# An example of an easy to understand undecidable problem

I am looking for an undecidable problem that I could give as an easy example in a presentation to the general public. I mean easy in the sense that the mathematics behind it can be described, well, without mathematics, that is with analogies and intuition, avoiding technicalities.

• The formula $\forall x, y \quad xy=xy$ is undecidable in the theory describing the usual group theory. – Watson Aug 29 '16 at 18:08

## 5 Answers

"Are these two real numbers (or functions, or grammars, or mathematical statements) equivalent?"
(See also word problem)

"Does this statement follow from these axioms?"
(Hilbert's Entscheidungsproblem)

"Does this computer program ever stop?"
"Does this computer program have any security vulnerabilities?"
"Does this computer program do <any non-trivial statement>?"
(The halting-problem, from which all semantic properties can be reduced)

"Can this set of domino-like tiles tile the plane?"
(See Tiling Problem)

"Does this Diophantine equation have an integer solution?"
(See Hilbert's Tenth Problem)

"Given two lists of strings, is there a list of indices such that the concatenations from both lists are equal?"
(See Post correspondence problem)

There is also a large list on wikipedia.

• Wikipedia's explanation of the proof of undecidability for the halting problem is really bad. – Daniel Nov 19 '13 at 3:23
• Regarding the first example, it's important to consider how you're specifying real numbers. For example, the first-order arithmetic of real numbers is complete; one can always decide, and do so algorithmically, whether two real numbers are equal when they're sepcified in the language of first-order real arithmetic. (also called real-closed fields) – Hurkyl Mar 26 at 4:31

I think the Post correspondence problem is a very good example of a simple undecideable problem that is also relatively unknown.

Given a finite set of string tuples

(a , bba) X
(ab,  aa) Y
(bba, bb) Z


the problem is to determine if there is a finite sequence of these tuples , allowing for repetition, such that the concatenation of the first half is equal to the concatenation of second half

(bba, bb) Z
(ab,  aa) Y
(bba, bb) Z
(a,  bba) X
------------ gives
(bbaabbbaa, bbaabbbaa)


The only big issue I have with this problem is that the only undecideability proof I know of falls back on simulating a Turing machine - it would be nice to find a more elementary alternate version.

• there is a typo (a,bba) instead of (a,baa) for tuple 1. but it is not allowed to change only one character – miracle173 Jan 3 '12 at 1:33
• @miracle173. Thanks! (btw, if I remember correctly you can add invisible html comments <!-- --> to bypass the edit size limit if you need to fix typos in the future) – hugomg Jan 3 '12 at 2:25
• The PCP can be reduced to the word problem for semigroups (and, more generally, for rewriting systems). This gives a more elementary proof of undecidability, if you are willing to accept that the word problem for semigroups/rewriting systems is undecidable! See Harju T., Karhumäki J. (1997) Morphisms. In: Rozenberg G., Salomaa A. (eds) Handbook of Formal Languages. Springer, Berlin, Heidelberg. – user1729 Apr 5 at 12:20

May be you want to check these:

MagicSquare on mathworld.wolfram

• Link rot happens. Imagine someone cannot click on any links, and consider how useful your answer will be. – ShreevatsaR Nov 10 '11 at 8:20
• @ShreevatsaR: Thanks for you suggestion. Do you prefer placing the text of the url instead? – NoChance Nov 10 '11 at 10:12
• Please do not post answers which force people to click on the links to know what they contain. See the simple way I used to avoid this by consulting the modified source of your post. – Did Nov 10 '11 at 10:20
• @DidierPiau: Thank you for your comment and clear instructions. – NoChance Nov 10 '11 at 13:06

"does this computer program ever stop?"

"does this equation have any solutions?" (of course you mean polynomial equation with integer solutions, but for a general public presentation you can probably get away with just "equation" and "solutions").

Maybe consider some Wang Tiles.