Help understand the notation

Question

I'm reading a computer science paper, this one: https://arxiv.org/abs/1608.03960. It uses certain notation in one of the illustrations (Figure 9, page 8 in the PDF version) that I don't understand how to interpret. Attaching here as well for better visibility:

$$ \text{EXEC} ~ \frac{\text{cmd}_1 : \text{CMD} \quad A_p, \text{cmd}_1 \implies A^\prime_p}{A_p, \langle \text{cmd}_1 ~ ; ~ \text{cmd}_2 ~ ; \ldots \rangle \implies A^\prime_p, \langle \text{cmd}_2 ~ ; \ldots \rangle} $$

I've tried to search for this online, but haven't come up with anything. I'd appreciate if someone explain it to me, particularly these questions:

1. Does this notation has an official or widely recognized name?

2. What's the relationships between expression above and below the horizontal line?

3. What's the meaning of $\Rightarrow$, long arrow?

4. What's the meaning of two expressions being separated by a comma, as in $A_p, cmd_1$? How is it different from two expressions separated by a blank space, as in $$\mathrm{cmd}_1, \mathrm{CMD}\ \ \ \ \ \ \ \ A_p, \mathrm{cmd}_1\,?$$

Here's another example from the paper:

$$ \text{VAR} ~\frac{x \in \text{dom}(A_p)}{A_p, x \implies A_p(x)}$$

5. What's the meaning of $dom(A_p)$ here?

Answer 1

Answering your questions one by one:

1. The rules are part of an operational semantics, which defines how to evaluate expressions.
2. The notation with the horizontal line is an inference rule: if all the logical statements above the line are true, then you can conclude that the statement below the line is also true.
3. The arrow is a predicate that indicates that the left hand side can be evaluated (or reduced) to the right hand side.
4. The expressions separated by a comma form a tuple; for example, $A_p, cmd_1$ is a pair consisting of $A_p$ and $cmd_1$.
5. $\mathrm{dom}(f)$ returns the domain of the function $f$, that is, the set of values $x$ for which $f(x)$ is defined.

In these rules, $A_p$ is a state object: a function that maps variable names to values. (Think of it a bit like the heap on which you can allocate objects, and where the variable names are pointers.) The VAR rule performs variable lookup in that state:

• If the statement above the line is true: $x \in \mathrm{dom}(A_p)$, that is, the function $A_p$ is defined for the argument $A_p(x)$,
• Then the statement below the line is true: $A_p, x \Longrightarrow A_p(x)$, that is, the expression $x$ in the state $A_p$ can be evaluated to $A_p(x)$, which is just the value of the function $A_p$ when you pass in the variable name $x$ as argument.

Hope that helps. For more details, lecture notes on operational semantics (e.g. these from Cambridge) will further explain the notation.

Answer 2

The expressions are in the general form of sequents, which are a way of writing down proofs. A rule of the form

$$\frac{\ X\ }{Y}$$

means that, if you can prove that $X$ is true, then you've proven that $Y$ is true. A word written to the left or right of the rule is just that rule's name. As such, proofs are read from the bottom. It's also possible to have multiple statements on the top line, as in $$\frac{\ X_1\quad X_2\ }{Y}\,.$$ This means that proving both $X_1$ and $X_2$ proves $Y$. The meaning is the same as $$\frac{\ X_1\land X_2\ }{Y}$$ but the point is that, in a full proof, you'll need to supply separate proofs of $X_1$ and $X_2$: $$\frac{\ \frac{\ \vdots\ }{X_1}\quad \frac{\ \vdots\ }{X_2}\ }{Y}\,.$$ Written out in a little more detail, you might have $$\frac{\ \frac{\ \vdots\ }{X_1}\quad \frac{\ \vdots\ }{X_2}\ }{\frac{\ X_1\land X_2\ }{Y}}\,,$$ which means the following. I have some proof of $X_1$ (which is just written as dots) and some proof of $X_2$ (ditto). This allows me to conclude that $X_1\land X_2$ is true and, from that, I can conclude $Y$. I think that covers your questions 1, 2 and half of 4. Your remaining questions are about the specific logic used in the paper, and I'll leave that to somebody with domain expertise.

Answer 3

Sorry for my inability to recall a better resource, plus I did not want to write out a textbook.

Here is a book which I learned the concept from using a hard copy of: Introduction to Bisimulation and Coinduction by Davide Sangiorgi

Luckily, I found a lecture slide which is based on that book. Take a look starting from page 56 titled "Mathematical induction".