algorithm,logic,tail-recursion,eiffel , Prove two algorithms are identical

Question:

Tag: algorithm,logic,tail-recursion,eiffel

I am supposed to show that two algorithms execute identical statements in identical order. One is a tail recursive version of the other. They are written in Eiffel.

``````tail_rec(x:instance_type):result_type is
local
y:instance_type;
do
if b(x) then
Result :=c(x)
else
y:=d(x);
Result:=tail_rec(y)
end
end
``````

Then the non-tail recursive version.

``````non_rec(x:instance_type):result_type is
do
from until b(x) loop
x:= d(x)
end;
Result:= c(x)
end
``````

where b(x), c(x), and d(x) are any functions of type BOOLEAN, result_type, and instance_type respectively.

How are these two algorithms similar and how do they execute identical statements in identical order?

By substituting all flow-of-control constructs with `goto`s, (essentially turning the code from `Eiffel` into pseudocode,) and allowing `if` statements to execute only `goto`s, it can be shown that both functions end up consisting of the exact same set of instructions.

Let me begin by copying the original `tail_rec` here for convenience:

``````tail_rec(x:instance_type):result_type is
local
y:instance_type;
do
if b(x) then
Result := c(x)
else
y := d(x);
Result := tail_rec(y)
end
end
``````

First, get rid of Eiffel's silly `Result :=` construct and replace it with `return` for convenience. (Otherwise, we are going to have to add more `goto`s, and frankly, the fewer the better.)

``````tail_rec(x:instance_type):result_type is
local
y:instance_type;
do
if b(x) then
return c(x)
end
y := d(x);
return tail_rec(y)
end
``````

Substitute the `if`-`then`-`end` with `if`-`then goto`:

``````tail_rec(x:instance_type):result_type is
local
y:instance_type;
do
if not b(x) then goto label1
return c(x)
label1:
y := d(x);
return tail_rec(y)
end
``````

Replace tail recursion with another `goto`:

``````tail_rec(x:instance_type):result_type is
do
label0:
if not b(x) then goto label1
return c(x)
label1:
x := d(x);
goto label0
end
``````

Replace `if not b(x)` with `if b(x)`:

``````tail_rec(x:instance_type):result_type is
do
label0:
if b(x) then goto label1
x := d(x);
goto label0
label1:
return c(x)
end
``````

Similar transformations to `tail_rec` should turn it into the exact same thing.

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