Given a unary string, check that its length is a multiple of every summand in a partition of $h$ with maximum least common multiple.


For every $h\geq 1$, landau$h$ (or just lan$h$) is defined over the unary alphabet $\{\mathtt0\}$. Its instances are all unary strings.

To determine positive and negative instances, we use a multiset $[x_1,x_2,\dots,x_k]$ of positive integers which add up to $h$,

$x_1,x_2,\dots,x_k\geq 1$   and   $x_1+x_2+\cdots+x_k=h$,
and maximize their least common multiple:
lcm$(x_1,x_2,\dots,x_k)$ is maximum possible.
(This is one of the partitions of $h$ which cause the corresponding value of Landau's function. If more than one exist, we pick one arbitrarily.)

Now, an instance is positive if its length $n$ is a multiple of every $x_j$:

for every $x_j$ there exists $\lambda$ such that $n=\lambda x_j$.
Otherwise, the instance is negative. (Equivalently, the instance is positive if its length is a multiple of the value of Landau's function on $h$.)

E.g., the positive instances of lan$7$ are the unary strings whose length is a multiple of $12$, since the partition of $7$ maximizing the least common multiple is $[x_1,x_2]=[3,4]$ (and thus Landau's function on $7$ returns $12$).


Introduced by Holzer Kutrib 2002, as an example of a problem in co-1N/uny\1N. Implicitly present already in Ott 1964, in an example of a unary 1NFA which starts repeating configurations after more than quadratically many steps.

The name “landau” is suggested by this site. huh?

See weak landau for the variant where $n$ must be a multiple of some $x_j$.

1N 1D re-1D re-1N co-1D rc-1D 1D/uny 1N/uny re-1D/uny re-1N/uny co-1D/uny rc-1D/uny
co-1N/uny 21D e1D r1D f1D 21N e1N r1N f1N 2RD eRD rRD fRD 2RN eRN rRN fRN 2SD eSD rSD fSD 2SN eSN rSN fSN 22D e2D r2D f2D 22N e2N r2N f2N re-21D re-e1D re-r1D re-f1D re-21N re-e1N re-r1N re-f1N re-2RD re-eRD re-rRD re-fRD re-2RN re-eRN re-rRN re-fRN re-2SD re-eSD re-rSD re-fSD re-2SN re-eSN re-rSN re-fSN re-22D re-e2D re-r2D re-f2D re-22N re-e2N re-r2N re-f2N co-21D co-e1D co-r1D co-f1D co-1N co-21N co-e1N co-r1N co-f1N co-2RD co-eRD co-rRD co-fRD co-RN co-2RN co-eRN co-rRN co-fRN co-2SD co-eSD co-rSD co-fSD co-SN co-2SN co-eSN co-rSN co-fSN co-22D co-e2D co-r2D co-f2D co-2N co-22N co-e2N co-r2N co-f2N rc-21D rc-e1D rc-r1D rc-f1D rc-1N rc-21N rc-e1N rc-r1N rc-f1N rc-2RD rc-eRD rc-rRD rc-fRD rc-RN rc-2RN rc-eRN rc-rRN rc-fRN rc-2SD rc-eSD rc-rSD rc-fSD rc-SN rc-2SN rc-eSN rc-rSN rc-fSN rc-22D rc-e2D rc-r2D rc-f2D rc-2N rc-22N rc-e2N rc-r2N rc-f2N 21D/uny e1D/uny r1D/uny f1D/uny 21N/uny e1N/uny r1N/uny f1N/uny 2RD/uny eRD/uny rRD/uny fRD/uny 2RN/uny eRN/uny rRN/uny fRN/uny 2SD/uny eSD/uny rSD/uny fSD/uny 2SN/uny eSN/uny rSN/uny fSN/uny 22D/uny e2D/uny r2D/uny f2D/uny 22N/uny e2N/uny r2N/uny f2N/uny re-21D/uny re-e1D/uny re-r1D/uny re-f1D/uny re-21N/uny re-e1N/uny re-r1N/uny re-f1N/uny re-2RD/uny re-eRD/uny re-rRD/uny re-fRD/uny re-2RN/uny re-eRN/uny re-rRN/uny re-fRN/uny re-2SD/uny re-eSD/uny re-rSD/uny re-fSD/uny re-2SN/uny re-eSN/uny re-rSN/uny re-fSN/uny re-22D/uny re-e2D/uny re-r2D/uny re-f2D/uny re-22N/uny re-e2N/uny re-r2N/uny re-f2N/uny co-21D/uny co-e1D/uny co-r1D/uny co-f1D/uny co-21N/uny co-e1N/uny co-r1N/uny co-f1N/uny co-2RD/uny co-eRD/uny co-rRD/uny co-fRD/uny co-RN/uny co-2RN/uny co-eRN/uny co-rRN/uny co-fRN/uny co-2SD/uny co-eSD/uny co-rSD/uny co-fSD/uny co-SN/uny co-2SN/uny co-eSN/uny co-rSN/uny co-fSN/uny co-22D/uny co-e2D/uny co-r2D/uny co-f2D/uny co-2N/uny co-22N/uny co-e2N/uny co-r2N/uny co-f2N/uny rc-21D/uny rc-e1D/uny rc-r1D/uny rc-f1D/uny rc-1N/uny rc-21N/uny rc-e1N/uny rc-r1N/uny rc-f1N/uny rc-2RD/uny rc-eRD/uny rc-rRD/uny rc-fRD/uny rc-RN/uny rc-2RN/uny rc-eRN/uny rc-rRN/uny rc-fRN/uny rc-2SD/uny rc-eSD/uny rc-rSD/uny rc-fSD/uny rc-SN/uny rc-2SN/uny rc-eSN/uny rc-rSN/uny rc-fSN/uny rc-22D/uny rc-e2D/uny rc-r2D/uny rc-f2D/uny rc-2N/uny rc-22N/uny rc-e2N/uny rc-r2N/uny rc-f2N/uny
complete for:
– no record –
also hard for:
– no record –
at least as hard as:
– no record –
lan$^r$:$\leq_\text{id}$ lan$^r$:$\leq^\text{t}_\text{h}$ lan$^r$:$\leq_\text{h}$ lan$^r$:$\leq^\text{lac}_\text{1D}$ lan$^r$:$\leq_\text{1D}$ lan$^r$:$\leq^\text{lac}_\text{RD}$ lan$^r$:$\leq_\text{RD}$ lan$^r$:$\leq^\text{lac}_\text{2D}$ lan$^r$:$\leq_\text{2D}$ lan$^r$:$\leq^\text{lac}_\text{1N}$ lan$^r$:$\leq_\text{1N}$
owl$^c$:$\leq^\text{t}_\text{h}$ owl$^{rc}$:$\leq^\text{t}_\text{h}$ sep$^c$:$\leq^\text{t}_\text{h}$ sep$^{rc}$:$\leq^\text{t}_\text{h}$ sp$^c$:$\leq^\text{t}_\text{h}$ sp$^{rc}$:$\leq^\text{t}_\text{h}$ twl$^c$:$\leq^\text{lac}_\text{1D}$ twl$^{rc}$:$\leq^\text{lac}_\text{1D}$
owl$^c$:$\leq^\text{t}_\text{h}$ owl$^{rc}$:$\leq^\text{t}_\text{h}$ sep$^c$:$\leq^\text{t}_\text{h}$ sep$^{rc}$:$\leq^\text{t}_\text{h}$ sp$^c$:$\leq^\text{t}_\text{h}$ sp$^{rc}$:$\leq^\text{t}_\text{h}$ twl$^c$:$\leq^\text{lac}_\text{1D}$ twl$^{rc}$:$\leq^\text{lac}_\text{1D}$ owl$^c$:$\leq_\text{h}$ owl$^c$:$\leq_\text{1D}$ owl$^c$:$\leq_\text{RD}$ owl$^c$:$\leq_\text{2D}$ owl$^c$:$\leq_\text{1N}$ owl$^{rc}$:$\leq_\text{h}$ owl$^{rc}$:$\leq_\text{1D}$ owl$^{rc}$:$\leq_\text{RD}$ owl$^{rc}$:$\leq_\text{2D}$ owl$^{rc}$:$\leq_\text{1N}$ sep$^c$:$\leq_\text{h}$ sep$^c$:$\leq_\text{1D}$ sep$^c$:$\leq_\text{RD}$ sep$^c$:$\leq_\text{2D}$ sep$^c$:$\leq_\text{1N}$ sep$^{rc}$:$\leq_\text{h}$ sep$^{rc}$:$\leq_\text{1D}$ sep$^{rc}$:$\leq_\text{RD}$ sep$^{rc}$:$\leq_\text{2D}$ sep$^{rc}$:$\leq_\text{1N}$ sp$^c$:$\leq_\text{h}$ sp$^c$:$\leq_\text{1D}$ sp$^c$:$\leq_\text{RD}$ sp$^c$:$\leq_\text{2D}$ sp$^c$:$\leq_\text{1N}$ sp$^{rc}$:$\leq_\text{h}$ sp$^{rc}$:$\leq_\text{1D}$ sp$^{rc}$:$\leq_\text{RD}$ sp$^{rc}$:$\leq_\text{2D}$ sp$^{rc}$:$\leq_\text{1N}$ twl$^c$:$\leq^\text{t}_\text{h}$ twl$^c$:$\leq_\text{h}$ twl$^c$:$\leq_\text{1D}$ twl$^c$:$\leq^\text{lac}_\text{RD}$ twl$^c$:$\leq_\text{RD}$ twl$^c$:$\leq^\text{lac}_\text{2D}$ twl$^c$:$\leq_\text{2D}$ twl$^c$:$\leq^\text{lac}_\text{1N}$ twl$^c$:$\leq_\text{1N}$ twl$^{rc}$:$\leq^\text{t}_\text{h}$ twl$^{rc}$:$\leq_\text{h}$ twl$^{rc}$:$\leq_\text{1D}$ twl$^{rc}$:$\leq^\text{lac}_\text{RD}$ twl$^{rc}$:$\leq_\text{RD}$ twl$^{rc}$:$\leq^\text{lac}_\text{2D}$ twl$^{rc}$:$\leq_\text{2D}$ twl$^{rc}$:$\leq^\text{lac}_\text{1N}$ twl$^{rc}$:$\leq_\text{1N}$