Here is a program to calculate the summation from X to Y of i using tail recursive logic in Prolog: predicates sum(integer, integer, integer) sum_aux(integer, integer, integer, integer) clauses sum(X, Y, Sum):- sum_aux(X, Y, Sum, 0). sum_aux(X, X, Sum, Sum):- !. sum_aux(X, Y, Sum, Acc):- NewX is X + 1, NewAcc is Acc + NewX, NewY is Y, sum_aux(NewX, NewY, Sum, NewAcc). This defines sum/3

1. Prolog
Section 4

2. Recursion
• A common method of simplification is to
divide a problem into subproblems of the
same type.
• Recursion is when a function calls itself.
• It is when a function (operation) exists on the
left and the right hand side of an equation.
• It should have a stop condition.

3. Factorial
Iterative
Recursive
X! = 1*2*3*…*X
= X*(X-1)*(X-2)*…*2*1
X!=
Factorial (X-1)
1
if X=0
X*(X-1)! otherwise

4. Factorial
Iterative prolog, there are no return values -> new
Recursive
•In
the result
int factorial(int X) variable forint factorial(int X)
{
}
factorial(X,Y) -> we’ll put the result in Y Basic step
{
int Y=1;
if(X==0)
•The if statements in C++ are converted into
for(int i=0;i<X;i++)
return 1;
rules in prolog
{
else
Y*=X-i;
return X*factorial(X-1);
}
}
General
return Y;
rule

5. Factorial
•We
Iterative can’t put an operation as a parameter to a
Recursive
rule. factorial(int X)
int factorial(int X)
int
factorial(X-1)  Z=X-1, factorial(Z)
{
{
int Y=1;
if(X==0)
•Bounded variables can’t be assigned a value
for(int i=0;i<X;i++)
return 1;
X=5
{
else
Y*=X-i;
return X*factorial(X-1);
If X is free
If X is bounded
}
} then this statement is
then this statement is
return Y;
assign statement
comparison statement,
}
and returns true or false

6. Factorial
(non-tailer recursion)
predicates
fact(integer,integer)
clauses
fact(X,Y):-X=0,Y=1.
fact(X,Y):- Z=X-1,
fact(Z,NZ),Y=NZ*X.
Recursive
int factorial(int X)
{
if(X==0)
return 1;
else
return
X*factorial(X-1);
}

7. Factorial
(non-tailer recursion)
Recursive
predicates
int factorial(int X)
fact(integer,integer)
Stack overflow {
clauses
(infinite loop)
if(X==0)
fact(0,1).
return 1;
else
return
X*factorial(X-1);
fact(X,Y):- Z=X-1,
fact(Z,NZ),Y=NZ*X.
}

8. Factorial
(non-tailer recursion)
predicates
fact(integer,integer)
clauses
fact(0,1):-!.
fact(X,Y):- Z=X-1,
fact(Z,NZ),Y=NZ*X.
Recursive
int factorial(int X)
{
if(X==0)
return 1;
else
return
X*factorial(X-1);
}

9. Tracing
fact(4,Y).
0
Y=24
Match with the second rule:
6
6
fact (4, 24 ):- Z=4-1=3, fact (3,NZ),Y=NZ*4.
Y
Match with the second rule:
fact (3,6):- Z=3-1=2, fact (2,NZ),Y=NZ*3.
Y
2
2
Match with the second rule:
fact (2,2):- Z=2-1=1, fact (1,NZ),Y=NZ*2.
Y
1
1
Match with the second rule:
fact (1,1):- Z=1-1=0, fact (0,NZ),Y=NZ*1.
Y
1
1
Match with the first rule:
fact (0,1).
Then Y=1

10. Tracing
fact(4,Y).
Y=24
Match with the second rule:
6
6
fact TheYproblem with non-tailer recursion is
(4, 24 ):- Z=4-1=3, fact (3,NZ),Y=NZ*4.
Match with the second rule: variable waiting for
that there is always a
fact (3,6):- Z=3-1=2, fact (2,NZ),Y=NZ*3. call
Y
2
2
the return value from the recursive
Match withto complete the computations
the second rule:
fact (2,2):- Z=2-1=1, fact (1,NZ),Y=NZ*2.
Y
1
1
Match with the second rule:
(lots of memory taken… not good!)
fact (1,1):- Z=1-1=0, fact (0,NZ),Y=NZ*1.
Y
1
1
Match with the first rule:
fact (0,1).
Then Y=1

11. Tailer recursion
• Put the recursive call at the tail.
• To convert from non-tailer to tailer recursion
we need:
– Auxiliary (helper) function
– Accumulator
• Instead of waiting for the returned value I’ll
accumulate the result, so each time I have a
recursive call I’ll be sending part of the result
until the computations are completed.

12. Factorial
(tailer recursion)
In C++ as if I’ll say:
Acc=1;
for(int i=X;i>0;i--)
{
Acc=Acc*i;
}

13. Factorial
(tailer recursion)
predicates
fact(integer,integer)
fact_aux(integer,integer,integer)
clauses
fact(X,F):-fact_aux(X,F,1).
fact_aux(0,F,Acc):- F=Acc,!.
fact_aux(X,F,Acc):NAcc=Acc*X, NX=X-1,
fact_aux(NX,F,NAcc).

14. Factorial
(tailer recursion)
predicates
fact(integer,integer)
fact_aux(integer,integer,integer)
clauses
fact(X,F):-fact_aux(X,F,1).
fact_aux(0,F,F):-!.
fact_aux(X,F,Acc):NAcc=Acc*X, NX=X-1,
fact_aux(NX,F,NAcc).

15. Factorial
(tailer recursion)
*
X
F
Acc
3
?
1
2
1
?
?
3
6
0
?
6
= NAcc
3
6
6
NX
2
1
0

16. Power
Iterative
Recursive
23 = 2*2*2
XN=X*X*…*X (N times)
XN =
X (N-1)
1
if N=0
X*XN-1 otherwise

17. Power
(non-tailer recursion)
predicates
power(integer,integer,integer)
clauses
power(X,0,1):-!.
power(X,N,P):-Z=N-1,
power(X,Z,NP),P=NP*X.
goal
power(2,4,X).

18. Tracing
power(2,4,P). P=16
Match with the second rule:
16
8
8
power(2,4, P ):-Z=4-1=3, power(2,3,NP),P=NP*2.
Match with the second rule:
4
4
power(2,3,P):-Z=3-1=2, power(2,2,NP),P=NP*2.
8
Match with the second rule:
power(2,2,P):- Z=2-1=1, power(2,1,NP),P=NP*2.
4
2
2
Match with the second rule:
power(2,1,P):- Z=1-1=0, power(2,0,NP),P=NP*2.
2
1
1
Match with the first rule:
power(2,0,1).
Then P=1

19. Power
(tailer recursion)
In C++ as if I’ll say:
Acc=1;//in case of multiplication initialize the accumulator
with 1 but in case of addition initialize it with zero
for(int i=N;i>0;i--)
{
Acc=Acc*???;
}

20. Power
(tailer recursion)
In C++ as if I’ll say:
Acc=1;//in case of multiplication initialize the accumulator
with 1 but in case of addition initialize it with zero
for(int i=N;i>0;i--)
{
Acc=Acc*X;
}

21. Power
(tailer recursion)
predicates
power(integer,integer,integer)
power_aux(integer,integer,integer,integer)
clauses
power(X,N,P):- power_aux(X,N,P,1).
power_aux(_,0,P,P):-!.
power_aux(X,N,P,Acc):Nacc=Acc*X, Z=N-1,
power_aux(X,Z,P,Nacc).

22. Power
(tailer recursion)
*
X
N
P
Acc
2
4
?
1
2
2
3
2
?
?
2
4
2
2
1
0
?
?
8
16
= NAcc
2
4
8
16
Z (new N)
3
2
1
0

23. Fibonacci
fib(0)=1
fib(1)=1
fib(X)=fib(X-1)+fib(X-2)
Ex:
X
F(X)
0
1
1
1
2
2
3
3
F(X)=F(X1)+F(X2)
4
5
5
8

24. Fibonacci
(non-tailer recursion)
predicates
fib(integer,integer)
clauses
fib(1,1):-!.
fib(0,1):-!.
fib(X,Y):M=X-1,N=X-2,
fib(M,B),fib(N,A),Y=A+B.

25. Fibonacci
(tailer recursion)
int fib (int X)
{
int first = 1;
int second = 1;
for(int i = X; i>1; i--)
{
int temp=first;
first = second;
second = temp + second;
}
return second;
}

26. Fibonacci
(tailer recursion)
predicates
fib(integer, integer)
fib_aux(integer, integer, integer, integer)
clauses
fib(X, Fib):fib_aux(X, Fib,1, 1).
fib_aux(1, Second,_, Second):-!.
fib_aux(X, Fib,First, Second):NewX = X - 1,
NewFirst = Second,
NewSecond = First + Second,
fib_aux(NewX, Fib, NewFirst, NewSecond).

27. Assignment
Write a program to calculate the summation
from X to Y of i