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I understand that there is a way to declare parametric datatypes in SMTLIB. Is there a way to define a function that takes in such type? For instance the standard doc has:
( declare - datatypes ( ( Pair 2) ) (
( par ( X Y ) ( ( pair ( first X ) ( second Y )) ))))
Now how can I declare a function that takes in a parametric Pair type?
SMTLib does not allow for parametric function definitions. Note that internal functions, like +, - etc., can be many-sorted/parametric, they work on Integers and Reals just fine, for instance. But user-defined functions aren't allowed to be many-sorted. That is, you can write a function that takes a (Pair Int Bool), but not one that takes (Pair a b) where a and b are type-variables; i.e., polymorphic.
This is explained in Section 4.1.5 of https://smtlib.cs.uiowa.edu/papers/smt-lib-reference-v2.6-r2021-05-12.pdf:
Well-sortedness checks, required for commands that use sorts or terms, are always done with respect to the current signature. It is an error to declare or define a symbol that is already in the current signature. This implies in particular that, contrary to theory function symbols, user-defined function symbols cannot be overloaded.(29)
And later on in Footnote 29, it says:
The motivation for not overloading user-defined symbols is to simplify their processing by a solver. This restriction is significant only for users who want to extend the signature of the theory used by a script with a new polymorphic function symbol—i.e., one whose rank would contain parametric sorts if it was a theory symbol. For instance, users who want to declare a “reverse” function on arbitrary lists, must define a different reverse function symbol for each (concrete) list sort used in the script. This restriction might be removed in future versions.
TLDR; No you cannot define parametric functions in SMTLib. But this might change in the future as the logic gets richer. The current workaround is the process of "monomorphisation," i.e, generate a new version of the function at each concrete-type that you use. This can be done by hand, or it can be automated by a higher-level tool that generates the SMTLib for you.
It's easy to create an array of functions and execute them in a loop.
It's easy to provide arguments in either a corresponding array of the same length or the array could be of tuples (fn, arg).
For 2, the loop is just
for fn_ar in arr # arr is [(myfunc, [1,2,3]), (func2, [10,11,12]), ...]
fn_ar[1](fn_ar[2])
end
Here is the problem: the arguments I am using are arrays of very large arrays. In #2, the argument that will be called with the function will be the current value of the array when the arg entry of the tuple is initially created. What I need is to provide the array names as the argument and defer evaluation of the arguments until the corresponding function is run in the loop body.
I could provide the arrays used as input as an expression and eval the expression in the loop to supply the needed arguments. But, eval can't eval in local scope.
What I did that worked (sort of) was to create a closure for each function that captured the arrays (which are really just a reference to storage). This works because the only argument to each function that varies in the loop body turns out to be the loop counter. The functions in question update the arrays in place. The array argument is really just a reference to the storage location, so each function executed in the loop body sees the latest values of the arrays. It worked. It wasn't hard to do. It is very, very slow. This is a known challenge in Julia.
I tried the recommended hints in the performance section of the manual. Make sure the captured variables are typed before they are captured so the JIT knows what they are. No effect on perf. The other hint is to put the definition of the curried function with the data for the closure in let block. Tried this. No effect on perf. It's possible I implemented the hints incorrectly--I can provide a code fragment if it helps.
But, I'd rather just ask the question about what I am trying to do and not muddy the waters with my past effort, which might not be going down the right path.
Here is a small fragment that is more realistic than the above:
Just a couple of functions and arguments:
(affine!, "(dat.z[hl], dat.a[hl-1], nnw.theta[hl], nnw.bias[hl])")
(relu!, "(dat.a[hl], dat.z[hl])")
Of course, the arguments could be wrapped as an expression with Meta.parse. dat.z and dat.a are matrices used in machine learning. hl indexes the layer of the model for the linear result and non-linear activation.
A simplified version of the loop where I want to run through the stack of functions across the model layers:
function feedfwd!(dat::Union{Batch_view,Model_data}, nnw, hp, ff_execstack)
for lr in 1:hp.n_layers
for f in ff_execstack[lr]
f(lr)
end
end
end
So, closures of the arrays is too slow. Eval I can't get to work.
Any suggestions...?
Thanks,
Lewis
I solved this with the beauty of function composition.
Here is the loop that runs through the feed forward functions for all layers:
for lr in 1:hp.n_layers
for f in ff_execstack[lr]
f(argfilt(dat, nnw, hp, bn, lr, f)...)
end
end
The inner function parameter to f called argfilt filters down from a generic list of all the inputs to return a tuple of arguments needed for the specific function. This also takes advantage of the beauty of method dispatch. Note that the function, f, is an input to argfilt. The types of functions are singletons: each function has a unique type as in typeof(relu!), for example. So, without any crazy if branching, method dispatch enables argfilt to return just the arguments needed. The performance cost compared to passing the arguments directly to a function is about 1.2 ns. This happens in a very hot loop that typically runs 24,000 times so that is 29 microseconds for the entire training pass.
The other great thing is that this runs in less than 1/10 of the time of the version using closures. I am getting slightly better performance than my original version that used some function variables and a bunch of if statements in the hot loop for feedfwd.
Here is what a couple of the methods for argfilt look like:
function argfilt(dat::Union{Model_data, Batch_view}, nnw::Wgts, hp::Hyper_parameters,
bn::Batch_norm_params, hl::Int, fn::typeof(affine!))
(dat.z[hl], dat.a[hl-1], nnw.theta[hl], nnw.bias[hl])
end
function argfilt(dat::Union{Model_data, Batch_view}, nnw::Wgts, hp::Hyper_parameters,
bn::Batch_norm_params, hl::Int, fn::typeof(relu!))
(dat.a[hl], dat.z[hl])
end
Background: I got here by reasoning that I could pass the same list of arguments to all of the functions: the union of all possible arguments--not that bad as there are only 9 args. Ignored arguments waste some space on the stack but it's teeny because for structs and arrays an argument is a pointer reference, not all of the data. The downside is that every one of these functions (around 20 or so) all need to have big argument lists. OK, but goofy: it doesn't make much sense when you look at the code of any of the functions. But, if I could filter down the arguments just to those needed, the function signatures don't need to change.
It's sort of a cool pattern. No introspection or eval needed; just functions.
I was curious that are there any ternary operator being used in programming language except ?: operator. And could found only 2 from wikipedia
Is it only operator we have been used? Are there any more than these?
Element update
Another useful class of ternary operator, especially in functional languages, is the "element update" operation. For example, OCaml expressions have three kinds of update syntax:
a.b<-c means the record a where field b has value c
a.(b)<-c means the array a where index b has value c
a.[b]<-c means the string a where index b has value c
Note that these are not "update" in the sense of "assignment" or "modification"; the original object is unchanged, and a new object is yielded that has the stated properties. Consequently, these operations cannot be regarded as a simple composition of two binary operators.
Similarly, the Isabelle theorem prover has:
a(|b := c|) meaning the record a where field b has value c
Array slice
Yet another sort of ternary operator is array slice, for example in Python we have:
a[b:c] meaning an array whose first element is a[b] and last element is a[c-1]
In fact, Python has a quaternary form of slice:
a[b:c:d] meaning an array whose elements are a[b + n*d] where n ranges from 0 to the largest value such that b + n*d < c
Bash/ksh variable substitution
Although quite obscure, bash has several forms of variable expansion (apparently borrowed from ksh) that are ternary:
${var:pos:len} is a maximum of len characters from $var, starting at pos
${var/Pattern/Replacement} is $var except the first substring within it that matches Pattern is replaced with Replacement
${var//Pattern/Replacement} is the same except all matches are replaced
${var/#Pattern/Replacement} is like the first case except Pattern has to match a prefix of $var
${var/%Pattern/Replacement} is like the previous except for matching a suffix
These are borderline in my opinion, being close to ordinary functions that happen to accept three arguments, written in the sometimes baroque style of shell syntax. But, I include them as they are entirely made of non-letter symbols.
Congruence modulo
In mathematics, an important ternary relation is congruence modulo:
a ≡ b (mod c) is true iff a and b both belong to the same equivalence class in c
I'm not aware of any programming language that has this, but programming languages often borrow mathematical notation, so it's possible it exists in an obscure language. (Of course, most programming languages have mod as a binary operator, allowing the above to be expressed as (a mod c) == (b mod c).) Furthermore, unlike the bash variable substitution syntax, if this were introduced in some language, it would not be specific to that language since it is established notation elsewhere, making it more similar to ?: in ubiquity.
Excluded
There are some categories of operator I've chosen to exclude from the category of "ternary" operator:
Operations like function application (a(b,c)) that could apply to any number of operators.
Specific named functions (e.g., f(a,b,c)) that accept three arguments, as there are too many of them for any to be interesting in this context.
Operations like SUM (Σ) or let that function as a binding introduction of a new variable, since IMO a ternary operator ought to act on three already-existing things.
One-letter operators in languages like sed that happen to accept three arguments, as these really are like the named function case, and the language just has a very terse naming convention.
Well it’s not a ternary operator per-say but I do think the three way comparison operator is highly underrated.
The ternary operator is appropriate when a computation has to take place, even if I cannot use the effect inside of an if/else statement or switch statement Consequently, 0 or the DEFAULT VALUE is treated as a DEFAULT VALUE when I try the computation.
The if/else or switch statements require me to enumerate every case that can take place and are only helpful if the ability to discriminate between a numeric value and a branching choice can help. In some cases, it is clear why it won't help if a condition test exists, simply because I am either too early or too late to do something about the condition, even though some other function is unable to test for the given condition.
The ternary operator forces a computation to have the effect that can pass through the code that follows it. Other types of condition tests resort to using, for instance, the && and || operators which can't guarantee a pass through.
I'm learning Elixir and wonder why it has two types of function definitions:
functions defined in a module with def, called using myfunction(param1, param2)
anonymous functions defined with fn, called using myfn.(param1, param2)
Only the second kind of function seems to be a first-class object and can be passed as a parameter to other functions. A function defined in a module needs to be wrapped in a fn. There's some syntactic sugar which looks like otherfunction(&myfunction(&1, &2)) in order to make that easy, but why is it necessary in the first place? Why can't we just do otherfunction(myfunction))? Is it only to allow calling module functions without parenthesis like in Ruby? It seems to have inherited this characteristic from Erlang which also has module functions and funs, so does it actually comes from how the Erlang VM works internally?
It there any benefit having two types of functions and converting from one type to another in order to pass them to other functions? Is there a benefit having two different notations to call functions?
Just to clarify the naming, they are both functions. One is a named function and the other is an anonymous one. But you are right, they work somewhat differently and I am going to illustrate why they work like that.
Let's start with the second, fn. fn is a closure, similar to a lambda in Ruby. We can create it as follows:
x = 1
fun = fn y -> x + y end
fun.(2) #=> 3
A function can have multiple clauses too:
x = 1
fun = fn
y when y < 0 -> x - y
y -> x + y
end
fun.(2) #=> 3
fun.(-2) #=> 3
Now, let's try something different. Let's try to define different clauses expecting a different number of arguments:
fn
x, y -> x + y
x -> x
end
** (SyntaxError) cannot mix clauses with different arities in function definition
Oh no! We get an error! We cannot mix clauses that expect a different number of arguments. A function always has a fixed arity.
Now, let's talk about the named functions:
def hello(x, y) do
x + y
end
As expected, they have a name and they can also receive some arguments. However, they are not closures:
x = 1
def hello(y) do
x + y
end
This code will fail to compile because every time you see a def, you get an empty variable scope. That is an important difference between them. I particularly like the fact that each named function starts with a clean slate and you don't get the variables of different scopes all mixed up together. You have a clear boundary.
We could retrieve the named hello function above as an anonymous function. You mentioned it yourself:
other_function(&hello(&1))
And then you asked, why I cannot simply pass it as hello as in other languages? That's because functions in Elixir are identified by name and arity. So a function that expects two arguments is a different function than one that expects three, even if they had the same name. So if we simply passed hello, we would have no idea which hello you actually meant. The one with two, three or four arguments? This is exactly the same reason why we can't create an anonymous function with clauses with different arities.
Since Elixir v0.10.1, we have a syntax to capture named functions:
&hello/1
That will capture the local named function hello with arity 1. Throughout the language and its documentation, it is very common to identify functions in this hello/1 syntax.
This is also why Elixir uses a dot for calling anonymous functions. Since you can't simply pass hello around as a function, instead you need to explicitly capture it, there is a natural distinction between named and anonymous functions and a distinct syntax for calling each makes everything a bit more explicit (Lispers would be familiar with this due to the Lisp 1 vs. Lisp 2 discussion).
Overall, those are the reasons why we have two functions and why they behave differently.
I don't know how useful this will be to anyone else, but the way I finally wrapped my head around the concept was to realize that elixir functions aren't Functions.
Everything in elixir is an expression. So
MyModule.my_function(foo)
is not a function but the expression returned by executing the code in my_function. There is actually only one way to get a "Function" that you can pass around as an argument and that is to use the anonymous function notation.
It is tempting to refer to the fn or & notation as a function pointer, but it is actually much more. It's a closure of the surrounding environment.
If you ask yourself:
Do I need an execution environment or a data value in this spot?
And if you need execution use fn, then most of the difficulties become much
clearer.
I may be wrong since nobody mentioned it, but I was also under the impression that the reason for this is also the ruby heritage of being able to call functions without brackets.
Arity is obviously involved but lets put it aside for a while and use functions without arguments. In a language like javascript where brackets are mandatory, it is easy to make the difference between passing a function as an argument and calling the function. You call it only when you use the brackets.
my_function // argument
(function() {}) // argument
my_function() // function is called
(function() {})() // function is called
As you can see, naming it or not does not make a big difference. But elixir and ruby allow you to call functions without the brackets. This is a design choice which I personally like but it has this side effect you cannot use just the name without the brackets because it could mean you want to call the function. This is what the & is for. If you leave arity appart for a second, prepending your function name with & means that you explicitly want to use this function as an argument, not what this function returns.
Now the anonymous function is bit different in that it is mainly used as an argument. Again this is a design choice but the rational behind it is that it is mainly used by iterators kind of functions which take functions as arguments. So obviously you don't need to use & because they are already considered arguments by default. It is their purpose.
Now the last problem is that sometimes you have to call them in your code, because they are not always used with an iterator kind of function, or you might be coding an iterator yourself. For the little story, since ruby is object oriented, the main way to do it was to use the call method on the object. That way, you could keep the non-mandatory brackets behaviour consistent.
my_lambda.call
my_lambda.call()
my_lambda_with_arguments.call :h2g2, 42
my_lambda_with_arguments.call(:h2g2, 42)
Now somebody came up with a shortcut which basically looks like a method with no name.
my_lambda.()
my_lambda_with_arguments.(:h2g2, 42)
Again, this is a design choice. Now elixir is not object oriented and therefore call not use the first form for sure. I can't speak for José but it looks like the second form was used in elixir because it still looks like a function call with an extra character. It's close enough to a function call.
I did not think about all the pros and cons, but it looks like in both languages you could get away with just the brackets as long as you make brackets mandatory for anonymous functions. It seems like it is:
Mandatory brackets VS Slightly different notation
In both cases you make an exception because you make both behave differently. Since there is a difference, you might as well make it obvious and go for the different notation. The mandatory brackets would look natural in most cases but very confusing when things don't go as planned.
Here you go. Now this might not be the best explanation in the world because I simplified most of the details. Also most of it are design choices and I tried to give a reason for them without judging them. I love elixir, I love ruby, I like the function calls without brackets, but like you, I find the consequences quite misguiding once in a while.
And in elixir, it is just this extra dot, whereas in ruby you have blocks on top of this. Blocks are amazing and I am surprised how much you can do with just blocks, but they only work when you need just one anonymous function which is the last argument. Then since you should be able to deal with other scenarios, here comes the whole method/lambda/proc/block confusion.
Anyway... this is out of scope.
I've never understood why explanations of this are so complicated.
It's really just an exceptionally small distinction combined with the realities of Ruby-style "function execution without parens".
Compare:
def fun1(x, y) do
x + y
end
To:
fun2 = fn
x, y -> x + y
end
While both of these are just identifiers...
fun1 is an identifier that describes a named function defined with def.
fun2 is an identifier that describes a variable (that happens to contain a reference to function).
Consider what that means when you see fun1 or fun2 in some other expression? When evaluating that expression, do you call the referenced function or do you just reference a value out of memory?
There's no good way to know at compile time. Ruby has the luxury of introspecting the variable namespace to find out if a variable binding has shadowed a function at some point in time. Elixir, being compiled, can't really do this. That's what the dot-notation does, it tells Elixir that it should contain a function reference and that it should be called.
And this is really hard. Imagine that there wasn't a dot notation. Consider this code:
val = 5
if :rand.uniform < 0.5 do
val = fn -> 5 end
end
IO.puts val # Does this work?
IO.puts val.() # Or maybe this?
Given the above code, I think it's pretty clear why you have to give Elixir the hint. Imagine if every variable de-reference had to check for a function? Alternatively, imagine what heroics would be necessary to always infer that variable dereference was using a function?
There's an excellent blog post about this behavior: link
Two types of functions
If a module contains this:
fac(0) when N > 0 -> 1;
fac(N) -> N* fac(N-1).
You can’t just cut and paste this into the shell and get the same
result.
It’s because there is a bug in Erlang. Modules in Erlang are sequences
of FORMS. The Erlang shell evaluates a sequence of
EXPRESSIONS. In Erlang FORMS are not EXPRESSIONS.
double(X) -> 2*X. in an Erlang module is a FORM
Double = fun(X) -> 2*X end. in the shell is an EXPRESSION
The two are not the same. This bit of silliness has been Erlang
forever but we didn’t notice it and we learned to live with it.
Dot in calling fn
iex> f = fn(x) -> 2 * x end
#Function<erl_eval.6.17052888>
iex> f.(10)
20
In school I learned to call functions by writing f(10) not f.(10) -
this is “really” a function with a name like Shell.f(10) (it’s a
function defined in the shell) The shell part is implicit so it should
just be called f(10).
If you leave it like this expect to spend the next twenty years of
your life explaining why.
Elixir has optional braces for functions, including functions with 0 arity. Let's see an example of why it makes a separate calling syntax important:
defmodule Insanity do
def dive(), do: fn() -> 1 end
end
Insanity.dive
# #Function<0.16121902/0 in Insanity.dive/0>
Insanity.dive()
# #Function<0.16121902/0 in Insanity.dive/0>
Insanity.dive.()
# 1
Insanity.dive().()
# 1
Without making a difference between 2 types of functions, we can't say what Insanity.dive means: getting a function itself, calling it, or also calling the resulting anonymous function.
fn -> syntax is for using anonymous functions. Doing var.() is just telling elixir that I want you to take that var with a func in it and run it instead of referring to the var as something just holding that function.
Elixir has a this common pattern where instead of having logic inside of a function to see how something should execute, we pattern match different functions based on what kind of input we have. I assume this is why we refer to things by arity in the function_name/1 sense.
It's kind of weird to get used to doing shorthand function definitions (func(&1), etc), but handy when you're trying to pipe or keep your code concise.
In elixir we use def for simply define a function like we do in other languages.
fn creates an anonymous function refer to this for more clarification
Only the second kind of function seems to be a first-class object and can be passed as a parameter to other functions. A function defined in a module needs to be wrapped in a fn. There's some syntactic sugar which looks like otherfunction(myfunction(&1, &2)) in order to make that easy, but why is it necessary in the first place? Why can't we just do otherfunction(myfunction))?
You can do otherfunction(&myfunction/2)
Since elixir can execute functions without the brackets (like myfunction), using otherfunction(myfunction)) it will try to execute myfunction/0.
So, you need to use the capture operator and specify the function, including arity, since you can have different functions with the same name. Thus, &myfunction/2.
For example, can I declare a metamethod for __index which takes two arguments, then do something like myuserdata[somearg1, somearg2]? Not that I want to use it or that I have any idea of why it would be useful, I'm just wondering if in my library that deals with binding c++ to Lua I can and should enforce a right (except for __call metamethod of course) number of arguments of the function bound as metamethod.
Lua doesn't enforce parameter counts on functions in general: http://ideone.com/kAynR
That said, the metamethods which map to syntax are bound by the syntax - for instance, Lua syntax doesn't allow ,-separated values inside [] indexing:
luac: ']' expected near ','
Lua functions can receive an arbitrary number of functions. This is true of your metamethods as well.
However, metatables don't let you change the arity of operators. A metamethod for unary operators (not, #, etc.) is only going to be passed on one argument. A metamethod for binary operators (+, %, ==, ^, .., etc.) is only going to be passed two arguments.
Table indexing operations are defined in the Lua grammar as taking one argument, so that's all you can use and all your metamethod for indexing will receive.
The only metamethod that may receive an arbitrary number of aruments is call.
No, the index operator can only take one argument, as in C++. While the function for the index operator can have as many arguments as you want, if you actually try to use more than one inside the [] you'll get an error.
It depends on the operator. So you neither can nor need to "enforce" that.