# A Pandoc Filter for Typesetting Operational Semantics

### 2017-03-23

Recently I decided to make my life a little easier by simplifying the laborious, tedious math typing that I had to do from time to time. For me this is most relevant for typesetting the formal semantics of programming languages, where proof trees are interwoven with a variety of programming constructs. Here’s my initial intention: To typeset something like (in big-step semantics) $\frac{ \stackrel{\mathcal{E}_0} {\langle b, \sigma \rangle \downarrow \textbf{true}} \qquad \stackrel{\mathcal{E}_1} {\langle c_0, \sigma \rangle \downarrow \sigma''} \qquad \stackrel{\mathcal{E}_2} {\langle \textbf{while } b \textbf{ do } c_0, \sigma'' \rangle \downarrow \sigma'} }{ \langle \textbf{while } b \textbf{ do } c_0, \sigma \rangle \downarrow \sigma'}$ Instead of the bloated TeX syntax:

  \frac{
\stackrel{\mathcal{E}_0}
{\langle b, \sigma \rangle \downarrow \textbf{true}} \qquad
\stackrel{\mathcal{E}_1}
{\langle c_0, \sigma \rangle \downarrow \sigma''} \qquad
\stackrel{\mathcal{E}_2}
{\langle \textbf{while } b \textbf{ do } c_0, \sigma'' \rangle \downarrow \sigma'}
}{
\langle \textbf{while } b \textbf{ do } c_0, \sigma \rangle \downarrow \sigma'}

I could just write it more neatly, intuitively, without all those hexes in magic backslashes and braces:

  <b, sig> ! true                 -- E_0
<c_0, sig> ! sig''              -- E_1
<while b do c_0, sig''> ! sig'  -- E_2
----
<while b do c_0, sig> ! sig'

But, of course, preserving the nice-looking outcome (from either TeX or MathJax).

Besides natural derivations, we also got a lot of evidently agreeable conventions for typesetting general mathematics: parentheses (brackets, braces) are almost always paired; “\left” and “\right” are very often desired since $$\left[\frac{1}{2}\right]$$ is a bit less awkward than $$[\frac{1}{2}]$$; when referring to a function name, “\operatorname” looks aesthetically better; etc. Furthermore, if we write down some math notations in plain text, clearly, “=>” has to be a “$$\Rightarrow$$” and “|->” has to be a “$$\mapsto$$”; “|-” means “$$\vdash$$” and “|=” means “$$\vDash$$”; a standalone letter “N” is just $$\mathbb{N}$$; etc. Taking such matters into consideration, there could be some handy alternative syntax (I call it “lazybones’ syntax”) for typesetting formulas, at least for a specific field someone specializes in, where these conventions are consistent:

Typesetting outcome Syntax
Provability $$\Gamma \vdash \varphi$$ TeX:
\Gamma \vdash \varphi
Lazybones’:
Gam |- phi
Validity $$\Gamma \vDash \varphi$$ TeX:
\Gamma \vDash \varphi
Lazybones’:
Gam |= phi
Big-step semantics $$\langle b_0 \land b_1, \sigma \rangle \downarrow t$$ TeX:
\langle b_0 \land b_1, \sigma
\rangle \downarrow t
Lazybones’:
<b_0 && b_1, sig> ! t
Small-step semantics $$\sigma \vdash b_0 \land b_1 \to^* t$$ TeX:
\sigma \vdash b_0 \land b_1
\to^* t
Lazybones’:
sig |- b_0 && b_1 ->* t
Hoare logic $$\vdash \{A\} \textbf{if } b \textbf{ then } c_0 \textbf{ else } c_1 \{B\}$$ TeX:
\vdash \{A\} \textbf{if } b
\textbf{ then } c_0
\textbf{ else } c_1 \{B\}
Lazybones’:
|- <[A]> if b then c_0
else c_1 <[B]>
Denotational semantics $$\mathcal{C}[\![X:=a]\!] = \lambda \sigma . \eta (\sigma[X \mapsto \mathcal{A}[\![a]\!] \sigma])$$ TeX:
\mathcal{C}[\![X:=a]\!] =
\lambda \sigma . \eta
(\sigma[X \mapsto
\mathcal{A}[\![a]\!] \sigma])
Lazybones’:
C[[X:=a]] = lam sig . eta
(sig[X |-> A[[a]] sig])

For simplicity, we require that all such transformations are regular, i.e., “lazybones’ syntax” may be translated into plain TeX code using merely substitutions of regular expressions.

As a basic example, let’s consider angle brackets, for which we desire to type in simply “<” and “>” instead of “\langle” and “\rangle”. To avoid any ambiguity (with normal less-than or greater-than sign), it is required that such brackets have no internal spacing, that is, we write “<x, y>” rather than “< x, y >”, but “1 < 2” instead of “1<2”. Once we fix the transformation rules, implementation would be quite straightforward in Haskell with Text.Regex, if we want a pandoc filter to handle these substitutions for everything embedded in TeX math mode:

subLAngle s =
subRegex (mkRegex "<([^[:blank:]])") s $"\\langle \\1" subRAngle s = subRegex (mkRegex "([^[:blank:]])>") s$ "\\1\\rangle "

semType m@(Math mathType s) = do
let s' = subLAngle $subRAngle s return$ Math mathType s'
semType x = return x

So, why bother implementing a pandoc filter rather than just defining some TeX macros? Two main reasons:

1. TeX code is nontrivial to write. And you can’t get rid of the overwhelming backslashes quite easily. It would be very hard, if not impossible, to achieve the same thing you’d do with a pandoc filter. (I would say that TeX is rather fine for typesetting, but only for typesetting; actual programming/manipulating macros in TeX seems very awkward for me.)
2. TeX macros are just… code in TeX, which means that they’re not designated to work for non-DVI originated publishing formats, such as HTML with MathJax. So if you do a lot of heavy lifting with TeX macros, they won’t play well for display on the Web (unless you’re willing to convert math formulas into pictures).

Surely you can write pandoc filters in any language as you like (officially Haskell and Python). Nothing like plain TeX (despite the fact that it’s Turing-complete): filters are easier to implement, unrestricted in the way that they interact with structured documents, with a great deal of cool supporting libraries even to blow up the earth. Although here in this case, we are only using some no-brainer pattern matching to release our fingers from heavy math typing.

A proof-of-concept of this is in semtype.hs. While undocumented, all tricks it does should be clear from the comments and regular expressions themselves.