Keywords

Statistics; Logic Programming; Artificial Intelligence

R interface

Most of the work can done using the three R functions query, submit, and clear. The functions consult, once, and findall are provided for convenience.

consult. In most applications, a number of Prolog facts and rules will be loaded into the system. To facilitate this recurrent task, the prolog directive consult/1 has been mirrored into R, consult(filename), with filename given as a string (or a list of strings if multiple files are to be consulted). Note that the full filename should be given, including the extension (e.g., “.pl”). The function returns TRUE on success, otherwise FALSE and an error message is shown.

query. The function query(call, options) is used to create a Prolog query (without invoking it yet). The first argument call is a regular R call that is created using R’s function call(name, ...). This call represents the Prolog predicate which will be queried in the later course. The creation of such predicates and Prolog terms is described below and can become quite cumbersome (see the examples in Section 4). The second argument, options, may be used for ad hoc modifications of the translation between R and Prolog, see the section below. The function returns TRUE on success. Note that the function does not check if the corresponding Prolog predicate exists (but see submit() below).

Only a single query can be opened at a given time. If a new query is created while another query is still open, a warning is shown and the other query is closed.

submit. Once a query has been created, it can be submitted using submit(). If the query fails, the return value is FALSE. If the query succeeds, a list of constraints is returned, with bindings for the variables that satisfy the query. Repeated calls to submit are possible, returning the different solutions of a query (until it eventually fails). Programmatically distinguishing between the different types of return values for success and failure (list vs. FALSE) is facilitated by the R function isFALSE(x).

clear. Closes the query. The name of the function is chosen to avoid name clashes with R’s own built-in function close. The function returns an invisible TRUE, even if there is no open query.

The following short R program illustrates a query to Prolog’s member/2 using Rolog’s syntax rules.

# member(1, [1, 2.0, a, "b", X])
query(call("member", 1L, list(1L, 2.0, quote(a), "b", expression(X), TRUE)))
#> [1] TRUE
#> attr(,"query")
#> [1] "member(1, [1, 2.0, a, b, X, true])"

# returns an empty list, stating that the query satisfied
submit()
#> list()

# returns a list, stating that the query is satisfied if X = 1
submit()
#> $X
#> [1] 1

# close the query
clear()

Listing 1.

A query to Prolog’s member/2 predicate.

once and findall. The function once(call, options) is a convenience function that acts as a shortcut for query(call, options), submit(), and clear(). Similarly, findall(call, options) abbreviates the commands query(call, options), repetition of submit() until failure, and clear(), returning a list collecting the the return value of the individual calls to submit.

Creating Prolog terms in R

Table 1 summarizes the rules for the translation from R to Prolog. Most rules work in both directions, but a few exceptions exist. For example, there is an empty atom in Prolog, but no empty symbol in R, so the empty atom is translated to a character string in R.

Table 1

Creating Prolog terms from R

Moreover, R is mostly vectorized, lacking support for scalar entities, that is, scalar entities are treated as vectors of length 1. Conversely, Prolog does not natively support vectors or matrices. The problem is solved in the following way:

• R vectors of length 0 are translated to Prolog’s empty list.
• R vectors of length 1 are translated to Prolog scalars.
• R vectors of length N > 1 are translated to Prolog terms #/N, %/N, $$/N, and !/N for floating points, integers, strings and logicals, respectively.

In the reverse direction, Prolog terms like #/N are translated back to R vectors of length N. This including the terms #/0 and #/1 that normally don’t occur. To summarize, the rules for translation are not fully symmetrical. A quick check for symmetry of the representation is obtained by a query to r_eval/2 (see also below, subsection Prolog interface):

once(call("r_eval", c(1, 2, NA, NaN, Inf), expression(X)))
#> $X
#> [1]   1   2  NA NaN Inf

Rolog options

A few package-specific options have been defined to allow some finetuning of the rules for translation between R and Prolog.

• realvec (string): Name of the prolog term for vectors of floats (default is #)
• intvec (string): same for vectors of integers (default is %)
• boolvec (string): same for vectors of logicals (default is !)
• charvec (string): same for vectors of character strings (default is $$). The single dollar cannot be used because it is the list operator in R.
• scalar (logical): if TRUE (default), R vectors of length 1 are translated to scalars in Prolog. If FALSE, R vectors are always translated to #/N etc., depending on the type.
• portray (logical): if TRUE (default), the result of query, once and findall includes an attribute with a representation of the query in Prolog.

The command rolog_options() returns a list with all the options. The options can be globally modified like e.g. this:

options(rolog.intvec="%%")

In a given query, the options can be set in the optional argument, e.g.

query(call("member", expression(X), list(1:3, 4:6)), list(intvec="%%"))

Prolog interface

Rolog offers some basic support to call R from Prolog, that is, connecting the two systems in the reverse direction. Two predicates can be used for this purpose, r_eval(Call) and r_eval(Function, Result). The former just invokes R with the command Call (ignoring the result); the latter evaluates Function and unifies the result with Result. Note that proper quoting of R functions is needed at the Prolog end, especially with R functions that start with uppercase letters or contain a dot in their name.

Two use cases for r_eval/2 are shown in Section 4 below.

Preprocessing in R

We have already seen above that raising even simple everyday Prolog queries such as member(X, [1, 2, 3, a, b]) require complicated R expressions such as

query(call("member", expression(X), list(1, 2, 3, quote("a"), quote(b)))

The R function as.rolog(call) is meant to simplify this a bit by translating symbols starting with a dot to Prolog variables, and calls like ""[1, 2, 3, a, b] to lists. The argument call is typically a quoted R call or symbol:

q <- quote(member(.X, ""[1, 2, 3, a, b]))
query(as.rolog(q))

Note that the name of the variable will still be X in the later course, not “dot-X”. A bit flexibility is lost because quote() treats the arguments a, b as symbols; to evaluate them (i.e., “unquote”), they can be put in parentheses:

a <- 4
b <- 5
q <- quote(member(.X, ""[1, 2, 3, a, (b)]))
query(as.rolog(q))

resulting in the query member(X, [1, 2, 3, a, 5]). More sophisticated work with quasi-quotations (i.e., “unquoting” expressions) is described in the “Advanced R” book (Wickham 2019).

Section 4 includes an example for mathematical rendering of R expressions. In that example, a preprocessing function is used to bring function calls with named arguments to a canonical form which is then handled in Prolog.

Postprocessing in R

This may again be a function that reverts some of the manipulations during preprocessing. For once() and submit(), such a function would operate on the bindings. For example, many Prolog programmers are used to operate with atoms instead of character strings, which is the preferred representation of symbolic information in R. The following simple example illustrates conversion of the results for a query like member(X, [a, b, c]) to strings.

stringify <- function(x)
{
  # replace Prolog variable by the value of an R variable with the same name
    if(is.name(x))
      return(as.character(x))
    
    # Recurse into lists and calls
    if(is.call(x))
      x[-1] <- lapply(x[-1], FUN=stringify)

    if(is.list(x))
      x <- lapply(x, FUN=stringify)

  # Leave the rest unchanged
    return(x)
}

# This may be the return value of a Prolog query
q <- quote(member(.X, ""[a, b, c]))
r <- findall(as.rolog(q))
stringify(r)
#> [[1]]
#> [[1]]$X
#> [1] "a"
#> 
#> 
#> [[2]]
#> [[2]]$X
#> [1] "b"
#> 
#> 
#> [[3]]
#> [[3]]$X
#> [1] "c"

Pre- and postprocessing in Prolog

Recent versions of SWI-Prolog support so-called dictionaries of the form Tag{Key1:Value1, Key2:Value2, ...}. The tag is typically an atom (but can be a variable, as well), the keys are unique atom or integers; the values can be anything. Suppose we have a Prolog predicate that does something with dicts, and we would like to query it from R. The simplest solution is a wrapper in Prolog that translates key-value pairs [Key1-Value1, Key2-Value2, ...] back and forth to dicts:

do_something_with_pairs(Pairs0, Pairs1) :-
    dict_pairs(Dict0, my_dict, Pairs0),
    do_something_with_dicts(Dict0, Dict1),
    dict_pairs(Dict1, my_dict, Pairs1).

do_something_with_pairs/2 can then be queried from R using, e.g., lists with named elements.

once(call("do_something_with_pairs", list(a=1, b=2), expression(X)))

In the code above, dict_pairs/2 takes the role of both preproc/2 and postproc/2 in Figure 1. It illustrates that complicated syntax on the R side when can be much simpler to do the conversion at the Prolog end.

A way to extend Prolog by add-ons (“packs”) are shown in the next section.

Hello, world

Prolog’s typical hello world example is a search through a directed acyclic graph (DAG), for example, a family tree like the one given in Listing 1.

parent(pam, bob).
parent(tom, bob).
parent(bob, ann).
parent(bob, pat).
parent(pat, jim).

ancestor(X, Z) :-
    parent(X, Z).

ancestor(X, Z) :-
    parent(X, Y),
    ancestor(Y, Z).

Listing 1

A family tree in Prolog (see also family.pl)

Listing 1 is included in the package and can be accessed from R using the function system.file(...). Within Prolog, the normal workflow is to consult the code with [family] and then to raise queries such as ancestor(X, jim), which returns, one by one, four solutions for the variable X. In R, we obtain the following results:

library(rolog)

# [family].
consult(system.file(file.path("pl", "family.pl"), package="rolog"))

# ancestor(X, jim).
query(call("ancestor", expression(X), quote(jim)))
#> [1] TRUE
#> attr(,"query")
#> [1] "ancestor(X, jim)"

# solutions for X, one by one
submit()
#> $X
#> pat
submit()
#> $X
#> pam
submit()
#> $X
#> tom
submit()
#> $X
#> bob
submit() # no more results (closing the query)
#> [1] FALSE
submit() # warning that no query is open
#> Warning in .submit(): submit: no open query.
#> [1] FALSE
# clear() # normally used to close a query

As stated above, consult() loads the facts and rules of Listing 1 into the database. query(expr) initializes the query expr, and the subsequent calls to submit return the conditions under which the query succeeds. In this example, the query succeeds if X is either pat, pam, tom or bob. A query is closed with clear(), or automatically if the query fails. Note that it is generally not possible to open two queries simultaneously, so opening a second query while another one is still open will raise a warning. If we are interested in just the first solution, we can use once(expr) as a shortcut to query(expr)-submit()-clear(). If we want to collect all solutions of a query with a finite set of solutions, we can use findall(expr).

A rather cumbersome aspect of Rolog is the construction of Prolog terms and queries. expression(X) encapsulates the variable X, R symbols from quote(jim) or as.symbol("jim") are translated to Prolog atoms, and Prolog compounds such as ancestor/2 are generated using call("ancestor", ...). A simplified syntax is provided by as.rolog(...) that accepts quoted expressions with dots indicating Prolog variables:

q = quote(ancestor(.X, jim))
findall(as.rolog(q))

Note that as.rolog(...) removes the dot from the variable name.

Backdoor test

A more useful application of DAGs is confounder adjustment in causal analysis (Greenland, Pearl, and Robins 1999; Barrett 2021). The Prolog file backdoor.pl is an implementation of Greenland et al.’s criteria for the backdoor test for d-separation in DAGs, with a predicate minimal/3 that searches for minimally sufficient sets of variables for confounder adjustment on the causal path between exposure and outcome.

# [backdoor].
consult(system.file(file.path("pl", "backdoor.pl"), package="rolog"))

# Figure 12 in Greenland et al.
add_node = function(N)
    invisible(once(call("assert", call("node", N))))

add_arrow = function(X, Y)
    invisible(once(call("assert", call("arrow", X, Y))))

add_node("a")
add_node("b")
add_node("c")
add_node("d") # outcome
add_node("e") # exposure
add_node("f")
add_node("u")

add_arrow("a", "d")
add_arrow("a", "f")
add_arrow("b", "d")
add_arrow("b", "f")
add_arrow("c", "d")
add_arrow("c", "f")
add_arrow("e", "d")
add_arrow("f", "e")
add_arrow("u", "a")
add_arrow("u", "b")
add_arrow("u", "c")

findall(call("minimal", "e", "d", expression(S)))
#> [[1]]
#> [[1]]$S
#> [[1]]$S[[1]]
#> [1] "a"
#> 
#> [[1]]$S[[2]]
#> [1] "b"
#> 
#> [[1]]$S[[3]]
#> [1] "c"
#> 
#> 
#> 
#> [[2]]
#> [[2]]$S
#> [[2]]$S[[1]]
#> [1] "f"

The query to minimal/3 returns two minimally sufficient sets of covariates for confounder adjustment (namely, {a, b, c} and {f}).

Definite clause grammars

One of the main driving forces of Prolog development was natural language processing (Dahl 1981). Therefore, the next example is an illustration of sentence parsing using so-called definite clause grammars. As Listing 3 shows, Rolog can access modules from SWI’s standard library (here, it is “dcg/basics.pl”).

:- use_module(library(dcg/basics)).

% Translate R string to code points and invoke phrase/2
sentence(Tree, Sentence) :-
    string_codes(Sentence, Codes),
    phrase(s(Tree), Codes).

% Simple grammar with sentences, noun, verb and participle phrases
s(s(NP, VP)) --> np(NP, C), blank, vp(VP, C).
np(NP, C) --> pn(NP, C).
np(np(Det, N), C) --> det(Det, C), blank, n(N, C).
np(np(Det, N, PP), C) --> det(Det, C), blank, n(N, C), blank, pp(PP).
vp(vp(V, NP), C) --> v(V, C), blank, np(NP, _).
vp(vp(V, NP, PP), C) --> v(V, C), blank, np(NP, _), blank, pp(PP).
pp(pp(P, NP)) --> p(P), blank, np(NP, _).

% Determiners, personal nouns, nouns, verbs and participles
det(det(a), sg) --> `a`.
det(det(the), _) --> `the`.
pn(pn(john), sg) --> `john`.
n(n(man), sg) --> `man`.
n(n(men), pl) --> `men`.
n(n(telescope), sg) --> `telescope`.
v(v(sees), sg) --> `sees`.
v(v(see), pl) --> `see`.
v(v(saw), _) --> `saw`.
p(p(with)) --> `with`.

Listing 2

Simple grammar and lexicon. sentence/2 preprocesses the R call.

As in the first example, we first consult a little Prolog program with a minimalistic grammar and lexicon (Listing 2), and then raise a query asking for the syntactic structure of “john saw a man with a telescope”. Closer inspection of the two results reveals the two possible meanings, “john saw a man who carries a telescope” versus “john saw a man through a telescope”. More Prolog examples of natural language processing are found in Blackburn and Bos (Blackburn and Bos 2005), including the resolution of anaphoric references and the extraction of semantic meaning.

# [telescope].
consult(system.file(file.path("pl", "telescope.pl"), package="rolog"))

# findall(sentence(Tree, "john saw a man with a telescope")).
findall(call("sentence", expression(Tree), "john saw a man with a telescope"))
#> [[1]]
#> [[1]]$Tree
#> s(pn(john), vp(v(saw), np(det(a), n(man), pp(p(with), np(det(a), 
#>     n(telescope))))))
#> 
#> 
#> [[2]]
#> [[2]]$Tree
#> s(pn(john), vp(v(saw), np(det(a), n(man)), pp(p(with), np(det(a), 
#>     n(telescope)))))

Installation of add-ons for Prolog

In description of the previous example, I noted in passing that Rolog can use the built-in libraries of SWI-Prolog (e.g., by calls to use_module/1). It is also possible to extend the installation by add-ons, including add-ons that require compilation, if the build tools (essentially, RTools) are properly configured. This is illustrated below by the demo add-on “environ” (J. Wielemaker 2012) that collects the current environment variables.

# pack_install(environ, [interactive(false)]).
once(call("pack_install", as.name("environ"), list(call("interactive", FALSE))))

# use_module(library(environ))
once(call("use_module", call("library", quote(environ))))

# environ(X)
once(call("environ", expression(X)))

The query then unifies X with a list with Key=Value terms. The purpose if this example is obviously not to mimick the built-in function Sys.getenv() from R, but to illustrate the installation and usage of Prolog extensions from within R.

Term manipulation

Prolog is homoiconic, that is, code is data. In this example, we make use of Prolog’s ability to match expressions against given patterns and modify these expressions according to a few predefined “buggy rules” (Brown and Burton 1978), inspired by recurrent mistakes in the statistics exams of my students. Consider the \(t\)-statistic for comparing an observed group average to a population mean:

\[ T = \frac{\overline{X} - \mu}{s / \sqrt{N}} \]

Some mistakes may occur in this calculation, for example, omission of the implicit parentheses around the numerator and the denominator when typing the numbers into a calculator, resulting to \(\overline{X} - \frac{\mu}{s} \div \sqrt{N}\), or forgetting the square root around \(N\), or both. Prolog code for the two buggy rules is given in Listing 3.

% Correct step from task to solution
expert(tratio(X, Mu, S, N), frac(X - Mu, S / sqrt(N))).

% Mistakes
buggy(frac(X - Mu, S / SQRTN), X - frac(Mu, S) / SQRTN).
buggy(sqrt(N), N).

% Apply expert and buggy rules
step(X, Y) :-
    expert(X, Y).

step(X, Y) :-
    buggy(X, Y).

% Enter expressions
step(X, Y) :-
    compound(X),
    mapargs(search, X, Y),
    dif(X, Y).

% Search through problem space
search(X, X).

search(X, Z) :-
    step(X, Y),
    search(Y, Z).

Listing 3

Manipulating terms in Prolog

The little e-learning system shown in Listing 2 (buggy.pl) is invoked using the R script below.

library(rolog)
consult(system.file(file.path("pl", "buggy.pl"), package="rolog"))

q <- quote(search(tratio(x, mu, s, n), .S))
findall(as.rolog(q))
#> [[1]]
#> [[1]]$S
#> tratio(x, mu, s, n)
#> 
#> 
#> [[2]]
#> [[2]]$S
#> frac(x - mu, s/sqrt(n))
#> 
#> 
#> [[3]]
#> [[3]]$S
#> x - frac(mu, s)/sqrt(n)
#> 
#> 
#> [[4]]
#> [[4]]$S
#> x - frac(mu, s)/n
#> 
#> 
#> [[5]]
#> [[5]]$S
#> frac(x - mu, s/n)
#> 
#> 
#> [[6]]
#> [[6]]$S
#> x - frac(mu, s)/n

The fourth and the sixth result are combinations of the two buggy rules (parenthesis, then square root, and the other way round). Some additional filters would be needed to eliminate trivial and redundant solutions (see, e.g., the chapter on generate-and-test in Sterling and Shapiro 1994).

It should be mentioned that R is homoiconic, too, and the Prolog code above can, in principle, be rewritten in R using non-standard evaluation techniques (Wickham 2019). Prolog’s inbuilt pattern matching algorithm simplifies things a lot, though. An important feature of such a term manipulation is that the evaluation of the term can be postponed; for example, there is no need to instantiate the variables X, Mu, s and N with given values before raising a query. This is especially helpful for variables that may represent larger sets of data in the later course.

Rendering mathematical expressions

The R extension of the markdown language (Xie, Dervieux, and Riederer 2020) enables reproducible statistical reports with nice typesetting in HTML, Microsoft Word, and Latex. However, so far, R expressions such as pbinom(k, N, p) are typeset as-is; prettier mathematical expressions such as \(P_\mathrm{Bi}(X \le k; N, p)\) require Latex commands like P_\mathrm{Bi}\left(X \le k; N, p\right), which are cumbersome to type in and hardly readable even for simple expressions. Below we make use of Prolog’s grammar rules for the automatic translation of R expressions to MathML. The result can then be used for calculations or it can be rendered on a web page. A limited set of rules for translation from R to MathML is found in pl/mathml.pl of the Rolog package. The relevant code snippets are shown in the listings below, along with their output.

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

# R interface to Prolog predicate r2mathml/2
mathml = function(term)
{
  t = once(call("r2mathml", term, expression(X)))
  cat(paste(t$X, collapse=""))
}

Listing 4

Using Rolog to generate MathML from R expressions

The first example is easy. At the Prolog end, there is a handler for pbinom/3 that translates the term into a pretty MathML syntax like P_bi(X <= k; N, pi).

term = quote(pbinom(k, N, p))

# Pretty print
mathml(term)

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

# Do some calculations with the same term
k = 10
N = 22
p = 0.4
eval(term)

[1] 0.77195

The next example is interesting because Prolog needs find out the name of the integration variable for sin. For that purpose, Rolog provides a predicate r_eval/2 that calls R from Prolog (i.e., the reverse direction, see also next example). Here, the predicate is used for the R function formalArgs(args(sin)), which returns the name of the function argument of sin, that is, x.

term = quote(integrate(sin, 0L, 2L*pi))
mathml(term)

$\underset{0}{\overset{2\pi }{\int }}\sin \left(x\right)dx$

eval(term)

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

Note that the Prolog end, the handler for integrate/3 is rather rigid; it accepts only these three arguments in that particular order, and without names, that is, integrate(sin, lower=0L, upper=2L * pi) would not print the desired result.

Therefore, pl/mathml.pl includes two handlers that accept terms with named arguments, integrate(f=Fn, lower=Lower, upper=Upper), as well as terms of the form $(integrate(Fn, Lower, Upper), value) that are needed for the evaluation below.

# Apply match.call to all components of a term
canonical <- function(term)
{
  if(is.call(term))
  {
    f <- match.fun(term[[1]])
    if(!is.primitive(f))
      term <- match.call(f, term)
    
    # Recurse into arguments
    term[-1] <- lapply(term[-1], canonical)
  }

  return(term)
}

# A custom function
g <- function(u)
{
  sin(u)
}

# Mixture of (partially) named and positional arguments in unusual order
term <- quote(2L * integrate(low=-Inf, up=Inf, g)$value)
mathml(canonical(term))

$2\cdot \underset{-\infty }{\overset{\infty }{\int }}g\left(u\right)du$

# It is a bit of a mystery that R knows the result of this integral.
eval(term)

[1] 0

The extra R function canonical() applies match.call() to non-primitive R calls, basically cleaning up the arguments and bringing them into the correct order.

Calling Prolog from R

The basic workflow of the bridge from R to Prolog is to (A) translate an R expression into a Prolog term (i.e., a predicate), (B) query the predicate, and then, (C) translate the result (i.e., the bindings of the variables) back to R. The reverse direction is straightforward, we start by translating a Prolog term to an R expression (i.e. Step C), evaluate the R expression, and then translate the result back to a Prolog term (Step A). Rolog provides two Prolog predicates for this purpose, r_eval(Expr) and r_eval(Expr, Res). The former is used to invoke an R expression Expr for its side effects (e.g., initializing a random number generator); it does not return a result. The latter is used to evaluate the R expression Expr and return the result Res. The code snippet in Listing 5 (r_eval.pl) illustrates this behavior.

r_seed(Seed) :-
    r_eval('set.seed'(Seed)).

r_norm(N, L) :-
    r_eval(rnorm(N), L).

# [r_eval].
consult(system.file(file.path("pl", "r_eval.pl"), package="rolog"))

# rnorm(3)
once(call("r_norm", 3L, expression(X)))
#> $X
#> [1]  0.91560562 -0.18800519 -0.03210339

Listing 5

Calling Prolog from R using r_eval/1 and r_eval/2. The R call set.seed is quoted because the dot is an operator in Prolog.

The example in Listing 5 is a bit trivial, basically illustrating the syntax and the workflow. More serious applications of r_eval/1,2 are illustrated in the example on mathematical expressions where r_eval/2 is used to obtain a names of a function argument, as well as in the next example on interval arithmetic, where r_eval/2 is used with monototonically behaving R functions.

Interval arithmetic

Let \(\langle\ell, u\rangle\) denote a number between \(\ell\) and \(u\), \(\ell\le u\). It is easily verified that the result of the difference \(\langle\ell_1, u_1\rangle - \langle\ell_2, u_2\rangle\) is somewhere in the interval \(\langle \ell_1 - u_2, u_1 - \ell_2\rangle\), and a number of rules exist for basic arithmetic operations and (piecewise) monotonically behaving functions (Hickey, Ju, and Emden 2001). For ratios, denominators with mixed sign yield two possible intervals, for example, \(\langle 1, 2\rangle / \langle -3, 3\rangle = \langle -\infty, 1/3\rangle \cup \langle 1/3, \infty\rangle\), as shown in Figure 4 in Hickey et al.’s article. The number of possible candidates increases if more complicated functions are involved, as unions of intervals themselves appear as arguments (e.g., if \(I_1 \cup I_2\) is added to \(I_3 \cup I_4\), the result is \(I_1 + I_3 \cup I_1 + I_4 \cup I_2 + I_3 \cup I_2 + I_4\)). As a consequence, calculations in interval arithmetic are non-deterministic in nature, and the number of possible results is not foreseeable and cannot, in general, be vectorized as is often done in R. Use cases for interval arithmetic are the limitations of floating-point representations in computer hardware, but intervals can also be used to represent the result of measurements with limited precision, or truncated intermediate results of students doing hand calculations. A few rules for basic interval arithmetic are found in pl/interval.pl; a few examples are shown below. Again, Prolog rings back to R via r_eval/2 to determine the result of dbinom(X, Size, Prob, Log).

# [interval].
consult(system.file(file.path("pl", "interval.pl"), package="rolog"))

# findall(1 ... 2 / -3 ... 3, Res).
q <- quote(int(`...`(1, 2) / `...`(-3, 3), .Res))
findall(as.rolog(q))
#> [[1]]
#> [[1]]$Res
#> ...(-Inf, -0.333333333333333)
#> 
#> 
#> [[2]]
#> [[2]]$Res
#> ...(0.333333333333333, Inf)

# t-ratio
D  = quote(`...`(5.7, 5.8))
mu = 4
s  = quote(`...`(3.8, 3.9))
N  = 24L
tratio = call("/", call("-", D, mu), call("/", s, call("sqrt", N)))
findall(call("int", tratio, expression(Res)))
#> [[1]]
#> [[1]]$Res
#> ...(2.13545259627251, 2.32056923000512)

# Binomial density
prob = quote(`...`(0.2, 0.3))
once(call("int", call("dbinom", 4L, 10L, prob, FALSE), expression(Res)))
#> $Res
#> ...(0.088080384, 0.200120949)

prob = quote(`...`(0.5, 0.6))
once(call("int", call("dbinom", 4L, 10L, prob, FALSE), expression(Res)))
#> $Res
#> ...(0.111476736, 0.205078125)

prob = quote(`...`(0.2, 0.6))
once(call("int", call("dbinom", 4L, 10L, prob, FALSE), expression(Res)))
#> $Res
#> ...(0.088080384, 0.250822656)

The slightly cumbersome syntax for entering an interval \(\langle \ell, u\rangle\) is due to the fact that the ellipsis is a reserved symbol in R and cannot be used as an infix operator. A way more powerful and comprehensive system for constraint logic programming over intervals is available as a Prolog pack (Workman 2021) and can easily be connected to R using, for example, the present package.