I am currently attempting to use VST to verify the correctness of a project which involves a global array of doubles. However, when attempting to access the array I have that the head of the array is given as a data_at statement while the rest of the array is given as a sepcon list of mapsto statements and there does not appear to be any way to prove field_compatible for elements beyond the head of the array.
Trying to access elements beyond offset_val 0 seems to inevitably involve proving a size_compatible statement. This is where I run into a problem. Since the alignment of tdouble is set to 4 and the size is set to 8, there seems to be a possibility that the head of the array is at Ptrofs.modulus - 12 making size_compatible false for the next element in the array. Am I going about this the wrong way?
I made a toy example with the same problem that I've mentioned above.
double dbls[] = {0.0, 1.1};
int main() {
double sum;
sum = dbls[0] + dbls[1];
return 0;
}
I will frame my answer in the form of a Coq development:
Require Import VST.floyd.proofauto.
Require Import VST.progs.foo.
#[export] Instance CompSpecs : compspecs. make_compspecs prog. Defined.
Definition Vprog : varspecs. mk_varspecs prog. Defined.
Definition main_spec :=
DECLARE _main
WITH gv : globals
PRE [] main_pre prog tt gv
POST [ tint ] main_post prog gv.
Definition Gprog : funspecs := [ ].
Lemma body_main: semax_body Vprog Gprog f_main main_spec.
Proof.
start_function.
(* Remark 1: it seems to be a bug in VST 2.11.1 (and earlier versions)
that the array is not packaged up into
(data_at Ews (tarray Tdouble 2) ...)
the way it ought to be. This seems to work better for integer
arrays, et cetera.*)
(* Remark 2: you are right to be concerned about alignment, but
VST addresses that issue correctly. Any extern global variable
in a C program, such as your [dbls] array, is aligned at the
biggest possible alignment requirement. VST expresses this
with the "headptr" predicate, and for any identifier id,
(gv id) is a headptr. So therefore, *)
assert_PROP (headptr (gv _dbls)) by entailer!.
(* and you can see above the line, H: headptr (gv _dbls). *)
Print headptr.
(* This shows that (gv _dbls) must be at offset zero within
some block, which guarantees alignment at any type.
One useful consequence is proved by the lemma
headptr_field_compatible: *)
Check headptr_field_compatible.
(* And now, let's apply that lemma: *)
pose proof headptr_field_compatible (tarray tdouble 2) nil _
H (eq_refl _) Logic.I ltac:(simpl; rep_lia).
(* So we see that as long as the pesky 'align_compatible_rec' is proved,
the pointer (gv _dbls) should be 'field_compatible' with the array
type that you want. And it's straightforward though tedious to prove
the 'align_compatible_rec' premise, as follows: *)
spec H0.
apply align_compatible_rec_Tarray; intros.
Search align_compatible_rec.
eapply align_compatible_rec_by_value; [ reflexivity | ].
apply Z.divide_add_r.
apply Z.divide_0_r.
apply Z.divide_mul_l.
apply Z.mod_divide; compute; intros; congruence.
(* Normally, VST users shouldn't have to do this 'by hand'.
We should fix the bug (failure to nicely package the precondition).
But in the interim, perhaps this gives what you need for a workaround.*)
I'm writing a quicksort function for an exercise. I already know of the 5-line functional quicksort; but I wanted to improve the partition by having it scan through the list once and return a pair of lists splitting the original list in half. So I wrote:
fun partition nil = (nil, nil)
| partition (pivot :: rest) =
let
fun part (lst, pivot, (lesseq, greater)) =
case lst of
[] => (lesseq, greater)
| (h::t) =>
if h <= pivot then part (t, pivot, (h :: lesseq, greater))
else part (t, pivot, (lesseq, h :: greater))
in
part (rest, pivot, ([pivot], []))
end;
This partitions well enough. It gives me a signature val partition = fn : int list -> int list * int list. It runs as expected.
It's when I use the quicksort below that things start to break.
fun quicksort_2 nil = nil
| quicksort_2 lst =
let
val (lesseq, greater) = partition lst
in
quicksort_2 lesseq # quicksort_2 greater
end;
I can run the above function if I eliminate the recursive calls to quicksort_2; but if I put them back in (to actually go and sort the thing), it will cease to run. The signature will be incorrect as well, giving me val quicksort_2 = fn : int list -> 'a list. The warning I receive when I call the function on a list is:
Warning: type vars not generalized because of value restriction are instantiated to dummy types (X1,X2,...)
What is the problem here? I'm not using any ref variables; the type annotation I've tried doesn't seem to help...
The main issue is that you're lacking the singleton list base case for your quicksort function. It ought to be
fun quicksort [ ] = [ ]
| quicksort [x] = [x]
| quicksort xs =
let
val (l, r) = partition xs
in
quicksort l # quicksort r
end
which should then have type int list -> int list given the type of your partition. We have to add this case as otherwise you'll never hit a base case and instead recurse indefinitely.
For some more detail on why you saw the issues you were having though:
The signature will be incorrect as well, giving me val quicksort_2 = fn : int list -> 'a list
This is because the codomain of your function was never restricted to be less general than 'a list. Taking a look at the possible branches in your original implementation we can see that in the nil branch you return nil (of most general type 'a list) and in the recursive case you get two 'a lists (per our assumptions thus far) and append them, resulting in an 'a list---this is fine so your type is not further restricted.
[Value Restriction Warning]
What is the problem here? I'm not using any ref variables
The value restriction isn't really related to refs (though can often arise when using them). Instead it is the prohibition that anything polymorphic at the top level must be a value by its syntax (and thus precludes the possibility that a computation is behind a type abstractor at the top level). Here it is because given xs : int list we (ignoring the value restriction) have quicksort_2 xs : 'a list---which would otherwise be polymorphic, but is not a syntactic value. Correspondingly it is value restricted.
In Isabelle, one can often hit proof goals where the intermediate type of terms is critical to a proof's correctness. For instance, consider the following lemma converting the nat 42 into an 'a word then back again:
theory Test
imports "~~/src/HOL/Word/Word"
begin
lemma "unat (of_nat 42) = 42"
...
Now the truth of this statement depends on the type of of_nat 42: if it is 32 word, then the statement is true, if it is a 2 word, however, the statement is false.
Unfortunately, I can't seem to get Isabelle to display this intermediate type to me.
I have tried the following:
declare [[show_types]]
declare [[show_sorts]]
local_setup {* Config.put show_all_types true *}
all of which just display:
unat (of_nat (42::nat)) = (42::nat)
In a pinch, one can do:
apply (tactic {* (fn t => (tracing (PolyML.makestring (prems_of t)); all_tac t)) *})
do get a raw dump of the term, but I was hoping there was a better way.
Is there a good way of displaying intermediate term types in a proof goal?
In Isabelle/jEdit you can always "control-hover" (i.e., keep the control-button pressed and hover the mouse) over a constant in order to get additional information. For of_nat in
lemma "unat (of_nat 42) = 42"
This results in
constant "Nat.semiring_1_class.of_nat"
:: nat => 'a word
Now you can recursively do the same on 'a of 'a word and will get
:: len
free type variable
which tells you that 'a is of sort len (by control-clicking len, you could directly jump to the definition of this type class, which is also quite handy).
So the answer to your question is: yes, control-hover in Isabelle/jEdit.
To make Isabelle show you the type of unat in this example, you want to declare the following:
declare [[show_types]]
declare [[show_sorts]]
declare [[show_consts]]
The last line prints the type of each constant used in the goal in the output window. This works both in jEdit and ProofGeneral.
There is a problem with this solution: if unat occurs multiple times with different types, it will print all these instances, but it will not tell you which type instance is which occurrence. Apart from the jEdit hover, I don't know of any solution for that, though.
Running the command:
setup {* Config.put_global show_all_types true *}
seems to do the trick.
The goal unat (of_nat 3) = 3 becomes the hideous (but complete):
goal (1 subgoal):
1. (Trueprop::bool => prop)
((op =::nat => nat => bool)
((unat::'a word => nat)
((of_nat::nat => 'a word)
((numeral::num => nat)
((num.Bit1::num => num) (num.One::num)))))
((numeral::num => nat)
((num.Bit1::num => num) (num.One::num))))
as desired.
It is interesting that a declare [[show_all_types]] does not work; the source looks like it should. Perhaps it is a bug in Isabelle2013?
It seems quite a few mainstream languages support function literals these days. They are also called anonymous functions, but I don't care if they have a name. The important thing is that a function literal is an expression which yields a function which hasn't already been defined elsewhere, so for example in C, &printf doesn't count.
EDIT to add: if you have a genuine function literal expression <exp>, you should be able to pass it to a function f(<exp>) or immediately apply it to an argument, ie. <exp>(5).
I'm curious which languages let you write function literals which are recursive. Wikipedia's "anonymous recursion" article doesn't give any programming examples.
Let's use the recursive factorial function as the example.
Here are the ones I know:
JavaScript / ECMAScript can do it with callee:
function(n){if (n<2) {return 1;} else {return n * arguments.callee(n-1);}}
it's easy in languages with letrec, eg Haskell (which calls it let):
let fac x = if x<2 then 1 else fac (x-1) * x in fac
and there are equivalents in Lisp and Scheme. Note that the binding of fac is local to the expression, so the whole expression is in fact an anonymous function.
Are there any others?
Most languages support it through use of the Y combinator. Here's an example in Python (from the cookbook):
# Define Y combinator...come on Gudio, put it in functools!
Y = lambda g: (lambda f: g(lambda arg: f(f)(arg))) (lambda f: g(lambda arg: f(f)(arg)))
# Define anonymous recursive factorial function
fac = Y(lambda f: lambda n: (1 if n<2 else n*f(n-1)))
assert fac(7) == 5040
C#
Reading Wes Dyer's blog, you will see that #Jon Skeet's answer is not totally correct. I am no genius on languages but there is a difference between a recursive anonymous function and the "fib function really just invokes the delegate that the local variable fib references" to quote from the blog.
The actual C# answer would look something like this:
delegate Func<A, R> Recursive<A, R>(Recursive<A, R> r);
static Func<A, R> Y<A, R>(Func<Func<A, R>, Func<A, R>> f)
{
Recursive<A, R> rec = r => a => f(r(r))(a);
return rec(rec);
}
static void Main(string[] args)
{
Func<int,int> fib = Y<int,int>(f => n => n > 1 ? f(n - 1) + f(n - 2) : n);
Func<int, int> fact = Y<int, int>(f => n => n > 1 ? n * f(n - 1) : 1);
Console.WriteLine(fib(6)); // displays 8
Console.WriteLine(fact(6));
Console.ReadLine();
}
You can do it in Perl:
my $factorial = do {
my $fac;
$fac = sub {
my $n = shift;
if ($n < 2) { 1 } else { $n * $fac->($n-1) }
};
};
print $factorial->(4);
The do block isn't strictly necessary; I included it to emphasize that the result is a true anonymous function.
Well, apart from Common Lisp (labels) and Scheme (letrec) which you've already mentioned, JavaScript also allows you to name an anonymous function:
var foo = {"bar": function baz() {return baz() + 1;}};
which can be handier than using callee. (This is different from function in top-level; the latter would cause the name to appear in global scope too, whereas in the former case, the name appears only in the scope of the function itself.)
In Perl 6:
my $f = -> $n { if ($n <= 1) {1} else {$n * &?BLOCK($n - 1)} }
$f(42); # ==> 1405006117752879898543142606244511569936384000000000
F# has "let rec"
You've mixed up some terminology here, function literals don't have to be anonymous.
In javascript the difference depends on whether the function is written as a statement or an expression. There's some discussion about the distinction in the answers to this question.
Lets say you are passing your example to a function:
foo(function(n){if (n<2) {return 1;} else {return n * arguments.callee(n-1);}});
This could also be written:
foo(function fac(n){if (n<2) {return 1;} else {return n * fac(n-1);}});
In both cases it's a function literal. But note that in the second example the name is not added to the surrounding scope - which can be confusing. But this isn't widely used as some javascript implementations don't support this or have a buggy implementation. I've also read that it's slower.
Anonymous recursion is something different again, it's when a function recurses without having a reference to itself, the Y Combinator has already been mentioned. In most languages, it isn't necessary as better methods are available. Here's a link to a javascript implementation.
In C# you need to declare a variable to hold the delegate, and assign null to it to make sure it's definitely assigned, then you can call it from within a lambda expression which you assign to it:
Func<int, int> fac = null;
fac = n => n < 2 ? 1 : n * fac(n-1);
Console.WriteLine(fac(7));
I think I heard rumours that the C# team was considering changing the rules on definite assignment to make the separate declaration/initialization unnecessary, but I wouldn't swear to it.
One important question for each of these languages / runtime environments is whether they support tail calls. In C#, as far as I'm aware the MS compiler doesn't use the tail. IL opcode, but the JIT may optimise it anyway, in certain circumstances. Obviously this can very easily make the difference between a working program and stack overflow. (It would be nice to have more control over this and/or guarantees about when it will occur. Otherwise a program which works on one machine may fail on another in a hard-to-fathom manner.)
Edit: as FryHard pointed out, this is only pseudo-recursion. Simple enough to get the job done, but the Y-combinator is a purer approach. There's one other caveat with the code I posted above: if you change the value of fac, anything which tries to use the old value will start to fail, because the lambda expression has captured the fac variable itself. (Which it has to in order to work properly at all, of course...)
You can do this in Matlab using an anonymous function which uses the dbstack() introspection to get the function literal of itself and then evaluating it. (I admit this is cheating because dbstack should probably be considered extralinguistic, but it is available in all Matlabs.)
f = #(x) ~x || feval(str2func(getfield(dbstack, 'name')), x-1)
This is an anonymous function that counts down from x and then returns 1. It's not very useful because Matlab lacks the ?: operator and disallows if-blocks inside anonymous functions, so it's hard to construct the base case/recursive step form.
You can demonstrate that it is recursive by calling f(-1); it will count down to infinity and eventually throw a max recursion error.
>> f(-1)
??? Maximum recursion limit of 500 reached. Use set(0,'RecursionLimit',N)
to change the limit. Be aware that exceeding your available stack space can
crash MATLAB and/or your computer.
And you can invoke the anonymous function directly, without binding it to any variable, by passing it directly to feval.
>> feval(#(x) ~x || feval(str2func(getfield(dbstack, 'name')), x-1), -1)
??? Maximum recursion limit of 500 reached. Use set(0,'RecursionLimit',N)
to change the limit. Be aware that exceeding your available stack space can
crash MATLAB and/or your computer.
Error in ==> create#(x)~x||feval(str2func(getfield(dbstack,'name')),x-1)
To make something useful out of it, you can create a separate function which implements the recursive step logic, using "if" to protect the recursive case against evaluation.
function out = basecase_or_feval(cond, baseval, fcn, args, accumfcn)
%BASECASE_OR_FEVAL Return base case value, or evaluate next step
if cond
out = baseval;
else
out = feval(accumfcn, feval(fcn, args{:}));
end
Given that, here's factorial.
recursive_factorial = #(x) basecase_or_feval(x < 2,...
1,...
str2func(getfield(dbstack, 'name')),...
{x-1},...
#(z)x*z);
And you can call it without binding.
>> feval( #(x) basecase_or_feval(x < 2, 1, str2func(getfield(dbstack, 'name')), {x-1}, #(z)x*z), 5)
ans =
120
It also seems Mathematica lets you define recursive functions using #0 to denote the function itself, as:
(expression[#0]) &
e.g. a factorial:
fac = Piecewise[{{1, #1 == 0}, {#1 * #0[#1 - 1], True}}] &;
This is in keeping with the notation #i to refer to the ith parameter, and the shell-scripting convention that a script is its own 0th parameter.
I think this may not be exactly what you're looking for, but in Lisp 'labels' can be used to dynamically declare functions that can be called recursively.
(labels ((factorial (x) ;define name and params
; body of function addrec
(if (= x 1)
(return 1)
(+ (factorial (- x 1))))) ;should not close out labels
;call factorial inside labels function
(factorial 5)) ;this would return 15 from labels
Delphi includes the anonymous functions with version 2009.
Example from http://blogs.codegear.com/davidi/2008/07/23/38915/
type
// method reference
TProc = reference to procedure(x: Integer);
procedure Call(const proc: TProc);
begin
proc(42);
end;
Use:
var
proc: TProc;
begin
// anonymous method
proc := procedure(a: Integer)
begin
Writeln(a);
end;
Call(proc);
readln
end.
Because I was curious, I actually tried to come up with a way to do this in MATLAB. It can be done, but it looks a little Rube-Goldberg-esque:
>> fact = #(val,branchFcns) val*branchFcns{(val <= 1)+1}(val-1,branchFcns);
>> returnOne = #(val,branchFcns) 1;
>> branchFcns = {fact returnOne};
>> fact(4,branchFcns)
ans =
24
>> fact(5,branchFcns)
ans =
120
Anonymous functions exist in C++0x with lambda, and they may be recursive, although I'm not sure about anonymously.
auto kek = [](){kek();}
'Tseems you've got the idea of anonymous functions wrong, it's not just about runtime creation, it's also about scope. Consider this Scheme macro:
(define-syntax lambdarec
(syntax-rules ()
((lambdarec (tag . params) . body)
((lambda ()
(define (tag . params) . body)
tag)))))
Such that:
(lambdarec (f n) (if (<= n 0) 1 (* n (f (- n 1)))))
Evaluates to a true anonymous recursive factorial function that can for instance be used like:
(let ;no letrec used
((factorial (lambdarec (f n) (if (<= n 0) 1 (* n (f (- n 1)))))))
(factorial 4)) ; ===> 24
However, the true reason that makes it anonymous is that if I do:
((lambdarec (f n) (if (<= n 0) 1 (* n (f (- n 1))))) 4)
The function is afterwards cleared from memory and has no scope, thus after this:
(f 4)
Will either signal an error, or will be bound to whatever f was bound to before.
In Haskell, an ad hoc way to achieve same would be:
\n -> let fac x = if x<2 then 1 else fac (x-1) * x
in fac n
The difference again being that this function has no scope, if I don't use it, with Haskell being Lazy the effect is the same as an empty line of code, it is truly literal as it has the same effect as the C code:
3;
A literal number. And even if I use it immediately afterwards it will go away. This is what literal functions are about, not creation at runtime per se.
Clojure can do it, as fn takes an optional name specifically for this purpose (the name doesn't escape the definition scope):
> (def fac (fn self [n] (if (< n 2) 1 (* n (self (dec n))))))
#'sandbox17083/fac
> (fac 5)
120
> self
java.lang.RuntimeException: Unable to resolve symbol: self in this context
If it happens to be tail recursion, then recur is a much more efficient method:
> (def fac (fn [n] (loop [count n result 1]
(if (zero? count)
result
(recur (dec count) (* result count))))))