# Questions tagged [loop-invariants]

Properties that hold before and after every execution of a loop. Used to prove correctness of algorithms.

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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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### 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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### 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 ...
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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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### 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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### 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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### 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 ...
1 vote
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### Proof of algorithm correctness

I am studying about algorithm correctness and have enctountered this problem. ...
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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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### 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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### 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. ...
1 vote
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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 ...
1 vote
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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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### 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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### 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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### 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 ...
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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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### 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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### 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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### 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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### Proof that this sorting algorithm sorts the input [duplicate]

I'm given this "sorting" algorithm and now I'm supposed to prove, that if given an array of integers of length $n$, sort(A,0,n-1) will sort it. ...
1 vote
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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 ...
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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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### 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--; } ...
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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 , ...
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### 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 $i$ ...
1 vote
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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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### 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 vote
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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: ...
1 vote
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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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### 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 $t=1,2,...$ $(1)\;\; \text{Read}\;x_t\in\mathbb{R}^n$ ...
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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 ...
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### 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 : ...
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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 =...
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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 ...
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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 ...
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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 ...
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### 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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### 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): ...
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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 ...
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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): ...
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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 ...
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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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### 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
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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 ...
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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 ...
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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 ...
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I am given the following invariant: Invariant The greatest $i$ keys of an Array are always in sorted order on the last $i$ indices of the Array. I am supposed to ...