.net.au> wrote:
>>> Hi,
>>>
>>> Presently I am going through AJT Davie's text "An Introduction to Functional Programming Systems Using Haskell".
>>>
>>> On page 84, regarding formal definitions of FREE and BOUND variables he gives Defn 5.2 as
>>>
>>> The variable X is free in the expression E in the following cases
>>>
>>> a) <omitted>
>>>
>>> b) If E is a combination E1 E2 then X is free in E if and only if X is free in E1 or X is free in E2
>>>
>>> c) <omitted>
>>>
>>> Then in Defn 5.3 he states
>>>
>>> The variable X is bound in the expression E in the following cases
>>>
>>> a) <omitted>
>>>
>>> b) If E is a combination E1 E2 then X is free in E if and only if X is free in E1 or X is free in E2.
>>>
>>> c) <omitted>
>>>
>>> Now, are these definitions correct? They seem to contradict each other....and they don't make much sense on their own either (try every combination of E1 and E2 for bound and free and you'll see what I mean). If it is correct then please give some examples of E1 and E2 showing exactly why. Personally I think that there's an error in the book.
>>>
>>> You can see the full text on Google Books (page 84)
>>>
>>>
>>> Mark Spezzano
>>>
>>>
>>> _______________________________________________
>>>
>>
>> _______________________________________________
>>
>>
>
>
> _______________________________________________
>
>

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