51 votes

Example of an algorithm that lacks a proof of correctness

Here is an algorithm for the identity function: Input: $n$ Check if the $n$th binary string encodes a proof of $0 > 1$ in ZFC, and if so, output $n+1$ Otherwise, output $n$ Most people suspect ...
Yuval Filmus's user avatar
19 votes
Accepted

Difference between Dependent type , refinement type and Hoare Logic

There is recent work by Paul-André Melliès and Noam Zeilberger that explores this. In particular the papers Functors are Type Refinement Systems and An Isbell Duality Theorem for Type Refinement ...
Derek Elkins left SE's user avatar
10 votes
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What was the major breakthrough between Hoare-Floyd logic and Scott–Strachey semantics?

I don't know why people didn't develop Hoare logics for lambda-calculi earlier. The first work to get this right was Honda et al's A Compositional Program Logic for Polymorphic Higher-Order Functions ...
Martin Berger's user avatar
10 votes

Example of an algorithm that lacks a proof of correctness

Most algorithms have not been proven correct in Hoare logic. The main reason is that such correctness proofs are extremely expensive as of Jan 2017, probably by several orders of magnitude in ...
Martin Berger's user avatar
9 votes

What is a predicate transformer?

The predicate transformer is just a formalization of the idea that you can produce a precondition given a program and its postcondition. For example, given a program ...
chepner's user avatar
  • 209
8 votes
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Relation between Hoare Type Theory and pointers

A pointer to a variable creates an alias. When the alias is modified, the corresponding variable is modified as well. Therefore, the rule for an assignment in Hoare's logic is not just update the ...
Alexander Kogtenkov's user avatar
8 votes
Accepted

What does it mean to "strengthen the precondition and weaken the postcondition" in Hoare logic?

Condition $A$ is stronger than condition $B$ if $A$ implies $B$. That is, if $B$ holds in all situations in which $A$ holds. Conversely, if $A$ is stronger than $B$, then $B$ is weaker than $A$. ...
David Richerby's user avatar
7 votes

Intuitive explanation of Hoare assignment axiom

Hoare Logic proceeds backwards. It is a method to compute a precondition such that the desired postcondition holds. In fact, the inference rules given in your standard Hoare Logic deductions compute ...
Lee's user avatar
  • 1,097
5 votes

Example of an algorithm that lacks a proof of correctness

Problem: Print "Yes" if every even number ≥ 4 is the sum of two primes, and "No" if there is an even number ≥ 4 that is not the sum of two primes. Algorithm: Print "Yes" Most people think that the ...
gnasher729's user avatar
  • 29.4k
5 votes

Example of an algorithm that lacks a proof of correctness

This is tied to the incompleteness of the underlying logic. Indeed, Hoare logic usually contains a weakening or "pre-post" rule $$ \dfrac{ P \implies P' \qquad \{P'\}c\{Q'\} \qquad Q' \implies Q' }{ \{...
chi's user avatar
  • 14.5k
5 votes
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Find the loop invariant of the given while loop

At the end of each iteration you have $$ \forall j: 0 \leq j < i \implies b[j] = a[j+1] $$ which you can prove by induction. Thus, when the algorithm terminates, $i$ has reached $n-1$ and you have ...
Sebastian Oberhoff's user avatar
4 votes
Accepted

Proving the loop invariant for a simple program in Hoare logic

Your invariant, together with the negation of the loop condition, is not strong enough to imply your postcondition. Try adding an additional conjunct to the invariant which, together with $\neg\ i<...
Klaus Draeger's user avatar
4 votes
Accepted

Hoare logic - total correctness of loops

They are equivalent, in the sense that every time you can apply the textbook rule you can also apply your own rule, and vice versa. The invariant for the two rules is similar, but not the same. ...
chi's user avatar
  • 14.5k
4 votes

Finding weakest precondition

Weakest precondition (WPC) can be computed with a procedure that takes your program, as well as the given postcondition (in this case, x=y), as inputs, and applies ...
ivcha's user avatar
  • 540
4 votes
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what is the 'x :=' part mean in a hoare triple?

Short answer, it's an assignment, but it's not part of Hoare logic. It means whatever it means in the programming language you're using. A Hoare triple in general looks like $\{P\}\; C\; \{Q\}$ (...
Luke Mathieson's user avatar
4 votes
Accepted

The difference between a Hoare Triple/Assertion and a Typed Function

You are putting your finger on angular stone of program verification. At a very rough and high level you can think a derivation in Hoare logic as proving a property, thing which can somehow be ...
Sn0w's user avatar
  • 364
4 votes
Accepted

Hoare logic, proving conjunction rule from basic rules, possible or not?

Your reasoning is fine as far as the derivability of this rule. To respond to the latter part of your last paragraph, if the rules you've listed are all the rules available, then those are the only ...
Derek Elkins left SE's user avatar
3 votes
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Continuation-passing style: what is meant by "CPS'ing"?

I had only a quick look at the paper, but I believe that they are referring to moving from $$ t_1 \to t_2 \cdots \to t_n $$ to $$ t_1 \to t_2 \cdots \to \lnot \lnot t_n $$ where $\lnot t = (t \...
chi's user avatar
  • 14.5k
3 votes
Accepted

How to prove the equivalence between Hoare and Floyd assignment axioms?

but when I try $F = G[v/e]$ then from $\exists v' (F[v/v'] \land v=e[v/v'])$ I can't obtain $G$. We can assume $v'$ not free in $G$. Then, we have $$ \begin{array}{ll} & \exists v' (F[v/v'] \...
chi's user avatar
  • 14.5k
3 votes
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Check whether loop invariants are correct?

Rather than telling you whether your specific invariants are correct, let me teach you the procedure for how you can check whether your invariants are correct on your own. Basically, you break it ...
D.W.'s user avatar
  • 158k
3 votes

Example of an algorithm that lacks a proof of correctness

Any algorithm that is correct but we don't know how long it takes to run can be transformed into an algorithm that stops in a guaranteed amount of time but we aren't sure if it is correct. For ...
Dan Brumleve's user avatar
3 votes

Intuitive explanation of Hoare assignment axiom

The precondition is in fact the weakest (liberal) precondition that guarantees a valid Hoare triple for that postcondition and assignment statement. ("liberal" because termination is not considered.) ...
Kai's user avatar
  • 855
3 votes
Accepted

Developing invariants for comparing two strings

I would not simplify $(2)$ at all. Just apply the rules for if, =, and command composition. Use weakening (pre- or post- rules) ...
chi's user avatar
  • 14.5k
3 votes
Accepted

Hoare-Logic: Requirements for imperfect data types

No, when using Hoare logic properly, you make sure to account for integer overflow etc. and model the full semantics of the programming language. It is possible to use Hoare logic along the lines you ...
D.W.'s user avatar
  • 158k
3 votes
Accepted

Understanding Hoare Logic Axioms

Here are some answers and hints to pursue your reflexions. $\mathtt{Assignment}$ What $\phi([x \leftarrow E])$ means. You said yourself that the semantic of the precondition confuses you that was ...
Sn0w's user avatar
  • 364
3 votes

The difference between a Hoare Triple/Assertion and a Typed Function

To be frank, it much more difficult to see how types and pre-/post-conditions are similar than how they differ. In Hoare Logic, for the Hoare triple $\{A\}\ f\ \{B\}$, $A$ and $B$ would be predicates ...
Derek Elkins left SE's user avatar
3 votes
Accepted

How to prove a side effect in a function

If the function has two outputs, a standard way to represent this is as a function $f: A \to (B \times C)$, i.e., $f$ outputs a pair of an AST and a symbol table. If the function updates an existing ...
D.W.'s user avatar
  • 158k
3 votes

How to prove a side effect in a function

It depends on how you model the system and what proof approach you're using. For early versions of Hoare Logic, there isn't really any notion of "scope", so there's absolutely nothing special you need ...
Derek Elkins left SE's user avatar
3 votes
Accepted

The Law of Excluded Miracle in the language of guarded commands

The answer I can provide is I was reusing the definition of weakest precondition from the IMP language, but in fact I need a stronger definition. This definition looks as follows: ...
user1868607's user avatar
  • 2,194
3 votes
Accepted

I cannot find an invariant for the following program

Figure out what the value of y is, depending on x, a, and c. Prove that your formula is correct before the first iteration, and prove that if it is true before an iteration then it is also true after ...
gnasher729's user avatar
  • 29.4k

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