# How to forlamlly define a minimal pair

• Oct 9th 2012, 03:38 AM
baxy77bax
How to forlamlly define a minimal pair
Hi,

i guess this is an easy one , but since i'm not a mathematician ....

Let a[i..j] and b[p..q] be two substring and let |a[i..j]| and |b[p..q]| be their lengths

How would i formally define : a pair (a[i..j],b[p..q]) is the minimal pair of substrings between all (a,b) substring pairs from 1 to 100 such that |b[p..q]|-|a[i..j]|=5.

my attempt :

$(a[i..j],b[p..q]) = min\{(a[i..j],b[p..q]) , from 1 to 100 : |b[p..q]|-|a[i..j]|=5\}$

does it say the same thing as in the sentence?

thnx

baxy
• Oct 9th 2012, 04:06 AM
emakarov
Re: How to forlamlly define a minimal pair
Quote:

Originally Posted by baxy77bax
How would i formally define : a pair (a[i..j],b[p..q]) is the minimal pair of substrings between all (a,b) substring pairs from 1 to 100 such that |b[p..q]|-|a[i..j]|=5.

When you say "the minimal pair of substrings," it implies that you can compare pairs of substrings and choose the minimal one. How do you compare pairs of substrings? Second, "substring pairs from 1 to 100" does not make sense because a substring pair cannot equal 1 or 100 or any number in between. A pair of substrings is not a number.
• Oct 9th 2012, 06:38 AM
baxy77bax
Re: How to forlamlly define a minimal pair
ok so the question is not if i can do it or not bur how would i write it down. But to answer your questions:

let say i have a string S = abababba

and i chpo it down to substrings (all possible)

s1 = aba
s2 = abab
s3 = abba
s4 = babab
...
now every substring has a length so a pair of substrings (s1,s3) = (3,4). therefore for every possible pair i will get a pair of their lengths. now i am looking for a pair of substrings with lengths when subtracted give a certain (absolute) value, let say 1. then what i have is a set of pairs (s2,s1),(s3,s1),(s4,s3),(s4,s2) and the minimum substring pair that satisfies my condition is (s2,s1),(s3,s1) (ok i have 2 minimum values but in a real case scenario this cannot ever happen since i have another condition that prevents it)

sorry if i was vague but when i talk to people that do this i usually end-up like an idiot for not knowing this so i did not want to look like one again (big fail again :D )

thnx
• Oct 9th 2012, 11:37 AM
emakarov
Re: How to forlamlly define a minimal pair
Quote:

Originally Posted by baxy77bax
now every substring has a length so a pair of substrings (s1,s3) = (3,4).

This should be written as (|s1|, |s3|) = (3, 4).

Quote:

Originally Posted by baxy77bax
therefore for every possible pair i will get a pair of their lengths. now i am looking for a pair of substrings with lengths when subtracted give a certain (absolute) value, let say 1. then what i have is a set of pairs (s2,s1),(s3,s1),(s4,s3),(s4,s2) and the minimum substring pair that satisfies my condition is (s2,s1),(s3,s1)

I still don't understand in what sense (s2, s1) is minimal. To say that some element of a set is minimal, you need to have an order on that set. What order do you have on pairs of strings?