Logic to test that 3 of 4 are True

Question

I want to return True if and only if 3 out of 4 boolean values are true.

The closest I've gotten is (x ^ y) ^ (a ^ b):

What should I do?

Answer 1

I suggest writing the code in a manner that indicates what you mean. If you want 3 values to be true, it seems natural to me that the value 3 appears somewhere.

For instance, in C++:

if ((int)a + (int)b + (int)c + (int)d == 3)
    ...

This is well defined in C++: the standard (§4.7/4) indicates that converting bool to int gives the expected values 0 or 1.

In Java and C#, you can use the following construct:

if ((a?1:0) + (b?1:0) + (c?1:0) + (d?1:0) == 3)
    ...

Answer 2

#1: Using a branching ?: 3 or 4 operations

A ^ B ? C & D : ( C ^ D ) & A

#2 Non-Branching, 7 operations

(A ^ B ^ C ^ D) & ((A & B) | (C & D))

Back when I use to profile everything, I found non-branching solutions were quite a bit quicker operation-for-operation as the CPU could predict the code path better, and execute more operations in tandem. There is about 50% less work in the branching statement here though.

Answer 3

If this had been Python, I would have written

if [a, b, c, d].count(True) == 3:

Or

if [a, b, c, d].count(False) == 1:

Or

if [a, b, c, d].count(False) == True:
# In Python True == 1 and False == 0

Or

print [a, b, c, d].count(0) == 1

Or

print [a, b, c, d].count(1) == 3

Or

if a + b + c + d == 3:

Or

if sum([a, b, c, d]) == 3:

All these work, since Booleans are subclasses of integers in Python.

if len(filter(bool, [a, b, c, d])) == 3:

Or, inspired by this neat trick,

data = iter([a, b, c, d])
if not all(data) and all(data):

Answer 4

Long but very simple, (disjuntive) normal form:

 (~a & b & c & d) | (a & ~b & c & d) | (a & b & ~c & d) | (a & b & c & ~d)

It may be simplified but that requires more thinking :P

Answer 5

Not sure it is simpler, but maybe.

((x xor y) and (a and b)) or ((x and y) and (a xor b))

Answer 6

If you want to use this logic in a programming language, my suggestion is

bool test(bool a, bool b, bool c, bool d){
    int n1 = a ? 1 : 0;
    int n2 = b ? 1 : 0;
    int n3 = c ? 1 : 0;
    int n4 = d ? 1 : 0;

    return n1 + n2 + n3 + n4 == 3;
}

Or if you want, you can put all of these in a single line:

return (a ? 1 : 0) + (b ? 1 : 0) + (C ? 1 : 0) + (d ? 1 : 0) == 3;

Also you can generalize this problem to n of m :

bool test(bool *values, int n, int m){
    int sum = 0;
    for(int i = 0; i < m; i += 1){
        sum += values[i] ? 1 : 0;
    }
    return sum == n;
}

Answer 7

This answer depends on the system of representation, but if 0 is the only value interpreted as false, and not(false) always returns the same numeric value, then not(a) + not(b) + not(c) + not(d) = not(0) should do the trick.

Answer 8

Keeping in mind that SO if for programming questions, rather than mere logical problems, the answer obviously depends on the choice of a programming language. Some languages support features that are uncommon to others.

For example, in C++ you could test your conditions with:

(a + b + c + d) == 3

This should be the fastest way to do the check in languages that support automatic (low-level) conversion from boolean to integer types. But again, there is no general answer for that problem.

Answer 9

The best I can do is ((x ^ y) ^ (a ^ b)) && ((a || x) && (b || y))

Answer 10

To check at least n out of all Boolean are true, ( n must be less than or equal to total number of Boolean :p)

if (((a ? 1:0) + (b ? 1:0 ) + (c ? 1:0) + (d ? 1:0 )) >= n) {
    // do the rest
}

Edit : After @Cruncher's comment

To check 3 boolean out of 4

if (((a ? 1:0) + (b ? 1:0 ) + (c ? 1:0) + (d ? 1:0 )) == 3) {
    // do the rest
}

Another one :

((c & d) & (a ^ b)) | ((a & b) & (c ^ d)) (Details)

Answer 11

((a xor b) xor (c xor d)) and ((a or b) and (c or d))

The fist expression searchs for 1 or 3 true's out of 4. The second one eliminates 0 or 1 (and sometimes 2) true's out of 4.

Answer 12

Java 8, filter out the false values, and count the remaining true values:

public static long count(Boolean... values) {
    return Arrays.stream(values).filter(t -> t).count();
}

Then you can use it as follows:

if (3 == count(a, b, c, d)) {
    System.out.println("There... are... THREE... lights!");
}

Easily generalizes to checking for n of m items being true.

Answer 13

Here's a way you could solve it in C# with LINQ:

bool threeTrue = new[] { a, b, x, y }.Count(x => x) == 3;

Answer 14

That is the symmetric Boolean function S₃(4). A symmetric Boolean function is a boolean function which only depends on the quantity of inputs set, but doesn't depend on which inputs they are. Knuth mentions functions of this type in section 7.1.2 in Volume 4 of The Art of Computer Programming.

S₃(4) can be computed with 7 operations as follows:

(x && y && (a || b)) ^ ((x || y) && a && b)

Knuth shows that this is optimal, meaning that you cannot do this in less than 7 operations using the normal operators: &&, || , ^, <, and >.

However if you want to use this in a language which uses 1 for true and 0 for false, you can also use addition easily:

x + y + a + b == 3

which makes your intention quite clear.

Answer 15

(a && b && (c xor d)) || (c && d && (a xor b))

From a pure logic point of view this is what I came up with.

By the pigeon hole principle, if exactly 3 are true, then either a and b is true, or c and d is true. Then its just a matter of anding each of those cases with exactly one of the other 2.

Wolfram truth table

Answer 16

If you use a logic visualization tool like Karnaugh Maps, you see that this is a problem where you can't avoid a full blown logic term if you want to write it in one if (...) line. Lopina showed it already, it's not possible to write it simpler. You can factor out a bit, but it will stay hard to read for you AND for the machine.

Counting solutions are not bad and they show what you are really after. How you do the counting efficiently depends on your programming language. The array solutions with Python oder LinQ are nice to look at, but beware, this is SLOW. Wolf's (a+b+x+y)==3 will work nicely and fast, but only if your language equates "true" with 1. If "true" is represented by -1, you will have to test for -3 :)

If your language uses true booleans, you could try to program it explicitly (I use != as XOR test):

if (a)
{
    if (b)
        return (x != y);    // a,b=true, so either x or y must be true
    else
        return (x && y);     // a=true, b=false, so x AND y must be true
}
else
{
    if (b)
        return (x && y);    // a=false, b=true, so x and y must be true
    else
        return false;       // a,b false, can't get 3 of 4
}

"x != y" works only if x,y are of a boolean type. If they are some other type where 0 is false and everything else is true, this can fail. Then use a boolean XOR, or ( (bool)x != (bool)y ), or write "if (x) return (y==false) else return (y==true);", which is a bit more work for the computer.

If your programming language provides the ternary ?: operator, you can shorten it to

if (a)
    return b ? (x != y) : (x && y);
else
    return b ? (x && y) : false;

which keeps a bit of readability, or cut it aggressively to

return a ? (b ? (x != y) : (x && y)) : (b ? (x && y) : false);

This code does exactly three logic tests (state of a, state of b, comparison of x and y) and should be faster than most of the other answers here. But you need to comment it, or you won't understand it after 3 months :)

Answer 17

There are a lot of good answers here; here is an alternate formulation which no one else has posted yet:

 a ? (b ? (c ^ d) : (c && d)) : (b && c && d)

Answer 18

Similar to the first answer, but pure Java:

int t(boolean b) {
    return (b) ? 1 : 0;
}

if (t(x) + t(y) + t(a) + t(b) == 3) return true;
return false;

I prefer counting them as integers because it makes for more readable code.

Answer 19

In Python, to see how many of an iterable of elements are True, use sum (it's quite straightforward):

Setup

import itertools

arrays = list(itertools.product(*[[True, False]]*4))

Actual Test

for array in arrays:
    print(array, sum(array)==3)

Output

(True, True, True, True) False
(True, True, True, False) True
(True, True, False, True) True
(True, True, False, False) False
(True, False, True, True) True
(True, False, True, False) False
(True, False, False, True) False
(True, False, False, False) False
(False, True, True, True) True
(False, True, True, False) False
(False, True, False, True) False
(False, True, False, False) False
(False, False, True, True) False
(False, False, True, False) False
(False, False, False, True) False
(False, False, False, False) False

Answer 20

If you're after the on-the-paper (non-programming) solution, then K-maps and Quine-McCluskey algorithms are what you're after, they help you minify your boolean function.

In your case, the result is

y = (x̄3 ^ x2 ^ x1 ^ x0) ∨ (x3 ^ x̄2 ^ x1 ^ x0) ∨ (x3 ^ x2 ^ x̄1 ^ x0) ∨ (x3 ^ x2 ^ x1 ^ x̄0)

If you want to do this programmatically, non-fixed amount of variables and a custom "threshold", then simply iterating thru a list of boolean values and counting occurrences of "true" is pretty simple and straightforward.

Answer 21

I want to return true if and only if 3 out of 4 boolean values are true.

Given the 4 boolean values, a, b, x, y, this task translates into the following C statement:

return (a+b+x+y) == 3;

Answer 22

((a^b)^(x^y))&((a|b)&(x|y))

is what you want. Basically I took your code and added checking if actually 3 are true and not 3 are false.

Answer 23

A programming question without an answer involving recursion? Inconceivable!

There are enough "exactly 3 out of 4 trues" answers, but here's a generalised (Java) version for "exactly m out of n trues" (otherwise recursion isn't really worth it) just because you can:

public static boolean containsTrues(boolean[] someBooleans,
    int anIndex, int truesExpected, int truesFoundSoFar) {
  if (anIndex >= someBooleans.length) {
    return truesExpected == truesFoundSoFar; // reached end
  }
  int falsesExpected = someBooleans.length - truesExpected;
  boolean currentBoolean = someBooleans[anIndex];
  int truesFound = truesFoundSoFar + (currentBoolean ? 1 : 0);
  if (truesFound > truesExpected) {
    return false;
  }
  if (anIndex - truesFound > falsesExpected) {
    return false; // too many falses
  }
  return containsTrues(someBooleans, anIndex + 1, truesExpected,
      truesFound);
}

This could be called with something like:

 boolean[] booleans = { true, false, true, true, false, true, true, false };
 containsTrues(booleans, 0, 5, 0);

which should return true (because 5 of 8 values were true, as expected). Not quite happy with the words "trues" and "falses", but can't think of a better name right now.... Note that the recursion stops when too many true or too many false values have been found.

Answer 24

Since readability is a big concern, you could use a descriptive function call (wrapping any of the suggested implementations). If this calculation needs to be done in multiple places, a function call is the best way to achieve reuse, and makes it clear exactly what you are doing.

bool exactly_three_true_from(bool cond1, bool cond2, bool cond3, bool cond4)
{
    //...
}

Answer 25

In PHP, making it more dynamic (just in case you change number of conditions, etc.):

$min = 6;
$total = 10;

// create our boolean array values
$arr = array_map(function($a){return mt_rand(0,1)>0;},range(1,$total));

// the 'check'
$arrbools = array_map(function($a){return (int)$a;},$arr);
$conditionMet = array_sum($arrbools)>=$min;

echo $conditionMet ? "Passed" : "Failed";

Answer 26

(((a AND b) OR (x AND y)) AND ((a XOR b) OR (x XOR y)))

While I could show that this is a good solution, Sam Hocevar's answer is easy both to write and understand later. In my book that makes it better.

Answer 27

Here is some c# code I just wrote because you have inspired me:

It takes any amount of arguments and will tell you if n of them are true.

    static bool boolTester(int n, params bool[] values)
    {
        int sum = 0;           

        for (int i = 0; i < values.Length; i++)
        {
            if (values[i] == true)
            {
                sum += 1;
            }                
        }
        if( sum == n)
        {
            return true;
        }            
        return false;                
    }

and you call it like so:

        bool a = true;
        bool b = true;
        bool c = true;
        bool d = false;            

        bool test = false;
        test = boolTester(3, a, b, c, d);

So you can now test 7/9 or 15/100 as you will.