Questions tagged [hoare-logic]
Questions about Hoare's logical framework for program correctness proofs and variants.
79
questions
21
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Difference between Dependent type , refinement type and Hoare Logic
I know little dependent type theory. From wikipedia :
A dependent type is a type whose definition depends on a value.
And from my Type theory course i recall that a dependent type is :
Family ...
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18
votes
5
answers
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Example of an algorithm that lacks a proof of correctness
We have Hoare logic. Why is it still possible that an algorithm is right but there is no proof that it's correct? Suppose the algorithm is expressed in C. Then we can argue step by step that it's ...
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12
votes
2
answers
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How to deal with arrays during Hoare-style correctness proofs
In the discussion around this question, Gilles mentions correctly that any correctness proof of an algorithm that uses arrays has to prove that there are no out-of-bounds array accesses; depending on ...
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8
votes
1
answer
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What was the major breakthrough between Hoare-Floyd logic and Scott–Strachey semantics?
I'm reading through a commentary on Milner's "The use of machines to assist in rigorous proof" by Mike Gordon. In this paper, he explains how LCF was born from the ideas of denotational semantics by ...
- 2,184
7
votes
1
answer
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Hoare logic - total correctness of loops
Consider a while loop of the form :
$\texttt{while (C) {S}}$
with $\texttt{C}$ the condition and $\texttt{S}$ the body of the loop.
Let $\texttt{I}$ and $\texttt{V}$ respectively be an invariant ...
- 73
6
votes
3
answers
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Intuitive explanation of Hoare assignment axiom
$\small\textit{''The obvious things are the most difficult to understand''}$
May be the question does not make sense, but let me ask it anyway.
The Hoare assignment axiom is
$$
\dfrac{}{\{Q[v \...
- 1,055
6
votes
1
answer
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Question about the formal proof of the inorder traversing
In Don Knuth's famous series of books, The Art of Computer Programming, section 2.3.1, he describes an algorithm to traverse binary tree in inorder, making use of an auxiliary stack:
T1 [Initialize.] ...
- 621
5
votes
3
answers
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Hoare triple for assignment P{x/E} x:=E {P}
I am trying to understand Hoare logic presented at Wikipedia,
Hoare logic at Wikipedia
Apparently, if I understand correctly, a Hoare triple $$\{P\}~ C ~\{Q\}$$ means
if P just before C, then Q ...
- 374
5
votes
2
answers
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What does it mean to "strengthen the precondition and weaken the postcondition" in Hoare logic?
Having learned a rough summary of Hoare logic (i.e. learning just the basic concept of Hoare triples and a few of the rules) I kept seeing a statement along these lines:
The rule of consquence ...
- 495
5
votes
1
answer
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What is a predicate transformer?
I'm reading Programming - The derivation of algorithms, and I want to understand the purpose of a predicate transformer. This is the excerpt (p. 14-15):
A more precise way in which constructs may be ...
- 183
5
votes
1
answer
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Relation between Hoare Type Theory and pointers
My understanding is that in Hoare Type Theory every imperative statement has a type of the form {Pre}res:T{Post} where T is the ...
- 53
5
votes
2
answers
666
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Why is the assignment rule the way it is in Hoare Logic?
Why is the assignment rule the way it is in Hoare Logic/Axiomatic Semantics?
I can't wrap my head around why the assignment rule is backwards from what I expected.
I understand Hoare logic is use to ...
- 3,000
5
votes
2
answers
189
views
proving program equivalence
I understand that the general problem of program equivalence is undecidable, but I'm wondering what approaches exist to tackle the problem? I am familiar with Hoare-style verification, but are there ...
- 51
4
votes
2
answers
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The difference between a Hoare Triple/Assertion and a Typed Function
I have been trying to wrap my head around applying Hoare Logic and am running into the question of how Hoare triples are any different from (simply) a typed function.
That is, say you have a typed ...
- 2,163
4
votes
1
answer
113
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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):
...
- 9,501
4
votes
1
answer
206
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Continuation-passing style: what is meant by "CPS'ing"?
I'm reading Dijkstra Monads for Free for a presentation I'll be doing and it's pretty meaty. One of the things that I keep running into is the term "CPS'ing". I've read a little bit on ...
- 357
4
votes
1
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How to find loop invariant from weakest precondition?
Consider this code:
Precondition:
Postcondition: rv == i <==> ∃i, 0 ≤ i ≤ a.length-1, a[i] == key
...
- 421
3
votes
1
answer
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Hoare logic, proving conjunction rule from basic rules, possible or not?
(This is HW.) Suppose I have these following proof rules given.
I am currently considering if I can prove the conjunction rule (given below) from the above given Hoare logic rules.
My answer would ...
- 189
3
votes
1
answer
494
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The Law of Excluded Miracle in the language of guarded commands
The definition of weakest precondition is familiar (let me use Isabelle's syntax here):
definition "wp c Q s ≡ ∃t. (c,s) ⇒ t ∧ Q t"
the weakest precondition ...
- 2,184
3
votes
1
answer
402
views
Why precondition strengtening is sound in Hoare logic
I have problem with understanding why precondition strengthening is sound rule in Hoare logic. The rule is:
$$
{P \implies Q, \{Q\} C \{X\} } \over {\{P\} C \{X\}}
$$
I really do not understand why ...
- 162
3
votes
1
answer
75
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What's the meaning of the $\top$ symbol in a Hoare triple?
I'm studying program verification and came across the following triple:
$$ \{\top\} \;P \; \{y=(x+1)\} $$
What's the meaning of the $\top$ symbol on the precondition? Does it mean $P$ can take any ...
- 43
3
votes
1
answer
41
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Formal semantics of a mutable/imperative stack
When introducing formal semantics for data structures, immutable stacks are a nice simple example :
$\mathit{is\_empty}(\mathit{create()})=\mathrm{True}$
$\mathit{is\_empty}(\mathit{push}(e, s))=\...
- 233
3
votes
0
answers
754
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 ...
- 131
3
votes
0
answers
96
views
Spot the formalism (some kind of process logic)
Consider the following specification technique.
A specification consists of a finite set of triples $\langle C, A, C' \rangle$,
where $A$ is the name of an action and $C, C'$ are conditions, that is,
...
- 5,359
2
votes
1
answer
348
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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2
votes
1
answer
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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:
...
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2
votes
2
answers
76
views
How to change the algorithm implication by changing the invariant?
I have this specification in GCL:
$[Ctx C: n\geqslant 0\ \wedge\ b:[0..n-1]\ \text{of int}$
$\{Q:\text{True}\}$
$sum,i:=0,0;$
$\{\text{Invariant}\ P: 0\leqslant i \leqslant n\ \wedge sum=\sum_{j=0}...
- 183
2
votes
1
answer
314
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Understanding Hoare Logic Axioms
Given these 5 axioms of Hoare Logic:
\begin{array}{cl}
\frac{}{\{\phi([x \leftarrow E])\}\ x := E\ \{\phi(x)\}} & \mathtt{Assignment}\\\\
\frac{\{\phi\}\ P_1\ \{\eta\} \quad \{\eta\}\ P_2\ \{\psi\...
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votes
1
answer
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Check whether loop invariants are correct?
I'm trying to prove some code is correct, using Hoare logic. How do I check whether my loop invariants are correct?
I'm asked to prove (using Hoare Logic) that the following program is valid:
...
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2
votes
1
answer
303
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Axiomatic Semantics and Postconditions
I'll preface this by saying that this IS a homework question.
However, when asked about how to solve it in class, (I believe) my professor was unable to complete it.
The question is:
Compute the ...
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2
votes
1
answer
49
views
Bottleneck in Hoare Logic unable to arrive at my {P} from {Q}
{Q} = {n>0}
C1 = i := 1;
C2 = c := 1;
C3 = p := 0;
{P} = {i<=n, p = fib(i-1), c = fib(i)}
My lack of understanding towards the rule of consequence in hoare logic is blocking me from find the ...
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2
votes
1
answer
385
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Hoare logic - invariant of loop [closed]
I am trying to prove partial corectness of following program:
...
user54001
2
votes
0
answers
160
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Hoare triple: Loop invariant and partial correctness
Below there is Hoare triple in which variable $a$ is an array of integers, $len$, $x, i$ are integer-valued variables, and $r$ is a Boolean-valued variable. I have to provide a loop invariant (using ...
- 21
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votes
0
answers
234
views
How to show equivalence of the Hoare assignment axiom vs Floyd assignment axiom?
I know from several sources that they are equivalent, but couldn't find a reasonable explanation. My first approach was to show their equivalence by using the logical consequence rule and both axioms, ...
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1
vote
2
answers
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How to prove a side effect in a function
I asked a question earlier about Saving to the Database, which was very general and about the requirements for a proof when you go through many layers of non-verified systems such as the network and ...
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1
vote
1
answer
77
views
what is the 'x :=' part mean in a hoare triple?
In Hoare logic, there's a thing called a Hoare triple, e.g.$$
\begin{array}{ccccc}
\{x = 2\} & & x := x+1 & & \{x = 3\}
\end{array}.
$$
What does '$x :=$' mean?
1
vote
1
answer
44
views
Hoare-Logic: Requirements for imperfect data types
Theoretically, Hoare-Logic let's one prove the correctness of an algorithm, given pre- and post-condition.
However, as far as I've seen it so far, one idealizes his data-types to a mathematical set ...
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1
vote
1
answer
238
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How to prove the equivalence between Hoare and Floyd assignment axioms?
How to show that these two axioms are equivalent:
1: $\{G[v/e]\} v:=e \{G\}$
2: $\{F\} v:=e \{\exists v' (F[v/v'] \land v=e[v/v'])\}$
I've tried with $G = \exists v' (F[v/v'] \land v=e[v/v']) $and ...
- 13
1
vote
1
answer
78
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Logic foundation for formal verification
What types of logic should one study as foundation before diving into the area of software verification? What I can think of are:
Hoare Logic (for proving correctness of imperative programs)
Linear ...
- 111
1
vote
1
answer
111
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How to determine the pre and post conditions of a program (Hoare-logic)
Problem
Below is a program named X:
y = 1;
while(y < x){
y = 10 * y;
}
What does this program X do? What are appropriate pre and post conditions?
(Assume ...
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1
vote
2
answers
3k
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how to solve Hoare logic problems
I'm having trouble proving Hoare logic questions as I'm not sure of the process that is taken to prove them. I understand that they're rules such as assignment axiom, pre-condition strengthening, post-...
1
vote
1
answer
359
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True Postcondition, with true Precondition
In my exam preparation I stumbled across the follwoing exercise regarding pre and postconditions:
As far as I understood the question, we need to express some condition for p, such that if that ...
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1
vote
1
answer
188
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Hoare's Axiom Scheme Precondition
I have a question about determining preconditions for Hoare's Axiom Scheme. For example, if we have P { x=2 } x==1 and we are trying to determine the precondition, ...
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1
vote
1
answer
184
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How to prove $c = a + b$ using Program Verification Techniques
I am trying to prove an elementary thing, but it seems at some point you get down to atoms where you can't prove anything else. This is why I am wondering about proving $c = a + b$, it seems like an ...
- 2,163
1
vote
1
answer
266
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How to use Hoare Logic to Prove this Assertion
Given this assertion in Hoare Logic:
\begin{align}
&\mathbf{\{p >= 0\}}\\
&s = 0 ; n = 1 ;\\
&\mathtt{while}\ (n <= p)\ \{\\
&\quad s = s + n ;\\
&\quad n = n + 1\\
&\}\\...
- 2,163
1
vote
1
answer
202
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 ...
user87320
1
vote
1
answer
152
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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: ...
- 167
1
vote
0
answers
27
views
Is true * true = true in Separation Logic?
I am trying to show that the following interference is unsound in terms of Separation Logic:
$$
(p_0 \implies p_1) \implies ((p_0 * q) \implies (p_1 * q))
$$
I came up with the following values for ...
1
vote
0
answers
112
views
How to prove program solves the problem?
I am preparing for the exam on Theory of Programming class. Now I am trying to solve the task from the sample paper:
Task description starts here
Given the following problem:
A problem is given by ...
- 69
1
vote
0
answers
73
views
Can we ignore the postcondition in the Hoare conditional rule when there is a return statement?
I'm proving the correctness of naive string matching using Hoare logic. I have the following pseudocode:
...
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