Questions tagged [loop-invariants]
Properties that hold before and after every execution of a loop. Used to prove correctness of algorithms.
84
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MergeSort's merge function loop invariant
I am reading a proof of correctness for the MergeSort Algorithm. This is the code for the MergeSort and the Merge function:
The correctness of the MergeSort function is easy to prove since the two <...
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1
answer
59
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Struggling to find loop invariant in power function
I am struggling to find a good loop invariant for the following function, which returns a^b where a is a real number and b is a natural number:
...
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1
answer
209
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Finding the loop invariant for Array Reversal
I've been assigned to find the loop invariant for the following code:
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1
answer
70
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Why is there a 'true' statement as precondition in the following loop invariant
Precondition: true
int i = 0;
while (i < a.length && a[i] != x) {
i++;
}
Postcondition: (∀ j : 0 ≤ j < i : a[j] ≠ x) ⋀ (i = a.length ∨ a[i] = x) }
As I read it, the program has no ...
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1
answer
58
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Are there invariants in text processing problems?
I have read that when programming it is good to identify relations -- invariants -- that should hold true throughout the program, and it is good to insert assertions throughout the code to check that ...
2
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1
answer
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What loop invariant can be used for this loop? (Algorithm to check if a sequence is stack permutable)
I've written a function that gets a permutation and checks if that permutation can be reached using a stack from an input sequence which is <1,2,3,...,n>. (we take elements from left)
For ...
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1
answer
45
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Proving loop invariant for a possibly nonterminating while loop
I started studying through Aho, Ullman - Foundations of Computer Science as a free time exercise. In the second chapter about loop invariants and inductive proofs, there is a starred exercise.
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2
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337
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Proving the correctness of an algorithm
What is the logic behind using a loop invariant proof for proving the correctness of an algorithm? How is it proved that using the loop invariant proof indeed proves the correctness of a loop?
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What could be a good loop invariant for this?
I am currently new and learning about loop invariants. I have come across this pseudocode where the goal is to shift the elements inside an array with N size in a clockwise direction by K steps.
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1
answer
80
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Does loop invariant has nothing to do with termination?
Consider the following program fragment:
var x, y: integer;
x := 1;
y := 0;
while y < x do
begin
x := 2*x;
y := y+1
end;
For the above fragment, which ...
2
votes
1
answer
254
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Proof of algorithm correctness
I am studying about algorithm correctness and have enctountered this problem.
...
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0
answers
1k
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loop invariant of selection sort algorithm
I am asked to write an C code Selection Sort algorithm and use loop invariant to show its correctness.
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1
answer
134
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What is loop invariant for this loop?
As far as i understand all loops have a loop invariants. My understanding is that loop invariant is an argument that is true at the beginning of the loop block as well as at the end of the loop.
For ...
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1
answer
93
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understanding loop invariants
I have a bit of experience with loop invariants but I'm not really clear on them. I'm trying to learn them through an Algorithm I have.
...
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2
answers
866
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quicksort invariant 3 conditions with loop invariant
in studying Quicksort using the book "Introduction to Algorithms" by Cormen, Leiserson, Rivest and Stein, they describe in order to show correctness, an invariant must hold for the 3 stages of the ...
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1
answer
118
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Hoare Logic for Factorial
I came across this hoare logic for factorials but I don't quite understand it. We multiply F and X but we're not adding up all values of F so how do we get the sum/factorial at the end?
Precondition: ...
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0
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174
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Loop invariant of a search algorithm
I have to come up with a proof of correctness of the following algorithm: GuardedSearch(A; v)
Input: an array A of n numbers and a number v
Output: an index i such that A[i] = v, or NotFound if no ...
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1
answer
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Is there a model of ZF¬C where some program always terminates but has no loop variant?
Wikipedia has a proof that every loop that terminates has a loop variant—a well-founded relation on the state space such that each iteration of the loop results in a state that is less than the ...
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1
answer
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I cannot find an invariant for the following program
I have the following:
(|$y=0; x=c$|) while(x > 0){y=y+a; x=x-1;} (|$y= a*c$|)
This seems like a fairly simple program and I can intuitively tell that the post ...
2
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2
answers
369
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Implementing Tarjan's strongly connected components algorithm in a language without exceptions or undefined behavior
I asked this question on Stack Overflow, but I have not obtained an actual answer to the question.
Tarjan's strongly connected components algorithm is stunningly beautiful, and inexpressible in a ...
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1
answer
47
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How do I "apply" a loop invariant?
A homework problem I have asks that I "state" a loop invariant, then "prove" it, then "apply" it. I get state and prove, but my textbook doesn't explicitly state how to "apply" a loop invariant.
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1
answer
59
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When are invariants true inside of a loop?
I am new to the idea of invariants and hope to find more information about it.
It can make an algorithm or loop show a high level of certainty / confidence that the code is correct, such as in the ...
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1
answer
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Loop invariant initialisation confusion
Consider the algorithm LastMatch below, which returns the offset (shift) of the last
occurrence of the pattern P in text T, or -1 if P does not occur in T:
...
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3
answers
394
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Dealing with test condition '=' for a while loop when determining a bound function/loop variant
The following is the definition of what a bound function for a while loop must satisfy:
The bound function is an integer-valued, total function of some of the inputs, variables and global data that ...
2
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1
answer
88
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How to prove a bound function for a sequence of numbers?
Let $G_n$ be defined by
$$G_n = \begin{cases} 1 & n=0 \\
2 & n = 1 \\
3 & n = 2 \\
4 & n = 3 \\
2G_{n-1}-2G_{n-3}+G_{n-4} & n\geq4
\end{cases}$$
How can I prove that $f(n) = n$...
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1
answer
96
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Loop Invariant Code Motion - am I missing something?
I've been implementing LICM for a project, and came upon a strange observation.
Let's say we have a loop
int i = 10;
while (i > 0) {
a = 2;
i--;
}
...
0
votes
1
answer
633
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Loop-Invariant for a program fragment
Q) Consider the following program fragment:
var x,y:integer;
x:=1; y:=0;
while y < x do
begin
x:=2∗x;
y:=y+1;
end;
For the above fragment , ...
0
votes
1
answer
153
views
Finding invariant when detecting a cycle
Let consider a connected graph $G = (V, E)$ which is not oriented. One way to detect a cycle in such a graph is :
Create an array : seen of size $\mid V \mid$ with seen[i] = false for all $i$
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2
answers
288
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Proving correctness of the Newton's Method for finding the square root of a number
I'm trying to prove the correctness of this simple square root calculation algorithm using SPARK:
...
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1
answer
459
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Find an invariant in the minimum algorithm
I have the following simple algorithm to find the smallest element of an array $A$ of numbers:
...
1
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2
answers
1k
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Iterative Fibonacci algorithm correctness proof, finding loop invariants
The algorithm take in an integer $n$ and outputs the $n$th number in the Fibonacci sequence ($F_n$). The sequence starts with $F_0$. I am trying to prove the correctness assuming valid input:
...
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1
answer
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How to understand out of bound in the following theoretical context?
Consider the following initialization step of loop invariant for merge procedure
Initialization: Prior to the first iteration of the loop, we have
$k=p$, so that the subarray $A[p .. k - 1]$ is ...
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0
answers
33
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Equivalent Algorithm with Sharman Morrison inversion
I am trying to invert a matrix using Woodbury identity. The inversion using Cholesky decomposition has the following pseudo-code:
For $t=1,2,...$
$(1)\;\; \text{Read}\;x_t\in\mathbb{R}^n$ ...
1
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0
answers
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Does code motion/hoisting happen on if statements/branches? [closed]
Consider the following C code:
for(int i=0; i<100000; i++) {
if (operation == 0) {
a[i] *= 2;
} else {
a[i] += 1;
}
}
Would a real ...
2
votes
1
answer
353
views
Loop Invariant summation
So I am stuck:
I have this algorithm from which I need to find a loop invariant but I just can't get my head around it :
...
1
vote
1
answer
203
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Finding a strong loop invariant
Pseudocode:
if n < 2
return false
while n != 1
if n % 2 != 0
return false
n = n/2
return true
The loop will terminate when n is odd. If n =...
2
votes
1
answer
2k
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Is Loop Invariant Proof a form of Induction?
As far as I see, what computer scientists refer to as loop invariant proofs are exact replicas of induction proof. Is it true? Can I state that loop invariant proof implies an induction? Is there a ...
3
votes
2
answers
381
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Invariant on "Find K Closest Elements" problem
I run across this problem:
Given a sorted array, two integers k and x, find the k closest ...
1
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1
answer
195
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Loop termination - Loop invariant
(x >= 0 && y >= 0)
q = 0;
r = x;
while ( r >=y ) {
r = r - y;
q = q + 1;
}
(x = q*y +r) && (r >= 0) && (r < y)
For ...
2
votes
1
answer
311
views
Find the loop invariant of the given while loop
I don't know how to find a loop invariant. I'm not sure where to start. Can anyone find the loop invariant of the given program and explain your method please.
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1
answer
80
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Inference rules for deriving invariants in Hoare logic
The following algorithm is supposed to compare two strings $S_1$ and $S_2$ ("/\" for empty string):
...
0
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0
answers
126
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Loop invariant for
The programme returns the number of digits of an integer $n>0$.
I still have some difficulties to understand the difference between the loop invariant condition and what the loop should actually ...
4
votes
1
answer
112
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Developing invariants for comparing two strings
The following algorithm is supposed to compare two strings $S_1$ and $S_2$ ("/\" for empty string):
...
0
votes
1
answer
569
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Loop invariant condition IsPrime program
I'm new to the concept of loop invariant and I'm trying to figure out the loop invariant for a program that returns if an integer is prime and, if not, one possible factorization.
My intuition is that ...
2
votes
1
answer
1k
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Proving correctness of an iterative Fibonacci algorithm
One of the questions in the problem sets that I'm struggling in is this specific number that asks me to prove an iterative Fibonacci algorithm. The algorithm is written below:
...
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0
answers
409
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Counting the number of occurences - loop invariant
I'm trying to come up with loop invariant for the following program.
k = a[0]
m = 1
p = 1
while p < n:
if a[p] == k:
m += 1
p += 1
return m
I ...
1
vote
1
answer
834
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How to get Loop invariants to prove program is correct in Hoare logic
start off stating the definition of a loop invariant. It is a condition that must be true at the start of the loop and must be true after each iteration of the loop. I understand what it is but i have ...
0
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2
answers
46
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Deterministic Nonfinite Automaton - Longest Block of Ones
How to Think About Algorithms - Jeff Edmonds
Exercise 2.2.2
The problem asks to find $\sum$, Q, δ, s, and F of this program if we see it as an automaton. Edmonds characterized this as deterministic ...
3
votes
0
answers
675
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Expression of the weakest precondition of a while loop
I am interested in computing weakest preconditions (WP) of loops.
If I refer to Wikipedia "Predicate Transformer semantics", the WP for e.g. total correction of a loop annotated with an invariant I ...
1
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0
answers
127
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Hoare correctness proof for a recursive definition of multiplication
Given the program:
{y=y0 ^ y>=0}
z=0;
while (y>0){
z=z+x; (1)
y=y-1;
}
{z=x*y0}
I am having trouble finding the ...