minicomplexity
function composition
definition

Given two partial functions on $[h]$, check that the value of the first one on the value of the second one on $0$ is $0$.

details

For every $h\geq 1$, function composition$h$ (or just fcomp$h$) is defined over the alphabet $([h]\rightharpoonup[h])$ of all partial functions from $[h]=\{0,1,\dots,h{-}1\}$ to $[h]$. Its instances are all strings of length $2$:

$f\;g$       with $f,g:[h]\rightharpoonup[h]$.
An instance is positive if $f(g(0))=0$. Otherwise, $f(g(0))$ is either undefined or different than $0$, and the instance is negative.

E.g., suppose $f=\mathtt{[303\text{-}4]}$, meaning that $f(0)=3$, $f(1)=0$, $f(2)=3$, $f(3)$ is undefined, and $f(4)=4$; and $g=\mathtt{[12\text{-}0\text{-}]}$, interpreted similarly. Then $fg$ is a positive instance of fcomp$5$, because $f(g(0))=f(1)=0$. The instance $gf$ is also positive, since $g(f(0))=g(3)=0$. In contrast, the instances $ff$ and $gg$ are negative, since $f(f(0))=f(3)$ is undefined and $g(g(0))=g(1)=2\neq0$. (Note that strings like $f$, $g$, $fff$, $fgf$ are not instances of the problem.)

notes

Introduced by Moore 1971 (also Moore 1969), as a problem in 2D\1D which witnesses the asymptotic value of the trade-off in the conversion from 2DFAs to 1DFAs. (Actually defined over a $4$-symbol alphabet, with the two functions encoded as inputs to a fixed ternary 1DFA.)

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

1D f1D/uny
1D f1D/uny co-1D 1D/uny 21D/uny e1D/uny r1D/uny 1N/uny 21N/uny e1N/uny r1N/uny f1N/uny RD/uny 2RD/uny eRD/uny rRD/uny fRD/uny RN/uny 2RN/uny eRN/uny rRN/uny fRN/uny SD/uny 2SD/uny eSD/uny rSD/uny fSD/uny SN/uny 2SN/uny eSN/uny rSN/uny fSN/uny 2D/uny 22D/uny e2D/uny r2D/uny f2D/uny 2N/uny 22N/uny e2N/uny r2N/uny f2N/uny re-1D/uny re-21D/uny re-e1D/uny re-r1D/uny re-f1D/uny re-1N/uny re-21N/uny re-e1N/uny re-r1N/uny re-f1N/uny re-RD/uny re-2RD/uny re-eRD/uny re-rRD/uny re-fRD/uny re-RN/uny re-2RN/uny re-eRN/uny re-rRN/uny re-fRN/uny re-SD/uny re-2SD/uny re-eSD/uny re-rSD/uny re-fSD/uny re-SN/uny re-2SN/uny re-eSN/uny re-rSN/uny re-fSN/uny re-2D/uny re-22D/uny re-e2D/uny re-r2D/uny re-f2D/uny re-2N/uny re-22N/uny re-e2N/uny re-r2N/uny re-f2N/uny co-1D/uny co-21D/uny co-e1D/uny co-r1D/uny co-f1D/uny co-1N/uny co-21N/uny co-e1N/uny co-r1N/uny co-f1N/uny co-RD/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-SD/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-2D/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-1D/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-RD/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-SD/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-2D/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
RD re-1D co-RD
RD re-1D co-RD 21D e1D r1D f1D 1N 21N e1N r1N f1N 2RD eRD rRD fRD RN 2RN eRN rRN fRN SD 2SD eSD rSD fSD SN 2SN eSN rSN fSN 2D 22D e2D r2D f2D 2N 22N e2N r2N f2N re-21D re-e1D re-r1D re-f1D re-1N re-21N re-e1N re-r1N re-f1N re-RD re-2RD re-eRD re-rRD re-fRD re-RN re-2RN re-eRN re-rRN re-fRN re-SD re-2SD re-eSD re-rSD re-fSD re-SN re-2SN re-eSN re-rSN re-fSN re-2D re-22D re-e2D re-r2D re-f2D re-2N 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-SD co-2SD co-eSD co-rSD co-fSD co-SN co-2SN co-eSN co-rSN co-fSN co-2D co-22D co-e2D co-r2D co-f2D co-2N co-22N co-e2N co-r2N co-f2N rc-1D rc-21D rc-e1D rc-r1D rc-f1D rc-1N rc-21N rc-e1N rc-r1N rc-f1N rc-RD rc-2RD rc-eRD rc-rRD rc-fRD rc-RN rc-2RN rc-eRN rc-rRN rc-fRN rc-SD rc-2SD rc-eSD rc-rSD rc-fSD rc-SN rc-2SN rc-eSN rc-rSN rc-fSN rc-2D rc-22D rc-e2D rc-r2D rc-f2D rc-2N rc-22N rc-e2N rc-r2N rc-f2N
complete for:
– no record –
also hard for:
– no record –
mem$^r$:$\leq^\text{t}_\text{h}$ mem$^{rc}$:$\leq^\text{t}_\text{h}$
mem$^r$:$\leq^\text{t}_\text{h}$ mem$^{rc}$:$\leq^\text{t}_\text{h}$ mem$^r$:$\leq_\text{h}$ mem$^{rc}$:$\leq_\text{h}$
fcomp$^c$:$\leq^\text{lac}_\text{1D}$ mem$^r$:$\leq^\text{lac}_\text{1D}$ mem$^{rc}$:$\leq^\text{lac}_\text{1D}$ proj$^r$:$\leq^\text{lac}_\text{1D}$ proj$^{rc}$:$\leq^\text{lac}_\text{1D}$
fcomp$^c$:$\leq^\text{lac}_\text{1D}$ mem$^r$:$\leq^\text{lac}_\text{1D}$ mem$^{rc}$:$\leq^\text{lac}_\text{1D}$ proj$^r$:$\leq^\text{lac}_\text{1D}$ proj$^{rc}$:$\leq^\text{lac}_\text{1D}$ fcomp$^c$:$\leq_\text{1D}$ fcomp$^c$:$\leq^\text{lac}_\text{RD}$ fcomp$^c$:$\leq_\text{RD}$ fcomp$^c$:$\leq^\text{lac}_\text{2D}$ fcomp$^c$:$\leq_\text{2D}$ fcomp$^c$:$\leq^\text{lac}_\text{1N}$ fcomp$^c$:$\leq_\text{1N}$ mem$^r$:$\leq_\text{1D}$ mem$^r$:$\leq^\text{lac}_\text{RD}$ mem$^r$:$\leq_\text{RD}$ mem$^r$:$\leq^\text{lac}_\text{2D}$ mem$^r$:$\leq_\text{2D}$ mem$^r$:$\leq^\text{lac}_\text{1N}$ mem$^r$:$\leq_\text{1N}$ mem$^{rc}$:$\leq_\text{1D}$ mem$^{rc}$:$\leq^\text{lac}_\text{RD}$ mem$^{rc}$:$\leq_\text{RD}$ mem$^{rc}$:$\leq^\text{lac}_\text{2D}$ mem$^{rc}$:$\leq_\text{2D}$ mem$^{rc}$:$\leq^\text{lac}_\text{1N}$ mem$^{rc}$:$\leq_\text{1N}$ proj$^r$:$\leq_\text{1D}$ proj$^r$:$\leq^\text{lac}_\text{RD}$ proj$^r$:$\leq_\text{RD}$ proj$^r$:$\leq^\text{lac}_\text{2D}$ proj$^r$:$\leq_\text{2D}$ proj$^r$:$\leq^\text{lac}_\text{1N}$ proj$^r$:$\leq_\text{1N}$ proj$^{rc}$:$\leq_\text{1D}$ proj$^{rc}$:$\leq^\text{lac}_\text{RD}$ proj$^{rc}$:$\leq_\text{RD}$ proj$^{rc}$:$\leq^\text{lac}_\text{2D}$ proj$^{rc}$:$\leq_\text{2D}$ proj$^{rc}$:$\leq^\text{lac}_\text{1N}$ proj$^{rc}$:$\leq_\text{1N}$
cowl:$\leq^\text{lac}_\text{1D}$ cowl$^r$:$\leq^\text{lac}_\text{1D}$ cowl$^c$:$\leq^\text{lac}_\text{1D}$ cowl$^{rc}$:$\leq^\text{lac}_\text{1D}$ disj:$\leq^\text{lac}_\text{1D}$ disj$^c$:$\leq^\text{lac}_\text{1D}$ fmatch:$\leq^\text{lac}_\text{1D}$ fmatch$^r$:$\leq^\text{lac}_\text{1D}$ fmatch$^c$:$\leq^\text{lac}_\text{1D}$ fmatch$^{rc}$:$\leq^\text{lac}_\text{1D}$ fpath:$\leq^\text{lac}_\text{1D}$ fpath$^r$:$\leq^\text{lac}_\text{1D}$ fpath$^c$:$\leq^\text{lac}_\text{1D}$ fpath$^{rc}$:$\leq^\text{lac}_\text{1D}$ fzmatch:$\leq^\text{lac}_\text{1D}$ fzmatch$^r$:$\leq^\text{lac}_\text{1D}$ fzmatch$^c$:$\leq^\text{lac}_\text{1D}$ fzmatch$^{rc}$:$\leq^\text{lac}_\text{1D}$ incl:$\leq^\text{lac}_\text{1D}$ incl$^r$:$\leq^\text{lac}_\text{1D}$ incl$^c$:$\leq^\text{lac}_\text{1D}$ incl$^{rc}$:$\leq^\text{lac}_\text{1D}$ msp:$\leq^\text{lac}_\text{1D}$ msp$^r$:$\leq^\text{lac}_\text{1D}$ msp$^c$:$\leq^\text{lac}_\text{1D}$ msp$^{rc}$:$\leq^\text{lac}_\text{1D}$ owl:$\leq^\text{t}_\text{h}$ owl$^r$:$\leq^\text{t}_\text{h}$ owl$^c$:$\leq^\text{t}_\text{h}$ owl$^{rc}$:$\leq^\text{t}_\text{h}$ olmem$^r$:$\leq^\text{lac}_\text{1D}$ olmem$^{rc}$:$\leq^\text{lac}_\text{1D}$ sep:$\leq^\text{t}_\text{h}$ sep:$\leq^\text{lac}_\text{1D}$ sep$^r$:$\leq^\text{t}_\text{h}$ sep$^r$:$\leq^\text{lac}_\text{1D}$ sep$^c$:$\leq^\text{t}_\text{h}$ sep$^c$:$\leq^\text{lac}_\text{1D}$ sep$^{rc}$:$\leq^\text{t}_\text{h}$ sep$^{rc}$:$\leq^\text{lac}_\text{1D}$ sp:$\leq^\text{t}_\text{h}$ sp:$\leq^\text{lac}_\text{1D}$ sp$^r$:$\leq^\text{t}_\text{h}$ sp$^r$:$\leq^\text{lac}_\text{1D}$ sp$^c$:$\leq^\text{t}_\text{h}$ sp$^c$:$\leq^\text{lac}_\text{1D}$ sp$^{rc}$:$\leq^\text{t}_\text{h}$ sp$^{rc}$:$\leq^\text{lac}_\text{1D}$ srtall:$\leq^\text{lac}_\text{1D}$ srtall$^c$:$\leq^\text{lac}_\text{1D}$ srtrc:$\leq^\text{lac}_\text{1D}$ srtrc$^c$:$\leq^\text{lac}_\text{1D}$
cowl:$\leq^\text{lac}_\text{1D}$ cowl$^r$:$\leq^\text{lac}_\text{1D}$ cowl$^c$:$\leq^\text{lac}_\text{1D}$ cowl$^{rc}$:$\leq^\text{lac}_\text{1D}$ disj:$\leq^\text{lac}_\text{1D}$ disj$^c$:$\leq^\text{lac}_\text{1D}$ fmatch:$\leq^\text{lac}_\text{1D}$ fmatch$^r$:$\leq^\text{lac}_\text{1D}$ fmatch$^c$:$\leq^\text{lac}_\text{1D}$ fmatch$^{rc}$:$\leq^\text{lac}_\text{1D}$ fpath:$\leq^\text{lac}_\text{1D}$ fpath$^r$:$\leq^\text{lac}_\text{1D}$ fpath$^c$:$\leq^\text{lac}_\text{1D}$ fpath$^{rc}$:$\leq^\text{lac}_\text{1D}$ fzmatch:$\leq^\text{lac}_\text{1D}$ fzmatch$^r$:$\leq^\text{lac}_\text{1D}$ fzmatch$^c$:$\leq^\text{lac}_\text{1D}$ fzmatch$^{rc}$:$\leq^\text{lac}_\text{1D}$ incl:$\leq^\text{lac}_\text{1D}$ incl$^r$:$\leq^\text{lac}_\text{1D}$ incl$^c$:$\leq^\text{lac}_\text{1D}$ incl$^{rc}$:$\leq^\text{lac}_\text{1D}$ msp:$\leq^\text{lac}_\text{1D}$ msp$^r$:$\leq^\text{lac}_\text{1D}$ msp$^c$:$\leq^\text{lac}_\text{1D}$ msp$^{rc}$:$\leq^\text{lac}_\text{1D}$ owl:$\leq^\text{t}_\text{h}$ owl$^r$:$\leq^\text{t}_\text{h}$ owl$^c$:$\leq^\text{t}_\text{h}$ owl$^{rc}$:$\leq^\text{t}_\text{h}$ olmem$^r$:$\leq^\text{lac}_\text{1D}$ olmem$^{rc}$:$\leq^\text{lac}_\text{1D}$ sep:$\leq^\text{t}_\text{h}$ sep:$\leq^\text{lac}_\text{1D}$ sep$^r$:$\leq^\text{t}_\text{h}$ sep$^r$:$\leq^\text{lac}_\text{1D}$ sep$^c$:$\leq^\text{t}_\text{h}$ sep$^c$:$\leq^\text{lac}_\text{1D}$ sep$^{rc}$:$\leq^\text{t}_\text{h}$ sep$^{rc}$:$\leq^\text{lac}_\text{1D}$ sp:$\leq^\text{t}_\text{h}$ sp:$\leq^\text{lac}_\text{1D}$ sp$^r$:$\leq^\text{t}_\text{h}$ sp$^r$:$\leq^\text{lac}_\text{1D}$ sp$^c$:$\leq^\text{t}_\text{h}$ sp$^c$:$\leq^\text{lac}_\text{1D}$ sp$^{rc}$:$\leq^\text{t}_\text{h}$ sp$^{rc}$:$\leq^\text{lac}_\text{1D}$ srtall:$\leq^\text{lac}_\text{1D}$ srtall$^c$:$\leq^\text{lac}_\text{1D}$ srtrc:$\leq^\text{lac}_\text{1D}$ srtrc$^c$:$\leq^\text{lac}_\text{1D}$ all:$\leq^\text{lac}_\text{1D}$ all:$\leq_\text{1D}$ all:$\leq^\text{lac}_\text{RD}$ all:$\leq_\text{RD}$ all:$\leq^\text{lac}_\text{2D}$ all:$\leq_\text{2D}$ all:$\leq^\text{lac}_\text{1N}$ all:$\leq_\text{1N}$ all$^r$:$\leq^\text{lac}_\text{1D}$ all$^r$:$\leq_\text{1D}$ all$^r$:$\leq^\text{lac}_\text{RD}$ all$^r$:$\leq_\text{RD}$ all$^r$:$\leq^\text{lac}_\text{2D}$ all$^r$:$\leq_\text{2D}$ all$^r$:$\leq^\text{lac}_\text{1N}$ all$^r$:$\leq_\text{1N}$ all$^c$:$\leq^\text{lac}_\text{1D}$ all$^c$:$\leq_\text{1D}$ all$^c$:$\leq^\text{lac}_\text{RD}$ all$^c$:$\leq_\text{RD}$ all$^c$:$\leq^\text{lac}_\text{2D}$ all$^c$:$\leq_\text{2D}$ all$^c$:$\leq^\text{lac}_\text{1N}$ all$^c$:$\leq_\text{1N}$ all$^{rc}$:$\leq^\text{lac}_\text{1D}$ all$^{rc}$:$\leq_\text{1D}$ all$^{rc}$:$\leq^\text{lac}_\text{RD}$ all$^{rc}$:$\leq_\text{RD}$ all$^{rc}$:$\leq^\text{lac}_\text{2D}$ all$^{rc}$:$\leq_\text{2D}$ all$^{rc}$:$\leq^\text{lac}_\text{1N}$ all$^{rc}$:$\leq_\text{1N}$ cowl:$\leq_\text{1D}$ cowl:$\leq^\text{lac}_\text{RD}$ cowl:$\leq_\text{RD}$ cowl:$\leq^\text{lac}_\text{2D}$ cowl:$\leq_\text{2D}$ cowl:$\leq^\text{lac}_\text{1N}$ cowl:$\leq_\text{1N}$ cowl$^r$:$\leq_\text{1D}$ cowl$^r$:$\leq^\text{lac}_\text{RD}$ cowl$^r$:$\leq_\text{RD}$ cowl$^r$:$\leq^\text{lac}_\text{2D}$ cowl$^r$:$\leq_\text{2D}$ cowl$^r$:$\leq^\text{lac}_\text{1N}$ cowl$^r$:$\leq_\text{1N}$ cowl$^c$:$\leq_\text{1D}$ cowl$^c$:$\leq^\text{lac}_\text{RD}$ cowl$^c$:$\leq_\text{RD}$ cowl$^c$:$\leq^\text{lac}_\text{2D}$ cowl$^c$:$\leq_\text{2D}$ cowl$^c$:$\leq^\text{lac}_\text{1N}$ cowl$^c$:$\leq_\text{1N}$ cowl$^{rc}$:$\leq_\text{1D}$ cowl$^{rc}$:$\leq^\text{lac}_\text{RD}$ cowl$^{rc}$:$\leq_\text{RD}$ cowl$^{rc}$:$\leq^\text{lac}_\text{2D}$ cowl$^{rc}$:$\leq_\text{2D}$ cowl$^{rc}$:$\leq^\text{lac}_\text{1N}$ cowl$^{rc}$:$\leq_\text{1N}$ ctwl:$\leq^\text{lac}_\text{1D}$ ctwl:$\leq_\text{1D}$ ctwl:$\leq^\text{lac}_\text{RD}$ ctwl:$\leq_\text{RD}$ ctwl:$\leq^\text{lac}_\text{2D}$ ctwl:$\leq_\text{2D}$ ctwl:$\leq^\text{lac}_\text{1N}$ ctwl:$\leq_\text{1N}$ ctwl$^r$:$\leq^\text{lac}_\text{1D}$ ctwl$^r$:$\leq_\text{1D}$ ctwl$^r$:$\leq^\text{lac}_\text{RD}$ ctwl$^r$:$\leq_\text{RD}$ ctwl$^r$:$\leq^\text{lac}_\text{2D}$ ctwl$^r$:$\leq_\text{2D}$ ctwl$^r$:$\leq^\text{lac}_\text{1N}$ ctwl$^r$:$\leq_\text{1N}$ ctwl$^c$:$\leq^\text{lac}_\text{1D}$ ctwl$^c$:$\leq_\text{1D}$ ctwl$^c$:$\leq^\text{lac}_\text{RD}$ ctwl$^c$:$\leq_\text{RD}$ ctwl$^c$:$\leq^\text{lac}_\text{2D}$ ctwl$^c$:$\leq_\text{2D}$ ctwl$^c$:$\leq^\text{lac}_\text{1N}$ ctwl$^c$:$\leq_\text{1N}$ ctwl$^{rc}$:$\leq^\text{lac}_\text{1D}$ ctwl$^{rc}$:$\leq_\text{1D}$ ctwl$^{rc}$:$\leq^\text{lac}_\text{RD}$ ctwl$^{rc}$:$\leq_\text{RD}$ ctwl$^{rc}$:$\leq^\text{lac}_\text{2D}$ ctwl$^{rc}$:$\leq_\text{2D}$ ctwl$^{rc}$:$\leq^\text{lac}_\text{1N}$ ctwl$^{rc}$:$\leq_\text{1N}$ disj:$\leq_\text{1D}$ disj:$\leq^\text{lac}_\text{RD}$ disj:$\leq_\text{RD}$ disj:$\leq^\text{lac}_\text{2D}$ disj:$\leq_\text{2D}$ disj:$\leq^\text{lac}_\text{1N}$ disj:$\leq_\text{1N}$ disj$^r$:$\leq^\text{lac}_\text{1D}$ disj$^r$:$\leq_\text{1D}$ disj$^r$:$\leq^\text{lac}_\text{RD}$ disj$^r$:$\leq_\text{RD}$ disj$^r$:$\leq^\text{lac}_\text{2D}$ disj$^r$:$\leq_\text{2D}$ disj$^r$:$\leq^\text{lac}_\text{1N}$ disj$^r$:$\leq_\text{1N}$ disj$^c$:$\leq_\text{1D}$ disj$^c$:$\leq^\text{lac}_\text{RD}$ disj$^c$:$\leq_\text{RD}$ disj$^c$:$\leq^\text{lac}_\text{2D}$ disj$^c$:$\leq_\text{2D}$ disj$^c$:$\leq^\text{lac}_\text{1N}$ disj$^c$:$\leq_\text{1N}$ disj$^{rc}$:$\leq^\text{lac}_\text{1D}$ disj$^{rc}$:$\leq_\text{1D}$ disj$^{rc}$:$\leq^\text{lac}_\text{RD}$ disj$^{rc}$:$\leq_\text{RD}$ disj$^{rc}$:$\leq^\text{lac}_\text{2D}$ disj$^{rc}$:$\leq_\text{2D}$ disj$^{rc}$:$\leq^\text{lac}_\text{1N}$ disj$^{rc}$:$\leq_\text{1N}$ fmatch:$\leq_\text{1D}$ fmatch:$\leq^\text{lac}_\text{RD}$ fmatch:$\leq_\text{RD}$ fmatch:$\leq^\text{lac}_\text{2D}$ fmatch:$\leq_\text{2D}$ fmatch:$\leq^\text{lac}_\text{1N}$ fmatch:$\leq_\text{1N}$ fmatch$^r$:$\leq_\text{1D}$ fmatch$^r$:$\leq^\text{lac}_\text{RD}$ fmatch$^r$:$\leq_\text{RD}$ fmatch$^r$:$\leq^\text{lac}_\text{2D}$ fmatch$^r$:$\leq_\text{2D}$ fmatch$^r$:$\leq^\text{lac}_\text{1N}$ fmatch$^r$:$\leq_\text{1N}$ fmatch$^c$:$\leq_\text{1D}$ fmatch$^c$:$\leq^\text{lac}_\text{RD}$ fmatch$^c$:$\leq_\text{RD}$ fmatch$^c$:$\leq^\text{lac}_\text{2D}$ fmatch$^c$:$\leq_\text{2D}$ fmatch$^c$:$\leq^\text{lac}_\text{1N}$ fmatch$^c$:$\leq_\text{1N}$ fmatch$^{rc}$:$\leq_\text{1D}$ fmatch$^{rc}$:$\leq^\text{lac}_\text{RD}$ fmatch$^{rc}$:$\leq_\text{RD}$ fmatch$^{rc}$:$\leq^\text{lac}_\text{2D}$ fmatch$^{rc}$:$\leq_\text{2D}$ fmatch$^{rc}$:$\leq^\text{lac}_\text{1N}$ fmatch$^{rc}$:$\leq_\text{1N}$ fpath:$\leq_\text{1D}$ fpath:$\leq^\text{lac}_\text{RD}$ fpath:$\leq_\text{RD}$ fpath:$\leq^\text{lac}_\text{2D}$ fpath:$\leq_\text{2D}$ fpath:$\leq^\text{lac}_\text{1N}$ fpath:$\leq_\text{1N}$ fpath$^r$:$\leq_\text{1D}$ fpath$^r$:$\leq^\text{lac}_\text{RD}$ fpath$^r$:$\leq_\text{RD}$ fpath$^r$:$\leq^\text{lac}_\text{2D}$ fpath$^r$:$\leq_\text{2D}$ fpath$^r$:$\leq^\text{lac}_\text{1N}$ fpath$^r$:$\leq_\text{1N}$ fpath$^c$:$\leq_\text{1D}$ fpath$^c$:$\leq^\text{lac}_\text{RD}$ fpath$^c$:$\leq_\text{RD}$ fpath$^c$:$\leq^\text{lac}_\text{2D}$ fpath$^c$:$\leq_\text{2D}$ fpath$^c$:$\leq^\text{lac}_\text{1N}$ fpath$^c$:$\leq_\text{1N}$ fpath$^{rc}$:$\leq_\text{1D}$ fpath$^{rc}$:$\leq^\text{lac}_\text{RD}$ fpath$^{rc}$:$\leq_\text{RD}$ fpath$^{rc}$:$\leq^\text{lac}_\text{2D}$ fpath$^{rc}$:$\leq_\text{2D}$ fpath$^{rc}$:$\leq^\text{lac}_\text{1N}$ fpath$^{rc}$:$\leq_\text{1N}$ fzmatch:$\leq_\text{1D}$ fzmatch:$\leq^\text{lac}_\text{RD}$ fzmatch:$\leq_\text{RD}$ fzmatch:$\leq^\text{lac}_\text{2D}$ fzmatch:$\leq_\text{2D}$ fzmatch:$\leq^\text{lac}_\text{1N}$ fzmatch:$\leq_\text{1N}$ fzmatch$^r$:$\leq_\text{1D}$ fzmatch$^r$:$\leq^\text{lac}_\text{RD}$ fzmatch$^r$:$\leq_\text{RD}$ fzmatch$^r$:$\leq^\text{lac}_\text{2D}$ fzmatch$^r$:$\leq_\text{2D}$ fzmatch$^r$:$\leq^\text{lac}_\text{1N}$ fzmatch$^r$:$\leq_\text{1N}$ fzmatch$^c$:$\leq_\text{1D}$ fzmatch$^c$:$\leq^\text{lac}_\text{RD}$ fzmatch$^c$:$\leq_\text{RD}$ fzmatch$^c$:$\leq^\text{lac}_\text{2D}$ fzmatch$^c$:$\leq_\text{2D}$ fzmatch$^c$:$\leq^\text{lac}_\text{1N}$ fzmatch$^c$:$\leq_\text{1N}$ fzmatch$^{rc}$:$\leq_\text{1D}$ fzmatch$^{rc}$:$\leq^\text{lac}_\text{RD}$ fzmatch$^{rc}$:$\leq_\text{RD}$ fzmatch$^{rc}$:$\leq^\text{lac}_\text{2D}$ fzmatch$^{rc}$:$\leq_\text{2D}$ fzmatch$^{rc}$:$\leq^\text{lac}_\text{1N}$ fzmatch$^{rc}$:$\leq_\text{1N}$ incl:$\leq_\text{1D}$ incl:$\leq^\text{lac}_\text{RD}$ incl:$\leq_\text{RD}$ incl:$\leq^\text{lac}_\text{2D}$ incl:$\leq_\text{2D}$ incl:$\leq^\text{lac}_\text{1N}$ incl:$\leq_\text{1N}$ incl$^r$:$\leq_\text{1D}$ incl$^r$:$\leq^\text{lac}_\text{RD}$ incl$^r$:$\leq_\text{RD}$ incl$^r$:$\leq^\text{lac}_\text{2D}$ incl$^r$:$\leq_\text{2D}$ incl$^r$:$\leq^\text{lac}_\text{1N}$ incl$^r$:$\leq_\text{1N}$ incl$^c$:$\leq_\text{1D}$ incl$^c$:$\leq^\text{lac}_\text{RD}$ incl$^c$:$\leq_\text{RD}$ 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