I wrote a simple Boot script for converting to and from Roman numerals using instaparse. Here's my EBNF grammar (roman.bnf):

number = thousands hundreds tens ones

I = 'I'
V = 'V'
X = 'X'
L = 'L'
C = 'C'
D = 'D'
M = 'M'

IV = 'IV'
IX = 'IX'
XL = 'XL'
XC = 'XC'
CD = 'CD'
CM = 'CM'

ones = V? I? I? I? | IV | IX
tens = L? X? X? X? | XL | XC
hundreds = D? C? C? C? | CD | CM
thousands = M? M? M?

And here's the Boot script itself (roman.clj):

#!/usr/bin/env boot

(set-env! :dependencies '[[instaparse "1.4.5"]])

(require '[boot.cli :as cli]
         '[instaparse.core :as insta])

(defn digits
  "Returns a lazy sequence of the digits of n in the specified base, starting
  with the least significant digit."
  [base n]
  (->> (iterate #(quot % base) n)
       (take-while pos?)
       (map #(mod % base))))

(defn int->roman
  "Returns the Roman numeral representation of n."
  (let [[ones tens hundreds thousands] (concat (digits 10 n) (repeat 0))]
    (->> (letfn [(digit [one five ten n]
                   (case n
                     4 [one five]
                     9 [one ten]
                     (cons (when (<= 5 n) five)
                           (repeat (mod n 5) one))))]
           [(repeat thousands "M")
            (digit "C" "D" "M" hundreds)
            (digit "X" "L" "C" tens)
            (digit "I" "V" "X" ones)])
         (apply concat)
         (apply str))))

(def parser (insta/parser (slurp "roman.bnf")))

(def symbols
  {:I 1 :V 5 :X 10 :L 50 :C 100 :D 500 :M 1000
   :IV 4 :IX 9 :XL 40 :XC 90 :CD 400 :CM 900})

(defn roman->int
  "Converts the Roman numeral string s to an integer."
  (let [parsed (parser s)]
    (if-some [failure (insta/get-failure parsed)]
      (->> (rest parsed)
           (mapcat rest)
           (map first)
           (map symbols)
           (apply +)))))

(cli/defclifn -main
  "Converts to and from Roman numerals."
  [t to ARABIC int "The number to convert to a Roman numeral."
   f from ROMAN str "The number to convert to an Arabic numeral."]
    (and to from) (println "Please choose only one conversion.")
    to (println (if (< 0 to 4000) (int->roman to) "Out of range."))
    (let [result (roman->int from)]
      ((if (insta/failure? result) print println) result))))

Here are some examples (test.sh):

./roman.clj -t 1
./roman.clj -t 52
./roman.clj -t 294
./roman.clj -t 2780
./roman.clj -f IX
./roman.clj -f LXIX
./roman.clj -f CCXCV
./roman.clj -f MMDCLXXXII
$ chmod +x roman.clj
$ chmod +x test.sh
$ ./test.sh

Obviously this is completely impractical due to Clojure's startup time, but meh.

Are there any improvements I can make? I'm especially interested in ways to make the grammar more elegant; for instance, I can't seem to find a good way to remove the repetition in I? I? I? etc., but if there is a way to genericize the maximum number of occurrences, that would be cool. Any improvements on the Clojure side of things would also be most welcome.


1 Answer 1


Looking at int->roman ...

  • It simply ignores digits that are too big. 10000 maps to ""! Surely better to repeat the biggest signifier as often as necessary.
  • The logic for the one five ten units for digit is in your head. We can derive them from a list of the symbols in increasing order. And it's easier to wrap them as a single argument to digit.
  • I've done the string concatenation within digit.
  • I've dumped the base argument and done the digit-ising inline.

For what it's worth ...

(def symbols "IVXLCDM")

(defn digit [[one five ten] n]
  (let [chars (case n
                4 [one five]
                9 [one ten]
                (cons (when (<= 5 n) five)
                      (repeat (mod n 5) one)))]
    (apply str chars)))

(defn int->roman
  "Returns the Roman numeral representation of n."
  {:pre [(integer? n) (pos? n) (odd? (count symbols))]}
  (let [triples (partition 3 2 symbols)
        heads (->> n (iterate #(quot % 10)))
        digits (map (fn [t i] (digit t (mod i 10))) triples heads)
        left-over (nth heads (count triples))
        left-overs (->> (last symbols)
                        (repeat left-over)
                        (apply str))]
    (->> (concat digits left-overs)
         (apply str))))

For example,

(int->roman 11976)

It could be made faster as a big fat reduce, but I prefer it slow and fairly clear. It gave me enough trouble!

  • \$\begingroup\$ Thanks! Your changes to digit are particularly helpful. I was (and still am) hesitant to convert the higher digits to M because I have seen specs for the Roman numeral system that forbid repeating any symbol more than three times in a row, and also because it can lead to memory overflow if not used very carefully. \$\endgroup\$
    – Sam Estep
    Apr 10, 2017 at 18:02

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