Questions tagged [loop-invariants]

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

65 questions
Filter by
Sorted by
Tagged with
33 views

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.
30 views

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 ...
15 views

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: ...
37 views

Proof that this sorting algorithm sorts the input

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. ...
85 views

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 ...
58 views

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 $f(n) = n$...
40 views

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--; } ...
77 views

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 , ...
37 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 $i$ ...
111 views

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: ...
51 views

Find an invariant in the minimum algorithm

I have the following simple algorithm to find the smallest element of an array $A$ of numbers: ...
236 views

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: ...
15 views

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 ...
27 views

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 $t=1,2,...$ $(1)\;\; \text{Read}\;x_t\in\mathbb{R}^n$ ...
34 views

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 ...
160 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 : ...
61 views

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 =...
218 views

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 ...
183 views

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 ...
86 views

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 ...
129 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. ...
61 views

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): ...
46 views

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 ...
90 views

Developing invariants for comparing two strings

The following algorithm is supposed to compare two strings $S_1$ and $S_2$ ("/\" for empty string): ...
143 views

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 ...
377 views

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: ...
136 views

Counting the number of occurences - loop invariant

I'm trying to come up with loop invariant for the following program. k = a m = 1 p = 1 while p < n: if a[p] == k: m += 1 p += 1 return m I ...
375 views

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 ...
36 views

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 ...
265 views

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 ...
64 views

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 ...
64 views

How to create an algorithm when given the invariant

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 ...
113 views

What is the loop invariant of the following function?

x = n; y = 0 while x >= b: x = x DIV b y = y + 1 return y This function takes in $n,b\in\mathbb{N}, n > 0, b > 1$, and returns $k\in\mathbb{N}$ ...
171 views

Loop invariant for adding 2 arrays storing binary value of integer

Question: Given 2 arrays of size $n$, $A = \langle a_1,a_2,\dots,a_n \rangle$ and $B = \langle b_1,b_2,\dots,b_n\rangle$, where $A$ and $B$ store the binary value of an integer. Find $A + B$ and ...
267 views

Loop invariant for a while loop

I need help understanding why the two statements I have underlined with red are correct. I am confused as to how we can get $n + 1$. And how $i \le n$ is true when it says $i < n$ in the while.
525 views

Loop invariant for a division algorithm

I'm having problems trying to understand the concept of loop invariants. I have the following code, where M and N are predefined constants. ...
272 views

Finding a loop invariant

the following pseudo code returns the nth Fibonacci number: ...
51 views

Verification condition in case of array theory

As far as i can understand the problem of checking safety property, can be reduce to solving for inductive invariants in VC's and there is following parts to it: Generate VC's from the program. Solve ...
817 views

Proving the loop invariant for a simple program in Hoare logic

I was studying loop invariant and came across Tomas Petricek's example. Here's the equivalent(I believe) program after I revised it a bit for proof in Hoare logic: ...
680 views

Finding the weakest precondition and invariant of while loop

I am trying to find the weakest precondition and invariant of this while loop. I have read and am trying to understand the concept of calculating preconditions based on postconditions. I was wondering ...
102 views

What does the algorithm calculate?

We have the following ...
742 views

Correctness of Dijkstra's algorithm

This question is about the correctness proof of Dijkstra's algorithm in the third edition of Introduction to Algorithms by Cormen et al. (pages 660–661). The proof makes a case that considering path \$...
1k views

Proof of correctness of algorithms (induction)

I am reading Algorithm's Design Manual by S.Skiena and I have a hard time understanding and proving the correctness of algorithms. I should use proof by induction and when we talk about summations and ...
297 views

Prove correctness of iterative LCM algorithm

I have been trying to prove the following algorithm, without success. Here is the C-Style pseudocode: ...
190 views

How to use the concept of loop invariant to reduce errors in loops?

Most of time while writing loops I usually write wrong boundary conditions(eg: wrong outcome) or my assumptions about loop terminations are wrong(eg: infinitely running loop). Here is an small example ...
842 views

How to find loop invariant from weakest precondition?

Consider this code: Precondition: Postcondition: rv == i <==> ∃i, 0 ≤ i ≤ a.length-1, a[i] == key ...
1k views

Finding a good loop invariant for a powering procedure

Consider the following algorithm for computing integer powers: ...
765 views

Proving correctness of a loop that calculates the sum of array

I am trying to mathematically prove that the following program is correct: ...