Re: [stack] Re: Jon Purdy: Why Concatenative Programming Matters

"William Tanksley, Jr" <[email protected]> Sat, 24 Mar 2012 14:14:36 -0700
Newsgroups	gmane.comp.lang.concatenative
Message-ID	<CAFTBfO4qDpsV2ftWTwtEbvYDzc7YgL8tAkUH-0gsxy0r6PiRMg@mail.gmail.com>
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Robbert van Dalen <[email protected]> wrote:
> William Tanksley, Jr wrote:
>> > 2) each flat (sub)expression can be reduced one step (or many steps, to reach fixpoint).
>> You later define "reduce" as "execute". I have to point out that some
>> expressions in any Turing-complete language never reach a fixed point
>> by execution.

> of course, never to reach fixed point is an 'infinite loop'.

Informally, yes. Formally, nontermination. There are many ways of
doing that which aren't as simple to discover as infinite loops.

> to reach fixed point is called the normal form of an expression.
> see: http://en.wikipedia.org/wiki/Term_rewriting_system

Okay, that's fine, then. I thought you were saying that every
expression has a normal form, and every expression will reduce to its
normal form no matter what path you take. The latter is definitely not
true.

>> Furthermore, you later assume that the reduced expression is itself
>> flat; that's not correct in my notation. It's actually possible to
>> build a formal logic where that does hold, but such a logic will be
>> both VERY complex and either incomplete or possible to express
>> contradictions in (per Godel).

> i always understood that a flat language must have this property:
> that *any* expression (albeit reduced) must be flat.

That depends on what you mean by "reduce". As long as you stick to
term rewriting, it's true. But I don't have a term rewriting system
yet for zeroone; my past attempts indicated that such a system would
be very complex, and would depend very much on the specific base I
chose (and I still haven't picked out a single base).

>> > 5) the reduction order of flat expressions doesn't matter - any reduction order will always yield the same (fixpoint) expression (confluence)
>> Whatever "reduce" means, it doesn't mean this. Sorry, but this would
>> imply that any two expressions of the same function will always reduce
>> to the same fixed form, thus allowing you to test in a finite amount
>> of time whether two functions are equivalent. That's impossible.

> the chosen reduction order (or strategy) doesn't imply at all that you can test two different expressions for equivalence.
> confluence means that you always end up with the same normal form, irrespective of reduction order.
> (when there are multiple ways rewrite an expression).

Unless I'm badly wrong, it should mean that you'll always end that way
*in theory*. In practice, the average solvable problem is much harder.

> actually, what you describe is very much the 'enchilada' way of doing conditionals.
> for example, the following expression:

You're right! I should have remembered that.

> R.

-Wm

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      <p>Robbert van Dalen &lt;<a href="mailto:robbert.van.dalen%40gmail.com">[email protected]</a>&gt; wrote:<br>
&gt; William Tanksley, Jr wrote:<br>
&gt;&gt; &gt; 2) each flat (sub)expression can be reduced one step (or many steps, to reach fixpoint).<br>
&gt;&gt; You later define &quot;reduce&quot; as &quot;execute&quot;. I have to point out that some<br>
&gt;&gt; expressions in any Turing-complete language never reach a fixed point<br>
&gt;&gt; by execution.<br>
<br>
&gt; of course, never to reach fixed point is an 'infinite loop'.<br>
<br>
Informally, yes. Formally, nontermination. There are many ways of<br>
doing that which aren't as simple to discover as infinite loops.<br>
<br>
&gt; to reach fixed point is called the normal form of an expression.<br>
&gt; see: <a href="http://en.wikipedia.org/wiki/Term_rewriting_system">http://en.wikipedia.org/wiki/Term_rewriting_system</a><br>
<br>
Okay, that's fine, then. I thought you were saying that every<br>
expression has a normal form, and every expression will reduce to its<br>
normal form no matter what path you take. The latter is definitely not<br>
true.<br>
<br>
&gt;&gt; Furthermore, you later assume that the reduced expression is itself<br>
&gt;&gt; flat; that's not correct in my notation. It's actually possible to<br>
&gt;&gt; build a formal logic where that does hold, but such a logic will be<br>
&gt;&gt; both VERY complex and either incomplete or possible to express<br>
&gt;&gt; contradictions in (per Godel).<br>
<br>
&gt; i always understood that a flat language must have this property:<br>
&gt; that *any* expression (albeit reduced) must be flat.<br>
<br>
That depends on what you mean by &quot;reduce&quot;. As long as you stick to<br>
term rewriting, it's true. But I don't have a term rewriting system<br>
yet for zeroone; my past attempts indicated that such a system would<br>
be very complex, and would depend very much on the specific base I<br>
chose (and I still haven't picked out a single base).<br>
<br>
&gt;&gt; &gt; 5) the reduction order of flat expressions doesn't matter - any reduction order will always yield the same (fixpoint) expression (confluence)<br>
&gt;&gt; Whatever &quot;reduce&quot; means, it doesn't mean this. Sorry, but this would<br>
&gt;&gt; imply that any two expressions of the same function will always reduce<br>
&gt;&gt; to the same fixed form, thus allowing you to test in a finite amount<br>
&gt;&gt; of time whether two functions are equivalent. That's impossible.<br>
<br>
&gt; the chosen reduction order (or strategy) doesn't imply at all that you can test two different expressions for equivalence.<br>
&gt; confluence means that you always end up with the same normal form, irrespective of reduction order.<br>
&gt; (when there are multiple ways rewrite an expression).<br>
<br>
Unless I'm badly wrong, it should mean that you'll always end that way<br>
*in theory*. In practice, the average solvable problem is much harder.<br>
<br>
&gt; actually, what you describe is very much the 'enchilada' way of doing conditionals.<br>
&gt; for example, the following expression:<br>
<br>
You're right! I should have remembered that.<br>
<br>
&gt; R.<br>
<br>
-Wm<br>
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