Related
I want to get the last element of a lazy but finite Seq in Raku, e.g.:
my $s = lazy gather for ^10 { take $_ };
The following don't work:
say $s[* - 1];
say $s.tail;
These ones work but don't seem too idiomatic:
say (for $s<> { $_ }).tail;
say (for $s<> { $_ })[* - 1];
What is the most idiomatic way of doing this while keeping the original Seq lazy?
What you're asking about ("get[ing] the last element of a lazy but finite Seq … while keeping the original Seq lazy") isn't possible. I don't mean that it's not possible with Raku – I mean that, in principle, it's not possible for any language that defines "laziness" the way Raku does with, for example, the is-lazy method.
If particular, when a Seq is lazy in Raku, that "means that [the Seq's] values are computed on demand and stored for later use." Additionally, one of the defining features of a lazy iterable is that it cannot know its own length while remaining lazy – that's why calling .elems on a lazy iterable throws an error:
my $s = lazy gather for ^10 { take $_ };
say $s.is-lazy; # OUTPUT: «True»
$s.elems; # THROWS: «Cannot .elems a lazy list onto a Seq»
Now, at this point, you might reasonably be thinking "well, maybe Raku doesn't know how long $s is, but I can tell that it has exactly 10 elements in it." And you're not wrong – with that code, $s is indeed guaranteed to have 10 elements. This means that, if you want to get the tenth (last) element of $s, you can do so with $s[9]. And accessing $s's tenth element like that won't change the fact that $s.is-lazy.
But, importantly, you can only do so because you know something "extra" about $s, and that extra info undoes a good chunk of the reason you might want a list to be lazy in practice.
To see what I mean, consider a very similar Seq
my $s2 = lazy gather for ^10 { last if rand > .95; take $_ };
say $s2.is-lazy; # OUTPUT: «True»
Now, $s2probably has 10 elements, but it might not – the only way to know is to iterate through it and find out. In turn, this means $s2[9] does not jump to the tenth element the way $s[9] did; it iterates through $s2 just like you'd need to. And, as a result, if you run $s2[9], then $s2 will no longer be lazy (i.e., $s2.is-lazy will return False).
And this is, in effect, what you did in the code in your question:
my $s = lazy gather for ^10 { take $_ };
say $s.is-lazy; # OUTPUT: «True»
say (for $s<> { $_ }).tail; # OUTPUT: «9»
say $s.is-lazy; # OUTPUT: «False»
Because Raku cannot ever know that it has reached the tail of a lazy Seq, the only way it could tell you the .tail is to fully iterate $s. And that necessarily means that $s is no longer lazy.
Two complications
It's worth mentioning two adjacent topics that aren't actually related but that are close enough that they trip some people up.
First, nothing I've said about lazy iterables not knowing their length precludes some non-lazy iterables from knowing their length. Indeed, a decent number of Raku types do both the Iterator role and the PredictiveIterator role – and the main point of a PredictiveIterator is that it does know how many elements it can produce without needing to produce/iterate them. But PredictiveIterators cannot be lazy.
The second potentially confusing topic is closely related to the first: while no PredictiveIterator can be lazy (that is, none will ever have an .is-lazy method that returns True), some PredictiveIterators have behavior that is very similar to laziness – and, in fact, may even be colloquially referred to as "lazy".
I can't do a great job explaining this distinction because, quite honestly, I don't fully understand it myself. But I can give you an example: the .lines method on an IO::Handle. It's certainly the case that reading the lines of a huge file behaves a lot like it's dealing with a lazy iterable. most obviously, you can process each line without ever having the whole file in memory. And the docs even say that "lines are read lazily" with the .lines method.
On the other hand:
my $l = 'some-file-with-100_000-lines.txt'.IO.lines;
say $l.is-lazy; # OUTPUT: «False»
say $l.iterator ~~ PredictiveIterator; # OUTPUT: «True»
say $l.elems; # OUTPUT: «100000»
So I'm not quite sure whether it's fair to say that $l "is a lazy iterable", but if it is, it's "lazy" in a different way than $s was.
I realize that was a lot, but I hope it is helpful. If you have a more specific use case in mind for laziness (I bet it wasn't gathering the numbers from zero to nine!), I'd be happy to address that more specifically. And if anyone else can fill in some of the details with .lines and other lazy-not-lazy PredictiveIterators, I'd really appreciate it!
Drop the lazy
Lazy sequences in Raku are designed to work well as is. You don't need to emphasize they're lazy by adding an explicit lazy.
If you add an explicit lazy, Raku interprets that as a request to block operations such as .tail because they will almost certainly immediately render laziness moot, and, if called on an infinite sequence, or even just a sufficiently large one, hang or OOM the program.
So, either drop the lazy, or don't invoke operations like .tail that will be blocked if you do.
Expanded version of my original answer
As noted by #ugexe, the idiomatic solution is to drop the lazy.
Quoting my answer to the SO About Laziness:
if a gather is asked if it's lazy, it returns False.
Aiui, something like the following applies:
Some lazy sequence producers may be actually or effectively infinite. If so, calling .tail etc on them will hang the calling program. Conversely, other lazy sequences perform fine when all their values are consumed in one go. How should Raku distinguish between these two scenarios?
A decision was made in 2015 to let value producing datatypes emphasize or deemphasize their laziness via their response to an .is-lazy call.
Returning True signals that a sequence is not only lazy but wants to be known to be lazy by consuming code that calls .is-lazy. (Not so much end-user code but instead built in consuming features such as # sigilled variables handling an assignment trying to determine whether or not to assign eagerly.) Built in consuming features take a True as a signal they ought block calls like .tail. If a dev knows this is overly conservative, they can add an eager (or remove an unneeded lazy).
Conversely, a datatype, or even a particular object instance, may return False to signal that it does not want to be considered lazy. This may be because the actual behaviour of a particular datatype or instance is eager, but it might instead be that it is lazy technically, but doesn't want a consumer to block operations such as .tail because it knows they will not be harmful, or at least prefers to have that be the default presumption. If a dev knows better (because, say, it hangs the program), or at least does not want to block potentially problematic operations, they can add a lazy (or remove an unneeded eager).
I think this approach works well, but it doc and error messages mentioning "lazy" may not have caught up with the shift made in 2015. So:
If you've been confused by some doc about laziness, please search for doc issues with "lazy" in them, or "laziness", and add comments to existing issues, or file a new doc issue (perhaps linking to this SO answer).
If you've been confused by a Rakudo error message mentioning laziness, please search for Rakudo issues with "lazy" in them, and tagged [LTA] (which means "Less Than Awesome"), and add comments, or file a new Rakudo issue (with an [LTA] tag, and perhaps a link to this SO answer).
Further discussion
the docs ... say “If you want to force lazy evaluation use the lazy subroutine or method. Binding to a scalar or sigilless container will also force laziness.”
Yes. Aiui this is correct.
[which] sounds like it implies “my $x := lazy gather { ... } is the same as my $x := gather { ... }”.
No.
An explicit lazy statement prefix or method adds emphasis to laziness, and Raku interprets that to mean it ought block operations like .tail in case they hang the program.
In contrast, binding to a variable alters neither emphasis nor deemphasis of laziness, merely relaying onward whatever the bound producer datatype/instance has chosen to convey via .is-lazy.
not only in connection with gather but elsewhere as well
Yes. It's about the result of .is-lazy:
my $x = (1, { .say; $_ + 1 } ... 1000);
my $y = lazy (1, { .say; $_ + 1 } ... 1000);
both act lazily ... but $x.tail is possible while $y.tail is not.
Yes.
An explicit lazy statement prefix or method forces the answer to .is-lazy to be True. This signals to a consumer that cares about the dangers of laziness that it should become cautious (eg rejecting .tail etc.).
(Conversely, an eager statement prefix or method can be used to force the answer to .is-lazy to be False, making timid consumers accept .tail etc calls.)
I take from this that there are two kinds of laziness in Raku, and one has to be careful to see which one is being used where.
It's two kinds of what I'll call consumption guidance:
Don't-tail-me If an object returns True from an .is-lazy call then it is treated as if it might be infinite. Thus operations like .tail are blocked.
You-can-tail-me If an object returns False from an .is-lazy call then operations like .tail are accepted.
It's not so much that there's a need to be careful about which of these two kinds is in play, but if one wants to call operations like tail, then one may need to enable that by inserting an eager or removing a lazy, and one must take responsibility for the consequences:
If the program hangs due to use of .tail, well, DIHWIDT.
If you suddenly consume all of a lazy sequence and haven't cached it, well, maybe you should cache it.
Etc.
What I would say is that the error messages and/or doc may well need to be improved.
I'm working on something which processes UTF-8 encoding, and I found myself asking the question:
What should I do when I encounter a byte which never occur inside a
UTF-8 encoded string?
i.e. 0x1111111X
For example, I'm writing a small snippet of code which looks at the current place in the stream of bytes, and tells you how many bytes are used to represent the code point at that place in the stream.
0x0XXXXXXX just 1
0x10XXXXXX oops, we are in a continuation byte,
search back upstream to find the leading byte
0x11XXXXXX count the
number of leading 1s, that's the answer
0x1111111X err, this is not
possible in UTF-8!!! what to do!?!?
I'm thinking of returning an error value, but wondering if I should, as a side effect, replace it with some more predictable error glyph (I mean the code point representing said glyph). And later when I do something more complicated, like jumping through the string and find that the leading byte does not have the correct number of continuation bytes after it... I'm thinking I should "fix" that up too.
Is it standard practice to leave wrongly encoded strings broken, or to change them and make them be wrong but at least play nice?
The most common way is to just throw a meaningful error if the input is not correct and stop.
There are a lot of good reasons to do so:
speed: if you try to fix errors this often cause your
function to be slower even on correct inputs
simplicity: your code can become really complicated if you try to fix any error
maintainability and correctness: it's just easier to ensure the function works correctly
when you stop whenever the input does not match the specification you are working with. Since you have only to check input according to specification.
purpose: any time you get to such a point like here you have to think about:
what is the purpose of my function? Why I came up with the idea to write it?
Also: a function fixcode which fixes the uft8 could be used also at an other place, so it makes total sense to separate fixing (purpose, simplicity, maintainability and correctness argument again).
Even if you expect an error, I would prefer to separate the encode and fixcode since
your can reuse fixcode in outer contexts.
If you are really thinking about fixing the utf8 code while encoding I would use a pattern like this:
try {
q = encode(s);
} catch(encodingerror) {
log(encodingerror);
t = fixcode(s);
q = encode(t);
}
I am developing a (large) package which does not load properly anymore.
This happened after I changed a single line of code.
When I attempt to load the package (with Needs), the package starts loading and then one of the setdelayed definitions “comes alive” (ie. Is somehow evaluated), gets trapped in an error trapping routine loaded a few lines before and the package loading aborts.
The error trapping routine with abort is doing its job, except that it should not have been called in the first place, during the package loading phase.
The error message reveals that the wrong argument is in fact a pattern expression which I use on the lhs of a setdelayed definition a few lines later.
Something like this:
……Some code lines
Changed line of code
g[x_?NotGoodQ]:=(Message[g::nogood, x];Abort[])
……..some other code lines
g/: cccQ[g[x0_]]:=True
When I attempt to load the package, I get:
g::nogood: Argument x0_ is not good
As you see the passed argument is a pattern and it can only come from the code line above.
I tried to find the reason for this behavior, but I have been unsuccessful so far.
So I decided to use the powerful Workbench debugging tools .
I would like to see step by step (or with breakpoints) what happens when I load the package.
I am not yet too familiar with WB, but it seems that ,using Debug as…, the package is first loaded and then eventually debugged with breakpoints, ect.
My problem is that the package does not even load completely! And any breakpoint set before loading the package does not seem to be effective.
So…2 questions:
can anybody please explain why these code lines "come alive" during package loading? (there are no obvious syntax errors or code fragments left in the package as far as I can see)
can anybody please explain how (if) is possible to examine/debug
package code while being loaded in WB?
Thank you for any help.
Edit
In light of Leonid's answer and using his EvenQ example:
We can avoid using Holdpattern simply by definying upvalues for g BEFORE downvalues for g
notGoodQ[x_] := EvenQ[x];
Clear[g];
g /: cccQ[g[x0_]] := True
g[x_?notGoodQ] := (Message[g::nogood, x]; Abort[])
Now
?g
Global`g
cccQ[g[x0_]]^:=True
g[x_?notGoodQ]:=(Message[g::nogood,x];Abort[])
In[6]:= cccQ[g[1]]
Out[6]= True
while
In[7]:= cccQ[g[2]]
During evaluation of In[7]:= g::nogood: -- Message text not found -- (2)
Out[7]= $Aborted
So...general rule:
When writing a function g, first define upvalues for g, then define downvalues for g, otherwise use Holdpattern
Can you subscribe to this rule?
Leonid says that using Holdpattern might indicate improvable design. Besides the solution indicated above, how could one improve the design of the little code above or, better, in general when dealing with upvalues?
Thank you for your help
Leaving aside the WB (which is not really needed to answer your question) - the problem seems to have a straightforward answer based only on how expressions are evaluated during assignments. Here is an example:
In[1505]:=
notGoodQ[x_]:=True;
Clear[g];
g[x_?notGoodQ]:=(Message[g::nogood,x];Abort[])
In[1509]:= g/:cccQ[g[x0_]]:=True
During evaluation of In[1509]:= g::nogood: -- Message text not found -- (x0_)
Out[1509]= $Aborted
To make it work, I deliberately made a definition for notGoodQ to always return True. Now, why was g[x0_] evaluated during the assignment through TagSetDelayed? The answer is that, while TagSetDelayed (as well as SetDelayed) in an assignment h/:f[h[elem1,...,elemn]]:=... does not apply any rules that f may have, it will evaluate h[elem1,...,elem2], as well as f. Here is an example:
In[1513]:=
ClearAll[h,f];
h[___]:=Print["Evaluated"];
In[1515]:= h/:f[h[1,2]]:=3
During evaluation of In[1515]:= Evaluated
During evaluation of In[1515]:= TagSetDelayed::tagnf: Tag h not found in f[Null]. >>
Out[1515]= $Failed
The fact that TagSetDelayed is HoldAll does not mean that it does not evaluate its arguments - it only means that the arguments arrive to it unevaluated, and whether or not they will be evaluated depends on the semantics of TagSetDelayed (which I briefly described above). The same holds for SetDelayed, so the commonly used statement that it "does not evaluate its arguments" is not literally correct. A more correct statement is that it receives the arguments unevaluated and does evaluate them in a special way - not evaluate the r.h.s, while for l.h.s., evaluate head and elements but not apply rules for the head. To avoid that, you may wrap things in HoldPattern, like this:
Clear[g,notGoodQ];
notGoodQ[x_]:=EvenQ[x];
g[x_?notGoodQ]:=(Message[g::nogood,x];Abort[])
g/:cccQ[HoldPattern[g[x0_]]]:=True;
This goes through. Here is some usage:
In[1527]:= cccQ[g[1]]
Out[1527]= True
In[1528]:= cccQ[g[2]]
During evaluation of In[1528]:= g::nogood: -- Message text not found -- (2)
Out[1528]= $Aborted
Note however that the need for HoldPattern inside your left-hand side when making a definition is often a sign that the expression inside your head may also evaluate during the function call, which may break your code. Here is an example of what I mean:
In[1532]:=
ClearAll[f,h];
f[x_]:=x^2;
f/:h[HoldPattern[f[y_]]]:=y^4;
This code attempts to catch cases like h[f[something]], but it will obviously fail since f[something] will evaluate before the evaluation comes to h:
In[1535]:= h[f[5]]
Out[1535]= h[25]
For me, the need for HoldPattern on the l.h.s. is a sign that I need to reconsider my design.
EDIT
Regarding debugging during loading in WB, one thing you can do (IIRC, can not check right now) is to use good old print statements, the output of which will appear in the WB's console. Personally, I rarely feel a need for debugger for this purpose (debugging package when loading)
EDIT 2
In response to the edit in the question:
Regarding the order of definitions: yes, you can do this, and it solves this particular problem. But, generally, this isn't robust, and I would not consider it a good general method. It is hard to give a definite advice for a case at hand, since it is a bit out of its context, but it seems to me that the use of UpValues here is unjustified. If this is done for error - handling, there are other ways to do it without using UpValues.
Generally, UpValues are used most commonly to overload some function in a safe way, without adding any rule to the function being overloaded. One advice is to avoid associating UpValues with heads which also have DownValues and may evaluate -by doing this you start playing a game with evaluator, and will eventually lose. The safest is to attach UpValues to inert symbols (heads, containers), which often represent a "type" of objects on which you want to overload a given function.
Regarding my comment on the presence of HoldPattern indicating a bad design. There certainly are legitimate uses for HoldPattern, such as this (somewhat artificial) one:
In[25]:=
Clear[ff,a,b,c];
ff[HoldPattern[Plus[x__]]]:={x};
ff[a+b+c]
Out[27]= {a,b,c}
Here it is justified because in many cases Plus remains unevaluated, and is useful in its unevaluated form - since one can deduce that it represents a sum. We need HoldPattern here because of the way Plus is defined on a single argument, and because a pattern happens to be a single argument (even though it describes generally multiple arguments) during the definition. So, we use HoldPattern here to prevent treating the pattern as normal argument, but this is mostly different from the intended use cases for Plus. Whenever this is the case (we are sure that the definition will work all right for intended use cases), HoldPattern is fine. Note b.t.w., that this example is also fragile:
In[28]:= ff[Plus[a]]
Out[28]= ff[a]
The reason why it is still mostly OK is that normally we don't use Plus on a single argument.
But, there is a second group of cases, where the structure of usually supplied arguments is the same as the structure of patterns used for the definition. In this case, pattern evaluation during the assignment indicates that the same evaluation will happen with actual arguments during the function calls. Your usage falls into this category. My comment for a design flaw was for such cases - you can prevent the pattern from evaluating, but you will have to prevent the arguments from evaluating as well, to make this work. And pattern-matching against not completely evaluated expression is fragile. Also, the function should never assume some extra conditions (beyond what it can type-check) for the arguments.
I'm writing an algorithm that detects clones in source code. E.g. if there is a block like:
for(int i = o; i <5; i++){
doSomething(abc);
}
...and if this block is repeated somewhere else in the source code it will be detected as a clone. The method I am using at the moment is to create hashes for lines/blocks and compare them with hashes of other lines/blocks in the same source to see if there are any matches.
Now, if the same block as above was to be repeated somewhere with only the argument of doSomething different, it would not be detected as a clone even though it would appear very much like a clone to you and me. My algorithm detects exact matches but doesn't detect matching blocks where only the argument is different.
Could anyone suggest any ways of getting around this issue? Thanks!
Here's a super-simple way, which might go too far in erasing information (i.e., might produce too many false positives): replace every identifier that isn't a keyword with some fixed name. So you'd get
for (int DUMMY = DUMMY; DUMMY<5; DUMMY++) {
DUMMY(DUMMY);
}
(assuming you really meant o rather than 0 in the initialization part of the for-loop).
If you get a huge number of false positives with this, you could then post-process them by, for instance, looking to see what fraction of the DUMMYs actually correspond to the same identifier in both halves of the match, or at least to identifiers that are consistent between the two.
To do much better you'll probably need to parse the code to some extent. That would be a lot more work.
Well if you're going todo something else then you're going to have to parse to code at least a bit. For example you could detect methods and then ignore the method arguments in your hash. Anyway I think it's always true that you need your program to understand the code better than 'just text blocks', and that might get awefuly complicated.
Related Questions: Benefits of using short-circuit evaluation, Why would a language NOT use Short-circuit evaluation?, Can someone explain this line of code please? (Logic & Assignment operators)
There are questions about the benefits of a language using short-circuit code, but I'm wondering what are the benefits for a programmer? Is it just that it can make code a little more concise? Or are there performance reasons?
I'm not asking about situations where two entities need to be evaluated anyway, for example:
if($user->auth() AND $model->valid()){
$model->save();
}
To me the reasoning there is clear - since both need to be true, you can skip the more costly model validation if the user can't save the data.
This also has a (to me) obvious purpose:
if(is_string($userid) AND strlen($userid) > 10){
//do something
};
Because it wouldn't be wise to call strlen() with a non-string value.
What I'm wondering about is the use of short-circuit code when it doesn't effect any other statements. For example, from the Zend Application default index page:
defined('APPLICATION_PATH')
|| define('APPLICATION_PATH', realpath(dirname(__FILE__) . '/../application'));
This could have been:
if(!defined('APPLICATION_PATH')){
define('APPLICATION_PATH', realpath(dirname(__FILE__) . '/../application'));
}
Or even as a single statement:
if(!defined('APPLICATION_PATH'))
define('APPLICATION_PATH', realpath(dirname(__FILE__) . '/../application'));
So why use the short-circuit code? Just for the 'coolness' factor of using logic operators in place of control structures? To consolidate nested if statements? Because it's faster?
For programmers, the benefit of a less verbose syntax over another more verbose syntax can be:
less to type, therefore higher coding efficiency
less to read, therefore better maintainability.
Now I'm only talking about when the less verbose syntax is not tricky or clever in any way, just the same recognized way of doing, but in fewer characters.
It's often when you see specific constructs in one language that you wish the language you use could have, but didn't even necessarily realize it before. Some examples off the top of my head:
anonymous inner classes in Java instead of passing a pointer to a function (way more lines of code).
in Ruby, the ||= operator, to evaluate an expression and assign to it if it evaluates to false or is null. Sure, you can achieve the same thing by 3 lines of code, but why?
and many more...
Use it to confuse people!
I don't know PHP and I've never seen short-circuiting used outside an if or while condition in the C family of languages, but in Perl it's very idiomatic to say:
open my $filehandle, '<', 'filename' or die "Couldn't open file: $!";
One advantage of having it all in one statement is the variable declaration. Otherwise you'd have to say:
my $filehandle;
unless (open $filehandle, '<', 'filename') {
die "Couldn't open file: $!";
}
Hard to claim the second one is cleaner in that case. And it'd be wordier still in a language that doesn't have unless
I think your example is for the coolness factor. There's no reason to write code like that.
EDIT: I have no problem with doing it for idiomatic reasons. If everyone else who uses a language uses short-circuit evaluation to make statement-like entities that everyone understands, then you should too. However, my experience is that code of that sort is rarely written in C-family languages; proper form is just to use the "if" statement as normal, which separates the conditional (which presumably has no side effects) from the function call that the conditional controls (which presumably has many side effects).
Short circuit operators can be useful in two important circumstances which haven't yet been mentioned:
Case 1. Suppose you had a pointer which may or may not be NULL and you wanted to check that it wasn't NULL, and that the thing it pointed to wasn't 0. However, you must not dereference the pointer if it's NULL. Without short-circuit operators, you would have to do this:
if (a != NULL) {
if (*a != 0) {
⋮
}
}
However, short-circuit operators allow you to write this more compactly:
if (a != NULL && *a != 0) {
⋮
}
in the certain knowledge that *a will not be evaluated if a is NULL.
Case 2. If you want to set a variable to a non-false value returned from one of a series of functions, you can simply do:
my $file = $user_filename ||
find_file_in_user_path() ||
find_file_in_system_path() ||
$default_filename;
This sets the value of $file to $user_filename if it's present, or the result of find_file_in_user_path(), if it's true, or … so on. This is seen perhaps more often in Perl than C, but I have seen it in C.
There are other uses, including the rather contrived examples which you cite above. But they are a useful tool, and one which I have missed when programming in less complex languages.
Related to what Dan said, I'd think it all depends on the conventions of each programming language. I can't see any difference, so do whatever is idiomatic in each programming language. One thing that could make a difference that comes to mind is if you had to do a series of checks, in that case the short-circuiting style would be much clearer than the alternative if style.
What if you had a expensive to call (performance wise) function that returned a boolean on the right hand side that you only wanted called if another condition was true (or false)? In this case Short circuiting saves you many CPU cycles. It does make the code more concise because of fewer nested if statements. So, for all the reasons you listed at the end of your question.
The truth is actually performance. Short circuiting is used in compilers to eliminate dead code saving on file size and execution speed. At run-time short-circuiting does not execute the remaining clause in the logical expression if their outcome does not affect the answer, speeding up the evaluation of the formula. I am struggling to remember an example. e.g
a AND b AND c
There are two terms in this formula evaluated left to right.
if a AND b evaluates to FALSE then the next expression AND c can either be FALSE AND TRUE or FALSE AND FALSE. Both evaluate to FALSE no matter what the value of c is. Therefore the compiler does not include AND c in the compiled format hence short-circuiting the code.
To answer the question there are special cases when the compiler cannot determine whether the logical expression has a constant output and hence would not short-circuit the code.
Think of it this way, if you have a statement like
if( A AND B )
chances are if A returns FALSE you'll only ever want to evaluate B in rare special cases. For this reason NOT using short ciruit evaluation is confusing.
Short circuit evaluation also makes your code more readable by preventing another bracketed indentation and brackets have a tendency to add up.