在 Mathematica 中，为什么递归函数中的替换不会终止？
想象一下，我在 Mathematica 中定义了一个递归阶乘，如下所示：
Clear[fact]
fact[0] = 1
fact[n_] := n fact[n - 1]
评估fact[10] 确认该函数有效并终止。
这是一个主要的例子，但它在这个问题中达到了它的目的。实际上，我的问题总体上与递归函数定义有关。
我预计评估以下替换也将终止：
x fact[x-1] /. x -> 2
唉，它遇到了递归深度限制：
$RecursionLimit::reclim: Recursion depth of 256 exceeded.
我预计会看到类似的内容：
2 fact[2-1]
或只是值
2
UPDATE：fact 的替代递归定义 确实按预期工作：
Clear[fact]
fact[n_] := If[n < 1, 1, n fact[n - 1]]
但是这个事实（双关语；-）让我更加神秘：为什么它的行为如此不同？
我的问题是双重的：
即使有内置的帮助并在网上搜索线索，我也无法解释为什么 Mathematica 显然坚持保留符号结果，而不是评估“中间”结果并很好地终止。谁敢提出可行的解释？
我如何说服 Mathematica 按照我的期望执行（除了使用 If[] 的替代方案）？
我真的很困惑这个问题，我真的希望有人能帮助我。
/吐温
Imagine I've defined a recursive factorial in Mathematica, like this:
Clear[fact]
fact[0] = 1
fact[n_] := n fact[n - 1]
Evaluating fact[10] confirms that the function works and terminates.
A bit of a staple example, but it serves its purpose in this question. Actually, my question pertains to recursive function definitions in general anyway.
I expected evaluating the following replacement to terminate as well:
x fact[x-1] /. x -> 2
Alas, it runs in to a recursion depth limit:
$RecursionLimit::reclim: Recursion depth of 256 exceeded.
I expected to see something like:
2 fact[2-1]
or just the value
2
UPDATE: An alternative recursive definition of fact does work as expected:
Clear[fact]
fact[n_] := If[n < 1, 1, n fact[n - 1]]
But this fact (pun intended ;-) makes it even more mysterious to me: Why does it behave so much differently?
My question is twofold:
Even with the built-in help and searching the net for clues, I can't explain why Mathematica insists in, apparently, keeping the symbolic result, rather than evaluating the 'intermediate' results and terminating nicely. Who ventures a viable explaination?
How can I convince Mathematica to perform according to my expections (Other than using the alternative using If[])?
I'm really puzzled by this one, and I really hope someone out there can help me out.
/Twan
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评论（3）
尝试 u[x_] := x;跟踪[x*u[x] /。 x-> 2]，它首先评估
x和u[x]。那么，在您的情况下，它首先尝试评估fact[x-1]，然后将x替换为 2，并达到递归限制。Attributes[ReplaceAll]显示它没有属性HoldFirst，因此它首先评估其第一个参数。例如，给出预期的
2，因为它保存第一个参数，替换，然后释放保留，如您所愿。另一种方法是
但不要这样做。
不过，肯定很快就会有人给出更好的解释。
编辑：为了回答关于为什么
不会导致无限递归的附加问题：以这种方式定义，
factif[x]计算结果为If[x1, 1, x factif[x - 1]]，因为无法计算x<1。因此，在尝试评估ReplaceAll的第一个参数之后，它仍保持这种形式，然后发生替换等。Trying
u[x_] := x; Trace[x*u[x] /. x -> 2], it first evaluatesxandu[x]. In your case, then, it first tries to evaluatefact[x-1]before replacingxby 2, and hits the recursion limit.Attributes[ReplaceAll]shows that it does not have attributeHoldFirst, so it starts by evaluating its first argument. For instance,gives the expected
2, as it holds the first argument, replaces, then releases the hold, as you intended.Another way to do it is
but don't do this.
Surely someone will give a better explanation soon, though.
EDIT: In response to the added question as to why
does not result in infinite recursion: defined this way,
factif[x]evaluates toIf[x < 1, 1, x factif[x - 1]], sincex<1cannot be evaluated. So it remains in this form after the attempted evaluation of the first argument ofReplaceAll, then the replacement occurs etc.这是因为您正在评估这一点：
在更换发生之前。只需执行
fact[x-1]即可得到错误。您可以像这样修复您的
fact函数：然后
x fact[x - 1] /。 x-> 2返回2这似乎是正确的。请记住，您的函数参数模式
fact[n_]是极其通用的。例如，它允许对诸如fact[Integrate[Sin[x], x]]之类的内容进行评估，这可能不是您想要的。使用fact[n_Integer] 更加精确，并且允许fact 函数按照您想要的方式运行。如果您想更好地定义此函数，您可以执行以下操作：
这样
fact["x"]之类的操作将失败并显示一条消息：This is because you're evaluating this:
before the replacement happens. Just do
fact[x-1]and you get the error.You can fix your
factfunction like so:Then
x fact[x - 1] /. x -> 2returns2which seems correct.Remember that your function argument pattern
fact[n_]is extremely general. For example it allows for something likefact[Integrate[Sin[x], x]]to evaluate, which is probably not something you intended. Usingfact[n_Integer]is much more precise, and will allow thefactfunction to act the way you want it to.If you want to define this function even better, you can do something like:
So that something like
fact["x"]will fail with a message:其他答案是正确的：
fact在其参数被替换之前评估。根本问题是您已经定义了事实并考虑了整数输入，并且没有为非整数输入提供终止条件。如果您这样做，则
fact将保持未计算状态，直到它具有与正整数匹配的内容为止。您可能需要将替换语句包装在
Evaluate中，然后在替换其参数后触发fact的定义。另一种方法可能是使用纯函数：
不应该过早地求值。
The other answers are correct:
factevaluates before its argument is replaced. The essential issue is that you have definedfactwith integer inputs in mind, and haven't provided a terminal condition for non-integer inputs. If you instead didThen
factwould be left unevaluated until it had something that matched a positive integer.You might need to wrap your replacement statement in
Evaluateto then fire the definition forfactafter replacing its argument.An alternative approach might be to use a pure function:
That shouldn't evaluate prematurely.