Rewrite recursive BNF rule with iteration - recursion

Look at the following recursive BNF rule
(1) X = Xa | b
This produces sentences like
X = b
X = ba
X = baa
X = baaa
...
This can be written as
(2) X = b a*
where the right hand side is not recursive
Now take a look at the following recursive BNF rule
(3) X = { X } | b
This produces sentences like
X = b
X = {b}
X = {{b}}
X = {{{b}}}
...
Is there some way to rewrite rule (3) in a non recursive way, analogous as we did when we rewrote rule (1) to rule (2).
Observe that X = {* b }* is no good since the parenthesis need to be balanced.

I do not know if the question above is possible to answer. The reason for the question above was that I wanted to avoid infinite loop in my parser (written in Java). One way was to insure that the BNF rules are not recursive, hence my question. But another way is to use the recursive rule, but avoid the infinite loop inside my (Java) program. Turns out that you can avoid loops by lazy instantiation.
For instance look at the following rules:
expression = term ('+' term)*;
term = factor ('*' factor)*;
factor = '(' expression ')' | Num;
expression() calls term(), which calls factor(), which calls expression(), thus we can end up with infinite loop. To avoid that we can use lazy instantiation, so instead of writing something like:
public Parser expression() {
expression = new ...
return expression;
}
we instead write:
public Parser expression() {
if (expression == null) {
expression = new ...
}
return expression;
}
Observe that you must declare expression as an instance variable to get this to work.

Related

iterating 2D array in Elixir

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.

How do I evaluate the function in only one of its variables in Scilab

How do I evaluate the function in only one of its variables, that is, I hope to obtain another function after evaluating the function. I have the following piece of code.
deff ('[F] = fun (x, y)', 'F = x ^ 2-3 * y ^ 2 + x * y ^ 3');
fun (4, y)
I hope to get 16-3y ^ 2 + 4y ^ 3
If what you want to do is to write x = f(4,y), and later just do x(2) to get -36, that is called partial application:
Intuitively, partial function application says "if you fix the first arguments of the function, you get a function of the remaining arguments".
This is a very useful feature, and very common Functional Programming Languages, such as Haskell, but even JS and Python now are able to do it. It is also possible to do this in MATLAB and GNU/Octave using anonymous functions (see this answer). In Scilab, however, this feature is not available.
Workround
Nonetheless, Scilab itself uses a workarounds to carry a function with its arguments without fully evaluating. You see this being used in ode(), fsolve(), optim(), and others:
Create a list containing the function and the arguments to partial evaluation: list(f,arg1,arg2,...,argn)
Use another function to evaluate such list and the last argument: evalPartList(list(...),last_arg)
The implementation of evalPartList() can be something like this:
function y = evalPartList(fList,last_arg)
//fList: list in which the first element is a function
//last_arg: last argument to be applied to the function
func = fList(1); //extract function from the list
y = func(fList(2:$),last_arg); //each element of the list, from second
//to last, becomes an argument
endfunction
You can test it on Scilab's console:
--> deff ('[F] = fun (x, y)', 'F = x ^ 2-3 * y ^ 2 + x * y ^ 3');
--> x = list(fun,4)
x =
x(1)
[F]= x(1)(x,y)
x(2)
4.
--> evalPartList(x,2)
ans =
36.
This is a very simple implementation for evalPartList(), and you have to be careful not to exceed or be short on the number of arguments.
In the way you're asking, you can't.
What you're looking is called symbolic (or formal) computational mathematics, because you don't pass actual numerical values to functions.
Scilab is numerical software so it can't do such thing. But there is a toolbox scimax (installation guide) that rely on a the free formal software wxmaxima.
BUT
An ugly, stupid but still sort of working solution is to takes advantages of strings :
function F = fun (x, y) // Here we define a function that may return a constant or string depending on the input
fmt = '%10.3E'
if (type(x)==type('')) & (type(y)==type(0)) // x is string is
ys = msprintf(fmt,y)
F = x+'^2 - 3*'+ys+'^2 + '+x+'*'+ys+'^3'
end
if (type(y)==type('')) & (type(x)==type(0)) // y is string so is F
xs = msprintf(fmt,x)
F = xs+'^2 - 3*'+y+'^2 + '+xs+'*'+y+'^3'
end
if (type(y)==type('')) & (type(x)==type('')) // x&y are strings so is F
F = x+'^2 - 3*'+y+'^2 + '+x+'*'+y+'^3'
end
if (type(y)==type(0)) & (type(x)==type(0)) // x&y are constant so is F
F = x^2 - 3*y^2 + x*y^3
end
endfunction
// Then we can use this 'symbolic' function
deff('F2 = fun2(y)',' F2 = '+fun(4,'y'))
F2=fun2(2) // does compute fun(4,2)
disp(F2)

Why does this happen when a user-defined R function does not return a value?

In the function shown below, there is no return. However, after executing it, I can confirm that the value entered d normally.
There is no return. Any suggestions in this regard will be appreciated.
Code
#installed plotly, dplyr
accumulate_by <- function(dat, var) {
var <- lazyeval::f_eval(var, dat)
lvls <- plotly:::getLevels(var)
dats <- lapply(seq_along(lvls), function(x) {
cbind(dat[var %in% lvls[seq(1, x)], ], frame = lvls[[x]])
})
dplyr::bind_rows(dats)
}
d <- txhousing %>%
filter(year > 2005, city %in% c("Abilene", "Bay Area")) %>%
accumulate_by(~date)
In the function, the last assignment is creating 'dats' which is returned with bind_rows(dats) We don't need an explicit return statement. Suppose, if there are two objects to be returned, we can place it in a list
In some languages like python, for memory efficiency, generators are used which will yield instead of creating the whole output in memory i.e. Consider two functions in python
def get_square(n):
result = []
for x in range(n):
result.append(x**2)
return result
When we run it
get_square(4)
#[0, 1, 4, 9]
The same function can be written as a generator. Instead of returning anything,
def get_square(n):
for x in range(n):
yield(x**2)
Running the function
get_square(4)
#<generator object get_square at 0x0000015240C2F9E8>
By casting with list, we get the same output
list(get_square(4))
#[0, 1, 4, 9]
There is always a return :) You just don't have to be explicit about it.
All R expressions return something. Including control structures and user-defined functions. (Control-structures are just functions, by the way, so you can just remember that everything is a value or a function call, and everything evaluates to a value).
For functions, the return value is the last expression evaluated in the execution of the function. So, for
f <- function(x) 2 + x
when you call f(3) you will invoke the function + with two parameters, 2 and x. These evaluate to 2 and 3, respectively, so `+`(2, 3) evaluates to 5, and that is the result of f(3).
When you call the return function -- and remember, this is a function -- you just leave the control-flow of a function early. So,
f <- function(x) {
if (x < 0) return(0)
x + 2
}
works as follows: When you call f, it will call the if function to figure out what to do in the first statement. The if function will evaluate x < 0 (which means calling the function < with parameters x and 0). If x < 0 is true, if will evaluate return(0). If it is false, it will evaluate its else part (which, because if has a special syntax when it comes to functions, isn't shown, but is NULL). If x < 0 is not true, f will evaluate x + 2 and return that. If x < 0 is true, however, the if function will evaluate return(0). This is a call to the function return, with parameter 0, and that call will terminate the execution of f and make the result 0.
Be careful with return. It is a function so
f <- function(x) {
if (x < 0) return;
x + 2
}
is perfectly valid R code, but it will not return when x < 0. The if call will just evaluate to the function return but not call it.
The return function is also a little special in that it can return from the parent call of control structures. Strictly speaking, return isn't evaluated in the frame of f in the examples above, but from inside the if calls. It just handles this special so it can return from f.
With non-standard evaluation this isn't always the case.
With this function
f <- function(df) {
with(df, if (any(x < 0)) return("foo") else return("bar"))
"baz"
}
you might think that
f(data.frame(x = rnorm(10)))
should return either "foo" or "bar". After all, we return in either case in the if statement. However, the if statement is evaluated inside with and it doesn't work that way. The function will return baz.
For non-local returns like that, you need to use callCC, and then it gets more technical (as if this wasn't technical enough).
If you can, try to avoid return completely and rely on functions returning the last expression they evaluate.
Update
Just to follow up on the comment below about loops. When you call a loop, you will most likely call one of the built-in primitive functions. And, yes, they return NULL. But you can write your own, and they will follow the rule that they return the last expression they evaluate. You can, for example, implement for in terms of while like this:
`for` <- function(itr_var, seq, body) {
itr_var <- as.character(substitute(itr_var))
body <- substitute(body)
e <- parent.frame()
j <- 1
while (j < length(seq)) {
assign(x = itr_var, value = seq[[j]], envir = e)
eval(body, envir = e)
j <- j + 1
}
"foo"
}
This function, will definitely return "foo", so this
for(i in 1:5) { print(i) }
evalutes to "foo". If you want it to return NULL, you have to be explicit about it (or just let the return value be the result of the while loop -- if that is the primitive while it returns NULL).
The point I want to make is that functions return the last expression they evaluate has to do with how the functions are defined, not how you call them. The loops use non-standard evaluation, so the last expression in the loop body you provide them might be the last value they evaluate and might not. For the primitive loops, it is not.
Except for their special syntax, there is nothing magical about loops. They follow the rules all functions follow. With non-standard evaluation it can get a bit tricky to work out from a function call what the last expression they will evaluate might be, because the function body looks like it is what the function evaluates. It is, to a degree, if the function is sensible, but the loop body is not the function body. It is a parameter. If it wasn't for the special syntax, and you had to provide loop bodies as normal parameters, there might be less confusion.

Evaluate expression with local variables

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.

Recursive function to repeat string in OCaml

I am absolute OCaml beginner. I want to create a function that repeats characters 20 times.
This is the function, but it does not work because of an error.
let string20 s =
let n = 20 in
s ^ string20 s (n - 1);;
string20 "u";;
I want to run like this
# string20 "u"
- : string = "uuuuuuuuuuuuuuuuuuuu"
Your function string20 takes one parameter but you are calling it recursively with 2 parameters.
The basic ideas are in there, but not quite in the right form. One way to proceed is to separate out the 2-parameter function as a separate "helper" function. As #PierreG points out, you'll need to delcare the helper function as a recursive function.
let rec string n s =
if n = 0 then "" else s ^ string (n - 1) s
let string20 = string 20
It is a common pattern to separate a function into a "fixed" part and inductive part. In this case, a nested helper function is needed to do the real recursive work in a new scope while we want to fix an input string s as a constant so we can use to append to s2. s2 is an accumulator that build up the train of strings over time while c is an inductor counting down to 1 toward the base case.
let repeat s n =
let rec helper s1 n1 =
if n1 = 0 then s1 else helper (s1 ^ s) (n1 - 1)
in helper "" n
A non-tail call versions is more straightforward since you won't need a helper function at all:
let rec repeat s n =
if n = 0 then "" else s ^ repeat s (n - 1)
On the side note, one very fun thing about a functional language with first-class functions like Ocaml is currying (or partial application). In this case you can create a function named repeat that takes two arguments n of type int and s of type string as above and partially apply it to either n or s like this:
# (* top-level *)
# let repeat_foo = repeat "foo";;
# repeat_foo 5;;
- : bytes = "foofoofoofoofoo" (* top-level output *)
if the n argument was labeled as below:
let rec repeat ?(n = 0) s =
if n = 0 then "" else s ^ repeat s (n - 1)
The order of application can be exploited, making the function more flexible:
# (* top-level *)
# let repeat_10 = repeat ~n:10;;
# repeat_10 "foo";;
- : bytes = "foofoofoofoofoofoofoofoofoofoo" (* top-level output *)
See my post Currying Exercise in JavaScript (though it is in JavaScript but pretty simple to follow) and this lambda calculus primer.
Recursive functions in Ocaml are defined with let rec
As pointed out in the comments you've defined your function to take one parameter but you're trying to recursively call with two.
You probably want something like this:
let rec stringn s n =
match n with
1 -> s
| _ -> s ^ stringn s (n - 1)
;;

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