ASwitchingLemmaTutorial
BenjaminRossman
UniversityofTorontoandNII,Tokyo
homagetoPaulBeame’sexcellent
ASwitchingLemmaPrimer
ASwitchingLemmaTutorial*
BenjaminRossman
UniversityofTorontoandNII,Tokyo
Overview
1.  BasicConcepts
–  modelsofcomputa?on(AC0,DNFs/CNFs,
decisiontrees)
–  randomrestric?onRp
2.  TheClassicSwitchingLemma(Hastad‘86)
3.  PARITYLowerBound
4.  AffineRestricGons
–  “Tsei?nexpanderswitchinglemma”(jointwith
Pitassi,ServedioandTan)
ModelsofComputa?on
•  AC0circuits/formulas
•  DNFs/CNFs(i.e.depth-2formulas)
•  Decisiontrees
•  CanonicaldecisiontreeofaDNF/CNF
AC0Circuits
size=#ofgates
depth=#layersofgates
∨
∧
∨
∨
∧
∧
∨
∨
∧
∨
∨
∨
∧
∨
∧
∨
∧
∨
x1 ¬x1 x2 ¬x2 x3 ¬x3 x4 ¬x4 x5 ¬x5
AC0Circuits
size=#ofgates
depth=#layersofgates
∨
∧
∨
∧
0refers
ThecomplexityclassAC
∨
∨
∨
∨
to(sequencesof)Boolean
func?ons{0,1}n→{0,1}
∧
∧
∧
∧
computablebypoly(n)-size,
constant-depthcircuits.
∨
∨
∨
∨
∧
∨
x1 ¬x1 x2 ¬x2 x3 ¬x3 x4 ¬x4 x5 ¬x5
Depth-2(DNFsandCNFs)
•  DNF=disjunc?venormalform(OR-ANDformula)
•  CNF=conjunc?venormalform(AND-ORformula)
•  width=max#ofvariablesinaterm/clause
∨
∞
k
∧
∧
∧
∧
∧
∧
……
∧
Depth-2(DNFsandCNFs)
•  k-DNF=width-kDNF
•  k-CNF=width-kCNF
∨
∞
k
∧
∧
∧
∧
∧
∧
……
∧
DecisionTrees
x3
0
depth3
1
x1
x7
0
1
x4
0
0
x7
1
0
1
x1
1
0
x8
1
0
1
10010110
DecisionTrees
Thedecision-treedepthofaBooleanfunc?on
f:{0,1}n→{0,1}
denotedDTdepth(f),istheminimumdepthofa
decisiontreethatcomputesf
•  DTdepth(PARITYn)=DTdepth(ANDn)=n
•  DTdepth(f)=0⇔fisconstant
DecisionTreetoDNF
∨(¬x3∧¬x1∧¬x4)
∨(¬x3∧x1∧x7)
∨(x3∧x7∧x1)
∨(x3∧x7∧x8)
x1
x3
0
x7
0
1
x4
0
1
0
x7
1
0
1
x1
1
0
x8
1
0
1
10010110
DecisionTreetoDNF
∨(¬x3∧¬x1∧¬x4)
x3
0
1
x1
x7
0
1
x4
0
0
x7
1
0
1
x1
1
0
x8
1
0
1
10010110
DecisionTreetoDNF
∨(¬x3∧¬x1∧¬x4)
∨(¬x3∧x1∧x7)
x3
0
1
x1
x7
0
1
x4
0
0
x7
1
0
1
x1
1
0
x8
1
0
1
10010110
DecisionTreetoDNF
∨(¬x3∧¬x1∧¬x4)
∨(¬x3∧x1∧x7)
∨(x3∧x7∧x1)
x3
0
1
x1
x7
0
1
x4
0
0
x7
1
0
1
x1
1
0
x8
1
0
1
10010110
DecisionTreetoDNF
∨(¬x3∧¬x1∧¬x4)
∨(¬x3∧x1∧x7)
∨(x3∧x7∧x1)
∨(x3∧x7∧x8)
x1
x3
0
x7
0
1
x4
0
1
0
x7
1
0
1
x1
1
0
x8
1
0
1
10010110
CanonicalDTofaDNF
∨(x1∧x2∧x3)
∨(¬x2∧x4)
∨(x1∧¬x3∧¬x7)
CanonicalDTofaDNF
∨(x1x2x3)
∨(¬x2x4)
∨(x1¬x3¬x7)
CanonicalDTofaDNF
∨(x1x2x3)
…
0
x1
…
0
1
x2
1
…
x3
0
…
1
1
CanonicalDTofaDNF
∨(x1x2x3)
∨(¬x2x4)
…
x1
0
x2
0
…
0
1
1
x2
…
1
1
0
x2
0
1
…
1
1
1
…
x4
0
1
x3
…
x4
0
x2
0
…
x4
0
1
1
1
1
CanonicalDTofaDNF
∨(x1x2x3)
∨(¬x2x4)
…
x1
0
x2
0
…
0
1
1
x2
…
1
1
0
x2
0
1
…
1
1
1
…
x4
0
1
x3
…
x4
0
x2
0
…
x4
0
1
1
1
1
CanonicalDTofaDNF
∨(x1x2x3)
∨(¬x2x4)
…
x1
0
x2
0
…
1
1
x2
0
…
x4
0
1
1
1
x4
0
…
1
1
x3
0
…
1
1
CanonicalDTofaDNF
∨(x1x2x3)
∨(¬x2x4)
∨(x1¬x3¬x7)
x1
0
x2
0
…
1
1
x4
1
…
x7
0
1
1
…
0
1
x3
0
x3
1
0
1
x2
0
…
x4
0
1
1
1
1
x7
0
1
1
…
CanonicalDTofaDNF
∨(x1x2x3)
∨(¬x2x4)
∨(x1¬x3¬x7)
x1
0
x2
0
0
1
0
0
1
1
0
0
1
1
x7
x3
1
x3
0
1
x4
0
1
x2
0
0
x4
0
1
1
1
1
x7
0
1
1
0
Restric?ons
a.k.a.func?ons[n]→{0,1,⋆}
a.k.a.subcubesof{0,1}n
Restric?ons
•  ConsideraBooleanfunc?on
f:{0,1}n→{0,1}
•  Arestric3on(w.r.t.thevariablesoff)isafunc?on
R:[n]→{0,1,⋆}
Restric?ons
•  ConsideraBooleanfunc?on
f:{0,1}n→{0,1}
•  Arestric3on(w.r.t.thevariablesoff)isafunc?on
R:[n]→{0,1,⋆}
[n]={1,…,n},whichweiden?fy
withthevariablesx1,…,xnoff(x)
Restric?ons
•  ConsideraBooleanfunc?on
f:{0,1}n→{0,1}
•  Arestric3on(w.r.t.thevariablesoff)isafunc?on
R:[n]→{0,1,⋆}
R(i)=0or1meansthatthe
variablexiisfixedto0or1
Restric?ons
•  ConsideraBooleanfunc?on
f:{0,1}n→{0,1}
•  Arestric3on(w.r.t.thevariablesoff)isafunc?on
R:[n]→{0,1,⋆}
R(i)=⋆meansthatthevariable
xiisfree(unrestricted)
Restric?ons
•  f:{0,1}n→{0,1}
•  R:[n]→{0,1,⋆}
•  ApplyingRtof,wegetaBooleanfunc?on
f↾R:{0,1}Stars(R)→{0,1}
R ⋆1⋆⋆10⋆1⋆100⋆⋆0⋆0⋆⋆⋆0⋆0
f↾R(01101001010011010101010)
f(01101001010011010101010)
Restric?ons
•  Restric?onsareappliedsyntac3callytoDNFs/
CNFs/decisiontrees/AC0circuits,etc.
Restric?nganAC0Circuit
•  ConsiderR={x1↦1}
∧
∨
∨
∨
∧
∧
∨
∨
∧
∨
∨
∨
∧
∨
∧
∨
∧
∨
x1 ¬x1 x2 ¬x2 x3 ¬x3 x4 ¬x4 x5 ¬x5
Restric?nganAC0Circuit
•  ConsiderR={x1↦1}
∧
i.e.,R(1)=1andR(i)=⋆foralli≠1
∨
∨
∨
∨
∨
∧
∧
∨
∨
∧
∨
∧
∨
∧
∨
∧
∨
x1 ¬x1 x2 ¬x2 x3 ¬x3 x4 ¬x4 x5 ¬x5
Restric?nganAC0Circuit
•  ConsiderR={x1↦1}
∧
∨
∧
∨
1 0
∨
∨
∧
1
∨
∧
∨
∨
∨
∧
∨
∧
∨
∧
∨
x2 ¬x2 x3 ¬x3 x4 ¬x4 x5 ¬x5
Restric?nganAC0Circuit
•  ConsiderR={x1↦1}
∧
∨
∧
∨
1 0
∨
∨
∧
1
∨
∧
∨
∨
∨
∧
∨
∧
∨
∧
∨
x2 ¬x2 x3 ¬x3 x4 ¬x4 x5 ¬x5
Restric?nganAC0Circuit
•  ConsiderR={x1↦1}
∧
∨
∧
∧
∨
∨
1 0
∨
∨
∧
∨
∨
∨
∧
∨
∧
∨
∧
∨
x2 ¬x2 x3 ¬x3 x4 ¬x4 x5 ¬x5
Restric?nganAC0Circuit
GateElimina9onMethod
•  ConsiderR={x1↦1}
N1+1/exp(d)lowerboundagainst
•  to⋆
depth-dcircuitsviadeterminis3c
∧
restric3ons[Chaudhuri&
Radhakrishnan’96]
∨
∧
∧
∨
∨
1 0
∨
∨
∧
∨
∨
∧
∨
∧
∨
∨
∧
∨
x2 ¬x2 x3 ¬x3 x4 ¬x4 x5 ¬x5
Restric?ngaDNF
R={x1↦1,x4↦0}
x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x7x8x9∨¬x4¬x7x9
Restric?ngaDNF
R={x1↦1,x4↦0}
1
0
1
0
1
x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x7x8x9∨¬x4¬x7x9
Restric?ngaDNF
R={x1↦1,x4↦0}
1
0
1
0
1
x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x7x8x9∨¬x4¬x7x9
Restric?ngaDNF
R={x1↦1,x4↦0}
1
0
1
0
1
x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x7x8x9∨¬x4¬x7x9
Restric?ngaDNF
R={x1↦1,x4↦0}
1
0
1
0
1
x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x7x8x9∨¬x4¬x7x9
↾R
x2¬x3∨x2x5∨x7x8x9∨¬x7x9
Restric?ngaDecisionTree
•  AgaincoR={x1↦1}
x3
0
1
x1
x7
0
1
x4
0
0
x7
1
0
1
x1
1
0
x8
1
0
1
10010110
Restric?ngaDecisionTree
•  AgaincoR={x1↦1}
x3
0
1
x1
x7
0
1
x4
0
0
x7
1
0
1
x1
1
0
x8
1
0
1
10010110
Restric?ngaDecisionTree
•  AgaincoR={x1↦1}
x3
0
x3
1
↾R
x1
1
x4
0
0
x7
1
0
x1
1
0
0
1
0
1
01
x8
1
x7
x7
x7
0
1
10010110
1
0
0
1
1 x
8
0
1
10
TheRandomRestric?onRp
Throughoutthistalk,
randomobjectsinboldface
TheRandomRestric?onRp
•  For0≤p≤1,letRpbetherandomrestric3onwith
⋆ withprob.p
Rp(i)= 0 withprob.(1-p)/2
1 withprob.(1-p)/2
independentlyforeachvariableindexi∈[n]
EffectofRp
•  RpsimplifiesBooleanfunc?onsrepresentedbysmall
decisiontrees/AC0circuits/deMorganformulas,…
[Subbotovskaya‘61,…]
•  CertainBooleanfunc?ons,likePARITY,maintain
theircomplexityunderRp
•  =>lowerbounds!
EffectofRponBooleanFunc?ons
•  RpkillsANDnwithveryhighprobability(forp≤1/2):
Pr[ANDn↾Rp≡0]≥1–(1/2)O(n)
EffectofRponBooleanFunc?ons
•  RpkillsANDnwithveryhighprobability(forp≤1/2):
Pr[ANDn↾Rp≡0]≥1–(1/2)O(n)
•  Ontheotherhand,foranyRwithkstars,
PARITYn↾R≡PARITYkor1–PARITYk
Inpar?cular,DTdepth(PARITYn↾Rp)≈pnwithhighprob
EffectofRponDecisionTrees
DecisionTreeSwitchingLemma
IfTisadepth-kdecisiontree,then
Pr[depth(T↾Rp)≥ℓ]≤(2pk)ℓ
EffectofRponDecisionTrees
DecisionTreeSwitchingLemma
IfTisadepth-kdecisiontree,then
Pr[depth(T↾Rp)≥ℓ]≤(2pk)ℓ
≤2–ℓwhenp≤1/4k
EffectofRponDNFs
DecisionTreeSwitchingLemma
IfTisadepth-kdecisiontree,then
Pr[depth(T↾Rp)≥ℓ]≤(2pk)ℓ
Hastad’sSwitchingLemma(1986)
IfFisak-DNF(i.e.OR∞ofdepth-kdecisiontrees),then
Pr[DTdepth(F↾Rp)≥ℓ]≤(5pk)ℓ
EffectofRponDNFs
DecisionTreeSwitchingLemma
IfTisadepth-kdecisiontree,then
ℓ
–ℓ
Pr[depth(T↾R
)≥ℓ]≤(2pk)
≤2 whenp≤1/10k
p
Hastad’sSwitchingLemma(1986)
IfFisak-DNF(i.e.OR∞ofdepth-kdecisiontrees),then
Pr[DTdepth(F↾Rp)≥ℓ]≤(5pk)ℓ
EffectofRponDNFs
Corollary(usualstatementoftheS.L.)
IfFisak-DNF,then
Pr[F↾Rpisnotequivalenttoanℓ-CNF]≤(5pk)ℓ
Hastad’sSwitchingLemma(1986)
IfFisak-DNF(i.e.OR∞ofdepth-kdecisiontrees),then
Pr[DTdepth(F↾Rp)≥ℓ]≤(5pk)ℓ
EffectofRponDNFsCNFs
Corollary(usualstatementoftheS.L.)
IfFisak-CNF,then
Pr[F↾Rpisnotequivalenttoanℓ-DNF]≤(5pk)ℓ
Hastad’sSwitchingLemma(1986)
IfFisak-CNF(i.e.AND∞ofdepth-kdecisiontrees),then
Pr[DTdepth(F↾Rp)≥ℓ]≤(5pk)ℓ
EffectofRponDNFs
∨
∞
k
∧
∧
∧
∧
∧
Rp
ℓ
∨
∨
……
∧
w.h.p.forsmallp
∧
∞
∨
∧
∨
∨
∨
……
∨
DepthReduc?on
∧
unbnd'd
fan-in
∨
k
∨
∨
∧
∧
∨
∨
k
∧
k
∨
∨
∨
∧
∧
∨
k
k
∨
∧
∨
k
DepthReduc?on
∧
ApplytheSwitchingLemmatoeachgate
(takeaunionboundoverfailureprobability)
∨
k
∨
∨
∧
∧
∨
∨
k
∧
k
∨
∨
∧
∧
∨
k
k
∨
∨
∧
∨
k
DepthReduc?on
∧
ApplytheSwitchingLemmatoeachgate
(takeaunionboundoverfailureprobability)
∨
ℓ
∨
∨
∨
∨
∧
∧
ℓ
∨
ℓ
∧
∨
∨
∨
∧
∧
ℓ
ℓ
∨
∨
ℓ
∧
DepthReduc?on
∧
∨
twolayers
of∨-gates
ℓ
∨
∨
∨
∨
∧
∧
ℓ
∨
ℓ
∧
∨
∨
∨
∨
∧
∧
ℓ
ℓ
∨
ℓ
∧
DepthReduc?on
∧
∨
∧
ℓ
∨
∨
∧
ℓ
ℓ
∧
∨
∨
∧
∧
ℓ
ℓ
ℓ
∧
DepthReduc?on
∧
∨
∧
∧
∨
∨
k
∨
∨
k
∧
∨
k
∧
∧
∨
∨
k
ApplyingtheS.L.to
theboAom-2layers
∨
∨
∧
∨
k
k
∧
∨
∧
ℓ
∨
∨
∧
ℓ
∧
ℓ
∨
∨
∧
ℓ
∧
ℓ
∧
ℓ
DepthReduc?on
∧
∨
∧
∧
∨
∨
k
∨
∨
k
∧
∨
k
∧
∧
∨
∨
k
ApplyingtheS.L.to
theboAom-2layers
∨
∨
∧
∨
k
k
∧
typically,k=ℓ≈n1/d
∨
∧
ℓ
∨
∨
∧
ℓ
∧
ℓ
∨
∨
∧
ℓ
∧
ℓ
∧
ℓ
DepthReduc?on
∧
unbnd'd
fan-in
∨
∧
depth-k
dec.trees
∨
∨
∧
∧
∨
∨
∧
∧
∧
Hastad’sSwitchingLemma(decisiontreeversion)
IfT1,…,Tmaredepth-kdecisiontrees,then
Pr[DTdepth((T1∧⋯∧Tm)↾Rp)≥ℓ]≤(5pk)ℓ
∧
unbnd'd
fan-in
∨
∧
depth-k
dec.trees
∨
∨
∧
∧
∨
∨
∧
∧
∧
Hastad’sSwitchingLemma(decisiontreeversion)
IfT1,…,Tmaredepth-kdecisiontrees,then
Pr[DTdepth((T1∧⋯∧Tm)↾Rp)≥ℓ]≤(5pk)ℓ
∧
∨
depth-ℓ
dec.trees
∨
∨
∨
∨
DepthReduc?on
∧
∨
∧
∨
∨
∧
∧
∧
applyingtheS.L.tothe
boAomlayerofgates
∨
∨
∧
∧
∧
∨
∨
∨
∨
∨
PARITYLowerBound
Hastad’86
Depthd+1circuitsforPARITYrequiresizeexp(Ω(n1/d))
Hastad’86
Depthd+1circuitsforPARITYrequiresizeexp(Ω(n1/d))
(Trivial)UpperBound
PARITYhasdepthd+1circuitsofsizeexp(O(n1/d))
2
n1/d
d
n1−1/d
n1−1/d
n1−1/d
Hastad’86
Depthd+1circuitsforPARITYrequiresizeexp(Ω(n1/d))
(Trivial)UpperBound
PARITYhasdepthd+1circuitsofsizeexp(O(n1/d))
d+1
n
PARITYLowerBound
∧
∨
∨
∨
∧
∧
∨
∨
∧
∨
∨
∨
∧
∨
∧
∨
∧
∨
x1 ¬x1 x2 ¬x2 x3 ¬x3 x4 ¬x4 x5 ¬x5
PARITYLowerBound
∧
∨
∨
∨
∧
∧
∨
∨
∧
∨
∨
∨
∧
∨
∧
∨
depth-1decisiontreesonnvariables
∧
∨
PARITYLowerBound
∧
∨
∨
∨
∧
∧
∨
∨
∨
∨
Rp-randomrestric?on
∧
∧
∧
𝝆1:[n]→{0,1,⋆}
∨
∨
∨
depth-1decisiontreesonnvariables
∧
∨
PARITYLowerBound
∧
∨
∨
∨
∨
∨
ApplytheSwitchingLemmatoeachbo|om-levelgate
Takeaunionboundoverfailureprobabili?es
∧
∧
∧
∧
∧
∨
∨
∨
∨
∨
depth-1decisiontreesonnvariables
∧
∨
PARITYLowerBound
∧
∨
∨
∨
∨
∨
ApplytheSwitchingLemmatoeachbo|om-levelgate
Takeaunionboundoverfailureprobabili?es
∧
∧
∧
∧
∧
∧
low-depthdecisiontreesonpnvariables
PARITYLowerBound
∧
•  1−O(ε)frac?onhavedepth0
•  O(ε)frac?onhavedepth1
∨
∨
∨
∨
•  O(ε2)frac?onhavedepth2
•  …
∧
∧ •  O(εℓ∧
∧
∧
)havedepthℓ
∨
∧
low-depthdecisiontreesonpnvariables
PARITYLowerBound
∧
∨
∧
∨
∨
∧
∧
∨
∨
∧
∧
∧
low-depthdecisiontreesonpnvariables
PARITYLowerBound
∧
Rp-randomrestric?on
∨
∨
∨
𝝆2:Stars(𝝆1)→{0,1,⋆}
∨
∧
∧
∧
∧
∧
∨
∧
low-depthdecisiontreesonpnvariables
PARITYLowerBound
∧
∨
∨
∨
∨
∨
low-depthdecisiontreesonp2nvariables
PARITYLowerBound
Rp-randomrestric?on
∧
𝝆3:Stars(𝝆2)→{0,1,⋆}
∨
∨
∨
∨
∨
low-depthdecisiontreesonp2nvariables
PARITYLowerBound
∧
low-depthdecisiontreesonp3nvariables
PARITYLowerBound
Rp-randomrestric?on
𝝆4:Stars(𝝆∧3)→{0,1,⋆}
low-depthdecisiontreesonp3nvariables
PARITYLowerBound
depth-0decisiontree
(i.e.constantfuncGon)
onp4nvariables
PARITYLowerBound
depth-0decisiontree
(i.e.constantfuncGon)
onp4nvariables
•  w.h.p.,outputisfixedto0or1a~errestric?ons
𝞺1𝞺2𝞺3𝞺4(equivalenttoRp4)
•  Ifp4n≫1,thisshowstheoriginalcircuitdoesnot
computePARITYn
PARITYLowerBound
∧
∨
∨
∨
∨
∨
Op?malchoiceofparametersgivesa?ght
exp(Ω(n1/d))lowerboundfordepthd+1circuits
∧
∧
∨
∨
∧
∨
∧
∨
∧
∨
∧
∨
x1 ¬x1 x2 ¬x2 x3 ¬x3 x4 ¬x4 x5 ¬x5
Formulasvs.Circuits
AC0Circuits
∧
∨
∨
∨
∧
∧
∨
∨
∧
∨
∨
∨
∧
∨
∧
∨
∧
∨
x1 ¬x1 x2 ¬x2 x3 ¬x3 x4 ¬x4 x5 ¬x5
AC0Formulas
∧
∨
∧
∧
∧
∨
∨
∧
∧
∧
∧
∧
∧
x3 ¬x7 x4 ¬x1 x3 ¬x2 x11 ¬x1 x2 ¬x5 ¬x4 x2 x1
Fact:Everydepthd+1circuitofsizeSisequivalent
toadepthd+1formulaofsizeatmostSd
∧
∧
PARITYLowerBound
Hastad’86
Depthd+1circuitsforPARITYrequiresizeexp(Ω(n1/d))
R.‘15
Depthd+1formulasforPARITYhavesizeexp(Ω(dn1/d))
PARITYLowerBound
Hastad’86
Depthd+1circuitsforPARITYrequiresizeexp(Ω(n1/d))
logn
Poly-sizecircuitsforPARITYhavedepth loglogn+O(1)
R.‘15
Depthd+1formulasforPARITYhavesizeexp(Ω(dn1/d))
Poly-sizeformulasforPARITYhavedepthΩ(logn)
TwoViewsofRp:[n]→{0,1,⋆}
(1)Independently,foreachcoordinatei∈[n],set
⋆ withprob.p
Rp(i)= 0 withprob.(1-p)/2
1 withprob.(1-p)/2
(2)  𝞂←{0,1}nuniformrandom“assignment”
i𝛕←[0,1]nuniformrandom“?me-stamp”
Rp(i)=
⋆ if𝛕i≤p
𝞂i if𝛕i>p
1
0
𝛕
𝞂01101001010011010111010
01101001010011010111010
𝞂←{0,1}nuniformrandom“assignment”
i𝛕←[0,1]nuniformrandom“?me-stamp”
⋆ if𝛕i≤p
Rp(i)=
𝞂i if𝛕i>p
1
0
𝛕
“?me”prunsfrom1to0
𝞂01101001010011010111010
01101001010011010111010
𝞂←{0,1}nuniformrandom“assignment”
i𝛕←[0,1]nuniformrandom“?me-stamp”
⋆ if𝛕i≤p
Rp(i)=
𝞂i if𝛕i>p
start?mep=1
1
0
𝛕
𝞂01101001010011010111010
01101001010011010111010
Rp⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆
𝞂←{0,1}nuniformrandom“assignment”
i𝛕←[0,1]nuniformrandom“?me-stamp”
⋆ if𝛕i≤p
Rp(i)=
𝞂i if𝛕i>p
start?mep=1
1
0
𝛕
𝞂01101001010011010111010
Rp⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆
𝞂←{0,1}nuniformrandom“assignment”
i𝛕←[0,1]nuniformrandom“?me-stamp”
⋆ if𝛕i≤p
Rp(i)=
𝞂i if𝛕i>p
p=0.99
1
0
𝛕
𝞂01101001010011010111010
01101001010011010111010
Rp⋆⋆⋆⋆⋆0⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆
𝞂←{0,1}nuniformrandom“assignment”
i𝛕←[0,1]nuniformrandom“?me-stamp”
⋆ if𝛕i≤p
Rp(i)=
𝞂i if𝛕i>p
p=0.5
1
0
𝛕
𝞂01101001010011010111010
01101001010011010111010
Rp⋆1⋆⋆10⋆1⋆100⋆⋆0⋆0⋆⋆⋆0⋆0
𝞂←{0,1}nuniformrandom“assignment”
i𝛕←[0,1]nuniformrandom“?me-stamp”
⋆ if𝛕i≤p
Rp(i)=
𝞂i if𝛕i>p
p=0.1
1
0
𝛕
𝞂01101001010011010111010
01101001010011010111010
Rp011⋆1001⋆100⋆10⋆01110⋆0
𝞂←{0,1}nuniformrandom“assignment”
i𝛕←[0,1]nuniformrandom“?me-stamp”
⋆ if𝛕i≤p
Rp(i)=
𝞂i if𝛕i>p
end?mep=0
1
0
𝛕
𝞂01101001010011010111010
01101001010011010111010
Rp01101001010011010111010
𝞂←{0,1}nuniformrandom“assignment”
i𝛕←[0,1]nuniformrandom“?me-stamp”
⋆ if𝛕i≤p
Rp(i)=
𝞂i if𝛕i>p
1
0
∧
∨
∧
∧
∨
∨
x1 ¬x1
∨
∨
x2 ¬x2
∧
∧
∨
x3 ¬x3
∨
∨
∨
∧
∨
x4 ¬x4
∧
∨
x5 ¬x5
1
p
0
∧
∨
∧
∧
∨
∨
x1 ¬x1
∨
∨
x2 ¬x2
∧
∧
Rp
∨
x3 ¬x3
∨
∨
∨
∧
∨
x4 ¬x4
∧
∨
x5 ¬x5
1
p
p2
0
∧
∨
∧
∧
∨
∨
x1 ¬x1
∨
∨
x2 ¬x2
∧R p2
Rp
∨
x3 ¬x3
∨
∨
∧
∨
∧
∨
x4 ¬x4
∧
∨
x5 ¬x5
1
p
p2
p3
0
∧
∨
∨
∧
∧
∨
∨
x1 ¬x1
x2 ¬x2
Rp∨
3
∧R p2
Rp
∨
x3 ¬x3
∨
∨
∧
∨
∧
∨
x4 ¬x4
∧
∨
x5 ¬x5
1
p
p2
p34
p
0
Rp∧
4
∨
∨
∧
∧
∨
∨
x1 ¬x1
x2 ¬x2
Rp∨
3
∧R p2
Rp
∨
x3 ¬x3
∨
∨
∧
∨
∧
∨
x4 ¬x4
∧
∨
x5 ¬x5
1
0
∧
∨
∧
∧
∧
∨
∨
∧
∧
∧
∧
∧
∧
1
•  Eachsub-formulaFwillhaveits
own“stopping?me”q(F)
•  q(F)isarandomvariablein[0,1],
whichisdeterminedby𝞂and𝛕
0
∧
∨
∧
∧
∧
∨
∨
∧
∧
∧
∧
∧
∧
1
q(G)
0
∧
∨
∨
∨
G
∧
∧
∧
∧
∧
∧
∧
∧
∧
1
q(H)
0
∧
∨
∨
∨
H
∧
∧
∧
∧
∧
∧
∧
∧
∧
1
q(I)
0
∧
∨
∨
∨
I
∧
∧
∧
∧
∧
∧
∧
∧
∧
1
q(J)
0
∧
∨
∨
∨
J
∧
∧
∧
∧
∧
∧
∧
∧
∧
1
0
q(F)
∧
F
∨
∧
∧
∧
∨
∨
∧
∧
∧
∧
∧
∧
1
p
0
∧
∨
∧
∧
∧
∨
∨
∧
∧
∧
∧
∧
∧
1
p
0
∧
∨
∧
Rp ∧
∧
∨
∨
∧
∧
∧
∧
∧
∧
1
p
0
∧
∨
∧
decisiontrees
∨
∨
∧
∧
∧
∧
1
p
0
∧
∨
∨
∨
k
∧
∧
∧
∧
∧
1
p
0
∧
F
∨
∨
∨
k
∧
∧
∧
∧
∧
1
p
0
q(F)
q(F)=p/14k
∧
F
∨
∨
∨
k
∧
∧
∧
∧
∧
Hastad’sSwitchingLemma
IfFisk-DNFork-CNF,then
Pr[DTdepth(f↾R1/14k)≥ℓ]≤e–ℓ
“StoppingTimeVersion”
ForeveryBooleanformulaF,
Pr[DTdepth(F↾Rq(F))≥ℓ]≤e–ℓ
TechnicalMainLemma(tailboundforq(F))
IfFhasdepthd+1,thenforall0≤λ≤1,
Cd
Pr[q(F)≤λ]≤size(F)×
exp(Ω(dλ−1/d))
•  TheconstantC<8isdefinedby
1+
1
X
i=0
exp(ei
1
1
(i + 1)e
2)
+
1 X
1
X
i=0 j=0
1
exp((j + 1)ei
1
(i + j + 2)e
2)
TechnicalMainLemma(tailboundforq(F))
IfFhasdepthd+1,thenforall0≤λ≤1,
Cd
Pr[q(F)≤λ]≤size(F)×
exp(Ω(dλ−1/d))
Corollary
Depthd+1formulasforPARITYhavesize
exp(Ω(dn1/d)–O(d))
ProofofHastad’s
SwitchingLemma
•  Fixak-DNFFandℓ≥1
SwitchingLemma:
Pr[depth(CanDT(F↾Rp))≥ℓ]=O(pk)ℓ
•  Fixak-DNFFandℓ≥1
•  BAD:={restric?onsR|depth(CanDT(F↾R))≥ℓ}
SwitchingLemma:
Pr[depth(CanDT(F↾Rp))≥ℓ]=O(pk)ℓ
•  Fixak-DNFFandℓ≥1
•  BAD:={restric?onsR|depth(CanDT(F↾R))≥ℓ}
•  Goal.Pr[Rp∈BAD]=O(pk)ℓ
SwitchingLemma:
Pr[depth(CanDT(F↾Rp))≥ℓ]=O(pk)ℓ
•  Fixak-DNFFandℓ≥1
•  BAD:={restric?onsR|depth(CanDT(F↾R))≥ℓ}
•  Goal.Pr[Rp∈BAD]=O(pk)ℓ
Keyidea.WeassociateeachR∈BADwitha
restric?onR*suchthat
•  Fixak-DNFFandℓ≥1
•  BAD:={restric?onsR|depth(CanDT(F↾R))≥ℓ}
•  Goal.Pr[Rp∈BAD]=O(pk)ℓ
Keyidea.WeassociateeachR∈BADwitha
restric?onR*suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
R* ⋆1⋆⋆10⋆1⋆100⋆⋆0⋆0⋆⋆⋆0⋆0
R* ⋆1⋆010⋆1⋆1001⋆0⋆0⋆0⋆0⋆0
(ℓ=3)
•  Fixak-DNFFandℓ≥1
•  BAD:={restric?onsR|depth(CanDT(F↾R))≥ℓ}
•  Goal.Pr[Rp∈BAD]=O(pk)ℓ
Keyidea.WeassociateeachR∈BADwitha
restric?onR*suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
inpar?cular,Pr[Rp=R*]/Pr[Rp=R]=((1-p)/2p)ℓ
•  Fixak-DNFFandℓ≥1
•  BAD:={restric?onsR|depth(CanDT(F↾R))≥ℓ}
•  Goal.Pr[Rp∈BAD]=O(pk)ℓ
Keyidea.WeassociateeachR∈BADwitha
restric?onR*suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
inpar?cular,Pr[Rp=R*]/Pr[Rp=R]=((1-p)/2p)ℓ
foranyrestric?on𝜌withsstars,
Pr[Rp=𝜌]=ps(1−p)n−s
•  Fixak-DNFFandℓ≥1
•  BAD:={restric?onsR|depth(CanDT(F↾R))≥ℓ}
•  Goal.Pr[Rp∈BAD]=O(pk)ℓ
Keyidea.WeassociateeachR∈BADwitha
restric?onR*suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
② themapR↦R*is(4k)ℓ-to-1
•  Fixak-DNFFandℓ≥1
•  BAD:={restric?onsR|depth(CanDT(F↾R))≥ℓ}
•  Goal.Pr[Rp∈BAD]=O(pk)ℓ
Keyidea.WeassociateeachR∈BADwitha
restric?onR*suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
② themapR↦R*is(4k)ℓ-to-1
anyrestric?on𝜌equalsR*for
atmost(4k)ℓdis?nctR∈BAD
Keyidea.WeassociateeachR∈BADwitha
restric?onR*suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
② themapR↦R*is(4k)ℓ-to-1
Keyidea.WeassociateeachR∈BADwitha
restric?onR*suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
② themapR↦R*is(4k)ℓ-to-1
Pr[Rp∈BAD]
=∑R∈BADPr[Rp=R] Keyidea.WeassociateeachR∈BADwitha
restric?onR*suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
② themapR↦R*is(4k)ℓ-to-1
Pr[Rp∈BAD]
=∑R∈BADPr[Rp=R]
=∑R∈BAD(2p/(1-p))ℓPr[Rp=R*]
Pr[Rp=R]=(2p/(1-p))ℓPr[Rp=R*]
Keyidea.WeassociateeachR∈BADwitha
restric?onR*suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
② themapR↦R*is(4k)ℓ-to-1
Pr[Rp∈BAD]
=∑R∈BADPr[Rp=R]
=∑R∈BAD(2p/(1-p))ℓPr[Rp=R*] ≤(4p)ℓ∑R∈BADPr[Rp=R*]
w.l.o.g.p≤1/2
Keyidea.WeassociateeachR∈BADwitha
restric?onR*suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
② themapR↦R*is(4k)ℓ-to-1
Pr[Rp∈BAD]
=∑R∈BADPr[Rp=R]
=∑R∈BAD(2p/(1-p))ℓPr[Rp=R*] ≤(4p)ℓ∑R∈BADPr[Rp=R*]
≤(4p)ℓ(4k)ℓPr[Rp∈{R*|R∈BAD}]
Keyidea.WeassociateeachR∈BADwitha
restric?onR*suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
② themapR↦R*is(4k)ℓ-to-1
Pr[Rp∈BAD]
=∑R∈BADPr[Rp=R]
=∑R∈BAD(2p/(1-p))ℓPr[Rp=R*] ≤(4p)ℓ∑R∈BADPr[Rp=R*]
≤(4p)ℓ(4k)ℓPr[Rp∈{R*|R∈BAD}]
Pr[…]≤1
Keyidea.WeassociateeachR∈BADwitha
restric?onR*suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
② themapR↦R*is(4k)ℓ-to-1
Pr[Rp∈BAD]
=∑R∈BADPr[Rp=R]
=∑R∈BAD(2p/(1-p))ℓPr[Rp=R*] ≤(4p)ℓ∑R∈BADPr[Rp=R*]
≤(4p)ℓ(4k)ℓPr[Rp∈{R*|R∈BAD}]
≤(16pk)ℓ
Q.E.D.
SwitchingLemma:
Pr[depth(CanDT(F↾Rp))≥ℓ]=O(pk)
]=O(pk)ℓ
Pr[Rp∈BAD]
=∑R∈BADPr[Rp=R]
=∑R∈BAD(2p/(1-p))ℓPr[Rp=R*] ≤(4p)ℓ∑R∈BADPr[Rp=R*]
≤(4p)ℓ(4k)ℓPr[Rp∈{R*|R∈BAD}]
≤(16pk)ℓ
Q.E.D.
SwitchingLemma:
Pr[depth(CanDT(F↾Rp))≥ℓ]=O(pk)
]=O(pk)ℓ
Pr[Rp∈BAD]
=∑R∈BADPr[Rp=R]
=∑R∈BAD(2p/(1-p))ℓPr[Rp=R*] ℓ
ℓ∑
morecarefulanalysisgives(5pk)
≤(4p)
Pr[R
=R*]
(w.l.o.g.p
R∈BAD
p
≤(4p)ℓ(4k)ℓPr[Rp∈{R*|R∈BAD}]
≤(16pk)ℓ
Q.E.D.
Keyidea.WeassociateeachR∈BADwitha
restric?onR*suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
② themapR↦R*is(4k)ℓ-to-1
Keyidea.WeassociateeachR∈BADwitha
pair(R*,Code(R))suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
② themapR↦R*is(4k)ℓ-to-1
③ 
④ 
Keyidea.WeassociateeachR∈BADwitha
pair(R*,Code(R))suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
② themapR↦R*is(4k)ℓ-to-1
③ Code(R)∈({0,1}2⨉[k])ℓ
④ 
Keyidea.WeassociateeachR∈BADwitha
pair(R*,Code(R))suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
② themapR↦R*is(4k)ℓ-to-1
③ Code(R)∈({0,1}2⨉[k])ℓ
④ 
inpar?cular,Code(R)has
(4k)ℓpossiblevalues
Keyidea.WeassociateeachR∈BADwitha
pair(R*,Code(R))suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
② themapR↦R*is(4k)ℓ-to-1
③ Code(R)∈({0,1}2⨉[k])ℓ
④ themapR↦(R*,Code(R))is1-to-1
Keyidea.WeassociateeachR∈BADwitha
pair(R*,Code(R))suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
② themapR↦R*is(4k)ℓ-to-1
③ Code(R)∈({0,1}2⨉[k])ℓ
④ themapR↦(R*,Code(R))is1-to-1
③&④imply②
Keyidea.WeassociateeachR∈BADwitha
pair(R*,Code(R))suchthat
① |Stars(R*)|=|Stars(R)|−ℓ
② themapR↦R*is(4k)ℓ-to-1
③ Code(R)∈({0,1}2⨉[k])ℓ
④ themapR↦(R*,Code(R))is1-to-1
intui?vely,Code(R)isa“recipe”
forinver?ngR↦(R*,Code(R))
givenknowledgeofR*
R↦(R*,Code(R))
k=3,ℓ=4
R↦(R*,Code(R))
F=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
k=3,ℓ=4
R↦(R*,Code(R))
F=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
R={x1↦1,x4↦0}
k=3,ℓ=4
R↦(R*,Code(R))
1
0
1
0
11
F↾R=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
R={x1↦1,x4↦0}
k=3,ℓ=4
R↦(R*,Code(R))
1
0
1
0
11
F↾R=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
R={x1↦1,x4↦0}
k=3,ℓ=4
R↦(R*,Code(R))
F↾R=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
x2
R={x1↦1,x4↦0}
0
1
x7
0
CanDT(F↾R)= 1
x3
1
0
1
0 1
x5
0
1
1
x7
0
1
1
0
k=3,ℓ=4
R↦(R*,Code(R))
F↾R=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
x2
R={x1↦1,x4↦0}
0
1
x7
R∈BAD,since
CanDT(F↾R)≥4
CanDT(F↾R)≥ℓ
0
1
x3
1
0
1
0 1
x5
0
1
1
x7
0
1
1
0
k=3,ℓ=4
R↦(R*,Code(R))
F↾R=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
x2
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
x2↦1,x3↦1,
x5↦0,x7↦1}
0
1
x7
0
1
x3
1
0
1
0 1
x5
0
1
1
x7
0
1
1
0
k=3,ℓ=4
R↦(R*,Code(R))
F↾R=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
x2
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
x2↦1,x3↦1,
x5↦0,x7↦1}
R*={x1↦1,x4↦0,
x2↦1,x3↦0,
x5↦1,x7↦0}
0
1
x7
0
1
x3
1
0
1
0 1
x5
0
1
1
x7
0
1
1
0
k=3,ℓ=4
R↦(R*,Code(R))
F↾R=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
x2
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
x2↦1,x3↦1,
x5↦0,x7↦1}
R*={x1↦1,x4↦0,
x2↦1,x3↦0,
x5↦1,x7↦0}
0
1
x7
0
1
x3
1
0
1
0 1
x5
0
1
1
x7
0
1
1
0
k=3,ℓ=4
R↦(R*,Code(R))
F↾R=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
x2
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
x2↦1,x3↦1,
x5↦0,x7↦1}
R*={x1↦1,x4↦0,
x2↦1,x3↦0,
x5↦1,x7↦0}
0
1
x7
0
1
x3
1
0
1
0 1
x5
0
1
1
x7
0
1
1
0
R↦(R*,Code(R))
k=3,ℓ=4
✓
F ↾ R=x x ¬x ¬x x x 1 2
3
∨
1 3 5
∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
x2
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
x2↦1,x3↦1,
x5↦0,x7↦1}
R*={x1↦1,x4↦0,
x2↦1,x3↦0,
x5↦1,x7↦0}
0
1
x7
0
1
x3
1
0
1
0 1
x5
0
1
1
x7
0
1
1
0
R↦(R*,Code(R))
k=3,ℓ=4
✓
F ↾ R=x x ¬x ¬x x x 1 2
3
∨
1 3 5
∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
x2
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
x2↦1,x3↦1,
x5↦0,x7↦1}
R*={x1↦1,x4↦0,
x2↦1,x3↦0,
x5↦1,x7↦0}
0
1
x7
0
1
x3
1
0
1
0 1
x5
0
1
1
x7
0
1
1
0
R↦(R*,Code(R))
k=3,ℓ=4
✓
F ↾ R=x x ¬x ¬x x x 1 2
3
∨
1 3 5
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
≈≈ 3↦1,
x2↦1,x
x5↦0,x7↦1}
R*={x1↦1,x4↦0,
x2↦1,x3↦0,
x5↦1,x7↦0}
∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
Code(R)says:
•  findthefirstsa?sfiedterm
ofF↾R*
•  the“longpath”beginswith
x2↦1,x3↦1
(i.e.var2ofterm↦1and
var3ofterm↦1)
•  …
k=3,ℓ=4
R↦(R*,Code(R))
1 0
1
F↾R=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
x2↦1,x3↦1,
x5↦0,x7↦1}
R*={x1↦1,x4↦0,
x2↦1,x3↦0,
x5↦1,x7↦0}
Fixvariables
accordingtothe
beginningofthe
longpath
k=3,ℓ=4
R↦(R*,Code(R))
F↾R=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
x2
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
x2↦1,x3↦1,
x5↦0,x7↦1}
R*={x1↦1,x4↦0,
x2↦1,x3↦0,
x5↦1,x7↦0}
0
1
x7
0
1
x3
1
0
1
0 1
x5
0
1
1
x7
0
1
1
0
k=3,ℓ=4
R↦(R*,Code(R))
F↾R=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
x2
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
x2↦1,x3↦1,
x5↦0,x7↦1}
R*={x1↦1,x4↦0,
x2↦1,x3↦0,
x5↦1,x7↦0}
0
1
x7
0
1
x3
1
0
1
0 1
x5
0
1
1
x7
0
1
1
0
k=3,ℓ=4
R↦(R*,Code(R))
F↾R=x1x2¬x3∨¬x1x3x5∨
✓
x ¬x x x x ¬x 2
4 5
∨
3 4
R*={x1↦1,x4↦0,
x2↦1,x3↦0,
x5↦1,x7↦0}
∨x1¬x4¬x7
x2
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
x2↦1,x3↦1,
x5↦0,x7↦1}
6
0
1
x7
0
1
x3
1
0
1
0 1
x5
0
1
1
x7
0
1
1
0
k=3,ℓ=4
R↦(R*,Code(R))
F↾R=x1x2¬x3∨¬x1x3x5∨
✓
x ¬x x x x ¬x 2
4 5
∨
3 4
R*={x1↦1,x4↦0,
x2↦1,x3↦0,
x5↦1,x7↦0}
∨x1¬x4¬x7
x2
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
x2↦1,x3↦1,
x5↦0,x7↦1}
6
0
1
x7
0
1
x3
1
0
1
0 1
x5
0
1
1
x7
0
1
1
0
k=3,ℓ=4
R↦(R*,Code(R))
✓
x ¬x x x x ¬x F↾R=x1x2¬x3∨¬x1x3x5∨
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
x2↦1,x3↦1,
x5↦0,x7↦1}
R*={x1↦1,x4↦0,
x2↦1,x3↦0,
x5↦1,x7↦0}
2
4 5
∨
3 4
6
∨x1¬x4¬x7
Code(R)nextsays:
•  findthenextsa?sfiedterm
ofF↾R*(overwri?ngx2↦1,x3↦1)
•  the“longpath”con?nues
x5↦0
(i.e.var3ofterm↦0)
•  …
k=3,ℓ=4
R↦(R*,Code(R))
0
F↾R=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
x2↦1,x3↦1,
x5↦0,x7↦1}
R*={x1↦1,x4↦0,
x2↦1,x3↦0,
x5↦1,x7↦0}
Fixvariables
accordingtothe
longpath
k=3,ℓ=4
R↦(R*,Code(R))
F↾R=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
x2
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
x2↦1,x3↦1,
x5↦0,x7↦1}
R*={x1↦1,x4↦0,
x2↦1,x3↦0,
x5↦1,x7↦0}
0
1
x7
0
1
x3
1
0
1
0 1
x5
0
1
1
x7
0
1
1
0
k=3,ℓ=4
R↦(R*,Code(R))
F=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
x2↦1,x3↦1,
x5↦0,x7↦1}
R*={x1↦1,x4↦0,
x2↦1,x3↦0,
x5↦1,x7↦0}
Code(R)∈({0,1}2⨉[k])ℓ
givenknowledgeofR*(andF),
followtheseinstruc?onsto
recoverR(andalongthewayR~)
k=3,ℓ=4
R↦(R*,Code(R))
F=x1x2¬x3∨¬x1x3x5∨x2¬x4x5∨x3x4¬x6∨x1¬x4¬x7
R={x1↦1,x4↦0}
R~={x1↦1,x4↦0,
x2↦1,x3↦1,
x5↦0,x7↦1}
R*={x1↦1,x4↦0,
x2↦1,x3↦0,
x5↦1,x7↦0}
Code(R)∈({0,1}2⨉[k])ℓ
givenknowledgeofR*(andF),
followtheseinstruc?onsto
recoverR(andalongthewayR~)
ü  R*hasℓfewer
starsthanR
ü  R↦(R*,Code(R))
is1-to-1
ADifferentApproach
ADifferentApproach
Hastad’sSwitchingLemma
IfFisak-DNF,then
Pr[depth(CanDT(F↾Rp))≥ℓ]=O(pk)ℓ
Weak(butflexible)SwitchingLemma
IfFisak-DNF,then
Pr[depth(CanDT(F)↾Rp)≥ℓ]=O(pk2k)ℓ
ShowsthatPARITYhasdepth-d
circuitsize
logn 2
d
( (() ))
exp Ω
Weak(butflexible)SwitchingLemma
IfFisak-DNF,then
Pr[depth(CanDT(F)↾Rp)≥ℓ]=O(pk2k)ℓ
Proofgeneralizestoaffine
restric3ons!
Weak(butflexible)SwitchingLemma
IfFisak-DNF,then
Pr[depth(CanDT(F)↾Rp)≥ℓ]=O(pk2k)ℓ
ADifferentApproach
1.  DecisionTreeSwitchingLemma
2.  k-ClippedDecisionTrees
3.  ArbitraryDistribu?onofStars
4.  SwitchingLemmaforAffineRestric3ons
5.  Tsei?nExpanderSwitchingLemma
ADifferentApproach
1.  DecisionTreeSwitchingLemma
2.  k-ClippedDecisionTrees
3.  ArbitraryDistribu?onofStars
4.  SwitchingLemmaforAffineRestric3ons
5.  Tsei?nExpanderSwitchingLemma
BinomialDistribu?on
•  ForafinitesetN,let
S⊆pN
Denote“Sisap-biasedrandomsubsetofN”,i.e.,
Pr[S=S]=p|S|(1−p)|N|−|S|
•  Foranaturalnumbernand0≤p≤1,thebinomial
randomvariableBin(n,p)isequivalentto|S|where
S⊆p[n]
DecisionTreeSwitchingLemma
IfTisadepth-kdecisiontree,then
Pr[depth(T↾Rp)≥ℓ]≤(2pk)ℓ
•  ConsiderarandombranchofT:
x3
0
0
x1
x2
0
0
1/4
x7
1
1/8
x2
1
x4
0
1
0
1
1/81/8
1
1/161/16
1
x7
0
1
1/81/8
•  ConsiderarandombranchofT:
x3
0
0
x1
x2
0
0
1/4
x7
1
1/8
x2
1
x4
0
1
0
1
1/81/8
1
1/161/16
1
x7
0
1
1/81/8
•  Let𝞫(T)⊆[n]bethesetofvariablesqueriedona
randombranchofT
x3
0
0
x1
x2
0
0
1/4
x7
1
1/8
x2
1
x4
0
1
0
1
1/81/8
1
1/161/16
1
x7
0
1
1/81/8
•  #𝞫(T)=thenumberofvariablesqueriedona
randombranchofT
x3
0
0
x1
x2
0
0
1/4
x7
1
1/8
x2
1
x4
0
1
0
1
1/81/8
1
1/161/16
1
x7
0
1
1/81/8
•  #𝞫(T)=thenumberofvariablesqueriedona
randombranchofT
x3
0
0
x1
1/8
x2
0
x2
0
1/4
x7
1
0
1
1
x4
0
1
1/81/8
1
1/161/16
Obs:Ifdepth(T)≥k,then
Pr[#𝞫(T)≥k]≥2−k
1
x7
0
1
1/81/8
Claim. Thefollowingrandomvariableshavethe
samedistribu?on:
1.  𝞫(T↾Rp) thesetofvariablesqueriedonarandombranchofT↾Rp
2.  S⊆p𝞫(T)
ap-biasedsubsetofthevars.onarandombranchofT
Claim. Thefollowingrandomvariableshavethe
samedistribu?on:
1.  𝞫(T↾Rp)
#𝞫(T↾Rp) 2.  S⊆p𝞫(T)
Bin(#𝞫(T),p)
thesetofvariablesqueriedonarandombranchofT↾Rp
ap-biasedsubsetofthevars.onarandombranchofT
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
T
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
Stars(Rp)
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
Rp
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
T↾Rp
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
T↾Rp
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
T↾Rp
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
T↾Rp
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
𝞫(T↾Rp)
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
𝞫(T↾Rp)
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
𝞫(T↾Rp)
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
𝞫(T↾Rp)
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
𝞫(T↾Rp)
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
T
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
𝞫(T)
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
𝞫(T)
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
S⊆p𝞫(T)
1.  𝞫(T↾Rp)
2.  S⊆p𝞫(T)
⇒
1.  #𝞫(T↾Rp)
2.  Bin(𝞫(T),p)
DecisionTreeSwitchingLemma
IfTisadepth-kdecisiontree,then
Pr[depth(T↾Rp)≥ℓ]≤(2pk)ℓ
DecisionTreeSwitchingLemma
IfTisadepth-kdecisiontree,then
Pr[depth(T↾Rp)≥ℓ]≤(2pk)ℓ
Pr[depth(T↾Rp)≥ℓ]
recall:depth(T’)≥ℓ⇒
−ℓ
Pr[#𝞫(T’)≥ℓ]≥2
DecisionTreeSwitchingLemma
IfTisadepth-kdecisiontree,then
Pr[depth(T↾Rp)≥ℓ]≤(2pk)ℓ
Pr[depth(T↾Rp)≥ℓ]
]
–ℓ] ≤PrRpp
[Pr𝞫(T↾R
[#𝞫(T↾R
)≥ℓ]≥2
p
p
p)
recall:depth(T’)≥ℓ⇒
−ℓ
Pr[#𝞫(T’)≥ℓ]≥2
DecisionTreeSwitchingLemma
IfTisadepth-kdecisiontree,then
Pr[depth(T↾Rp)≥ℓ]≤(2pk)ℓ
Pr[depth(T↾Rp)≥ℓ]
–ℓ] ≤PrRpp
[Pr𝞫(T↾R
[#𝞫(T↾R
)≥ℓ]≥2
p
p
p)
DecisionTreeSwitchingLemma
IfTisadepth-kdecisiontree,then
Pr[depth(T↾Rp)≥ℓ]≤(2pk)ℓ
Pr[depth(T↾Rp)≥ℓ]
–ℓ]
≤PrRpp
[Pr𝞫(T↾R
[#𝞫(T↾R
)≥ℓ]≥2
p
p
p)
≤2ℓExRpp
[Pr𝞫(T↾R
pp)[#𝞫(T↾Rp)≥ℓ]]
Markov’sinequality
DecisionTreeSwitchingLemma
IfTisadepth-kdecisiontree,then
Pr[depth(T↾Rp)≥ℓ]≤(2pk)ℓ
Pr[depth(T↾Rp)≥ℓ]
–ℓ]
≤PrRpp
[Pr𝞫(T↾R
[#𝞫(T↾R
)≥ℓ]≥2
p
p
p)
≤2ℓExRpp
[Pr𝞫(T↾R
pp)[#𝞫(T↾Rp)≥ℓ]]
DecisionTreeSwitchingLemma
IfTisadepth-kdecisiontree,then
Pr[depth(T↾Rp)≥ℓ]≤(2pk)ℓ
Pr[depth(T↾Rp)≥ℓ]
–ℓ]
≤PrRpp
[Pr𝞫(T↾R
[#𝞫(T↾R
)≥ℓ]≥2
p
p
p)
≤2ℓPr[#𝞫(T↾Rp)≥ℓ]
DecisionTreeSwitchingLemma
IfTisadepth-kdecisiontree,then
Pr[depth(T↾Rp)≥ℓ]≤(2pk)ℓ
Pr[depth(T↾Rp)≥ℓ]
–ℓ]
≤PrRpp
[Pr𝞫(T↾R
[#𝞫(T↾R
)≥ℓ]≥2
p
p
p)
≤2ℓPr[#𝞫(T↾Rp)≥ℓ]
=2ℓPr[Bin(#𝞫(T),p)≥ℓ]
#𝞫(T↾Rp)andBin(#𝞫(T),p)
havethesamedistribu?on
DecisionTreeSwitchingLemma
IfTisadepth-kdecisiontree,then
Pr[depth(T↾Rp)≥ℓ]≤(2pk)ℓ
Pr[depth(T↾Rp)≥ℓ]
–ℓ]
≤PrRpp
[Pr𝞫(T↾R
[#𝞫(T↾R
)≥ℓ]≥2
p
p
p)
≤2ℓPr[#𝞫(T↾Rp)≥ℓ]
=2ℓPr[Bin(#𝞫(T),p)≥ℓ]
≤2ℓPr[Bin(k,p)≥ℓ]
#𝞫(T)≤depth(T)≤k
DecisionTreeSwitchingLemma
IfTisadepth-kdecisiontree,then
Pr[depth(T↾Rp)≥ℓ]≤(2pk)ℓ
Pr[depth(T↾Rp)≥ℓ]
–ℓ]
≤PrRpp
[Pr𝞫(T↾R
[#𝞫(T↾R
)≥ℓ]≥2
p
p
p)
≤2ℓPr[#𝞫(T↾Rp)≥ℓ]
=2ℓPr[Bin(#𝞫(T),p)≥ℓ]
≤2ℓPr[Bin(k,p)≥ℓ]
≤2ℓ(pk)ℓ
unionbound
DecisionTreeSwitchingLemma
IfTisadepth-kdecisiontree,then
Pr[depth(T↾Rp)≥ℓ]≤(2pk)ℓ
Pr[depth(T↾Rp)≥ℓ]
–ℓ]
≤PrRpp
[Pr𝞫(T↾R
[#𝞫(T↾R
)≥ℓ]≥2
p
p
p)
≤2ℓPr[#𝞫(T↾Rp)≥ℓ]
=2ℓPr[Bin(#𝞫(T),p)≥ℓ]
≤2ℓPr[Bin(k,p)≥ℓ]
≤2ℓ(pk)ℓ
Q.E.D.
ADifferentApproach
1.  DecisionTreeSwitchingLemma
2.  k-ClippedDecisionTrees
3.  ArbitraryDistribu?onofStars
4.  SwitchingLemmaforAffineRestric3ons
5.  Tsei?nExpanderSwitchingLemma
Recall:CanonicalDTofak-DNF
∨(x1x2x3)
∨(¬x2x4)
∨(x1¬x3¬x7)
∨…
x1
0
x2
0
…
1
1
x4
1
…
x7
0
1
1
…
0
1
x3
0
x3
1
0
1
x2
0
…
x4
0
1
1
1
1
x7
0
1
1
…
Recall:CanonicalDTofak-DNF
∨(x1x2x3)
∨(¬x2x4)
∨(x1¬x3¬x7)
∨…
x1
0
x2
0
…
1
1
x4
1
Obs:Everynodehas0
distance≤kfromaleaf!
x7 …
1
1
…
0
1
x3
0
x3
1
0
1
x2
0
…
x4
0
1
1
1
1
x7
0
1
1
…
Defini?on:Adecisiontreeisk-clippedifeverynode
hasdistance≤kfromaleaf
Defini?on:Adecisiontreeisk-clippedifeverynode
hasdistance≤kfromaleaf
Lemma:IfFisak-DNF(ork-CNF),then
CanDT(F)isk-clipped
Defini?on:Adecisiontreeisk-clippedifeverynode
hasdistance≤kfromaleaf
Lemma:IfFisak-DNF(ork-CNF),then
CanDT(F)isk-clipped
Lemma: IfTisk-clipped,then
(
)
#𝞫(T)
k)ℓ
Ex≤O(k2
ℓ
Defini?on:Adecisiontreeisk-clippedifeverynode
hasdistance≤kfromaleaf
#𝞫(T)stochas?callydominatedby
Lemma:IfFisak-DNF(ork-CNF),then
thenumberofunbiasedcoinflips
CanDT(F)isk-clipped
un?lseeingkconsecu?veheads
Lemma: IfTisk-clipped,then
(
)
#𝞫(T)
k)ℓ
Ex≤O(k2
ℓ
k-ClippedDecisionTreeSwitchingLemma
IfTisak-clippeddecisiontree,then
Pr[depth(T↾Rp)≥ℓ]=O(pk2k)ℓ
k-ClippedDecisionTreeSwitchingLemma
IfTisak-clippeddecisiontree,then
Pr[depth(T↾Rp)≥ℓ]=O(pk2k)ℓ
Pr[depth(T↾Rp)≥ℓ]
–ℓ]
≤PrRpp[Pr
[#𝞫(T↾R
)≥ℓ]≥2
p
𝞫(T↾Rp )
p
≤2ℓPr[#𝞫(T↾Rp)≥ℓ]
=2ℓPr[Bin(#𝞫(T),p)≥ℓ] sameasbefore
k-ClippedDecisionTreeSwitchingLemma
IfTisak-clippeddecisiontree,then
Pr[depth(T↾Rp)≥ℓ]=O(pk2k)ℓ
Pr[depth(T↾Rp)≥ℓ]
–ℓ]
≤PrRpp[Pr
[#𝞫(T↾R
)≥ℓ]≥2
p
𝞫(T↾Rp )
p
≤2ℓPr[#𝞫(T↾Rp)≥ℓ]
=2ℓPr[Bin(#𝞫(T),p)≥ℓ] (
)
#𝞫(T) ℓ
ℓ
≤2 Ex[p ]
ℓ
unionbound
k-ClippedDecisionTreeSwitchingLemma
IfTisak-clippeddecisiontree,then
Pr[depth(T↾Rp)≥ℓ]=O(pk2k)ℓ
Lemma: IfTisk-clipped,then
#𝞫(T)
Pr[depth(T↾Rp)≥ℓ]
k)ℓ
Ex≤O(k2
ℓ[#𝞫(T↾R )≥ℓ]≥2–ℓ]
≤Pr
[Pr
Rpp
𝞫(T↾Rpp)
p
≤2ℓPr[#𝞫(T↾Rp)≥ℓ]
=2ℓPr[Bin(#𝞫(T),p)≥ℓ] (
)
(
)
#𝞫(T) ℓ
ℓ
≤2 Ex[p ]
ℓ
≤2ℓpℓO(k2k)ℓ
k-ClippedDecisionTreeSwitchingLemma
IfTisak-clippeddecisiontree,then
Pr[depth(T↾Rp)≥ℓ]=O(pk2k)ℓ
Pr[depth(T↾Rp)≥ℓ]
–ℓ]
≤PrRpp[Pr
[#𝞫(T↾R
)≥ℓ]≥2
p
𝞫(T↾Rp )
p
≤2ℓPr[#𝞫(T↾Rp)≥ℓ]
=2ℓPr[Bin(#𝞫(T),p)≥ℓ] (
)
#𝞫(T) ℓ
ℓ
≤2 Ex[p ]
ℓ
≤2ℓpℓO(k2k)ℓ
Q.E.D.
ADifferentApproach
1.  DecisionTreeSwitchingLemma
2.  k-ClippedDecisionTrees
3.  ArbitraryDistribuGonofStars
4.  SwitchingLemmaforAffineRestric3ons
5.  Tsei?nExpanderSwitchingLemma
•  LetS⊆[n]beanarbitrarydistribu?onof“stars”
•  LetS⊆[n]beanarbitrarydistribu?onof“stars”
•  Letu∈unif{0,1}n
•  LetS⊆[n]beanarbitrarydistribu?onof“stars”
•  Letu∈unif{0,1}n
•  Definerandomrestric?on𝜌S,u:[n]→{0,1,⋆}by
⋆ ifi∈S,
𝜌S,u(i)=
ui otherwise.
•  LetS⊆[n]beanarbitrarydistribu?onof“stars”
•  Letu∈unif{0,1}n
•  Definerandomrestric?on𝜌S,u:[n]→{0,1,⋆}by
⋆ ifi∈S,
𝜌S,u(i)=
ui otherwise.
IfS⊆p[n],then𝜌S,uhasthe
samedistribu?onasRp
•  LetS⊆[n]beanarbitrarydistribu?onof“stars”
Defini?on:Forp∈[0,1],wesaythatSisp-bounded
ifPr[J⊆S]≤p|J|foreveryJ⊆[n]
Thefollowing
areequivalent:
1.  𝞫(T↾𝜌S,u)
2.  𝞫(T)∩S
𝞫(T)∩S
Thefollowing
areequivalent:
𝞫(T)∩S
1.  𝞫(T↾𝜌S,u)
2.  𝞫(T)∩S
infact,this
equivalence
validforfixed
S∈Supp(S)
Thefollowing
areequivalent:
𝞫(T)∩S
1.  𝞫(T↾𝜌S,u)
2.  𝞫(T)∩S
IfSisp-bounded,then
Pr[|𝞫(T)∩S|≥ℓ]
(
)
#𝞫(T) ℓ
≤Ex[
p ]
ℓ
Thefollowing
𝞫(T)∩S
areequivalent:Pr[J⊆S]≤pℓ
1.  𝞫(T↾𝜌S,u) foreveryJ⊆[n]ofsize|J|=ℓ
2.  𝞫(T)∩S
IfSisp-bounded,then
Pr[|𝞫(T)∩S|≥ℓ]
(
)
#𝞫(T) ℓ
≤Ex[
p ]
ℓ
Theorem(SwitchingLemmafor𝜌S,u)
IfSisp-boundedandTisak-clippedDT,then
Pr[depth(T↾𝜌S,u)≥ℓ]=O(pk2k)ℓ
Theorem(SwitchingLemmafor𝜌S,u)
IfSisp-boundedandTisak-clippedDT,then
Pr[depth(T↾𝜌S,u)≥ℓ]=O(pk2k)ℓ
Pr[depth(T↾𝜌S,u)≥ℓ]
≤PrS,u[PrV(T↾𝜌S,u)[#𝞫(T↾𝜌S,u)≥ℓ]≥2–ℓ]
≤2ℓPr[#𝞫(T↾𝜌S,u)≥ℓ]
=2ℓPr[|𝞫(T)∩S|≥ℓ]
(
)
#𝞫(T) ℓ
ℓ
≤2 Ex[p ]
ℓ
≤2ℓpℓO(k2k)ℓ
Q.E.D.
ADifferentApproach
1.  DecisionTreeSwitchingLemma
2.  k-ClippedDecisionTrees
3.  ArbitraryDistribu?onofStars
4.  SwitchingLemmaforAffineRestric9ons
5.  Tsei?nExpanderSwitchingLemma
•  Restric?ons[n]→{0,1,⋆}arein1-1correspondence
withsubcubesof{0,1}n
•  WhataboutthecomplexityofF↾A(e.g.forFak-DNF)
forarandomaffinesubspaceofA⊆{0,1}n?
•  Restric?ons[n]→{0,1,⋆}arein1-1correspondence
withsubcubesof{0,1}n
•  WhataboutthecomplexityofF↾A(e.g.forFak-DNF)
forarandomaffinesubspaceofA⊆{0,1}n?
A=V+u
whereVisalinearsubspaceof{0,1}n
•  Restric?ons[n]→{0,1,⋆}arein1-1correspondence
withsubcubesof{0,1}n
•  WhataboutthecomplexityofF↾A(e.g.forFak-DNF)
forarandomaffinesubspaceofA⊆{0,1}n?
AffineSwitchingLemma
IfFisak-DNFandAisa“p-bounded”random
affinesubsetof{0,1}n,then
Pr[DTdepth(F↾A)≥ℓ]=O(pk2k)ℓ
ApplyinganAffineRestric?on
•  ConsideranaffinesetA⊆{0,1}n
•  Forf:{0,1}n→{0,1},wedefine:
f↾A:A→{0,1},
DTdepth(f↾A)=min.depthofadecisiontreeT
suchthatf(x)=T(x)forallx∈A
•  Wealsohaveasyntac3copera?onT↦T↾Aon
decisiontrees
ApplyinganAffineRestric?on
•  Example:A={x∈{0,1}n|x1≠x3}
x3
0
x3
1
↾A
x1
1
x4
0
0
x7
1
0
x1
1
0
0
1
0
1
01
x8
1
x7
x7
x7
0
1
10010110
1
0
0
1
0 x
8
0
1
10
•  LetS⊆[n]beanarbitrarydist.of“stars”
•  Letu∈unif{0,1}n
•  Considerrandomrestric?on𝜌S,u(i)=
⋆ ifi∈S,
ui otherwise.
•  LetS⊆[n]beanarbitrarydist.of“stars”
LetVbeanarbitrarydist.onlinearsubspacesof{0,1}n
•  Letu∈unif{0,1}n
•  Considerrandomrestric?on𝜌S,u(i)=
⋆ ifi∈S,
ui otherwise.
•  LetS⊆[n]beanarbitrarydist.of“stars”
LetVbeanarbitrarydist.onlinearsubspacesof{0,1}n
•  Letu∈unif{0,1}n
•  Considerrandomrestric?on𝜌S,u(i)=
ConsiderrandomaffinespaceV+u
⋆ ifi∈S,
ui otherwise.
•  LetS⊆[n]beanarbitrarydist.of“stars”
LetVbeanarbitrarydist.onlinearsubspacesof{0,1}n
•  Letu∈unif{0,1}n
•  Considerrandomrestric?on𝜌S,u(i)=
⋆ ifi∈S,
ui otherwise.
ConsiderrandomaffinespaceV+u
Visthearrangementof“stars”
•  LetS⊆[n]beanarbitrarydist.of“stars”
LetVbeanarbitrarydist.onlinearsubspacesof{0,1}n
•  Letu∈unif{0,1}n
•  Considerrandomrestric?on𝜌S,u(i)=
ConsiderrandomaffinespaceV+u
ugivesthe“assignmentto
non-stars”
⋆ ifi∈S,
ui otherwise.
•  LetS⊆[n]beanarbitrarydist.of“stars”
LetVbeanarbitrarydist.onlinearsubspacesof{0,1}n
•  Letu∈unif{0,1}n
•  Considerrandomrestric?on𝜌S,u(i)=
⋆ ifi∈S,
ui otherwise.
ConsiderrandomaffinespaceV+u
{b1,…,bn}=standardbasisfor{0,1}n
ifV=Span{bi|i∈S},thenV+uis
thesubcubedefinedby𝜌S,u
Defini?on
ArandomsetS⊆[n]isp-boundedif,forallJ⊆[n],
Pr[J⊆S]≤p|J|
Defini?on
ArandomlinearsubspaceV⊆{0,1}nisp-boundedif,
forallJ⊆[n],
Pr[Vsha|ersJ]≤p|J|
Defini?on
ArandomsetS⊆[n]isp-boundedif,forallJ⊆[n],
Pr[J⊆S]≤p|J|
Defini?on
ArandomlinearsubspaceV⊆{0,1}nisp-boundedif,
forallJ⊆[n],
Pr[Vsha|ersJ]≤p|J|
∀I⊆J∃v∈Vsuchthat
•  vi=1foralli∈I
•  vj=0forallj∈J∖I
Defini?on
Vsha|ersJ⊆[n]
ArandomsetS⊆[n]isp-boundedif,forallJ⊆[n],
⟺
Pr[J⊆S]≤p|J|
Jisindependentinthe
matroidassociatedwithV
Defini?on
ArandomlinearsubspaceV⊆{0,1}nisp-boundedif,
forallJ⊆[n],
Pr[Vsha|ersJ]≤p|J|
∀I⊆J∃v∈Vsuchthat
•  vi=1foralli∈I
•  vj=0forallj∈J∖I
Defini?on
ArandomsetS⊆[n]isp-boundedif,forallJ⊆[n],
Pr[J⊆S]≤p|J|
Defini?on
ArandomlinearsubspaceV⊆{0,1}nisp-boundedif,
forallJ⊆[n],
Pr[Vsha|ersJ]≤p|J|
equivalentifV=Span{bi|i∈S}
SupposeA=V+uwhere
•  Visap-boundedrandomlinearsubspaceof{0,1}n
•  u∈unif{0,1}n
TheoremForeveryk-clippeddecisiontreeT,
Pr[depth(T↾A)≥ℓ]=O(pk2k)ℓ
essen?allythesameproofas
theSwitchingLemmafor𝜌S,u
SupposeA=V+uwhere
•  Visap-boundedrandomlinearsubspaceof{0,1}n
•  u∈unif{0,1}n
TheoremForeveryk-clippeddecisiontreeT,
Pr[depth(T↾A)≥ℓ]=O(pk2k)ℓ
Corollary(AffineSwitchingLemma)
Foreveryk-DNFF,Pr[DTdepth(F↾A)≥ℓ]=O(pk2k)ℓ
SupposeA=V+uwhere
•  Visap-boundedrandomlinearsubspaceof{0,1}n
•  u∈unif{0,1}n
TheoremForeveryk-clippeddecisiontreeT,
Pr[depth(T↾A)≥ℓ]=O(pk2k)ℓ
Corollary(AffineSwitchingLemma)
Foreveryk-DNFF,Pr[DTdepth(F↾A)≥ℓ]=O(pk2k)ℓ
Conjecture:O(pk)ℓ
ForeveryfixedV(and
randomu∈{0,1}n),the
followingareequivalent:
1.  𝞫(T↾(V+u))
#𝞫(T↾(V+u))
2.  “greedybasis”of
𝞫(T)intheV-matroid
rankV(𝞫(T))
SupposeA=V+uwhere
•  Visap-boundedrandomlinearsubspaceof{0,1}n
•  u∈unif{0,1}n
Finalgeneraliza?on:
Replacethe“ambientspace”{0,1}n
withanarbitraryaffinespaceB
SupposeA=V+uwhere
•  Visap-boundedrandomlinearsubspaceofB
•  u∈unifB
Finalgeneraliza?on:
Replacethe“ambientspace”{0,1}n
withanarbitraryaffinespaceB
SupposeA=V+uwhere
•  Visap-boundedrandomlinearsubspaceofB
•  u∈unifB
AisarandomaffinesubspaceofB
SupposeA=V+uwhere
•  Visap-boundedrandomlinearsubspaceofB
•  u∈unifB
Theorem
ForeveryB-independentk-clippeddecisiontreeT,
Pr[depth(T↾A)≥ℓ]=O(pk2k)ℓ
SupposeA=V+uwhere
•  Visap-boundedrandomlinearsubspaceofB
•  u∈unifB
Theorem
ForeveryB-independentk-clippeddecisiontreeT,
Pr[depth(T↾A)≥ℓ]=O(pk2k)ℓ
Everybranch𝛽⊆[n]ofTis“sha|ered”byB
(i.e.𝛽isindependentinthematroidassociatedwith
Lin(B)={x−y|x,y∈B})
ADifferentApproach
1.  DecisionTreeSwitchingLemma
2.  k-ClippedDecisionTrees
3.  ArbitraryDistribu?onofStars
4.  SwitchingLemmaforAffineRestric3ons
5.  TseiGnExpanderSwitchingLemma
Tsei?nExpanderSwitchingLemma
Theorem(Pitassi-R.-Servedio-Tan‘16)
Depth-dAC0FregeproofsofTsei?n(3-regular
2
expander)requiresizeexp(Ω(() ))
logn
d
Theorem(Beame-Pitassi-Impagliazzo‘93,Ben-Sasson‘02)
Depth-dAC0FregeproofsofTsei?n(3-regular
expander)requiresizeexp(Ω(
))
d
1/2
n
Tsei?nExpanderSwitchingLemma
Theorem(Pitassi-R.-Servedio-Tan‘16)
Depth-dAC0FregeproofsofTsei?n(3-regular
2
expander)requiresizeexp(Ω(() ))
logn
d
Theorem(Beame-Pitassi-Impagliazzo‘93,Ben-Sasson‘02)
0FregeproofsofTsei?n(3-regular
ImprovingtheAFFINESWITCHING
Depth-dAC
1/d))
LEMMAwouldyieldexp(Ω(n
d
expander)requiresizeexp(Ω(n1/2))
SupposeA=V+uwhere
•  Visap-boundedrandomlinearsubspaceof{0,1}n
•  u∈unif{0,1}n
AffineSwitchingLemmafork-DNFs
Foreveryk-DNFF,
Pr[DTdepth(F↾A)≥ℓ]=O(pk2k)ℓ
Conjecture:O(pk)ℓ
StayTuned…
•  ToniPitassi:ExpanderS.L.&AC0-Fregelowerbound
•  JohanHastad,AvishayTal:Correla?onboundsfrom
improvedswitchinglemmas
•  JohanHastad,RoccoServedio:Randomprojec?ons,
depthhierarchytheorem
•  SrikanthSrinivasan:Adap?verandomrestric?ons
(againstAC0withfewthresholdgates)
•  …
Thanks!