I'm a Java rookie and in School we got some homework. I shall explain boolean terms but I don't understand one of them.
The question is
Why is this expression always true?
!!(a||!a)
I understand the part in the brackets but what do the two exclamation marks in front of it?
If the first a = true --> !a = not true --> !! ( double negation = true?) a = true and the second !a = not true --> !!a = true --> !!!a = not true
and if I'm right , why is this expression alsways true? That beats me.
Could any of you explain that to me?
Thanks for your help!
First off, !! expression cancels out: it’s the same as expression, because the negation of a negation is the original value.
So we’re left with a || ! a, which is a disjunction. So the result is true if at least one of the sub-expressions a or !a is true.
And lastly, a is true if a is true (duh). And ! a is true if a is false. Thus, regardless of the value of a, the overall expression is true.
If "a" is TRUE add "!a" is FALSE
Knowing a is true and !a is false (! is for the negation; in this case, negating true -> false)
If a||!a means true or false, and from that expression you get true...
You can see it like this:
!!(true).
What is the result of a double negation? True.
Then, true(true) = (aka a||!a), which finally makes your expression !!(a||!a) always true.
If "a" is FALSE and "!a" is TRUE
Knowing a is false and !a is true (! is for the negation; in this case, negating false -> true)
If a||!a means false or true, and from that expression you get true...
You can see it like this:
!!(true).
What is the result of a double negation? True.
Then, true(true) = (aka a||!a), which finally makes your expression !!(a||!a) always true.
!!(a||!a)
Using one of the axioms of Boolean Algebra, a double negation cancels out everything.
so it would be equivalent to say (a||!a).
Assuming, we set a = true; many programing languages use lazy evaluation of conditional statements, the conditional statement would evaluate to true and execute whatever was inside the body of the if statement.
If we set a = false , then the right hand side of the conditional statement becomes true... as !a is the inverse of a meaning if a == false, !a == true.
The following makes one or more literals true:
a,b,c :- condition.
a;b;c :- condition.
In the above a,b,c,condition is a valid model, but also (a,condition), (a,b,condition), etc. I want all of a, b, c to be true, always, if condition is true.
I can write the following to force a,b,c to be always true together.
:- condition, a, not b.
:- condition, b, not c.
:- condition, c, not a.
but these become very verbose and bug prone for complex rules.
You can write
3 { a ; b ; c } 3 :- condition.
That means that at least 3 and at most 3 atoms in the curly brackets are defined as true if the condition is true.
The constraints which you wrote have a very different meaning, for example the first constraint requires that some other rule must not define truth of condition and a unless there is also a rule that defines b to be true.
Given an arbitrary list of booleans, what is the most elegant way of determining that exactly one of them is true?
The most obvious hack is type conversion: converting them to 0 for false and 1 for true and then summing them, and returning sum == 1.
I'd like to know if there is a way to do this without converting them to ints, actually using boolean logic.
(This seems like it should be trivial, idk, long week)
Edit: In case it wasn't obvious, this is more of a code-golf / theoretical question. I'm not fussed about using type conversion / int addition in PROD code, I'm just interested if there is way of doing it without that.
Edit2: Sorry folks it's a long week and I'm not explaining myself well. Let me try this:
In boolean logic, ANDing a collection of booleans is true if all of the booleans are true, ORing the collection is true if least one of them is true. Is there a logical construct that will be true if exactly one boolean is true? XOR is this for a collection of two booleans for example, but any more than that and it falls over.
You can actually accomplish this using only boolean logic, although there's perhaps no practical value of that in your example. The boolean version is much more involved than simply counting the number of true values.
Anyway, for the sake of satisfying intellectual curiosity, here goes. First, the idea of using a series of XORs is good, but it only gets us half way. For any two variables x and y,
x ⊻ y
is true whenever exactly one of them is true. However, this does not continue to be true if you add a third variable z,
x ⊻ y ⊻ z
The first part, x ⊻ y, is still true if exactly one of x and y is true. If either x or y is true, then z needs to be false for the whole expression to be true, which is what we want. But consider what happens if both x and y are true. Then x ⊻ y is false, yet the whole expression can become true if z is true as well. So either one variable or all three must be true. In general, if you have a statement that is a chain of XORs, it will be true if an uneven number of variables are true.
Since one is an uneven number, this might prove useful. Of course, checking for an uneven number of truths is not enough. We additionally need to ensure that no more than one variable is true. This can be done in a pairwise fashion by taking all pairs of two variables and checking that they are not both true. Taken together these two conditions ensure that exactly one if the variables are true.
Below is a small Python script to illustrate the approach.
from itertools import product
print("x|y|z|only_one_is_true")
print("======================")
for x, y, z in product([True, False], repeat=3):
uneven_number_is_true = x ^ y ^ z
max_one_is_true = (not (x and y)) and (not (x and z)) and (not (y and z))
only_one_is_true = uneven_number_is_true and max_one_is_true
print(int(x), int(y), int(z), only_one_is_true)
And here's the output.
x|y|z|only_one_is_true
======================
1 1 1 False
1 1 0 False
1 0 1 False
1 0 0 True
0 1 1 False
0 1 0 True
0 0 1 True
0 0 0 False
Sure, you could do something like this (pseudocode, since you didn't mention language):
found = false;
alreadyFound = false;
for (boolean in booleans):
if (boolean):
found = true;
if (alreadyFound):
found = false;
break;
else:
alreadyFound = true;
return found;
After your clarification, here it is with no integers.
bool IsExactlyOneBooleanTrue( bool *boolAry, int size )
{
bool areAnyTrue = false;
bool areTwoTrue = false;
for(int i = 0; (!areTwoTrue) && (i < size); i++) {
areTwoTrue = (areAnyTrue && boolAry[i]);
areAnyTrue |= boolAry[i];
}
return ((areAnyTrue) && (!areTwoTrue));
}
No-one mentioned that this "operation" we're looking for is shortcut-able similarly to boolean AND and OR in most languages. Here's an implementation in Java:
public static boolean exactlyOneOf(boolean... inputs) {
boolean foundAtLeastOne = false;
for (boolean bool : inputs) {
if (bool) {
if (foundAtLeastOne) {
// found a second one that's also true, shortcut like && and ||
return false;
}
foundAtLeastOne = true;
}
}
// we're happy if we found one, but if none found that's less than one
return foundAtLeastOne;
}
With plain boolean logic, it may not be possible to achieve what you want. Because what you are asking for is a truth evaluation not just based on the truth values but also on additional information(count in this case). But boolean evaluation is binary logic, it cannot depend on anything else but on the operands themselves. And there is no way to reverse engineer to find the operands given a truth value because there can be four possible combinations of operands but only two results. Given a false, can you tell if it is because of F ^ F or T ^ T in your case, so that the next evaluation can be determined based on that?.
booleanList.Where(y => y).Count() == 1;
Due to the large number of reads by now, here comes a quick clean up and additional information.
Option 1:
Ask if only the first variable is true, or only the second one, ..., or only the n-th variable.
x1 & !x2 & ... & !xn |
!x1 & x2 & ... & !xn |
...
!x1 & !x2 & ... & xn
This approach scales in O(n^2), the evaluation stops after the first positive match is found. Hence, preferred if it is likely that there is a positive match.
Option 2:
Ask if there is at least one variable true in total. Additionally check every pair to contain at most one true variable (Anders Johannsen's answer)
(x1 | x2 | ... | xn) &
(!x1 | !x2) &
...
(!x1 | !xn) &
(!x2 | !x3) &
...
(!x2 | !xn) &
...
This option also scales in O(n^2) due to the number of possible pairs. Lazy evaluation stops the formula after the first counter example. Hence, it is preferred if its likely there is a negative match.
(Option 3):
This option involves a subtraction and is thus no valid answer for the restricted setting. Nevertheless, it argues how looping the values might not be the most beneficial solution in an unrestricted stetting.
Treat x1 ... xn as a binary number x. Subtract one, then AND the results. The output is zero <=> x1 ... xn contains at most one true value. (the old "check power of two" algorithm)
x 00010000
x-1 00001111
AND 00000000
If the bits are already stored in such a bitboard, this might be beneficial over looping. Though, keep in mind this kills the readability and is limited by the available board length.
A last note to raise awareness: by now there exists a stack exchange called computer science which is exactly intended for this type of algorithmic questions
It can be done quite nicely with recursion, e.g. in Haskell
-- there isn't exactly one true element in the empty list
oneTrue [] = False
-- if the list starts with False, discard it
oneTrue (False : xs) = oneTrue xs
-- if the list starts with True, all other elements must be False
oneTrue (True : xs) = not (or xs)
// Javascript
Use .filter() on array and check the length of the new array.
// Example using array
isExactly1BooleanTrue(boolean:boolean[]) {
return booleans.filter(value => value === true).length === 1;
}
// Example using ...booleans
isExactly1BooleanTrue(...booleans) {
return booleans.filter(value => value === true).length === 1;
}
One way to do it is to perform pairwise AND and then check if any of the pairwise comparisons returned true with chained OR. In python I would implement it using
from itertools import combinations
def one_true(bools):
pairwise_comp = [comb[0] and comb[1] for comb in combinations(bools, 2)]
return not any(pairwise_comp)
This approach easily generalizes to lists of arbitrary length, although for very long lists, the number of possible pairs grows very quickly.
Python:
boolean_list.count(True) == 1
OK, another try. Call the different booleans b[i], and call a slice of them (a range of the array) b[i .. j]. Define functions none(b[i .. j]) and just_one(b[i .. j]) (can substitute the recursive definitions to get explicit formulas if required). We have, using C notation for logical operations (&& is and, || is or, ^ for xor (not really in C), ! is not):
none(b[i .. i + 1]) ~~> !b[i] && !b[i + 1]
just_one(b[i .. i + 1]) ~~> b[i] ^ b[i + 1]
And then recursively:
none(b[i .. j + 1]) ~~> none(b[i .. j]) && !b[j + 1]
just_one(b[i .. j + 1] ~~> (just_one(b[i .. j]) && !b[j + 1]) ^ (none(b[i .. j]) && b[j + 1])
And you are interested in just_one(b[1 .. n]).
The expressions will turn out horrible.
Have fun!
That python script does the job nicely. Here's the one-liner it uses:
((x ∨ (y ∨ z)) ∧ (¬(x ∧ y) ∧ (¬(z ∧ x) ∧ ¬(y ∧ z))))
Retracted for Privacy and Anders Johannsen provided already correct and simple answers. But both solutions do not scale very well (O(n^2)). If performance is important you can stick to the following solution, which performs in O(n):
def exact_one_of(array_of_bool):
exact_one = more_than_one = False
for array_elem in array_of_bool:
more_than_one = (exact_one and array_elem) or more_than_one
exact_one = (exact_one ^ array_elem) and (not more_than_one)
return exact_one
(I used python and a for loop for simplicity. But of course this loop could be unrolled to a sequence of NOT, AND, OR and XOR operations)
It works by tracking two states per boolean variable/list entry:
is there exactly one "True" from the beginning of the list until this entry?
are there more than one "True" from the beginning of the list until this entry?
The states of a list entry can be simply derived from the previous states and corresponding list entry/boolean variable.
Python:
let see using example...
steps:
below function exactly_one_topping takes three parameter
stores their values in the list as True, False
Check whether there exists only one true value by checking the count to be exact 1.
def exactly_one_topping(ketchup, mustard, onion):
args = [ketchup,mustard,onion]
if args.count(True) == 1: # check if Exactly one value is True
return True
else:
return False
How do you want to count how many are true without, you know, counting? Sure, you could do something messy like (C syntax, my Python is horrible):
for(i = 0; i < last && !booleans[i]; i++)
;
if(i == last)
return 0; /* No true one found */
/* We have a true one, check there isn't another */
for(i++; i < last && !booleans[i]; i++)
;
if(i == last)
return 1; /* No more true ones */
else
return 0; /* Found another true */
I'm sure you'll agree that the win (if any) is slight, and the readability is bad.
It is not possible without looping. Check BitSet cardinality() in java implementation.
http://fuseyism.com/classpath/doc/java/util/BitSet-source.html
We can do it this way:-
if (A=true or B=true)and(not(A=true and B=true)) then
<enter statements>
end if
My question uses Java as an example, but I guess it applies to probably all.
Is there any practical difference between the XOR operator (^ in Java) and the not-equal-to operator (!= in Java), when comparing booleans?
I evaluated things here, but I just kept wondering (seems weird, two things equal)... and didn't find anything on the net. Just one discussion in some forum that ended quickly without any result.
For Boolean values, they mean the same thing - although there's a compound assignment operator for XOR:
x ^= y;
There's no equivalent compound assignment operator for inequality.
As for why they're both available - it would be odd for XOR not to be available just because it works the same way as inequality. It should logically be there, so it is. For non-Boolean types the result is different because it's a different result type, but that doesn't mean it would make sense to remove XOR for boolean.
As stated in the Java Language Specification:
The result of != is false if the operands are both true or both false; otherwise, the result is true. Thus != behaves the same as ^ (§15.22.2) when applied to boolean operands.
In addition if you try looking at bytecode of a simple snippet:
void test(boolean b1, boolean b2) {
boolean res1 = b1^b2;
boolean res2 = b1!=b2;
}
you obtain:
test(ZZ)V
L0
LINENUMBER 45 L0
ILOAD 1
ILOAD 2
IXOR
ISTORE 3
L1
LINENUMBER 46 L1
ILOAD 1
ILOAD 2
IXOR
ISTORE 4
L2
LINENUMBER 47 L2
RETURN
L3
This assures that, in addition to the same semantics, there's no any actual practical difference in implementation. (you can also see that internally ints are used to store boolean values)
Yes, you can use XOR to test booleans for (in)equality, although the code is less intuitive: if (x ^ y) versus if (x != y).
With boolean values, there should be no difference. You should choose whichever is more appropriate for your sense of operation.
Example:
bool oldChoice = ...;
bool newChoice = ...;
if (oldChoice != newChoice)
...
Here XOR would give the same result, but will not reflect the real code intention.
They should be essentially the same in this case.
There's a big difference, XOR works at bit-level, keeping differences as ones,
so 0b0011 xor 0b1101 => 0b1110
regards,
//t