Please help me with how to count even numbers in a list in Prolog. I am a beginner, just started learning Prolog yesterday. I know to count the elements in the list is
mylen([H|Lc],N) :- mylen(Lc,M),N is M+1.
mylen([],0).
And I think defining even number maybe helpful in this case, and I guess the code is maybe something like:
even(n):-
N rem 2 =:= 0.
Can you help me with putting these two parts together, so my code counts even numbers? I know I also need to add a counter, but I have no idea of how to do this in Prolog.
Thank you so very much for you help!
Currently, you have two rules:
(1) The number of elements in the empty list is 0
my_len([], 0).
(2) The number of elements in the list [H|Lc] is N if the number of elements in the list Lc is M and N is M+1
my_len([H|Lc], N) :- my_len(Lc, M), N is M+1.
You're already armed with a predicate which is true if a number is even, and false if it is not: N is even if the remainder of N when divided by 2 is 0:
even(N) :- N rem 2 =:= 0.
Now you can piece it together. The number of even elements in an empty list is still zero. So you keep rule (1). Your rule (2) will need to change as it will need to check if the head of the list is even. You can do this with with two rules in Prolog which take care of the two different cases (the list head is even, or the list head is odd):
(2a) The number of even elements in list [H|Lc] is N if H is even, and the number of even elements in list Lc is M, and N is M+1.
(2b) The number of even elements in list [H|Lc] is N if H is not even (or H is odd), and the number of even elements in list Lc is N. [Notice that N doesn't change if H is odd.]
I'll leave the rendering of these two rules into Prolog as an exercise. You can use the Prolog negation functor, \+ to test if a number is not even, by \+ even(N). Or you can define an odd(N) :- N rem 2 =:= 1. predicate to use for that case.
If your Prolog system offers clpfd, you can use the meta-predicate tcount/3 in combination with even_truth/2. Here's how:
?- tcount(even_truth, [1,2,3,5,7,9,10], N_even).
N_even = 2.
Both predicates (tcount/3 and even_truth/2) are monotone and preserve logical-purity! This makes them very robust and enables you
to always get logically sound answers.
Consider the following more general query:
?- Xs = [_,_,_], tcount(even_truth, Xs, N_even).
Xs = [_A,_B,_C], N_even = 0, _A mod 2 #= 1, _B mod 2 #= 1, _C mod 2 #= 1 ;
Xs = [_A,_B,_C], N_even = 1, _A mod 2 #= 1, _B mod 2 #= 1, _C mod 2 #= 0 ;
Xs = [_A,_B,_C], N_even = 1, _A mod 2 #= 1, _B mod 2 #= 0, _C mod 2 #= 1 ;
Xs = [_A,_B,_C], N_even = 2, _A mod 2 #= 1, _B mod 2 #= 0, _C mod 2 #= 0 ;
Xs = [_A,_B,_C], N_even = 1, _A mod 2 #= 0, _B mod 2 #= 1, _C mod 2 #= 1 ;
Xs = [_A,_B,_C], N_even = 2, _A mod 2 #= 0, _B mod 2 #= 1, _C mod 2 #= 0 ;
Xs = [_A,_B,_C], N_even = 2, _A mod 2 #= 0, _B mod 2 #= 0, _C mod 2 #= 1 ;
Xs = [_A,_B,_C], N_even = 3, _A mod 2 #= 0, _B mod 2 #= 0, _C mod 2 #= 0.
Related
I would like to make a predicat reverse(N,Result) in Prolog.
For example:
reverse(12345,Result).
Result = 54321.
I have to use tail-recursion. I can use *, +, - and divmod/4 and that's all.I can't use list.
I can reverse a number < 100 but I don't find how to finish my code, I can't complete my code to reverse integers bigger than 100 correctly.
reverse(N,N):-
N <10,
N>0.
reverse(N,Result):-
N > 9,
iter(N,0,Result).
iter(N,Ac,Result):-
N < 100, !,
divmod(N,10,Q,R),
R1 is R*10,
Result is Q + R1.
Can I have some help please ?
Thanks you in advance.
I suggest the use of CLP(FD), since it offers declarative reasoning over integer arithmetic and a lot of Prolog systems provide it. Concerning the digit-reversal, I recommend you take a look at entry A004086 in The On-Line Encyclopedia of Integer Sequences. In the paragraph headed FORMULA, you'll find, among others, the following formulae:
a(n) = d(n,0) with d(n,r) = if n=0 then r else d(floor(n/10),r*10+(n mod 10))
These can be translated into a predicates by adding an additional argument for the reversed number. First let's give it a nice declarative name, say digits_reversed/2. Then the relation can be expressed using #>/2, #=/2, (/)/2, +/2, mod/2 and tail-recursion:
:- use_module(library(clpfd)).
digits_reversed(N,X) :-
digits_reversed_(N,X,0).
digits_reversed_(0,R,R).
digits_reversed_(N,X,R) :-
N #> 0,
N0 #= N/10,
R1 #= R*10 + (N mod 10),
digits_reversed_(N0,X,R1).
Note that digits_reversed/2 correspond to a(n) and digits_reversed_/3 corresponds to d(n,r) in the above formulae. Now let's query the predicate with the example from your post:
?- digits_reversed(12345,R).
R = 54321 ;
false.
The predicate can also be used in the other direction, that is ask What number has been reversed to obtain 54321? However, since leading zeros of numbers are omitted one reversed number has infinitely many original numbers:
?- digits_reversed(N,54321).
N = 12345 ;
N = 123450 ;
N = 1234500 ;
N = 12345000 ;
N = 123450000 ;
N = 1234500000 ;
N = 12345000000 ;
N = 123450000000 ;
.
.
.
Even the most general query yields solutions but you'll get residual goals as an answer for numbers with more than one digit:
?- digits_reversed(N,R).
N = R, R = 0 ; % <- zero
N = R,
R in 1..9 ; % <- other one-digit numbers
N in 10..99, % <- numbers with two digits
N mod 10#=_G3123,
N/10#=_G3135,
_G3123 in 0..9,
_G3123*10#=_G3159,
_G3159 in 0..90,
_G3159+_G3135#=R,
_G3135 in 1..9,
R in 1..99 ;
N in 100..999, % <- numbers with three digits
N mod 10#=_G4782,
N/10#=_G4794,
_G4782 in 0..9,
_G4782*10#=_G4818,
_G4818 in 0..90,
_G4818+_G4845#=_G4842,
_G4845 in 0..9,
_G4794 mod 10#=_G4845,
_G4794 in 10..99,
_G4794/10#=_G4890,
_G4890 in 1..9,
_G4916+_G4890#=R,
_G4916 in 0..990,
_G4842*10#=_G4916,
_G4842 in 0..99,
R in 1..999 ;
.
.
.
To get actual numbers with the query above, you have to restrict the range of N and label it after the predicate has posted the arithmetic constraints:
?- N in 10..20, digits_reversed(N,R), label([N]).
N = 10,
R = 1 ;
N = R, R = 11 ;
N = 12,
R = 21 ;
N = 13,
R = 31 ;
N = 14,
R = 41 ;
N = 15,
R = 51 ;
N = 16,
R = 61 ;
N = 17,
R = 71 ;
N = 18,
R = 81 ;
N = 19,
R = 91 ;
N = 20,
R = 2 ;
false.
If for some reason you don't want a constraints based solution, or if you using a Prolog system not supporting constraints, an alternative solution is:
reverse_digits(N, M) :-
( integer(N) ->
reverse_digits(N, 0, M)
; integer(M),
reverse_digits(M, 0, N)
).
reverse_digits(0, M, M) :- !.
reverse_digits(N, M0, M) :-
N > 0,
R is N div 10,
M1 is M0 * 10 + N mod 10,
reverse_digits(R, M1, M).
This solution can be used with either argument bound to an integer and leaves no spurious choice-points:
?- reverse_digits(12345, M).
M = 54321.
?- reverse_digits(N, 12345).
N = 54321.
?- reverse_digits(12345, 54321).
true.
But note that this solution, unlike the constraints based solution, cannot be used as a generator of pairs of integers that satisfy the relation:
?- reverse_digits(N, M).
false.
reverseNumber(N,R):-reverse_acc(N,0,R).
reverse_acc(0,Acc,Acc).
reverse_acc(N,Acc,R):- C is N mod 10, N1 is N div 10,
Acc1 is Acc * 10 + C,
reverse_acc(N1, Acc1,R).
I'm still new to Prolog, and I've encountered an error I have no idea how to fix.
I've written a simple exponentiation program that looks like this:
exp(b, 0, R) :- R is 1. % non-recursive case: exponent is 0
exp(0, e, R) :- R is 0. % non-recursive case: base is 0
exp(Base, Exponent, Result) :- % recurse if base and exponent are non-negative
Base >= 0,
Exponent >= 0,
E1 is Exponent-1,
exp(Base, E1, R1),
Result is Base*R1.
This compiles fine, but when I run it and give it a query like, say, exp(2, 4, X). I'm met with the following output:
?- exp(2, 4, X).
false.
Is there something I've done wrong? Or is it a matter of formatting the result in some way I'm unaware of?
You are confusing variables with atoms. It works as expected if you simple change the two nonrecusive clauses to:
exp(_, 0, 1).
exp(0, _, 0).
In fact, I recommend to change the whole program to use CLP(FD) constraints throughout:
exp(_, 0, 1).
exp(0, _, 0).
exp(Base, Exponent, Result):-
Base #>= 0,
Exponent #>= 0,
E1 #= Exponent-1,
exp(Base, E1, R1),
Result #= Base*R1.
Now for example the following at least yields a solution:
?- exp(2, X, 16).
X = 4
whereas we previously had:
?- exp(2, X, 16).
>=/2: Arguments are not sufficiently instantiated
Note also the most general query:
?- exp(X, Y, Z).
Y = 0,
Z = 1 ;
X = Z, Z = 0 ;
X = Z,
Y = 1,
Z in 0..sup ;
X = Z, Z = 0,
Y in 0..sup,
_G801+1#=Y,
_G801 in -1..sup .
I'm trying to implement a method to work as follows foo(5) = 5^4 + 4^3 + 3^2 + 2^1 + 1^0 = 701 using recursion. I've been trying to follow the logic but I keep getting errors. can someone guide me?
(define (foo n) ; size-n problem
( cond ( (= (- n 1) 0 ) ; stopping condition
0 ); return value
(else (+ ( expt n (- n 1) ) ( foo (- n 1) ) ) ))) ; size-m problems
If you tagged the question correctly, you want to answer this in Prolog, but your code fragment suggests you use lisp (or a language that I don't know).
In Prolog you write predictes. For your problem, there are two cases:
The case where N is less than or equal to zero, which is zero:
foo(N,0) :-
N =< 0,
!.
The inductive case when N is greater than 0. In that case we calculate foo for N-1 and add up N^(N-1):
foo(N,S) :-
N1 is N-1,
foo(N1,T),
S is T+N^N1.
You can simply write program containing the two cases:
foo(N,0) :-
N =< 0,
!.
foo(N,S) :-
N1 is N-1,
foo(N1,T),
S is T+N^N1.
And test it as follows:
?- foo(-1,S).
S = 0.
?- foo(0,S).
S = 0.
?- foo(1,S).
S = 1.
?- foo(2,S).
S = 3.
?- foo(3,S).
S = 12.
?- foo(5,S).
S = 701.
You can akso make th is predicate more safer from looping by adding a check for the inductive case:
foo(N,0) :-
N =< 0,
!.
foo(N,S) :-
N > 0,
N1 is N-1,
foo(N1,T),
S is T+N^N1.
Or you can further boost the predicate's performance using an accumulator:
foo(N,S) :-
foo(N,0,S).
foo(N,S,T) :-
N > 0,
!,
N1 is N-1,
Q is S+N^N1,
foo(N1,Q,T).
foo(N,S,S) :-
N =< 0.
This version also checks first whether N > 0 before N =< 0 because it is a much more likely scenario: after one N =< 0 we stop recursion whereas the N > 0 will be called N-1 times. Using an accumulator enables a optimization technique called tail recursion.
I've got stuck on this problem :
Given a, b and c three
natural numbers (such that 1<= a, b, c <= 10^9), you are supposed to find the last digit of the number a^b^c."
What I've firstly thought was the O(log n) algorithm for raising a at power n.
int acc=1; //accumulator
while(n>0) {
if(n%2==1)
acc*=a;
a=a*a;
n/=2;
}
Obviously, some basic math might help, like the "last digit" stuff :
Last_digit(2^n) = Last_digit(2^(n%4))
Where n%4 is the remainder of the division n/4
In a nutshell, I've tried to combine these, but I couldn't get on the good way.
Some help would really be apreciated.
The problem is that b^c may be very large. So you want to reduce it before using the standard modular exponentiation.
You can remark that a^(b^c) MOD 10 can have a maximum of 10 different values.
Because of the pigeonhole principle, there will be a number p such that for some r:
a^r MOD 10 = a^(p+r) MOD 10
p <= 10
r <= 10
This implies that for any q:
a^r MOD 10 = a^r*a^p MOD 10
= (a^r*a^p)*a^p MOD 10
= ...
= a^(r+q*p) MOD 10
For any n = s+r+q*p, with s < p you have:
a^n MOD 10 = a^s*a^(r+q*p) MOD 10
= a^s*a^r MOD 10
= a^((n-r) MOD p)*a^r MOD 10
You can just replace n= (b^c) in the previous equation.
You will only compute (b^c-r) MOD p where p <= 10 which is easily done and then compute a^((b^c-r) MOD p)*a^r MOD 10.
Like I mentioned in my comments, this really doesn't have much to do with smart algorithms. The problem can be reduced completely using some elementary number theory. This will yield an O(1) algorithm.
The Chinese remainder theorem says that if we know some number x modulo 2 and modulo 5, we know it modulo 10. So finding a^b^c modulo 10 can be reduced to finding a^b^c modulo 2 and a^b^c modulo 5. Fermat's little theorem says that for any prime p, if p does not divide a, then a^(p-1) = 1 (mod p), so a^n = a^(n mod (p-1)) (mod p). If p does divide a, then obviously a^n = 0 (mod p) for any n > 0. Note that x^n = x (mod 2) for any n>0, so a^b^c = a (mod 2).
What remains is to find a^b^c mod 5, which reduces to finding b^c mod 4. Unfortunately, we can use neither the Chinese remainder theorem, nor Fermat's little theorem here. However, mod 4 there are only 4 possibilities for b, so we can check them separately. If we start with b = 0 (mod 4) or b = 1 (mod 4), then of course b^c = b (mod 4). If we have b = 2 (mod 4) then it is easily seen that b^c = 2 (mod 4) if c = 1, and b^c = 0 (mod 4) if c > 1. If b = 3 (mod 4) then b^c = 3 if c is even, and b^c = 1 if c is odd. This gives us b^c (mod 4) for any b and c, which then gives us a^b^c (mod 5), all in constant time.
Finally with a^b^c = a (mod 2) we can use the Chinese remainder theorem to find a^b^c (mod 10). This requires a mapping between (x (mod 2), y (mod 5)) and z (mod 10). The Chinese remainder theorem only tells us that this mapping is bijective, it doesn't tell us how to find it. However, there are only 10 options, so this is easily done on a piece of paper or using a little program. Once we find this mapping we simply store it in an array, and we can do the entire calculation in O(1).
By the way, this would be the implementation of my algorithm in python:
# this table only needs to be calculated once
# can also be hard-coded
mod2mod5_to_mod10 = [[0 for i in range(5)] for j in range(2)]
for i in range(10):
mod2mod5_to_mod10[i % 2][i % 5] = i
[a,b,c] = [int(input()) for i in range(3)]
if a % 5 == 0:
abcmod5 = 0
else:
bmod4 = b % 4
if bmod4 == 0 or bmod4 == 1:
bcmod4 = bmod4
elif bmod4 == 2:
if c == 1:
bcmod4 = 2
else:
bcmod4 = 0
else:
if c % 2 == 0:
bcmod4 = 1
else:
bcmod4 = 3
abcmod5 = ((a % 5)**bcmod4) % 5
abcmod2 = a % 2
abcmod10 = mod2mod5_to_mod10[abcmod2][abcmod5]
print(abcmod10)
I'm struggling with this code right now. I want to determine whether an integer is divsible by 11. From what I have read, an integer is divisible to 11 when the sum (one time +, one time -) of its digits is divisible by 11.
For example: 56518 is divisible by 11, because 8-1+5-6+5 = 11, and 11 is divisible by 11.
How can i write this down in Haskell? Thanks in advance.
A number x is divisible by y if it's remainder when divided by y is 0. So you can just do
divisibleBy11 x = x `rem` 11 == 0
ifan I'm sure you know that in real life you would use mod or rem for this simple example, but the algorithm you are asking about is interesting. Here's a fun way to do it that emphasizes the functional nature of Haskell:
digits = map (`mod` 10) . takeWhile (> 0) . iterate (`div` 10)
divisible11 = (== 0) . head . dropWhile (>= 11) . iterate (reduce11 . digits)
where
reduce11 [] = 0
reduce11 (d:ds) = foldl combine d $ zip (cycle [(-), (+)]) ds
combine d (op, d') = d `op` d'
Surely, div and mod are faster, but why not? I assume the problem is converting a number to a list of digits:
toDigits = map (read . (:[])) . show
56518 is converted to a String "56518", and each symbol in the string (every digit) is converted to a string itself with map (:[]), at this point we have ["5","6","5","1","8"], and we read every single-digit string as an integer value: [5,6,5,1,8]. Done.
Now we can calculate the sum of digits this way:
sumDigits x = sum (zipWith (*) (cycle [1,-1]) (reverse (toDigits x)))
cycle [1,-1] makes an infinite list [1, -1, 1, -1, ...], which we pair with the reversed list of digits (toDigit x), and multiply elements of every pair. So we have [8, -1, 5, -6, 5] and its sum.
Now we can do it recursively:
isDivisible x
| x == 11 || x == 0 = True
| x < 11 = False
| x > 11 = isDivisible (sumDigits x)
How about...
mod11 n | n < 0 = 11 - mod11 (-n)
| n < 11 = n
| otherwise = mod11 $ (n `mod` 10) - (n `div` 10)