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.

term <- quote(1 + -2L + a + abc + "a" + phi + Phi + varphi + roof(b)[i, j]^2L)
math(term)

$1.00+-2+a+\mathrm{abc}+\text{a}+\phi +\Phi +\varphi +{\widehat{b}}_{ij}^2$

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

$3.142+\mathrm{nan}+\mathrm{na}+T+F+\infty +\left(-\infty \right)$

An expression such as 1 + -2 may be considered unsatisfactory from an aesthetical 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).

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 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.

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

${\mathbf{b}}_{\mathbf{x}5}+{\mathbf{b}}_{\mathit{x}}+\mathit{ab}+42$

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

$\tilde{a}+\overline{X}+\boxed{c}+\cancel{d}+\phantom{e}+f^{\prime }$

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

Operators and parentheses

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

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

$a-\left[\left(b+c\right)\right]-de+f\cdot \left(g+h\right)+i/j+k^{\left(l+m\right)}+{\left(no\right)}^{p+q}$

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

$a\cdot b+\frac{1}{c\left(d+e\right)}+{\displaystyle \frac{1}{g\times h}}$

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):

term <- quote(a^(b + c))
paste(term)

## [1] "^"       "a"       "(b + c)"

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.

term <- quote(mean(X) %+-% 1.96 * s / sqrt(N))
math(term)

$\left(\overline{X}\pm 1.96\right)\cdot s/\sqrt{N}$

term <- quote('%+-%'(mean(X), 1.96 * s / sqrt(N))) # functional form of '%+-%'
term <- quote(mean(X) %+-% {1.96 * s / sqrt(N)})   # the same
math(term)

$\overline{X}\pm 1.96s/\sqrt{N}$

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 (2000) for graphical annotations and arrows in R figures.

Builtin functions

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

term <- quote(sin(x) + sin(x)^2L + cos(pi/2L) + tan(2L*pi) * expm1(x))
math(term)

$\sin x+{\left(\sin x\right)}^2+\cos \left(\pi /2\right)+\tan \left(2\pi \right)\cdot \left(\exp x-1\right)$

term <- quote(choose(N, k) + abs(x) + sqrt(x) + floor(x) + exp(frac(x, y)))
math(term)

$\left(\genfrac{}{}{0ex}{}{N}{k}\right)+\left|x\right|+\sqrt{x}+\lfloor x\rfloor +\exp \left(\frac{x}{y}\right)$

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

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).

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

math(sgn)

$\{\begin{array}{}0\text{,}\text{if}x=0-1\text{,}\text{if}x<01\text{,}\text{if}x>0\end{array}$

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

$\mathrm{sgn}x=\{\begin{array}{}0\text{,}\text{if}x=0-1\text{,}\text{if}x<01\text{,}\text{if}x>0\end{array}$

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.

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

$S_Y^2=\frac{1}{N}\cdot {\sum \left(Y_i-\overline{Y}\right)}^2$

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

$\log {\prod }_{i=1}^N{L}_i={\sum }_{i=1}^N\log L_i$

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.

term <- quote(integrate(sin, 0L, 2L*pi))
math(term)

$\underset{0}{\overset{2\pi }{\int }}\sin xdx$

eval(term)

## 2.221501e-16 with absolute error < 4.4e-14

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.

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

## integrate(sin, 0L, 2L * pi)

math(canonical(term))

$\underset{0}{\overset{2\pi }{\int }}\sin xdx$

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.

v <- 1:3
math(call("t", v))

${\left(123\right)}^{\text{T}}$

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

$\left(\begin{array}{lll}11& 13& 15\\ 12& 14& 16\end{array}\right)+\left(\begin{array}{lll}21& 23& 25\\ 22& 24& 26\end{array}\right)$

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 quote 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.

hook(successes, k)
hook(quote(Ntotal), quote(N), quote=FALSE)
hook(prob, pi)
term <- quote(dbinom(successes, Ntotal, prob))
math(term)

$P_{\text{Bi}}\left(X=k;N,\pi \right)$

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.

hook(pbinom(.K, .N, .P), sum_over(dbinom(i, .N, .P), i=0L, .K))
math(term)

$P_{\text{Bi}}\left(X=k;N,\pi \right)$

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”.

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)

$t={\displaystyle \frac{{\overline{X}}_{\text{A}}-{\overline{X}}_{\text{B}}}{\sqrt{s_{\text{pool}}^2\cdot \left(\frac{1}{n_{\text{A}}}+\frac{1}{n_{\text{B}}}\right)}}}$, with $s_{\text{pool}}^2=\frac{\left(n_{\text{A}}-1\right)\cdot s_{\text{A}}^2+\left(n_{\text{B}}-1\right)\cdot s_{\text{B}}^2}{n_{\text{A}}+n_{\text{B}}-2}$ denoting the pooled variance.

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.

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))

## [1] -3.157427

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:

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

${\displaystyle \frac{\overline{D}\cancel{-{\mu }_0}}{s/\sqrt{N}}}$

math(t, flags=list(error="fix"))

${\displaystyle \frac{\overline{D}\boxed{-{\mu }_0}}{s/\sqrt{N}}}$

The R function omit_right(a + b) uses non-standard evaluation techniques (e.g., Wickham 2019) 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.

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

Implementation details

For convenience, the translation of R expressions to mathematical output is achieved through a Prolog interpreter from R package rolog (Gondan 2022). Prolog is a classical logic programming language with many applications in expert systems, computer linguistics and symbolic artificial intelligence. The main strength of Prolog is its concise representation of facts and rules for knowledge and grammar, as well as its efficient built-in search engine for closed world domains. As it is well-known, R is a statistical programming language for data analysis and statistical modeling. Whereas Prolog is weak in statistical computation, but strong in symbolic manipulation, the converse may be said for the R language. The rolog package bridges this gap by providing an interface to a SWI-Prolog distribution (Wielemaker et al. 2012) in R. The communication between the two systems is mainly in the form of queries from R  to Prolog, but two predicates allow Prolog to ring back and evaluate 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.

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.

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 Anderson, Heiss, and Sumners 2023). This can in two ways, using either the hooks described above, or a new math/2 macro at the Prolog end.

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 how to build a linear equation of the form \(Y = b_0 + b_1X_1 + ...\) using the Prolog macros from mathml.

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).

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)

$\mathrm{EOT}=b_0+b_{\mathrm{T0}}\mathrm{T0}+b_{\mathrm{Therapy}}\mathrm{Therapy}+\epsilon$

\(n\)-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 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, Borchers 2022), 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.

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

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

$\mathrm{nthroot}\left(2,3\right)$

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>.

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.

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

term <- quote(nthroot(a * (b + c), 3L)^2L)
math(term)

${\left[\sqrt[3]{a\cdot \left(b+c\right)}\right]}^2$

term <- quote(a^(1L/3L) + a^{1L/3L} + a^(1.0/3L))
math(term)

$\sqrt[3]{a}+a^{1/3}+a^{\left(1.00/3\right)}$

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.