IdentityTesting&LowerBounds - Tel Aviv Universityshpilka/wact2016/program/slides/BenLeeVolk.pdf ·...

Identity Testing & Lower Boundsfor

Read-k Oblivious ABPs

Ben Lee Volk

Joint withMatthew AndersonMichael A. Forbes

Ramprasad SaptharishiAmir Shpilka

Read-Once Oblivious ABPs

x9 − 1

x9 + 2

x9 + 9

x2 + 1

3x2 · · · ...t

2x7 − 2

x7 − 4

x9 − 1

x9 + 2

x9 + 9

x2 + 1

3x2 · · · ...t

2x7 − 2

x7 − 4

• Each s→ t path computes multiplication of edge labels• Program computes the sum of those over all s→ t paths• Read Once: Each var appears in one layer

x9 − 1

x9 + 2

x9 + 9

x2 + 1

3x2 ...t

2x7 − 2

x7 − 4Width

• Each s→ t path computes multiplication of edge labels• Program computes the sum of those over all s→ t paths• Read Once: Each var appears in one layer

x9 − 1

x9 + 2

x9 + 9

x2 + 1

3x2 ...t

2x7 − 2

x7 − 4Width

Equivalently: f is the (1, 1) entry of the iterated matrix productn∏

Mi(xπ(i))

Some Things You’ve All Heard AboutWe know a lot about ROABPs :)

• Exponential lower bounds [Nisan]• Poly-time white-box PIT [Raz-Shpilka]• Quasipoly-size hitting sets even when order is unknown

[Forbes-Shpilka, Forbes-Shpilka-Saptharishi,Agrawal-Gurjar-Korwar-Saxena]

... and also PIT for sums of ROABPs and bounded-widthROABPs (all in this workshop).

This talk is about read-k oblivious ABPs.(def: same as before except that now every variable appears in atmost k layers)

• Exponential lower bounds [Nisan]

• Poly-time white-box PIT [Raz-Shpilka]• Quasipoly-size hitting sets even when order is unknown

• Exponential lower bounds [Nisan]• Poly-time white-box PIT [Raz-Shpilka]

• Quasipoly-size hitting sets even when order is unknown[Forbes-Shpilka, Forbes-Shpilka-Saptharishi,Agrawal-Gurjar-Korwar-Saxena]

This talk is about read-k oblivious ABPs.

(def: same as before except that now every variable appears in atmost k layers)

Reading k Times

• Generalizes sum of k ROABPs (PIT by [Gurjar, Korwar,Saxena, Thierauf])

• Generalizes read-k formulas (PIT by [Anderson,van Melkbeek, Volkovich])

• Well-studied boolean analogFor read-k oblivious boolean branching programs:

• exp(n/2k) lower bounds [Okolnishnikova,Borodin-Razborov-Smolensky] even for randomized andnon-deterministic variants

• PRG with seed length ps for size-s programs[Impagliazzo-Meka-Zuckerman]

Reading k Times

• Well-studied boolean analog

For read-k oblivious boolean branching programs:• exp(n/2k) lower bounds [Okolnishnikova,

Borodin-Razborov-Smolensky] even for randomized andnon-deterministic variants

Reading k Times

Read-k Oblivious ABPs

LowerBound: There is a polynomial f ∈ VP that requiresread-k oblivious ABPs of width exp(n/kk).

PIT: There is a white-box* PIT algorithm for read-k obliviousABPs, of running time exp(n1−1/2k−1

).*only the order in which the variables appear is important

Read-k Oblivious ABPsLowerBound: There is a polynomial f ∈ VP that requiresread-k oblivious ABPs of width exp(n/kk).

Evaluation DimensionReminder: f ∈ F[x1, . . . , xn], S ⊆ [n].

eval-dimS,S( f ) = dim span�

f |xS=α | α ∈ F|S|

Characterizes ROABP complexity:Theorem [Nisan]: f has ROABP of width w in variable orderx1, x2, . . . , xn iff for every i ∈ [n],

eval-dim[i],[i]( f )≤ w.

(same as rank of partial derivative matrix)

f |xS=α | α ∈ F|S|

Warm-up: 2-pass ABPSame as ROABP but with two “passes”:

x1 x2 · · · xn−1 xn x1 x2 · · · xn−1 xn

f =�M1

1 (x1)M12 (x2) · · ·M1

n (xn) ·M21 (x1)M

22 (x2) · · ·M2

n (xn)�(1,1)

x1 x2 · · · xn−1 xn x1 x2 · · · xn−1 xn

f =�M1

1 (x1)M12 (x2) · · ·M1

n (xn) ·M21 (x1)M

22 (x2) · · ·M2

n (xn)�(1,1)

Fixing x1 = α1:�N1(α1)M

12 (x2) · · ·M1

n (xn) · N2(α1)M22 (x2) · · ·M2

n (xn)�(1,1)

x1 x2 · · · xn−1 xn x1 x2 · · · xn−1 xn

f =�M1

1 (x1)M12 (x2) · · ·M1

n (xn) ·M21 (x1)M

22 (x2) · · ·M2

n (xn)�(1,1)

Fixing x1 = α1, x2 = α2:�N1(α1,α2)M

13 (x3) · · ·M1

n (xn) · N2(α1,α2)M23 (x3) · · ·M2

n (xn)�(1,1)

x1 x2 · · · xn−1 xn x1 x2 · · · xn−1 xn

f =�M1

1 (x1)M12 (x2) · · ·M1

n (xn) ·M21 (x1)M

22 (x2) · · ·M2

n (xn)�(1,1)

Fixing x1, x2, . . . , x i:

f |x[i]=α =�N1(α1, ...,αi)M

1i+1(x i+1) · · ·M1

n (xn)

N2(α1, ...,αi)M2i+1(x i+1) · · ·M2

n (xn)�(1,1)

x1 x2 · · · xn−1 xn x1 x2 · · · xn−1 xn

f =�M1

1 (x1)M12 (x2) · · ·M1

n (xn) ·M21 (x1)M

22 (x2) · · ·M2

n (xn)�(1,1)

Fixing x1, x2, . . . , x i:

f |x[i]=α =�N1(α1, ...,αi)M

1i+1(x i+1) · · ·M1

n (xn)

N2(α1, ...,αi)M2i+1(x i+1) · · ·M2

n (xn)�(1,1)

Every restriction determined by N1, N2 that have w2 entries.

x1 x2 · · · xn−1 xn x1 x2 · · · xn−1 xn

f =�M1

1 (x1)M12 (x2) · · ·M1

n (xn) ·M21 (x1)M

22 (x2) · · ·M2

n (xn)�(1,1)

Fixing x1, x2, . . . , x i:

f |x[i]=α =�N1(α1, ...,αi)M

1i+1(x i+1) · · ·M1

n (xn)

N2(α1, ...,αi)M2i+1(x i+1) · · ·M2

n (xn)�(1,1)

Every restriction determined by N1, N2 that have w2 entries.So eval-dim[i],[i]( f )≤ w4.

x1 x2 · · · xn−1 xn x1 x2 · · · xn−1 xn

f =�M1

1 (x1)M12 (x2) · · ·M1

n (xn) ·M21 (x1)M

22 (x2) · · ·M2

n (xn)�(1,1)

Fixing x1, x2, . . . , x i:

f |x[i]=α =�N1(α1, ...,αi)M

1i+1(x i+1) · · ·M1

n (xn)

N2(α1, ...,αi)M2i+1(x i+1) · · ·M2

n (xn)�(1,1)

Every restriction determined by N1, N2 that have w2 entries.So eval-dim[i],[i]( f )≤ w4. =⇒ f has width w4 ROABP.

Generalize: k-pass ABPTheorem: If f is computed by a width-w k-pass ABP in variableorder x1, x2, . . . , xn, then for every i ∈ [n], eval-dim[i],[i]( f )≤ w2k.

In particular, f is computed by a ROABP of width w2k.

=⇒ Exp. lower bounds and quasi-poly PIT for k-pass ABPs.

Up next: 2-pass, different order.

this is already exponentially more powerful than ROABPs and evensums of ROABPs: ∃ a polynomial computed by a 2-pass ABP withdifferent orders that requires exponential width when computed as asum of ROABPs.

this is already exponentially more powerful than ROABPs and evensums of ROABPs:

∃ a polynomial computed by a 2-pass ABP withdifferent orders that requires exponential width when computed as asum of ROABPs.

2-pass, different order

x1 x2 · · · xn−1 xn x8 xn · · · x2 xn/2

y1, . . . , ypn

x1 x2 · · · xn−1 xn x8 xn · · · x2 xn/2

y1, . . . , ypn

Theorem [Erdős-Szekeres]: Every sequence of n integers hasa monotone subsequence of length pn.

x1 x2 · · · xn−1 xn x8 xn · · · x2 xn/2

y1, . . . , ypn

Theorem [Erdős-Szekeres]: Every sequence of n integers hasa monotone subsequence of length pn.

x1 x2 · · · xn−1 xn x8 xn · · · x2 xn/2

y1, . . . , ypn

Theorem [Erdős-Szekeres]: Every sequence of n integers hasa monotone subsequence of length pn.Think of the ABP as computing a polynomial in the y vars overF(y) (i.e. all others vars are now “constants”)

x1 x2 · · · xn−1 xn x8 xn · · · x2 xn/2

y1, . . . , ypn

Theorem [Erdős-Szekeres]: Every sequence of n integers hasa monotone subsequence of length pn.Think of the ABP as computing a polynomial in the y vars overF(y) (i.e. all others vars are now “constants”)What you get is a 2-pass ABP over y vars.

x1 x2 · · · xn−1 xn x8 xn · · · x2 xn/2

y1, . . . , ypn

Theorem [Erdős-Szekeres]: Every sequence of n integers hasa monotone subsequence of length pn.Think of the ABP as computing a polynomial in the y vars overF(y) (i.e. all others vars are now “constants”)What you get is a 2-pass ABP over y vars. In other words,ignoring y, for every i ∈ [pn], eval-dim[i],[i]( f )≤ w4.

PIT for 2-pass, different orderPIT algorithm:

1. Find monotone subsequence y of length pn

2. Plug-in hitting set for width w4 ROABPs to y

3. Repeat with y(plugging in a fresh copy of the hitting set each time)

Running Time: In total, ≈pn copies of a nlog n size hitting set=⇒≈ n

Naturally generalizes to k passes with different orders.

PIT for k-pass, different ordersBy repeatedly applying the Erdős-Szekeres theorem, we can finda subsequence of size n1/2k−1 which is monotone in each of the kpasses.

Same algorithm gives nn1−1/2k−1

hitting set.

This is still not a general read-k oblivious ABP!

hitting set.

Read-Twice oblivious ABPs

Begin by applying Erdős-Szekeres.

x1 x2 x3 x4 x1 x2 x5 x6 x3 · · ·

Monotone sequences are not disjoint...BUT we can find a large set of the variables such that theresulting sequence is “regularly-interleaving”:

first X1

second X1

first X2

second X2

· · ·first X t

second X t

Read-Twice oblivious ABPsBegin by applying Erdős-Szekeres.

x1 x2 x3 x4 x1 x2 x5 x6 x3 · · ·

first X1

second X1

first X2

second X2

· · ·first X t

second X t

x1 x2 x3 x4 x1 x2 x5 x6 x3 · · ·

Monotone sequences are not disjoint...

BUT we can find a large set of the variables such that theresulting sequence is “regularly-interleaving”:

first X1

second X1

first X2

second X2

· · ·first X t

second X t

x1 x2 x3 x4 x1 x2 x5 x6 x3 · · ·

first X1

second X1

first X2

second X2

· · ·first X t

second X t

x1 x2 x3 x4 x1 x2 x5 x6 x3 · · ·

first X1

second X1

first X2

second X2

· · ·first X t

second X t

Regularly Interleaving SubsequencesThis structure is enough to carry out the original argument: withrespect to the variables y in the regularly interleaving sequence(|y| ≈ pn), the evaluation dimension is at most w4.

Generalizes to read-k: apply Erdős-Szekeres to every sequenceand make every pair regularly-interleaving.

Wrap-up: PIT algorithm with running time exp(n1−1/2k−1) for

read-k oblivious ABPs.

Regularly Interleaving SubsequencesThis structure is enough to carry out the original argument: withrespect to the variables y in the regularly interleaving sequence(|y| ≈ pn), the evaluation dimension is at most w4.

Generalizes to read-k: apply Erdős-Szekeres to every sequenceand make every pair regularly-interleaving.

Wrap-up: PIT algorithm with running time exp(n1−1/2k−1) for

read-k oblivious ABPs.

Lower Bounds for Read-k

• These arguments are sufficient to get a lower bound ofroughly exp(n1/2k

• But actually, for a lower bound we don’t need to show thatfor every prefix [i] the eval-dimension is small: it’s enough toshow it is small for some prefix [i]

• That is, to show that if f is computed by a read-k obliviousABP, then there is i such that eval-dim[i],[i]( f )≤ w2k

• This is very close to being true

Exponential Lower BoundClaim: We can fix n/10 variables and partition the remaining tosubsets S, T with |S|, |T | ≥ n/kk and eval-dimS,T ( f )≤ w2k

Proof: Partition program into r contiguous blocks.

By averaging, ∃k blocks that contain all reads of n/�r

�vars.

Call them S and fix all other vars in those blocks.

�vars.

Call them S and fix all other vars in those blocks.T = all remaining variables. Now compute eval-dimS,T usingprevious arguments.

�vars.

Call them S and fix all other vars in those blocks.T = all remaining variables. Now compute eval-dimS,T usingprevious arguments.if r = 10k2 we fix at most n/10 vars and |S| ≥ n/kk.

�vars.

Call them S and fix all other vars in those blocks.T = all remaining variables. Now compute eval-dimS,T usingprevious arguments.if r = 10k2 we fix at most n/10 vars and |S| ≥ n/kk.what’s left is to find a polynomial such that eval-dimS,T ≥ 2min{|S|,|T |}

Summary

LowerBound: An exp(n/kk) lower bound on any read-koblivious ABP computing some polynomial f ∈ VP.

PIT: A white-box PIT algorithm for read-k oblivious ABPs, withrunning time exp(n1−1/2k−1

SummaryLowerBound: An exp(n/kk) lower bound on any read-koblivious ABP computing some polynomial f ∈ VP.

Open Problems

• Faster PIT algorithm• A complete black-box test (no dependence on order)• “Tighter” lower bounds (e.g. a hierarchy theorem for read-k

ABPs)• Non-oblivious? (open even for k = 1)• Connections with pseudorandomness for boolean branching

programs?

Thank You

Open Problems

• Faster PIT algorithm

• A complete black-box test (no dependence on order)• “Tighter” lower bounds (e.g. a hierarchy theorem for read-k

programs?

Thank You

Open Problems

• Faster PIT algorithm• A complete black-box test (no dependence on order)

• “Tighter” lower bounds (e.g. a hierarchy theorem for read-kABPs)

• Non-oblivious? (open even for k = 1)• Connections with pseudorandomness for boolean branching

programs?

Thank You

Open Problems

• Non-oblivious? (open even for k = 1)• Connections with pseudorandomness for boolean branching

programs?

Thank You

Open Problems

ABPs)• Non-oblivious? (open even for k = 1)

• Connections with pseudorandomness for boolean branchingprograms?

Thank You

Open Problems

programs?

Thank You

IdentityTesting&LowerBounds - Tel Aviv Universityshpilka/wact2016/program/slides/BenLeeVolk.pdf ·...

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