# [Haskell-cafe] Period of a sequence

Steffen Schuldenzucker sschuldenzucker at uni-bonn.de
Mon Jun 27 10:32:57 CEST 2011

```
On 06/26/2011 04:16 PM, michael rice wrote:
> MathWorks has the function seqperiod(x) to return the period of sequence
> x. Is there an equivalent function in Haskell?

Could you specify what exactly the function is supposed to do? I am
pretty sure that a function like

seqPeriod :: (Eq a) => [a] -> Maybe Integer  -- Nothing iff non-periodic

cannot be written. If "sequences" are represented by the terms that
define them (or this information is at least accessible), chances might
be better, but I would still be interested how such a function works.
The problem seems undecidable to me in general.

On finite lists (which may be produced from infinite ones via 'take'), a
naive implementation could be this:

>
> import Data.List (inits, cycle, isPrefixOf)
> import Debug.Trace
>
> -- Given a finite list, calculate its period.
> -- The first parameter controls what is accepted as a generator. See
below.
> -- Set it to False when looking at chunks from an infinite sequence.
> listPeriod :: (Eq a) => Bool -> [a] -> Int
> listPeriod precisely xs = case filter (generates precisely xs) (inits
xs) of
>     -- as (last \$ init xs) == xs, this will always suffice.
>     (g:_) -> length g  -- length of the *shortest* generator
>
> -- @generates prec xs g@ iff @g@ generates @xs@ by repitition. If
@prec@, the
> -- lengths have to match, too. Consider
> --
> -- >>> generates True [1,2,3,1,2,1,2] [1,2,3,1,2]
> -- False
> --
> -- >>> generates False [1,2,3,1,2,1,2] [1,2,3,1,2]
> -- True
> generates :: (Eq a) => Bool -> [a] -> [a] -> Bool
> generates precisely xs g = if null g
>     then null xs
>     else (not precisely || length xs `mod` length g == 0)
>       && xs `isPrefixOf` cycle g
>

-- Steffen

```