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If I have a word that consists of letters I, V, X, L, C, D, M, how can I tell whether it is a valid roman numeral? For example, how do I tell that IXXL is not valid?

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    You can draw a syntax diagram. It should break down neatly into four successive phases, for thousands, hundreds, tens and units, the last three being similar in structure. – 2017-01-16

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To ease the description you can consider the pairs "IV", "IX", "XL", "XC", "CD" and "CM" as single symbols.

That is, the roman numerals are sequences made of these symbols: I, IV, V, IX, X, XL, L, XC, C, CD, D, CM, M.

The sequences must hold these rules:

  • The symbols I, X, C, M can be repeated up to three consecutive times. Other symbols must not be repeated.
  • In each sequence, the symbols appear in decreasing order.
  • If a symbol with two letters occurs, none of these two letters occur after it, with the followinf exceptions: after XL and XC there can be IX; and after CD and CM there can be XC.

This way, your example is not legal because IX < XL, or because after IX there must not be any letter X.

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    "If a symbol with two letters occurs, none of these two letters occur after it." is not right. For example 99 - XCIX. After 'XC' there occurs 'X'. – 2017-01-16
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    These are “modern” rules: in Roman inscriptions one can find “IIXX” for 18 (Latin *duodeviginti*), possibly because “IIXX” takes a bit less space than “XVIII”. Repetition of more than three of the main symbols I C M was also allowed. Sticking to a consistent set of rules was not of a concern for the ancient Romans, who were not very proficient in arithmetic. – 2017-01-16
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    @JaroslawMatlak The pair “IX” should be considered as a single symbol. Probably the third rule should be better phrased. – 2017-01-16
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    What about pairs like XM or IC? – 2017-01-17
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    @JaroslawMatlak Thank you for your pointing. I have changed the third rule. – 2017-01-17
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    @spiderface These pairs are not allowed, at least in the modern standard. To write 990. the correct expression is CMXC. – 2017-01-17