X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week3_what_is_computation.mdwn;h=47748a16e29a50f1aefa5b0219f7ceb209f92b65;hp=7575abccf610f5a6038d9850dc94e6cfd8806798;hb=ee79965dbd07b39cd3ab7d489ba87b8cc5e2b68b;hpb=b38fa6022c267d7bd66496828194f04117ff0130 diff --git a/topics/_week3_what_is_computation.mdwn b/topics/_week3_what_is_computation.mdwn index 7575abcc..47748a16 100644 --- a/topics/_week3_what_is_computation.mdwn +++ b/topics/_week3_what_is_computation.mdwn @@ -9,10 +9,10 @@ This equation can be interpreted as expressing the thought that the complex expression `3 + 4` evaluates to `7`. The evaluation of the expression computing a sum. There is a clear sense in which the expression `7` is simpler than the expression `3 + 4`: `7` is -syntactically simple, and `3 + 4` is syntactically complex. +syntactically simple, and `3 + 4` is syntactically complex. Now let's take this folk notion of computation, and put some pressure -on it. +on it. ##Church arithmetic## @@ -64,14 +64,14 @@ But now consider multiplication: Is the final result simpler? This time, the answer is not so straightfoward. Compare the starting expression with the final expression: - * 3 4 + * 3 4 (\lrf.l(rf))(\fz.f(f(fz)))(\fz.f(f(f(fz)))) ~~> 12 (\fz.f(f(f(f(f(f(f(f(f(f(f(fz)))))))))))) And if we choose different numbers, the result is even less clear: - * 3 6 + * 3 6 (\lrf.l(rf))(\fz.f(f(fz)))(\fz.f(f(f(f(f(fz)))))) ~~> 18 (\fz.f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(f(fz)))))))))))))))))) @@ -99,7 +99,7 @@ that reduce to that term. In the arithmetic example, there is only one number that corresponds to the sum of 3 and 4 (namely, 7). But there are many sums that add -up to 7: 3+4, 4+3, 5+2, 2+5, 6+1, 1+6, etc. +up to 7: 3+4, 4+3, 5+2, 2+5, 6+1, 1+6, etc. So the unevaluated expression contains information that is missing from the evaluated value: information about *how* that value was @@ -124,7 +124,7 @@ pathological examples where the results do not align so well: In this example, reduction returns the exact same lambda term. There is no simplification at all. - (\x.xxx)(\x.xxx) ~~> ((\x.xxxx)(\x.xxxx)(\x.xxxx)) + (\x.xxx)(\x.xxx) ~~> ((\x.xxxx)(\x.xxxx)(\x.xxxx)) Even worse, in this case, the "reduced" form is longer and more complex by any measure.