Why does the denotational semantics for a while loop have a existence quantifier?

Question

I was going through these notes and they have the following operator on partial functions:

$$ \mathcal F^{k}(\bot)(\sigma) = \left\{ \begin{array}{ll} \alpha( [\![s]\!]\sigma ) & [\![b]\!]\sigma=true \\ \sigma & [\![b]\!]\sigma=false \\ \bot & [\![b]\!]\sigma=\bot \end{array} \right. $$

and supposedly it gives the following definition somehow (which ends up being the right denotation for while loops)

$$ \mathcal F^{k}(\bot)(\sigma) = \left\{ \begin{array}{ll} [\![s]\!]^i & \mbox{if $ \exists i \in [0,k)$ s.t. $ [\![b]\!] [\![s]\!]^i \sigma = false,\ [\![b]\!] [\![s]\!]^j \sigma = true,\ j \in [0,i) $}\\ \bot & \mbox{ o.w. } \end{array} \right. $$

the proof by induction is shouldn't be hard. However, it feels that it doesn't really answer the question for me intuitively because I could do the induction mechanically without understanding what is going on. Why isn't the definition something like:

$$ f(\sigma,i) = [\![s]\!]^i\sigma$$

then the induction is easy:

$$ f(\sigma,0) = [\![s]\!]^0\sigma = \sigma$$

then the inductive step:

$$ f(\sigma,k+1) = f \circ f{\circ}^k(\sigma)= f \circ[\![s]\!]^{k}\sigma$$

but also notice $f = [\![s]\!]^1\sigma$ by strong induction:

$$ f(\sigma,k+1) = [\![s]\!]^{k+1}\sigma$$

as required. But that doesn't seem at all to be the sort of argument being used (assuming what I wrote is right). Did I miss something?

Answer 1

The existential quantifier is misplaced. This is due to lack of reasonable notation for what needs to be expressed, namely: "if there is $i$ safisfying condition $C$, then use that $i$", where we make sure that at most one such $i$ exists. It would be better to write (and by the way, you might appreciate LaTeX's cases environment): $$ \mathcal F^{k}(\bot)(\sigma) = \begin{cases} [\![s]\!]^i & \text{if $i < k$ and $[\![b]\!] [\![s]\!]^i \sigma = \text{false}$ and, for all $j < i$, $[\![b]\!] [\![s]\!]^j \sigma = true$}\\ \bot & \text{otherwise} \end{cases} $$ It may be helpful to look at a simpler definition. Suppose we want a function $f : \mathbb{N} \to \mathbb{N}$ which maps an odd number to zero and an even number to half of itself. The following is a reasonable definition: \begin{align*} f(2 n) &= n\\ f(2 n + 1) &= 0 \end{align*} as is the following (where $m$ and $n$ range over natural numbers): $$ f(m) = \begin{cases} n & \text{if $m = 2 n$,}\\ 0 & \text{otherwise.} \end{cases}$$ The point is that precisely one branch applies, and in a unique way, as there can be at most one $n$ satisfying the condition $m = 2 n$. However, the following is not a reasonable definition: $$ f(m) = \begin{cases} n & \text{if $\exists n \in \mathbb{N} \,.\, m = 2 n$,}\\ 0 & \text{otherwise.} \end{cases}$$ The problem is that the $n$ outside the $\exists$ quantifier is completely unspecified. This is so because the $n$ inside the $\exists$ quantifier is bound and can therefore be renamed at will: $$ f(m) = \begin{cases} n & \text{if $\exists k \in \mathbb{N} \,.\, m = 2 k$,}\\ 0 & \text{otherwise.} \end{cases}$$ (And we do not rename the outer $n$ because it is not in the scope of $\exists$.) This makes it clear that the existential quantifier has no place there. It is important to understand variable bindings. In the above definition of $f$, the variable $n$ is bound in the entire case, not just in the condition.

From a very formal point of view, the map $f$ is defined here as a relation explaining how an argument of $f$ is related to its value. A definition of the form $$f(m) = \begin{cases} e_1(m,n) & \text{if $\phi_1(m,n)$} \\ e_2(m,n) & \text{if $\phi_2(m,n)$} \end{cases} $$ is equivalent to $$f(m) = \text{the $v$ such that $\psi(m,v)$} $$ where $$\psi(m,v) \iff (\exists n \,.\, \phi_1(m,n) \land v = e_1(m,n)) \lor (\exists n \,.\, \phi_2(m,n) \land v = e_2(m,n))$$ Now the quantifiers are justified. By the way, we could have used $\forall$ instead: $$\psi(m,v) \iff (\forall n \,.\, \phi_1(m,n) \Rightarrow v = e_1(m,n)) \land (\forall n \,.\, \phi_2(m,n) \Rightarrow v = e_2(m,n)).$$

Answer 2

The intuition is that the loop terminates at the $i$th iteration if the loop condition is false at that iteration but true at all earlier ones. If the loop condition is never false, then the loop doesn't terminate and the denotation is $\bot$.

However, it's written down wrongly. What it says is (and please pardon me for throwing away all the notation and working by analogy):

• The answer is $i$ if there exists some $i$ such that the loop condition becomes false at the $i$th iteration;
• otherwise, the answer is $\bot$.

The problem is that the variable $i$ only has meaning within the scope of "there exists some $i$ such that...", and it's referred to outside that scope. So what's written is equivalent to "The answer is $i$ if there exists some $q$ such that the loop condition becomes false at the $q$th iteration", to which the obvious response is "What's $i$?"

What it should say is:

• If there is some $i$ such that the loop condition becomes false at the $i$th iteration, the answer is $i$;
• otherwise, the answer is $\bot$.