---
title: "mathml: Translate R expressions to MathML and LaTeX"
author: "Matthias Gondan and Irene Alfarone
         (Universität Innsbruck, Department of Psychology)"
date: "2024-01-30"
output: rmarkdown::html_vignette
bibliography: bibliography.bibtex
preamble:
- \usepackage{cancel}
- \usepackage{amsmath}
Vignette: >
  %\VignetteIndexEntry{mathml: Translate R expressions to MathML and LaTeX}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include=FALSE}
library(mathml)
```

The R\ extension of the markdown language [@Xie2018;@rmarkdown] enables
reproducible statistical reports with nice typesetting in HTML, Microsoft Word,
and LaTeX. Moreover, since recently [@R, version 4.3], R\'s manual pages 
include support for mathematical expressions [@Sarkar2022;@Viechtbauer2022],
which is already a big improvement. However, except for special cases 
such as regression models [@equatiomatic] and R's own plotmath annotation, 
rules for the mapping of built-in language elements to their mathematical 
representation are still lacking. So far, R\ expressions such 
as `pbinom(k, N, p)` are printed as they are and pretty mathematical formulae 
such as \(P_{\mathrm{Bi}}(X \le k; N, p)\) require explicit LaTeX commands
like `P_{\mathrm{Bi}}\left(X \le k; N, p\right)`. Except for very basic use
cases, these commands are tedious to type and their source code is hard to
read.

The present R\ package defines a set of rules for the automatic translation of
R\ expressions to mathematical output in RMarkdown documents [@Xie2020] and
Shiny Apps [@Chang2022]. The translation is done by an embedded Prolog
interpreter that maps nested expressions recursively to MathML and
LaTeX/MathJax, respectively. User-defined hooks enable extension of the set of
rules, for example, to represent specific R\ elements by custom mathematical
signs.

The main feature of the package is that the same R\ expressions and equations
can be used for both mathematical typesetting and calculations. This saves time
and potentially reduces mistakes, as will be illustrated below.

Similar to other high-level programming languages, R is homoiconic, that is,
R\ commands (i.e., R\ "calls") are, themselves, symbolic data structures that
can be created, parsed and modified. Because the default response of the
R\ interpreter is to evaluate a call and return its result, this property is not
transparent to the general user. There exists, however, a number of built-in
R\ functions (e.g., `quote()`, `call()` etc.) that allow the user to create R\ calls
which can be stored in regular variables and then, for example, evaluated at
a later stage or in a specific environment [@Wickham2019]. The present package
includes a set of rules that translate such calls to a mathematical
representation in MathML and LaTeX. For a first illustration of the _mathml_
package, we consider the binomial probability.

```{r}
term <- quote(pbinom(k, N, p))
term
```

The term is quoted to avoid its immediate evaluation (which would raise an error
anyway since the variables `k`, `N`, `p` have not yet been defined). Experienced
readers will remember that the quoted expression above is a short form for

```r
term <- call("pbinom", as.name("k"), as.name("N"), as.name("p"))
```

As can be seen from the output, to the variable `term` is not assigned the 
result of the calculation, yet an R\ call [see, e.g., @Wickham2019, for details 
on "non-standard evaluation"], which can eventually be evaluated with `eval()`,

```{r}
k <- 10
N <- 22
p <- 0.4
eval(term)
```

The R\ package _mathml_ can now be used to render the call in MathML, that
is the dialect for mathematical elements on HTML webpages or in MathJax/LaTeX,
as shown below (some of the curly braces are not really needed in this simple
example, but are necessary in edge cases).

```{r}
library(mathml)
mathjax(term)
```

Some of the curly braces are not really needed in the LaTeX output, but are
necessary in edge cases. The package also includes a function `mathout()` that
wraps a call to `mathml()` for HTML output and `mathjax()` for LaTeX output.
Moreover, the function `math(x)` adds the class `"math"` to its argument, such
that a special knitr printing function is
invoked [see the vignette on custom print methods in @knitr]. An RMarkdown
code chunk with `mathout(term)` thus produces:

````{r, echo=FALSE}
math(term)
````

Similarly, `inline()` produces inline
output, `` `r "\x60r inline(term)\x60"` `` yields `r inline(term)`.

# Package _mathml_ in practice

_mathml_ is an R\ package for pretty mathematical representation of
R\ functions and objects in data analysis, scientific reports and interactive
web content. The currently supported features are listed below, roughly
following the order proposed by Murrell and Ihaka [-@murrell2000].

## Basic elements

_mathml_ handles the basic elements of everyday mathematical expressions,
such as numbers, Latin and Greek letters, multi-letter identifiers, accents,
subscripts, and superscripts. 
 
```{r}
term <- quote(1 + -2L + a + abc + "a" + phi + Phi + varphi + roof(b)[i, j]^2L)
math(term)

term <- quote(round(3.1415, 3L) + NaN + NA + TRUE + FALSE + Inf + (-Inf))
math(term)
```

An expression such as `1 + -2` may be considered unsatisfactory from an
aesthetic perspective. It is correct R\ syntax, though, and is reproduced
accordingly, without the parentheses. Parentheses around negated numbers or
symbols can be added as shown above for `+ (-Inf)`. If `round` is not given,
R's default number of decimals from `getOption("digits")` is used (which is way
too large in the author's opinion---in line with the unfortunate practice of
statistics programs).

To avoid name clashes with package _stats_, `roof()` is used instead of `hat()`
to put a hat on a symbol (see next section for further decorations). Note that
an R\ function `roof()` does not exist in base R, it is provided by the package
for convenience and points to the identity function.

## Decorations

The package offers some support for different fonts and colors as well as accents and
boxes etc. Internally, these decorations are implemented as identity functions,
so they can be introduced into R expressions without side-effects.

```{r}
term <- quote(bold(b[x, 5L]) + bold(b[italic(x)]) + italic(ab) + italic(42L))
math(term)

term <- quote(tilde(a) + mean(X) + boxed(c) + cancel(d) + phantom(e) + prime(f))
math(term)
```

Note that the font styles only affect the display of identifiers, whereas
numbers, character strings etc. are left untouched.

Colors can be specified as color names, RGB-values (in decimal or hexadecimal format), 
HSL-values and HWB-values.

```{r}
term <- quote(color("red", tilde(a)) + color("rgb(255, 0, 0)", mean(X)) + 
        color("#FF0000", boxed(c)) + color("hsl(0, 100%, 50%)", cancel(d)) + 
        color("hwb(0 0% 0%)", prime(f)))
math(term)
```

## Operators and parentheses

Arithmetic operators and parentheses are translated as they are, as illustrated
below.

```{r}
term <- quote(a - ((b + c)) - d*e + f*(g + h) + i/j + k^(l + m) + (n*o)^{p + q})
math(term)

term <- quote(dot(a, b) + frac(1L, nodot(c, d + e)) + dfrac(1L, times(g, h)))
math(term)
```

For multiplications involving only numbers and symbols, the multiplication sign
is omitted. This heuristic does not always produce the desired result;
therefore, _mathml_ defines alternative R\ functions `dot()`, `nodot()`,
and `times()`. These functions calculate a product and produce the respective
multiplication signs. Similarly, `frac()` and `dfrac()` can be used for small
and large fractions.

For standard operators with known precedence, _mathml_ is generally able to
detect if parentheses are needed; for example, parentheses are automatically
placed around `d + e` in the `nodot`-example. However, we note unnecessary
parentheses around `l + m` above. Thes parentheses are a consequence
of `quote(a^(b + c))` actually producing a nested R\ call of the
form `'^'(a, (b + c))` instead of `'^'(a, b + c)`:

```{r}
term <- quote(a^(b + c))
paste(term)
```

For the present purpose, this feature is unfortunate because extra parentheses
around `b + c` are not needed. The preferred result is obtained by using the
functional form `quote('^'(k, l + m))` of the power, or curly braces as a
workaround (see `p + q` above).

## Custom operators

Whereas in standard infix operators, the parentheses typically follow the rules
for precedence, undesirable results may be obtained in custom operators.

```{r}
term <- quote(mean(X) %+-% 2L * s / sqrt(N))
math(term)
term <- quote('%+-%'(mean(X), 2L * s / sqrt(N))) # functional form of '%+-%'
term <- quote(mean(X) %+-% {2L * s / sqrt(N)})   # the same
math(term)
```

The example is a reminder that it is not possible to define the precedence of
custom operators in R, and that expressions with such operators are evaluated
strictly from left to right. Again, the problem can be worked around by the
functional form of the operator, or a curly brace to hide the parenthesis but
enforce the correct operator precedence.

More operators are shown in Table\ 1, including the
suggestions by Murrell and Ihaka [-@murrell2000] for graphical annotations and
arrows in R\ figures.

```{r custom-operators, echo=FALSE}
op1 <- list(
  "A\\ %*%\\ B"=quote(A %*% B),
  "A\\ %.%\\ B"=quote(A %.% B),
  "A\\ %x%\\ B"=quote(A %x% B),
  "A\\ %/%\\ B"=quote(A %/% B),
  "A\\ %%\\ B"=quote(A %% B),
  "A\\ &\\ B"=quote(A & B),
  "A\\ |\\ B"=quote(A | B),
  "xor(A,\\ B)"=quote(xor(A, B)),
  "!A"=quote(!A),
  "A\\ ==\\ B"=quote(A == B),
  "A\\ <-\\ B"=quote(A <- B))

m1 <- lapply(op1, FUN=mathout, flags=list(cat=FALSE))

op1 <- names(op1)
if(knitr::is_latex_output())
  op1 <- sapply(op1, FUN=knitr:::escape_latex)
if(knitr::is_html_output())
  op1 <- sapply(op1, FUN=xfun::html_escape, attr=TRUE)

op2 <- list(
  "A\\ !=\\ B"=quote(A != B),
  "A\\ ~ B"=quote(A ~ B),
  "A\\ %~~%\\ B"=quote(A %~~% B),
  "A\\ %==%\\ B"=quote(A %==% B),
  "A\\ %=~%\\ B"=quote(A %=~% B),
  "A\\ %prop%\\ B"=quote(A %prop% B),
  "A\\ %in%\\ B"=quote(A %in% B),
  "intersect(A,\\ B)"=quote(intersect(A, B)),
  "union(A,\\ B)"=quote(union(A, B)),
  "crossprod(A,\\ B)"=quote(crossprod(A, B)),
  "is.null(A)"=quote(is.null(A)))

m2 <- lapply(op2, FUN=mathout, flags=list(cat=FALSE))

op2 <- names(op2)
if(knitr::is_latex_output())
  op2 <- sapply(op2, FUN=knitr:::escape_latex)
if(knitr::is_html_output())
  op2 <- sapply(op2, FUN=xfun::html_escape, attr=TRUE)

op3 <- list(
  "A\\ %<->%\\ B"=quote(A %<->% B),
  "A\\ %->%\\ B"=quote(A %->% B),
  "A\\ %<-%\\ B"=quote(A %<-% B),
  "A\\ %up%\\ B"=quote(A %up% B),
  "A\\ %down%\\ B"=quote(A %down% B),
  "A\\ %<=>%\\ B"=quote(A %<=>% B),
  "A\\ %=>%\\ B"=quote(A %=>% B),
  "A\\ %<=%\\ B"=quote(A %<=% B),
  "A\\ %dblup%\\ B"=quote(A %dblup% B),
  "A\\ %dbldown%\\ B"=quote(A %dbldown% B),
  " "="")

m3 <- lapply(op3, FUN=mathout, flags=list(cat=FALSE))

op3 <- names(op3)
if(knitr::is_latex_output())
  op3 <- sapply(op3, FUN=knitr:::escape_latex)
if(knitr::is_html_output())
  op3 <- sapply(op3, FUN=xfun::html_escape, attr=TRUE)

t <- cbind(Operator=op1, Output=m1,
  Operator=op2, Output=m2,
  Operator=op3, Arrow=m3)

knitr::kable(t, caption="Table 1. Custom operators in mathml",
  row.names=FALSE, escape=FALSE)
```

## Builtin functions

There is support for most functions from package _base_, with adequate use and
omission of parentheses.

```{r}
term <- quote(sin(x) + sin(x)^2L + cos(pi/2L) + tan(2L*pi) * expm1(x))
math(term)
term <- quote(choose(N, k) + abs(x) + sqrt(x) + floor(x) + exp(frac(x, y)))
math(term)
```

A few more examples are shown in Table\ 2, including functions from _stats_. 

```{r base-stats, echo=FALSE}
op1 <- list(
  "sin(x)"=quote(sin(x)),
  "cosh(x)"=quote(cosh(x)),
  "tanpi(alpha)"=quote(tanpi(alpha)),
  "asinh(x)"=quote(asinh(x)),
  "log(p)"=quote(log(p)),
  "log1p(x)"=quote(log1p(x)),
  "logb(x,\\ e)"=quote(logb(x, e)),
  "exp(x)"=quote(exp(x)),
  "expm1(x)"=quote(expm1(x)),
  "choose(n,\\ k)"=quote(choose(n, k)),
  "lchoose(n,\\ k)"=quote(lchoose(n, k)),
  "factorial(n)"=quote(factorial(n)), 
  "lfactorial(n)"=quote(lfactorial(n)),
  "sqrt(x)"=quote(sqrt(x)),
  "mean(X)"=quote(mean(X)),
  "abs(x)"=quote(abs(x)))
m1 <- lapply(op1, FUN=mathout, flags=list(cat=FALSE))

op1 <- names(op1)
if(knitr::is_latex_output())
  op1 <- sapply(op1, FUN=knitr:::escape_latex)
if(knitr::is_html_output())
  op1 <- sapply(op1, FUN=xfun::html_escape, attr=TRUE)

op2 <- list(
  "dbinom(k,\\ N,\\ pi)"=quote(dbinom(k, N, pi)),
  "pbinom(k,\\ N,\\ pi)"=quote(pbinom(k, N, pi)),
  "qbinom(p,\\ N,\\ pi)"=quote(qbinom(p, N, pi)),
  "dpois(k,\\ lambda)"=quote(dpois(k, lambda)),
  "ppois(k,\\ lambda)"=quote(ppois(k, lambda)),
  "qpois(p,\\ lambda)"=quote(qpois(p, lambda)),
  "dexp(x,\\ lambda)"=quote(dexp(x, lambda)),
  "pexp(x,\\ lambda)"=quote(pexp(x, lambda)),
  "qexp(p,\\ lambda)"=quote(qexp(p, lambda)),
  "dnorm(x,\\ mu,\\ sigma)"=quote(dnorm(x, mu, sigma)),
  "pnorm(x,\\ mu,\\ sigma)"=quote(pnorm(x, mu, sigma)),
  "qnorm(alpha/2L)"=quote(qnorm(alpha/2L)),
  "1L\\ -\\ pchisq(x,\\ 1L)"=quote(1L - pchisq(x, 1L)),
  "qchisq(1L\\ -\\ alpha,\\ 1L)"=quote(qchisq(1L-alpha, 1L)),
  "pt(t,\\ N\\ -\\ 1L)"=quote(pt(t, N-1L)),
  "qt(alpha/2L,\\ N\\ -\\ 1L)"=quote(qt(alpha/2L, N-1L)))
m2 <- lapply(op2, FUN=mathout, flags=list(cat=FALSE))

op2 <- names(op2)
if(knitr::is_latex_output())
  op2 <- sapply(op2, FUN=knitr:::escape_latex)
if(knitr::is_html_output())
  op2 <- sapply(op2, FUN=xfun::html_escape, attr=TRUE)

t <- cbind(Function=op1, Output=m1, Function=op2, Output=m2)
knitr::kable(t, caption="Table 2. R functions from _base_ and _stats_",
  row.names=FALSE, escape=FALSE)
```

## Custom functions

For self-written functions, the matter is somewhat more complicated. For a function
such as `g <- function(...) ...`, the name _g_ is not transparent to R, because
only the function body is represented. We can still display functions in the
form `head(x) = body` if we embed the object to be shown into a
call `"<-"(head, body)`.


```{r}
sgn <- function(x)
{
  if(x == 0L) return(0L)
  if(x < 0L) return(-1L)
  if(x > 0L) return(1L)
}

math(sgn)
math(call("<-", quote(sgn(x)), sgn))
```

As shown in the example, we can still display functions in the
form `head(x) = body` if we embed the object to be shown into a
call `"<-"(head, body)`.

The function body is generally a nested R\ call of the form `'{'(L)`, with `L`
being a list of commands (the semicolon, not necessary in R, is translated to a
newline). As illustrated in the example, _mathml_ provides limited support for
control structures such as `if`.

## Indices and powers

Indices in square brackets are rendered as subscripts, powers are rendered as
superscript. Moreover, _mathml_ defines the
functions `sum_over(x, from, to)`, and `prod_over(x, from, to)` that simply
return their first argument. The other two arguments serve as
decorations (_to_ is optional), for example, for summation and product signs.

```{r}
term <- quote(S[Y]^2L <- frac(1L, N) * sum(Y[i] - mean(Y))^2L)
math(term)

term <- quote(log(prod_over(L[i], i==1L, N)) <- sum_over(log(L[i]), i==1L, N))
math(term)
```

## Ringing back to R

R\'s `integrate` function takes a number of arguments, the most important ones
being the function to integrate, and the lower and the upper bound of the
integration.

```{r}
term <- quote(integrate(sin, 0L, 2L*pi))
math(term)
eval(term)
```

For mathematical typesetting in the form
of \(\int f(x)\, dx\), _mathml_ needs to find out the name of the
integration variable. For that purpose, the underlying Prolog bridge provides a
predicate `r_eval/3` that calls R\ from Prolog. In the example above, this
predicate evaluates `formalArgs(args(sin))`, which returns the names of the
arguments of `sin`, namely, `x`.

Note that in the example above, the quoted term is an abbreviation
for `call("integrate", quote(sin), ...)`, with `sin` being an R\ symbol, not a
function. While the R\ function `integrate()` can handle both symbols and
functions, _mathml_ needs the symbol because it is unable to determine the
function name of custom functions.

## Names and order of arguments

One of R's great features is the possibility to refer to function arguments by
their names, not only by their position in the list of arguments. At the other
end, Prolog does not have such a feature. Therefore, the Prolog handlers for
R\ calls are rather rigid, for example, `integrate/3` accepts exactly three
arguments in a particular order and without names, that
is, `integrate(lower=0L, upper=2L*pi, sin)`, would not print the desired result.

To "canonicalize" function calls with named arguments and arguments in unusual
order, _mathml_ provides an auxiliary R\ function `canonical(f, drop)` that
reorders the argument list of calls to known R\ functions and, 
if `drop=TRUE` (which is the default), also removes the names of the arguments.

```{r}
term <- quote(integrate(lower=0L, upper=2L*pi, sin))
canonical(term)
```

```{r}
math(canonical(term))
```

This function can be used to feed mixtures of partially named and positional
arguments into the renderer. For details, see the R\ function `match.call()`.

## Matrices and Vectors

Of course, _mathml_ also supports matrices and vectors.

```{r}
v <- 1:3
math(call("t", v))

A <- matrix(data=11:16, nrow=2, ncol=3)
B <- matrix(data=21:26, nrow=2, ncol=3)
term <- call("+", A, B)
math(term)
```

Note that the seemingly more convenient `term <- quote(A + B)` yields \(A + B\) in
the output---instead of the desired matrix representation. This behavior is
expected because quotation of R calls also quotes the components of the
call (here, _A_ and _B_).

## Short mathematical names for R symbols

In typical R\ functions, variable names are typically longer than just single
letters, which may yield unsatisfactory results in the mathematical output.

```{r}
term <- quote(dbinom(successes, Ntotal, prob))
hook(successes, k)
hook(quote(Ntotal), quote(N), quote=FALSE)
hook(prob, pi)
math(term)
hook(prob, p) # update hook
math(term)
```

To improve the situation, _mathml_ provides a simple hook that can be used
to replace elements (e.g., verbose variable names) of the code by concise
mathematical symbols, as illustrated in the example. To simplify notation,
the `quote` flag of `hook()` defaults to TRUE, and `hook()` uses non-standard
evaluation to unpack its arguments. If quote is FALSE, as shown above, the user
has to provide the quoted expressions. Care should be taken to
avoid recursive hooks such as `hook(s, s["A"])` that endlessly replace
the \(s\) from \(s_{\mathrm{A}}\) as
in \(s_{\mathrm{A}_{\mathrm{A}_{\mathrm{A}\cdots}}}\).

The hooks can also be used for more complex elements such as R\ calls, with
dotted symbols representing Prolog variables.

```{r}
term <- quote(pbinom(successes, Ntotal, prob))
hook(pbinom(.K, .N, .P), sum_over(dbinom(i, .N, .P), i=0L, .K))
math(term)
```

The replacement can be checked with `hooked(term)`, and the old behavior is
restored with `unhook(term)`.

```{r}
hooked(term)
unhook(pbinom(.K, .N, .P))
math(term)
```

For custom functions, the function `hook_fn` adds a hook with the function body.
 
```{r}
square <- function(x)
{ 
  return(x^2L) 
}

hook_fn(square)
term <- quote(square(a))
math(term)
```

## Abbreviations

We consider the \(t\)-statistic for independent samples with equal variance. To
avoid clutter in the equation, the pooled variance \(s^2_{\mathrm{pool}}\) is
abbreviated, and a comment is given with the expression
for \(s^2_{\mathrm{pool}}\). For this purpose, _mathml_ provides a
function `denote(abbr, expr, info)`, with `expr` actually being
evaluated, `abbr` being rendered, plus a comment of the
form "with `expr` denoting `info`".

```{r}
hook(m_A, mean(X)["A"]) ; hook(s2_A, s["A"]^2L) ;
hook(n_A, n["A"])
hook(m_B, mean(X)["B"]) ; hook(s2_B, s["B"]^2L)
hook(n_B, n["B"]) ; hook(s2_p, s["pool"]^2L)
term <- quote(t <- dfrac(m_A - m_B, 
    sqrt(denote(s2_p, frac((n_A - 1L)*s2_A + (n_B - 1L)*s2_B, n_A + n_B - 2L),
                "the pooled variance.") * (frac(1L, n_A) + frac(1L, n_B)))))
math(term)
```

The term is evaluated below. `print()` is needed because the return value of an
assignment of the form `t <- dfrac(...)` is not visible in R.

```{r}
m_A <- 1.5; s2_A <- 2.4^2; n_A <- 27; m_B <- 3.9; s2_B <- 2.8^2; n_B <- 20
print(eval(term))
```

## Context-dependent rendering

Consider an educational scenario in which we want to highlight a certain
element of a term, for example, that a student has forgotten to subtract the
null hypothesis in a \(t\)-ratio:

```{r}
t <- quote(dfrac(omit_right(mean(D) - mu[0L]), s / sqrt(N)))
math(t, flags=list(error="highlight"))
math(t, flags=list(error="fix"))
```

The R function `omit_right(a + b)` uses non-standard evaluation
techniques [e.g., @Wickham2019] to return only the left part an operation,
and cancels the right part. This may not always be desired, for example, when
illustrating how to fix the mistake.

For this purpose, the functions `mathml()` or `mathjax()` have an optional
argument `flags` which is a list with named elements. In this example, we use
this argument to tell _mathml_ how to render such erroneous
expressions using the flag `error` which is one of asis, highlight, fix, or
ignore. For more examples, see Table 3.

```{r mistakes, echo=FALSE}
op1 <- list(
  "omit_left(a\\ +\\ b)"=quote(omit_left(a + b)),
  "omit_right(a\\ +\\ b)"=quote(omit_right(a + b)),
  "list(quote(a),\\ quote(omit(b)))"=list(quote(a), quote(omit(b))),
  "add_left(a\\ +\\ b)"=quote(add_left(a + b)),
  "add_right(a\\ +\\ b)"=quote(add_right(a + b)),
  "list(quote(a),\\ quote(add(b)))"=list(quote(a), quote(add(b))),
  "instead(a,\\ b)\\ +\\ c"=quote(instead(a, b) + c))

asis <- lapply(op1, FUN=mathout, flags=list(cat=FALSE, error="asis"))
high <- lapply(op1, FUN=mathout, flags=list(cat=FALSE, error="highlight"))
fix  <- lapply(op1, FUN=mathout, flags=list(cat=FALSE, error="fix"))
igno <- lapply(op1, FUN=mathout, flags=list(cat=FALSE, error="ignore"))

op1 <- names(op1)
if(knitr::is_latex_output())
  op1 <- sapply(op1, FUN=knitr:::escape_latex)
if(knitr::is_html_output())
  op1 <- sapply(op1, FUN=xfun::html_escape, attr=TRUE)

t <- cbind(Operation=op1, 
  "error\\ =\\ asis"=asis, highlight=high, fix=fix, ignore=igno)
knitr::kable(t, caption="Table 3. Highlighting elements of a term",
  row.names=FALSE, escape=FALSE)
```

Further customization requires the assertion of new Prolog rules `math/2`,
`ml/3`, `jax/3`, as shown in the Appendix.

# Conclusion

This package allows R\ to render its terms in pretty mathematical equations. It
extends the current features of R\ and existing packages for displaying
mathematical formulas in R\ [@murrell2000], but most
importantly, _mathml_ bridges the gap between computational needs,
presentation of results, and their reproducibility. The package supports both
MathML and LaTeX/MathJax for use in RMarkdown documents, presentations and
Shiny App webpages.

Researchers or teachers can already use RMarkdown to conduct analyses and show
results, and _mathml_ smoothes this process and allows for integrated
calculations and output. As shown in the case study of the previous
section, _mathml_ can help to improve data analyses and statistical reports
from an aesthetic perspective, as well as regarding reproducibility of
research.

Furthermore, the package may also allow for a better detection of possible
mistakes in R\ programs. Similar to most programming languages [@green1977],
R\ code is notoriously hard to read, and the poor legibility of the language is
one of the main sources of mistakes. For illustration, we consider again
Equation\ 10 in Schwarz [-@schwarz1994].

````{r}
hook(mu_A, mu["A"])
hook(mu_B, mu["B"])
hook(sigma_A, sigma["A"])
hook(sigma_B, sigma["B"])

f1 <- function(tau)
{ dfrac(c, mu_A) + (dfrac(1L, mu_A) - dfrac(1L, mu_A + mu_B) * 
    ((mu_A*tau - c) * pnorm(dfrac(c - mu_A*tau, sqrt(sigma_A^2L*tau)))
      - (mu_A*tau + c) * exp(dfrac(2L*mu_A*tau, sigma_A^2L))
        * pnorm(dfrac(-c - mu_A*tau, sqrt(sigma_A^2L*tau)))))
}

math(f1)
````

The first version has a wrong parenthesis, which is barely visible in the code,
whereas in the mathematical representation, the wrong curly brace is immediately
obvious (the correct version is shown below for comparison).

````{r}
f2 <- function(tau)
{ dfrac(c, mu_A) + (dfrac(1L, mu_A) - dfrac(1L, mu_A + mu_B)) * 
    ((mu_A*tau - c) * pnorm(dfrac(c - mu_A*tau, sqrt(sigma_A^2L*tau)))
      - (mu_A*tau + c) * exp(dfrac(2L*mu_A*tau, sigma_A^2L))
        * pnorm(dfrac(-c - mu_A*tau, sqrt(sigma_A^2L*tau))))
}

math(f2)
````

As the reader may know from their own experience, missed parentheses are frequent
causes of wrong results and errors that are hard to locate in programming code.
This particular example shows that mathematical rendering can help to
substantially reduce the amount of careless errors in programming.

One limitation of the package is the lack of a convenient way to insert line
breaks. This is mostly due to lacking support by MathML and LaTeX renderers. 
For example, in its current stage, the LaTeX package breqn [@breqn] is mostly
a proof of concept. Moreover, _mathml_ only works in one direction,
that is, it is not possible to translate from LaTeX or HTML back to R 
[see @latex2r, for an example].

The package _mathml_ is available for R\ version 4.2 and later, and can be
easily installed using the usual `install.packages("mathml")`. At its present
stage, it supports output in HTML, LaTeX, and Microsoft 
Word [via pandoc, @pandoc]. The source code of the package is found
at https://github.com/mgondan/mathml.

# Appendix A: For package developers

If you use `mathml` for your own package, please do not "Import" `mathml` in
your DESCRIPTION, but "Depend" on it.

```
Package: onmathml
Type: Package
Title: A package that uses mathml
...
Depends: 
    R (>= 4.3),
    mathml (>= 1.3)
```

It's not entirely clear why this is needed.

# Appendix B: Customizing the package

## Implementation details

For convenience, the translation of the R expressions is achieved through a
Prolog interpreter provided by another R\ package _rolog_ [@rolog]. If a
version of SWI-Prolog [@swipl] is found on the system, _rolog_ connects
to it. Alternatively, the SWI-Prolog runtime libraries can be conveniently
accessed by installing the R\ package _rswipl_ [@rswipl]. Prolog is a
classical logic programming language with many applications in expert systems,
computer linguistics and symbolic artificial intelligence. The strength of
Prolog lies in its concise representation of facts and rules for knowledge and
grammar, as well as its efficient built-in search engine for closed world
domains. Whereas Prolog is weak in statistical computation, but strong in
symbolic manipulation, the converse may be said for the R\ language. _rolog_
bridges this gap by providing an interface to a SWI-Prolog
distribution [@swipl] in R. The communication between the two systems is mainly
in the form of queries from R\ to Prolog, but two Prolog functions allow ring
back and evaluation of terms in R. 

The proper term for a Prolog "function" is predicate, and it is typically
written with name and arity (i.e., number of arguments), separated by a forward
slash. Thus, at the Prolog end, a predicate `math/2` translates the call 
`pbinom(K, N, Pi)` into a "function" `fn/2` with the name `P_Bi`, one argument 
`X =< K`, and the two parameters `N` and `Pi`.

```prolog
math(pbinom(K, N, Pi), M)
 => M = fn(subscript('P', "Bi"), (['X' =< K] ; [N, Pi])).
```

`math/2` operates like a "macro" that translates one mathematical 
element (here, `pbinom(K, N, Pi)`) to a different mathematical
element, namely `fn(Name, (Args ; Pars))`. The low-level predicate `ml/3` is
used to convert these basic elements to MathML.

```prolog
ml(Flags, fn(Name, (Args ; Pars)), M)
 => ml(Flags, Name, N),
    ml(Flags, paren(list(op(;), [list(op(','), Args), list(op(','), Pars)])), X),
    M = mrow([N, mo(&(af)), X]).
```

The relevant rule for `ml/3` builds the MathML entity `mrow([N, mo(&(af)), X])`,
with `N` representing the name of the function and `X` its arguments and
parameters, enclosed in parentheses. A corresponding rule `jax/3` does the same
for MathJax/LaTeX. A list of flags can be used for context-sensitive
translation (see, e.g., the section on errors above).

Several ways exist for translating new R\ terms to their mathematical
representation. We have already seen above how to use "hooks" to translate long
variable names from R to compact mathematical signs, as well as functions such
as cumulative probabilities \(P(X \le k)\) to different representations
like \(\sum_{i=0}^k P(X = i)\). Obviously, the hooks require that there already
exists a rule to translate the target representation into MathML and MathJax.

In this appendix we describe a few more ways to extend the set of translations
according to a user's needs. As stated in the background section, the Prolog end
provides two classes of rules for translation, macros `math/2,3,4` mirroring the
R\ hooks mentioned above, and the low-level predicates `ml/3` and `jax/3` that
create proper MathML and LaTeX terms.

## Linear models

To render the model equation of a linear model such
as `lm(EOT ~ T0 + Therapy, data=d)` in mathematical form, it is sufficient to
map the `Formula` in `lm(Formula, Data)` to its respective equation 
[see also @equatiomatic]. This can in two ways, using either the hooks 
described above, or a new `math/2` macro at the Prolog end.

````r
hook(lm(.Formula, .Data), .Formula)
````

The hook is simple, but is a bit limited because only R's tilde-form
of linear models is shown, and it only works for a call with exactly two
arguments.

Below is an example of how to build a linear equation of the
form \(Y = b_0 + b_1X_1 + ...\) using the Prolog macros from _mathml_.

````prolog
math_hook(LM, M) :-
    compound(LM),
    LM =.. [lm, ~(Y, Sum) | _Tail],
    summands(Sum, Predictors),
    findall(subscript(b, X) * X, member(X, Predictors), Terms),
    summands(Model, Terms),
    M = (Y == subscript(b, 0) + Model + epsilon).
````

The predicate `summands/2` unpacks an expression `A + B + C` to a
list `[C, B, A]` and vice-versa (see the file `lm.pl` for details).

````{r}
rolog::consult(system.file(file.path("pl", "lm.pl"), package="mathml"))

term <- quote(lm(EOT ~ T0 + Therapy, data=d, na.action=na.fail))
math(term)
````

## \(n\)-th root

Base R does not provide a function like `cuberoot(x)` or `nthroot(x, n)`, and
the present package does not support the respective representation. To obtain a
cube root, a programmer would typically type `x^(1/3)` or better `x^{1/3}` (see
the practice section why the curly brace is preferred in an exponent), resulting
in \(x^{1/3}\) which may still not match everyone's taste. Here we describe the
steps needed to represent the \(n\)-th root as \(\sqrt[n]x\).

We assume that `nthroot(x, n)` is available in the current
namespace [manually defined, or from R package _pracma_, @pracma], so that the
names of the arguments and their order are accessible to `canonical()` if
needed. As we can see below, _mathml_ uses a default
representation `name(arguments)` for such unknown functions.

```{r}
nthroot <- function(x, n)
  x^{1L/n}

term <- canonical(quote(nthroot(n=3L, 2L)))
math(term)
```

A proper MathML term is obtained by `mlx/3` (the x in mlx indicates that it is
an extension and is prioritized over the default ml/3 rules). `mlx/3`
recursively invokes `ml/3` for translating the function arguments _X_ and _N_,
and then constructs the correct MathML entity `<mroot>...</mroot>`.

```prolog
mlx(nthroot(X, N), M, Flags) :-
    ml(X, X1, Flags),
    ml(N, N1, Flags),
    M = mroot([X1, N1]).
```

The explicit unification `M = ...` in the last line serves to avoid clutter in
the head of `mlx/3`. The Prolog file `nthroot.pl` also includes the respective
rule for LaTeX and can be consulted from the package folder via the underlying
package _rolog_.

```{r}
rolog::consult(system.file(file.path("pl", "nthroot.pl"), package="mathml"))

options(digits = 2)
term <- quote(nthroot(a * (b + c), 3L)^2L)
math(term)
term <- quote(a^(1L/3L) + a^{1L/3L} + a^(1.0/3L))
math(term)
```

The file `nthroot.pl` includes three more statements `precx/3` and `parenx/3`,
as well as a `math_hook/2` macro. The first sets the operator precedence of the
cubic root above the power, thereby putting a parentheses around nthroot
in \((\sqrt[3]{\ldots})^2\). The second tells the system to increase the counter
of the parentheses below the root, such that the outer parenthesis becomes a
square bracket.

The last rule maps powers like `a^(1L/3L)` to `nthroot/3`, as shown in the
first summand. Of course, _mathml_ is not a proper computer algebra system. As
is illustrated by the other terms in the sum, such macros are limited to purely
syntactical matching, and terms like `a^{1L/3L}` with the curly brace
or `a^(1.0/3L)` with a floating point number in the numerator are not detected.

## P-values

With `pval/2`, a value specified as first argument can be displayed with the
following rounding rules:  

* 0.1 < value < 1 : two digits  
* 0.001 < value < 0.1 : three digits  
* value < 0.001 : output defaults to "< 0.001". 

The identifier, typically \(p\) or \(P\), is provided as the second argument.

```{r}
library(mathml)
rolog::consult(system.file(file.path("pl", "pval.pl"), package="mathml"))

term <- quote(pval(0.539, P))
math(term)

term <- quote(pval(0.0137, p))
math(term)

term <- quote(pval(0.0003, P))
math(term)
```

The file `prolog/pval.pl` illustrates how to distinguish the above cases in
Prolog. Two nested `math_hook` macros are used to decide if an equation sign
is needed or the less-than sign, and to determine the number of decimal places
shown.

The file also includes the convenience function `tstat/1` that rounds to two
digits, and `percent/1` that displays a number as a
percentage (e.g., 0.05 as 5%).

# Acknowledgment

Supported by the Erasmus+ program of the European
Commission (2019-1-EE01-KA203-051708).

# References
