I'm having troubles to define the objective fucntion in a SMT problem with z3py.
Long story, short, I have to optimize the placing of smaller blocks inside a board that has fixed width but variable heigth.
I have an array of coordinates (represented by an array of integers of length 2) and a list of integers (representing the heigth of the block to place).
# [x,y] list of integer variables
P = [[Int("x_%s" % (i + 1)), Int("y_%s" % (i + 1))]
for i in range(blocks)]
y = [int(b) for a, b in data[2:]]
I defined the objective function like this:
obj= Int(max([P[i][1] + y[i] for i in range(blocks)]))
It calculates the max height of the board given the starting coordinate of the blocks and their heights.
I know it could be better, but I think the problem would be the same even with a different definition.
Anyway, if I run my code, the following error occurs on the line of the objective function:
" raise Z3Exception("Symbolic expressions cannot be cast to concrete Boolean values.") "
While debugging I've seen that is P[i][1] that gives an error and I think it's because the program reads "y_i + 3" (for example) and they can't be added togheter.
Point is: it's obvious that the objective function depends on the variables of the problem, so how can I get rid of this error? Is there another place where I should define the objective function so it waits to have the P array instantiated before doing anything?
Full code:
from z3 import *
from math import ceil
width = 8
blocks = 4
x = [3,3,5,5]
y = [3,5,3,5]
height = ceil(sum([x[i] * y[i] for i in range(blocks)]) / width) + 1
# [blocks x 2] list of integer variables
P = [[Int("x_%s" % (i + 1)), Int("y_%s" % (i + 1))]
for i in range(blocks)]
# value/ domain constraint
values = [And(0 <= P[i][0], P[i][0] <= width - 1, 0 <= P[i][1], P[i][1] <= height - 1)
for i in range(blocks)]
obj = Int(max([P[i][1] + y[i] for i in range(blocks)]))
board_problem = values # other constraints I've not included for brevity
o = Optimize()
o.add(board_problem)
o.minimize(obj)
if (o.check == 'unsat'):
print("The problem is unsatisfiable")
else:
print("Solved")
The problem here is that you're calling Python's max on symbolic values, which is not designed to work for symbolic expressions. Instead, define a symbolic version of max and use that:
# Return maximum of a vector; error if empty
def symMax(vs):
m = vs[0]
for v in vs[1:]:
m = If(v > m, v, m)
return m
obj = symMax([P[i][1] + y[i] for i in range(blocks)])
With this change your program will go through and print Solved when run.
I have tried to write a function that takes in notes in MIDI form (C2,A4,Bb6) and returns their respective frequencies in hertz. I'm not sure what the best method of doing this should be. I am torn between two approaches. 1) a list based one where I can switch on an input and return hard-coded frequency values given that I may only have to do this for 88 notes (in the grand piano case). 2) a simple mathematical approach however my math skills are a limitation as well as converting the input string into a numerical value. Ultimately I've been working on this for a while and could use some direction.
You can use a function based on this formula:
The basic formula for the frequencies of the notes of the equal
tempered scale is given by
fn = f0 * (a)n
where
f0 = the frequency of one fixed note which must be defined. A common choice is setting the A above middle C (A4) at f0 = 440 Hz.
n = the number of half steps away from the fixed note you are. If you are at a higher note, n is positive. If you are on a lower note, n is negative.
fn = the frequency of the note n half steps away. a = (2)1/12 = the twelth root of 2 = the number which when multiplied by itself 12 times equals 2 = 1.059463094359...
http://www.phy.mtu.edu/~suits/NoteFreqCalcs.html
In Objective-C, this would be:
+ (double)frequencyForNote:(Note)note withModifier:(Modifier)modifier inOctave:(int)octave {
int halfStepsFromA4 = note - A;
halfStepsFromA4 += 12 * (octave - 4);
halfStepsFromA4 += modifier;
double frequencyOfA4 = 440.0;
double a = 1.059463094359;
return frequencyOfA4 * pow(a, halfStepsFromA4);
}
With the following enums defined:
typedef enum : int {
C = 0,
D = 2,
E = 4,
F = 5,
G = 7,
A = 9,
B = 11,
} Note;
typedef enum : int {
None = 0,
Sharp = 1,
Flat = -1,
} Modifier;
https://gist.github.com/NickEntin/32c37e3d31724b229696
Why don't you use a MIDI pitch?
where f is the frequency, and d the MIDI data.
In idris 0.9.17.1,
with inspiration from https://wiki.haskell.org/Prime_numbers,
I've written the following code for generating prime numbers
module Main
concat: List a -> Stream a -> Stream a
concat [] ys = ys
concat (x :: xs) ys = x :: (concat xs ys)
generate: (Num a, Ord a) => (start:a) -> (step:a) -> (max:a) -> List a
generate start step max = if (start < max) then start :: generate (start + step) step max else []
mutual
sieve: Nat -> Stream Int -> Int -> Stream Int
sieve k (p::ps) x = concat (start) (sieve (k + 1) ps (p * p)) where
fs: List Int
fs = take k (tail primes)
start: List Int
start = [n | n <- (generate (x + 2) 2 (p * p - 2)), (all (\i => (n `mod` i) /= 0) fs)]
primes: Stream Int
primes = 2 :: 3 :: sieve 0 (tail primes) 3
main:IO()
main = do
printLn $ take 10 primes
In the REPL, if I write take 10 primes, the REPL correctly shows [2, 3, 5, 11, 13, 17, 19, 29, 31, 37] : List Int
But if I try :exec, nothing happen and if I try to compile ans execute the program I get Segmentation fault: 11
Can someone help me to debug this problem ?
Your concat function can be made lazy to fix this. Just change its type to
concat : List a -> Lazy (Stream a) -> Stream a
This will do it.
Note:
To get all primes, change the < inside the generate function into <=
(Currently some are missing, e.g. 7 and 23).
I need to find whether a number is divisible by 3 without using %, / or *. The hint given was to use atoi() function. Any idea how to do it?
The current answers all focus on decimal digits, when applying the "add all digits and see if that divides by 3". That trick actually works in hex as well; e.g. 0x12 can be divided by 3 because 0x1 + 0x2 = 0x3. And "converting" to hex is a lot easier than converting to decimal.
Pseudo-code:
int reduce(int i) {
if (i > 0x10)
return reduce((i >> 4) + (i & 0x0F)); // Reduces 0x102 to 0x12 to 0x3.
else
return i; // Done.
}
bool isDiv3(int i) {
i = reduce(i);
return i==0 || i==3 || i==6 || i==9 || i==0xC || i == 0xF;
}
[edit]
Inspired by R, a faster version (O log log N):
int reduce(unsigned i) {
if (i >= 6)
return reduce((i >> 2) + (i & 0x03));
else
return i; // Done.
}
bool isDiv3(unsigned i) {
// Do a few big shifts first before recursing.
i = (i >> 16) + (i & 0xFFFF);
i = (i >> 8) + (i & 0xFF);
i = (i >> 4) + (i & 0xF);
// Because of additive overflow, it's possible that i > 0x10 here. No big deal.
i = reduce(i);
return i==0 || i==3;
}
Subtract 3 until you either
a) hit 0 - number was divisible by 3
b) get a number less than 0 - number wasn't divisible
-- edited version to fix noted problems
while n > 0:
n -= 3
while n < 0:
n += 3
return n == 0
Split the number into digits. Add the digits together. Repeat until you have only one digit left. If that digit is 3, 6, or 9, the number is divisible by 3. (And don't forget to handle 0 as a special case).
While the technique of converting to a string and then adding the decimal digits together is elegant, it either requires division or is inefficient in the conversion-to-a-string step. Is there a way to apply the idea directly to a binary number, without first converting to a string of decimal digits?
It turns out, there is:
Given a binary number, the sum of its odd bits minus the sum of its even bits is divisible by 3 iff the original number was divisible by 3.
As an example: take the number 3726, which is divisible by 3. In binary, this is 111010001110. So we take the odd digits, starting from the right and moving left, which are [1, 1, 0, 1, 1, 1]; the sum of these is 5. The even bits are [0, 1, 0, 0, 0, 1]; the sum of these is 2. 5 - 2 = 3, from which we can conclude that the original number is divisible by 3.
A number divisible by 3, iirc has a characteristic that the sum of its digit is divisible by 3. For example,
12 -> 1 + 2 = 3
144 -> 1 + 4 + 4 = 9
The interview question essentially asks you to come up with (or have already known) the divisibility rule shorthand with 3 as the divisor.
One of the divisibility rule for 3 is as follows:
Take any number and add together each digit in the number. Then take that sum and determine if it is divisible by 3 (repeating the same procedure as necessary). If the final number is divisible by 3, then the original number is divisible by 3.
Example:
16,499,205,854,376
=> 1+6+4+9+9+2+0+5+8+5+4+3+7+6 sums to 69
=> 6 + 9 = 15 => 1 + 5 = 6, which is clearly divisible by 3.
See also
Wikipedia/Divisibility rule - has many rules for many divisors
Given a number x.
Convert x to a string. Parse the string character by character. Convert each parsed character to a number (using atoi()) and add up all these numbers into a new number y.
Repeat the process until your final resultant number is one digit long. If that one digit is either 3,6 or 9, the origional number x is divisible by 3.
My solution in Java only works for 32-bit unsigned ints.
static boolean isDivisibleBy3(int n) {
int x = n;
x = (x >>> 16) + (x & 0xffff); // max 0x0001fffe
x = (x >>> 8) + (x & 0x00ff); // max 0x02fd
x = (x >>> 4) + (x & 0x000f); // max 0x003d (for 0x02ef)
x = (x >>> 4) + (x & 0x000f); // max 0x0011 (for 0x002f)
return ((011111111111 >> x) & 1) != 0;
}
It first reduces the number down to a number less than 32. The last step checks for divisibility by shifting the mask the appropriate number of times to the right.
You didn't tag this C, but since you mentioned atoi, I'm going to give a C solution:
int isdiv3(int x)
{
div_t d = div(x, 3);
return !d.rem;
}
bool isDiv3(unsigned int n)
{
unsigned int n_div_3 =
n * (unsigned int) 0xaaaaaaab;
return (n_div_3 < 0x55555556);//<=>n_div_3 <= 0x55555555
/*
because 3 * 0xaaaaaaab == 0x200000001 and
(uint32_t) 0x200000001 == 1
*/
}
bool isDiv5(unsigned int n)
{
unsigned int n_div_5 =
i * (unsigned int) 0xcccccccd;
return (n_div_5 < 0x33333334);//<=>n_div_5 <= 0x33333333
/*
because 5 * 0xcccccccd == 0x4 0000 0001 and
(uint32_t) 0x400000001 == 1
*/
}
Following the same rule, to obtain the result of divisibility test by 'n', we can :
multiply the number by 0x1 0000 0000 - (1/n)*0xFFFFFFFF
compare to (1/n) * 0xFFFFFFFF
The counterpart is that for some values, the test won't be able to return a correct result for all the 32bit numbers you want to test, for example, with divisibility by 7 :
we got 0x100000000- (1/n)*0xFFFFFFFF = 0xDB6DB6DC
and 7 * 0xDB6DB6DC = 0x6 0000 0004,
We will only test one quarter of the values, but we can certainly avoid that with substractions.
Other examples :
11 * 0xE8BA2E8C = A0000 0004, one quarter of the values
17 * 0xF0F0F0F1 = 10 0000 0000 1
comparing to 0xF0F0F0F
Every values !
Etc., we can even test every numbers by combining natural numbers between them.
A number is divisible by 3 if all the digits in the number when added gives a result 3, 6 or 9. For example 3693 is divisible by 3 as 3+6+9+3 = 21 and 2+1=3 and 3 is divisible by 3.
inline bool divisible3(uint32_t x) //inline is not a must, because latest compilers always optimize it as inline.
{
//1431655765 = (2^32 - 1) / 3
//2863311531 = (2^32) - 1431655765
return x * 2863311531u <= 1431655765u;
}
On some compilers this is even faster then regular way: x % 3. Read more here.
well a number is divisible by 3 if all the sum of digits of the number are divisible by 3. so you could get each digit as a substring of the input number and then add them up. you then would repeat this process until there is only a single digit result.
if this is 3, 6 or 9 the number is divisable by 3.
Here is a pseudo-algol i came up with .
Let us follow binary progress of multiples of 3
000 011
000 110
001 001
001 100
001 111
010 010
010 101
011 000
011 011
011 110
100 001
100 100
100 111
101 010
101 101
just have a remark that, for a binary multiple of 3 x=abcdef in following couples abc=(000,011),(001,100),(010,101) cde doest change , hence, my proposed algorithm:
divisible(x):
y = x&7
z = x>>3
if number_of_bits(z)<4
if z=000 or 011 or 110 , return (y==000 or 011 or 110) end
if z=001 or 100 or 111 , return (y==001 or 100 or 111) end
if z=010 or 101 , return (y==010 or 101) end
end
if divisible(z) , return (y==000 or 011 or 110) end
if divisible(z-1) , return (y==001 or 100 or 111) end
if divisible(z-2) , return (y==010 or 101) end
end
C# Solution for checking if a number is divisible by 3
namespace ConsoleApplication1
{
class Program
{
static void Main(string[] args)
{
int num = 33;
bool flag = false;
while (true)
{
num = num - 7;
if (num == 0)
{
flag = true;
break;
}
else if (num < 0)
{
break;
}
else
{
flag = false;
}
}
if (flag)
Console.WriteLine("Divisible by 3");
else
Console.WriteLine("Not Divisible by 3");
Console.ReadLine();
}
}
}
Here is your optimized solution that every one should know.................
Source: http://www.geeksforgeeks.org/archives/511
#include<stdio.h>
int isMultiple(int n)
{
int o_count = 0;
int e_count = 0;
if(n < 0)
n = -n;
if(n == 0)
return 1;
if(n == 1)
return 0;
while(n)
{
if(n & 1)
o_count++;
n = n>>1;
if(n & 1)
e_count++;
n = n>>1;
}
return isMultiple(abs(o_count - e_count));
}
int main()
{
int num = 23;
if (isMultiple(num))
printf("multiple of 3");
else
printf(" not multiple of 3");
return 0;
}
I came up with the following solution to format an integer (bytesize of a file). Is there any better/shorter solution? I esacially don't like the float_as_string() part.
human_filesize(Size) ->
KiloByte = 1024,
MegaByte = KiloByte * 1024,
GigaByte = MegaByte * 1024,
TeraByte = GigaByte * 1024,
PetaByte = TeraByte * 1024,
human_filesize(Size, [
{PetaByte, "PB"},
{TeraByte, "TB"},
{GigaByte, "GB"},
{MegaByte, "MB"},
{KiloByte, "KB"}
]).
human_filesize(Size, []) ->
integer_to_list(Size) ++ " Byte";
human_filesize(Size, [{Block, Postfix}|List]) ->
case Size >= Block of
true ->
float_as_string(Size / Block) ++ " " ++ Postfix;
false ->
human_filesize(Size, List)
end.
float_as_string(Float) ->
Integer = trunc(Float), % Part before the .
NewFloat = 1 + Float - Integer, % 1.<part behind>
FloatString = float_to_list(NewFloat), % "1.<part behind>"
integer_to_list(Integer) ++ string:sub_string(FloatString, 2, 4).
Edit: Fixed bug round() -> trunc()
human_filesize(Size) -> human_filesize(Size, ["B","KB","MB","GB","TB","PB"]).
human_filesize(S, [_|[_|_] = L]) when S >= 1024 -> human_filesize(S/1024, L);
human_filesize(S, [M|_]) ->
io_lib:format("~.2f ~s", [float(S), M]).
Note that this returns an iolist. If you need a string, you can convert that to binary and that to string.