Related
I am new to Elixir language and I am having some issues while writing a piece of code.
What I am given is a 2D array like
list1 = [
[1 ,2,3,4,"nil"],
[6,7,8,9,10,],
[11,"nil",13,"nil",15],
[16,17,"nil",19,20] ]
Now, what I've to do is to get all the elements that have values between 10 and 20, so what I'm doing is:
final_list = []
Enum.each(list1, fn row ->
Enum.each(row, &(if (&1 >= 10 and &1 <= 99) do final_list = final_list ++ &1 end))
end
)
Doing this, I'm expecting that I'll get my list of numbers in final_list but I'm getting blank final list with a warning like:
warning: variable "final_list" is unused (there is a variable with the same name in the context, use the pin operator (^) to match on it or prefix this variable with underscore if it is not meant to be used)
iex:5
:ok
and upon printing final_list, it is not updated.
When I try to check whether my code is working properly or not, using IO.puts as:
iex(5)> Enum.each(list1, fn row -> ...(5)> Enum.each(row, &(if (&1 >= 10 and &1 <= 99) do IO.puts(final_list ++ &1) end))
...(5)> end
...(5)> )
The Output is:
10
11
13
15
16
17
19
20
:ok
What could I possibly be doing wrong here? Shouldn't it add the elements to the final_list?
If this is wrong ( probably it is), what should be the possible solution to this?
Any kind of help will be appreciated.
As mentioned in Adam's comments, this is a FAQ and the important thing is the message "warning: variable "final_list" is unused (there is a variable with the same name in the context, use the pin operator (^) to match on it or prefix this variable with underscore if it is not meant to be used)" This message actually indicates a very serious problem.
It tells you that the assignment "final_list = final_list ++ &1" is useless since it just creates a local variable, hiding the external one. Elixir variables are not mutable so you need to reorganize seriously your code.
The simplest way is
final_list =
for sublist <- list1,
n <- sublist,
is_number(n),
n in 10..20,
do: n
Note that every time you write final_list = ..., you actually declare a new variable with the same name, so the final_list you declared inside your anonymous function is not the final_list outside the anonymous function.
I'm writing a genetic program in order to test the fitness of randomly generated expressions. Shown here is the function to generate the expression as well a the main function. DIV and GT are defined elsewhere in the code:
function create_single_full_tree(depth, fs, ts)
"""
Creates a single AST with full depth
Inputs
depth Current depth of tree. Initially called from main() with max depth
fs Function Set - Array of allowed functions
ts Terminal Set - Array of allowed terminal values
Output
Full AST of typeof()==Expr
"""
# If we are at the bottom
if depth == 1
# End of tree, return function with two terminal nodes
return Expr(:call, fs[rand(1:length(fs))], ts[rand(1:length(ts))], ts[rand(1:length(ts))])
else
# Not end of expression, recurively go back through and create functions for each new node
return Expr(:call, fs[rand(1:length(fs))], create_single_full_tree(depth-1, fs, ts), create_single_full_tree(depth-1, fs, ts))
end
end
function main()
"""
Main function
"""
# Define functional and terminal sets
fs = [:+, :-, :DIV, :GT]
ts = [:x, :v, -1]
# Create the tree
ast = create_single_full_tree(4, fs, ts)
#println(typeof(ast))
#println(ast)
#println(dump(ast))
x = 1
v = 1
eval(ast) # Error out unless x and v are globals
end
main()
I am generating a random expression based on certain allowed functions and variables. As seen in the code, the expression can only have symbols x and v, as well as the value -1. I will need to test the expression with a variety of x and v values; here I am just using x=1 and v=1 to test the code.
The expression is being returned correctly, however, eval() can only be used with global variables, so it will error out when run unless I declare x and v to be global (ERROR: LoadError: UndefVarError: x not defined). I would like to avoid globals if possible. Is there a better way to generate and evaluate these generated expressions with locally defined variables?
Here is an example for generating an (anonymous) function. The result of eval can be called as a function and your variable can be passed as parameters:
myfun = eval(Expr(:->,:x, Expr(:block, Expr(:call,:*,3,:x) )))
myfun(14)
# returns 42
The dump function is very useful to inspect the expression that the parsers has created. For two input arguments you would use a tuple for example as args[1]:
julia> dump(parse("(x,y) -> 3x + y"))
Expr
head: Symbol ->
args: Array{Any}((2,))
1: Expr
head: Symbol tuple
args: Array{Any}((2,))
1: Symbol x
2: Symbol y
typ: Any
2: Expr
[...]
Does this help?
In the Metaprogramming part of the Julia documentation, there is a sentence under the eval() and effects section which says
Every module has its own eval() function that evaluates expressions in its global scope.
Similarly, the REPL help ?eval will give you, on Julia 0.6.2, the following help:
Evaluate an expression in the given module and return the result. Every Module (except those defined with baremodule) has its own 1-argument definition of eval, which evaluates expressions in that module.
I assume, you are working in the Main module in your example. That's why you need to have the globals defined there. For your problem, you can use macros and interpolate the values of x and y directly inside the macro.
A minimal working example would be:
macro eval_line(a, b, x)
isa(a, Real) || (warn("$a is not a real number."); return :(throw(DomainError())))
isa(b, Real) || (warn("$b is not a real number."); return :(throw(DomainError())))
return :($a * $x + $b) # interpolate the variables
end
Here, #eval_line macro does the following:
Main> #macroexpand #eval_line(5, 6, 2)
:(5 * 2 + 6)
As you can see, the values of macro's arguments are interpolated inside the macro and the expression is given to the user accordingly. When the user does not behave,
Main> #macroexpand #eval_line([1,2,3], 7, 8)
WARNING: [1, 2, 3] is not a real number.
:((Main.throw)((Main.DomainError)()))
a user-friendly warning message is provided to the user at parse-time, and a DomainError is thrown at run-time.
Of course, you can do these things within your functions, again by interpolating the variables --- you do not need to use macros. However, what you would like to achieve in the end is to combine eval with the output of a function that returns Expr. This is what the macro functionality is for. Finally, you would simply call your macros with an # sign preceding the macro name:
Main> #eval_line(5, 6, 2)
16
Main> #eval_line([1,2,3], 7, 8)
WARNING: [1, 2, 3] is not a real number.
ERROR: DomainError:
Stacktrace:
[1] eval(::Module, ::Any) at ./boot.jl:235
EDIT 1. You can take this one step further, and create functions accordingly:
macro define_lines(linedefs)
for (name, a, b) in eval(linedefs)
ex = quote
function $(Symbol(name))(x) # interpolate name
return $a * x + $b # interpolate a and b here
end
end
eval(ex) # evaluate the function definition expression in the module
end
end
Then, you can call this macro to create different line definitions in the form of functions to be called later on:
#define_lines([
("identity_line", 1, 0);
("null_line", 0, 0);
("unit_shift", 0, 1)
])
identity_line(5) # returns 5
null_line(5) # returns 0
unit_shift(5) # returns 1
EDIT 2. You can, I guess, achieve what you would like to achieve by using a macro similar to that below:
macro random_oper(depth, fs, ts)
operations = eval(fs)
oper = operations[rand(1:length(operations))]
terminals = eval(ts)
ts = terminals[rand(1:length(terminals), 2)]
ex = :($oper($ts...))
for d in 2:depth
oper = operations[rand(1:length(operations))]
t = terminals[rand(1:length(terminals))]
ex = :($oper($ex, $t))
end
return ex
end
which will give the following, for instance:
Main> #macroexpand #random_oper(1, [+, -, /], [1,2,3])
:((-)([3, 3]...))
Main> #macroexpand #random_oper(2, [+, -, /], [1,2,3])
:((+)((-)([2, 3]...), 3))
Thanks Arda for the thorough response! This helped, but part of me thinks there may be a better way to do this as it seems too roundabout. Since I am writing a genetic program, I will need to create 500 of these ASTs, all with random functions and terminals from a set of allowed functions and terminals (fs and ts in the code). I will also need to test each function with 20 different values of x and v.
In order to accomplish this with the information you have given, I have come up with the following macro:
macro create_function(defs)
for name in eval(defs)
ex = quote
function $(Symbol(name))(x,v)
fs = [:+, :-, :DIV, :GT]
ts = [x,v,-1]
return create_single_full_tree(4, fs, ts)
end
end
eval(ex)
end
end
I can then supply a list of 500 random function names in my main() function, such as ["func1, func2, func3,.....". Which I can eval with any x and v values in my main function. This has solved my issue, however, this seems to be a very roundabout way of doing this, and may make it difficult to evolve each AST with each iteration.
Is there a nifty way to iterate over combinations of keys in a dictionary?
my dictionary has values like:
[1] => [1,2], [2,3] => [15], [3] => [6,7,8], [4,9,11] => [3], ...
what I need to do is fetch all combinations of keys that are of length 1:n where n might be fx 3
So as in the example above, I would want to iterate over
[[1], [3], [2,3], [[1],[1,2]], [[3],[2,3]], [4,9,11]]
I know I could just collect the keys, but my dictionary is rather large and I am in the middle of redesigning the entire algorithm because it starts swapping insanely when n > 3, reducing efficiency terribly
tl;dr is there a way to create a combinatoric iterator from a dictionary without collect-ing the dictionary?
The following is a straight forward implementation, which tries to minimize a bit on going through the dictionary. Additionally it uses OrderedDict so holding key indices makes sense (since Dicts don't promise consistent key iteration each time and thus meaningful key indexing).
using Iterators
using DataStructures
od = OrderedDict([1] => [1,2], [2,3] => [15], [3] => [6,7,8], [4,9,11] => [3])
sv = map(length,keys(od)) # store length of keys for quicker calculations
maxmaxlen = sum(sv) # maximum total elements in good key
for maxlen=1:maxmaxlen # replace maxmaxlen with lower value if too slow
#show maxlen
gsets = Vector{Vector{Int}}() # hold good sets of key _indices_
for curlen=1:maxlen
foreach(x->push!(gsets,x),
(x for x in subsets(collect(1:n),curlen) if sum(sv[x])==maxlen))
end
# indmatrix is necessary to run through keys once in next loop
indmatrix = zeros(Bool,length(od),length(gsets))
for i=1:length(gsets) for e in gsets[i]
indmatrix[e,i] = true
end
end
# gkeys is the vector of vecotrs of keys i.e. what we wanted to calculate
gkeys = [Vector{Vector{Int}}() for i=1:length(gsets)]
for (i,k) in enumerate(keys(od))
for j=1:length(gsets)
if indmatrix[i,j]
push!(gkeys[j],k)
end
end
end
# do something with each set of good keys
foreach(x->println(x),gkeys)
end
Is this more efficient that what you currently have? It would also be better to put the code in a function or turn it into a Julia task which produces the next keys set each iteration.
--- UPDATE ---
Using the answer about iterators from tasks in https://stackoverflow.com/a/41074729/3580870
An improved iterator-ified version is:
function keysubsets(n,d)
Task() do
od = OrderedDict(d)
sv = map(length,keys(od)) # store length of keys for quicker calculations
maxmaxlen = sum(sv) # maximum total elements in good key
for maxlen=1:min(n,maxmaxlen) # replace maxmaxlen with lower value if too slow
gsets = Vector{Vector{Int}}() # hold good sets of key _indices_
for curlen=1:maxlen
foreach(x->push!(gsets,x),(x for x in subsets(collect(1:n),curlen) if sum(sv[x])==maxlen))
end
# indmatrix is necessary to run through keys once in next loop
indmatrix = zeros(Bool,length(od),length(gsets))
for i=1:length(gsets) for e in gsets[i]
indmatrix[e,i] = true
end
end
# gkeys is the vector of vecotrs of keys i.e. what we wanted to calculate
gkeys = [Vector{Vector{Int}}() for i=1:length(gsets)]
for (i,k) in enumerate(keys(od))
for j=1:length(gsets)
if indmatrix[i,j]
push!(gkeys[j],k)
end
end
end
# do something with each set of good keys
foreach(x->produce(x),gkeys)
end
end
end
Which now enables iterating over all keysubsets up to combined size 4 in this way (after running the code from the other StackOverflow answer):
julia> nt2 = NewTask(keysubsets(4,od))
julia> collect(nt2)
10-element Array{Array{Array{Int64,1},1},1}:
Array{Int64,1}[[1]]
Array{Int64,1}[[3]]
Array{Int64,1}[[2,3]]
Array{Int64,1}[[1],[3]]
Array{Int64,1}[[4,9,11]]
Array{Int64,1}[[1],[2,3]]
Array{Int64,1}[[2,3],[3]]
Array{Int64,1}[[1],[4,9,11]]
Array{Int64,1}[[3],[4,9,11]]
Array{Int64,1}[[1],[2,3],[3]]
(the definition of NewTask from the linked StackOverflow answer is necessary).
I found this project on GitHub; it was the only search term returned for "nimrod matrix". I took the bare bones of it and changed it a little bit so that it compiled without errors, and then I added the last two lines to build a simple matrix, and then output a value, but the "getter" function isn't working for some reason. I adapted the instructions for adding properties found here, but something isn't right.
Here is my code so far. I'd like to use the GNU Scientific Library from within Nimrod, and I figured that this was the first logical step.
type
TMatrix*[T] = object
transposed: bool
dataRows: int
dataCols: int
data: seq[T]
proc index[T](x: TMatrix[T], r,c: int): int {.inline.} =
if r<0 or r>(x.rows()-1):
raise newException(EInvalidIndex, "matrix index out of range")
if c<0 or c>(x.cols()-1):
raise newException(EInvalidIndex, "matrix index out of range")
result = if x.transposed: c*x.dataCols+r else: r*x.dataCols+c
proc rows*[T](x: TMatrix[T]): int {.inline.} =
## Returns the number of rows in the matrix `x`.
result = if x.transposed: x.dataCols else: x.dataRows
proc cols*[T](x: TMatrix[T]): int {.inline.} =
## Returns the number of columns in the matrix `x`.
result = if x.transposed: x.dataRows else: x.dataCols
proc matrix*[T](rows, cols: int, d: openarray[T]): TMatrix[T] =
## Constructor. Initializes the matrix by allocating memory
## for the data and setting the number of rows and columns
## and sets the data to the values specified in `d`.
result.dataRows = rows
result.dataCols = cols
newSeq(result.data, rows*cols)
if len(d)>0:
if len(d)<(rows*cols):
raise newException(EInvalidIndex, "insufficient data supplied in matrix constructor")
for i in countup(0,rows*cols-1):
result.data[i] = d[i]
proc `[][]`*[T](x: TMatrix[T], r,c: int): T =
## Element access. Returns the element at row `r` column `c`.
result = x.data[x.index(r,c)]
proc `[][]=`*[T](x: var TMatrix[T], r,c: int, a: T) =
## Sets the value of the element at row `r` column `c` to
## the value supplied in `a`.
x.data[x.index(r,c)] = a
var m = matrix( 2, 2, [1,2,3,4] )
echo( $m[0][0] )
This is the error I get:
c:\program files (x86)\nimrod\config\nimrod.cfg(36, 11) Hint: added path: 'C:\Users\H127\.babel\libs\' [Path]
Hint: used config file 'C:\Program Files (x86)\Nimrod\config\nimrod.cfg' [Conf]
Hint: system [Processing]
Hint: mat [Processing]
mat.nim(48, 9) Error: type mismatch: got (TMatrix[int], int literal(0))
but expected one of:
system.[](a: array[Idx, T], x: TSlice[Idx]): seq[T]
system.[](a: array[Idx, T], x: TSlice[int]): seq[T]
system.[](s: string, x: TSlice[int]): string
system.[](s: seq[T], x: TSlice[int]): seq[T]
Thanks you guys!
I'd like to first point out that the matrix library you refer to is three years old. For a programming language in development that's a lot of time due to changes, and it doesn't compile any more with the current Nimrod git version:
$ nimrod c matrix
...
private/tmp/n/matrix/matrix.nim(97, 8) Error: ']' expected
It fails on the double array accessor, which seems to have changed syntax. I guess your attempt to create a double [][] accessor is problematic, it could be ambiguous: are you accessing the double array accessor of the object or are you accessing the nested array returned by the first brackets? I had to change the proc to the following:
proc `[]`*[T](x: TMatrix[T], r,c: int): T =
After that change you also need to change the way to access the matrix. Here's what I got:
for x in 0 .. <2:
for y in 0 .. <2:
echo "x: ", x, " y: ", y, " = ", m[x,y]
Basically, instead of specifying two bracket accesses you pass all the parameters inside a single bracket. That code generates:
x: 0 y: 0 = 1
x: 0 y: 1 = 2
x: 1 y: 0 = 3
x: 1 y: 1 = 4
With regards to finding software for Nimrod, I would like to recommend you using Nimble, Nimrod's package manager. Once you have it installed you can search available and maintained packages. The command nimble search math shows two potential packages: linagl and extmath. Not sure if they are what you are looking for, but at least they seem more fresh.
It is by now a well known theorem of the lambda calculus that any function taking two or more arguments can be written through currying as a chain of functions taking one argument:
# Pseudo-code for currying
f(x,y) -> f_curried(x)(y)
This has proven to be extremely powerful not just in studying the behavior of functions but in practical use (Haskell, etc.).
Functions returning values, however, seem to not be discussed. Programmers typically deal with their inability to return more than one value from a function by returning some meta-object (lists in R, structures in C++, etc.). It has always struck me as a bit of a kludge, but a useful one.
For instance:
# R code for "faking" multiple return values
uselessFunc <- function(dat) {
model1 <- lm( y ~ x , data=dat )
return( list( coef=coef(model1), form=formula(model1) ) )
}
Questions
Does the lambda calculus have anything to say about a multiplicity of return values? If so, do any surprising conclusions result?
Similarly, do any languages allow true multiple return values?
According to the Wikipedia page on lambda calculus:
Lambda calculus, also written as λ-calculus, is a formal system for function
definition, function application and recursion
And a function, in the mathematical sense:
Associates one quantity, the argument of the function, also known as the input,
with another quantity, the value of the function, also known as the output
So answering your first question no, lambda calculus (or any other formalism based on mathematical functions) can not have multiple return values.
For your second question, as far as I know, programming languages that implement multiple return values do so by packing multiple results in some kind of data structure (be it a tuple, an array, or even the stack) and then unpacking it later - and that's where the differences lie, as some programming languages make the packing/unpacking part transparent for the programmer (for instance Python uses tuples under the hood) while other languages make the programmer do the job explicitly, for example Java programmers can simulate multiple return values to some extent by packing multiple results in a returned Object array and then extracting and casting the returned result by hand.
A function returns a single value. This is how functions are defined in mathematics. You can return multiple values by packing them into one compound value. But then it is still a single value. I'd call it a vector, because it has components. There are vector functions in mathematics there, so there are also in programming languages. The only difference is the support level from the language itself and does it facilitate it or not.
Nothing prevents you from having multiple functions, each one returning one of the multiple results that you would like to return.
For example, say, you had the following function in python returning a list.
def f(x):
L = []
for i in range(x):
L.append(x * i)
return L
It returns [0, 3, 6] for x=3 and [0, 5, 10, 15, 20] for x=5. Instead, you can totally have
def f_nth_value(x, n):
L = []
for i in range(x):
L.append(x * i)
if n < len(L):
return L[n]
return None
Then you can request any of the outputs for a given input, and get it, or get None, if there aren't enough outputs:
In [11]: f_nth_value(3, 0)
Out[11]: 0
In [12]: f_nth_value(3, 1)
Out[12]: 3
In [13]: f_nth_value(3, 2)
Out[13]: 6
In [14]: f_nth_value(3, 3)
In [15]: f_nth_value(5, 2)
Out[15]: 10
In [16]: f_nth_value(5, 5)
Computational resources may be wasted if you have to do some of the same work, as in this case. Theoretically, it can be avoided by returning another function that holds all the results inside itself.
def f_return_function(x):
L = []
for i in range(x):
L.append(x * i)
holder = lambda n: L[n] if n < len(L) else None
return holder
So now we have
In [26]: result = f_return_function(5)
In [27]: result(3)
Out[27]: 15
In [28]: result(4)
Out[28]: 20
In [29]: result(5)
Traditional untyped lambda calculus is perfectly capable of expressing this idea. (After all, it is Turing complete.) Whenever you want to return a bunch of values, just return a function that can give the n-th value for any n.
In regard to the second question, python allows for such a syntax, if you know exactly, just how many values the function is going to return.
def f(x):
L = []
for i in range(x):
L.append(x * i)
return L
In [39]: a, b, c = f(3)
In [40]: a
Out[40]: 0
In [41]: b
Out[41]: 3
In [42]: c
Out[42]: 6
In [43]: a, b, c = f(2)
---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
<ipython-input-43-5480fa44be36> in <module>()
----> 1 a, b, c = f(2)
ValueError: need more than 2 values to unpack
In [44]: a, b, c = f(4)
---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
<ipython-input-44-d2c7a6593838> in <module>()
----> 1 a, b, c = f(4)
ValueError: too many values to unpack
Lastly, here is an example from this Lisp tutorial:
;; in this function, the return result of (+ x x) is not assigned so it is essentially
;; lost; the function body moves on to the next form, (* x x), which is the last form
;; of this function body. So the function call only returns (* 10 10) => 100
* ((lambda (x) (+ x x) (* x x)) 10)
=> 100
;; in this function, we capture the return values of both (+ x x) and (* x x), as the
;; lexical variables SUM and PRODUCT; using VALUES, we can return multiple values from
;; a form instead of just one
* ((lambda (x) (let ((sum (+ x x)) (product (* x x))) (values sum product))) 10)
=> 20 100
I write this as a late response to the accepted answer since it is wrong!
Lambda Calculus does have multiple return values, but it takes a bit to understand what returning multiple values mean.
Lambda Calculus has no inherent definition of a collection of stuff, but it does allow you to invent it using products and church numerals.
pure functional JavaScript will be used for this example.
let's define a product as follows:
const product = a => b => callback => callback(a)(b);
then we can define church_0, and church_1 aka true, false, aka left, right, aka car, cdr, aka first, rest as follows:
const church_0 = a => b => a;
const church_1 = a => b => b;
let's start with making a function that returns two values, 20, and "Hello".
const product = a => b => callback => callback(a)(b);
const church_0 = a => b => a;
const church_1 = a => b => b;
const returns_many = () => product(20)("Hello");
const at_index_zero = returns_many()(church_0);
const at_index_one = returns_many()(church_1);
console.log(at_index_zero);
console.log(at_index_one);
As expected, we got 20 and "Hello".
To return more than 2 values, it gets a bit tricky:
const product = a => b => callback => callback(a)(b);
const church_0 = a => b => a;
const church_1 = a => b => b;
const returns_many = () => product(20)(
product("Hello")(
product("Yes")("No")
)
);
const at_index_zero = returns_many()(church_0);
const at_index_one = returns_many()(church_1)(church_0);
const at_index_two = returns_many()(church_1)(church_1)(church_0);
console.log(at_index_zero);
console.log(at_index_one);
console.log(at_index_two);
As you can see, a function can return an arbitrary number of return values, but to access these values, a you cannot simply use result()[0], result()[1], or result()[2], but you must use functions that filter out the position you want.
This is mindblowingly similar to electrical circuits, in that circuits have no "0", "1", "2", "3", but they do have means to make decisions, and by abstracting away our circuitry with byte(reverse list of 8 inputs), word(reverse list of 16 inputs), in this language, 0 as a byte would be [0, 0, 0, 0, 0, 0, 0, 0] which is equivalent to:
const Byte = a => b => c => d => e => f => g => h => callback =>
callback(a)(b)(c)(d)(e)(f)(g)(h);
const Byte_one = Byte(0)(0)(0)(0)(0)(0)(0)(1); // preserves
const Bit_zero = Byte_one(b7 => b6 => b5 => b4 => b3 => b2 => b1 => b0 => b0);
After inventing a number, we can make an algorithm to, given a byte-indexed array, and a byte representing index we want from this array, it will take care of the boilerplate.
Anyway, what we call arrays is nothing more than the following, expressed in higher level to show the point:
// represent nested list of bits(addresses)
// to nested list of bits(bytes) interpreted as strings.
const MyArray = function(index) {
return (index == 0)
? "0th"
: (index == 1)
? "first"
: "second"
;
};
except it doesnt do 2^32 - 1 if statements, it only does 8 and recursively narrows down the specific element you want. Essentially it acts exactly like a multiplexor(except the "single" signal is actually a fixed number of bits(coproducts, choices) needed to uniquely address elements).
My point is that is Arrays, Maps, Associative Arrays, Lists, Bits, Bytes, Words, are all fundamentally functions, both at circuit level(where we can represent complex universes with nothing but wires and switches), and mathematical level(where everything is ultimately products(sequences, difficult to manage without requiring nesting, eg lists), coproducts(types, sets), and exponentials(free functors(lambdas), forgetful functors)).