Hitting Sets for UPT Circuits Ramprasad Saptharishi and Anamay Tengse - - PowerPoint PPT Presentation

Hitting Sets for UPT Circuits

Ramprasad Saptharishi and Anamay Tengse

TIFR, Mumbai, India

6th March 2018

Non-commutative models

+ × 3 × × −0.5 × × x1 x2 x3 x4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5 ℓ6 ℓ7 s t ◮ x1x2x1 = x1x1x2

monomials ∼ words

◮ Introduced by Nisan [N91] ◮ Circuits:

• No. of nodes

◮ ABPs:

Width, No. of layers

Non-commutative models

+ × 3 × × −0.5 × × x1 x2 x3 x4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5 ℓ6 ℓ7 s t ◮ x1x2x1 = x1x1x2

monomials ∼ words

◮ Introduced by Nisan [N91] ◮ Circuits:

• No. of nodes

◮ ABPs:

Width, No. of layers

◮ ABPs Circuits

[N91]

Non-commutative models

+ × 3 × × −0.5 × × x1 x2 x3 x4 ℓ1 ℓ2 ℓ3 ℓ4 ℓ5 ℓ6 ℓ7 s t ◮ x1x2x1 = x1x1x2

monomials ∼ words

◮ Introduced by Nisan [N91] ◮ Circuits:

• No. of nodes

◮ ABPs:

Width, No. of layers

◮ ABPs Circuits

[N91]

Homogeneous circuits: Each gate is homogeneous Homogeneous ABPs: Each of the ℓis are homogeneous

Hitting sets for Non-commutative circuits

Given a non-commutative circuit class C ⊆ Fx, a set of matrices H is called a hitting set for C if a nonzero C ∈ C evaluates to a nonzero value on at least one input from H. Note: Variables from x can be thought of as matrices with commuting variables from y as entries. Strategy: Substitute univariates of low degree, interpolate. φ : y → F[t].

Hitting sets for Non-commutative circuits

Given a non-commutative circuit class C ⊆ Fx, a set of matrices H is called a hitting set for C if a nonzero C ∈ C evaluates to a nonzero value on at least one input from H. Note: Variables from x can be thought of as matrices with commuting variables from y as entries. Strategy: Substitute univariates of low degree, interpolate. φ : y → F[t]. For this talk:

◮ Non-commutative circuits, ABPs ◮ WLOG models will be homogeneous

Parse Trees and Unambiguity

Parse tree: Start from root, one child of +, all children of ×

+ × × × × × x1 x2 x3 x4 (a) + × × (b) + × × (c)

Parse Trees and Unambiguity

Parse tree: Start from root, one child of +, all children of ×

+ × × × × × x1 x2 x3 x4 (a) + × × (b) + × × (c) ◮ Unambiguous or Unique Parse Tree (UPT) [LMP16]

all parse trees have the same shape.

◮ ABPs UPT Circuits [LMP16]

ABPs as UPT circuits

ℓ1 ℓ2 ℓ3 ℓ4 ℓ5 ℓ6 ℓ7 s t

ABPs as UPT circuits

ℓ1 ℓ2 ℓ3 ℓ4 ℓ5 ℓ6 ℓ7 s t + × × + + × × × ℓ1 ℓ3 ℓ6 ℓ4 ℓ2 ℓ5 ℓ7

ABPs as UPT circuits

ℓ1 ℓ2 ℓ3 ℓ4 ℓ5 ℓ6 ℓ7 s t + × × + + × × × ℓ1 ℓ3 ℓ6 ℓ4 ℓ2 ℓ5 ℓ7 + × + ×

ABPs as UPT circuits

ℓ1 ℓ2 ℓ3 ℓ4 ℓ5 ℓ6 ℓ7 s t + × × + + × × × ℓ1 ℓ3 ℓ6 ℓ4 ℓ2 ℓ5 ℓ7 + × + × ◮ ABPs are UPT circuits with left-skew tree.

Properties of UPT circuits [LMP16]

• 1. WLOG each gate appears in a fixed position in the tree.

Can be done with a d blow-up.

• 2. Natural notion of width of a position.
• No. of gates appearing in that position.

Analogous to width of an ABP.

• 3. All product gates are position disjoint.

Consequence of 1. Similar to edges in different layer segments in an ABP.

Properties of UPT circuits [LMP16]

• 1. WLOG each gate appears in a fixed position in the tree.

Can be done with a d blow-up.

• 2. Natural notion of width of a position.
• No. of gates appearing in that position.

Analogous to width of an ABP.

• 3. All product gates are position disjoint.

Consequence of 1. Similar to edges in different layer segments in an ABP. Plan:

◮ Overview of hitting sets for ABPs. ◮ Extend ideas to UPT circuits.

Quick survey

◮ Nisan’s characterization for ABPs [N91].

Quick survey

◮ Nisan’s characterization for ABPs [N91]. ◮ [RS05] gave White box PIT for ABPs in poly(n).

Quick survey

◮ Nisan’s characterization for ABPs [N91]. ◮ [RS05] gave White box PIT for ABPs in poly(n). ◮ [FS13] Known order hitting sets for ROABPs in nO(log n) .

Quick survey

◮ Nisan’s characterization for ABPs [N91]. ◮ [RS05] gave White box PIT for ABPs in poly(n). ◮ [FS13] Known order hitting sets for ROABPs in nO(log n) . ◮ [AGKS15] Unknown order hitting sets for ROABPs in nO(log n).

Basis Isolating Weight Assignment (BIWA).

Quick survey

◮ Nisan’s characterization for ABPs [N91]. ◮ [RS05] gave White box PIT for ABPs in poly(n). ◮ [FS13] Known order hitting sets for ROABPs in nO(log n) . ◮ [AGKS15] Unknown order hitting sets for ROABPs in nO(log n).

Basis Isolating Weight Assignment (BIWA).

◮ Hitting sets for models related to ROABPs

◮ Constant width, known order in poly(n) [GKS16]. ◮ Sum of c ROABPs: white box in poly(n), hitting sets nO(log n)

[GKST15] (Nisan’s characterization + BIWA).

Quick survey

◮ Nisan’s characterization for ABPs [N91]. ◮ [RS05] gave White box PIT for ABPs in poly(n). ◮ [FS13] Known order hitting sets for ROABPs in nO(log n) . ◮ [AGKS15] Unknown order hitting sets for ROABPs in nO(log n).

Basis Isolating Weight Assignment (BIWA).

◮ Hitting sets for models related to ROABPs

◮ Constant width, known order in poly(n) [GKS16]. ◮ Sum of c ROABPs: white box in poly(n), hitting sets nO(log n)

[GKST15] (Nisan’s characterization + BIWA).

◮ [LMP16] introduced UPT circuits

◮ Extend Nisan’s characterization ◮ White box in poly(n)

Quick survey

◮ Nisan’s characterization for ABPs [N91]. ◮ [RS05] gave White box PIT for ABPs in poly(n). ◮ [FS13] Known order hitting sets for ROABPs in nO(log n) . ◮ [AGKS15] Unknown order hitting sets for ROABPs in nO(log n).

Basis Isolating Weight Assignment (BIWA).

◮ Hitting sets for models related to ROABPs

◮ Constant width, known order in poly(n) [GKS16]. ◮ Sum of c ROABPs: white box in poly(n), hitting sets nO(log n)

[GKST15] (Nisan’s characterization + BIWA).

◮ [LMP16] introduced UPT circuits

◮ Extend Nisan’s characterization ◮ White box in poly(n)

◮ [LLS17] extend white box results of [GKST15].

Quick survey

◮ Nisan’s characterization for ABPs [N91]. ◮ [RS05] gave White box PIT for ABPs in poly(n). ◮ [FS13] Known order hitting sets for ROABPs in nO(log n) . ◮ [AGKS15] Unknown order hitting sets for ROABPs in nO(log n).

Basis Isolating Weight Assignment (BIWA).

◮ Hitting sets for models related to ROABPs

◮ Constant width, known order in poly(n) [GKS16]. ◮ Sum of c ROABPs: white box in poly(n), hitting sets nO(log n)

[GKST15] (Nisan’s characterization + BIWA).

◮ [LMP16] introduced UPT circuits

◮ Extend Nisan’s characterization ◮ White box in poly(n)

◮ [LLS17] extend white box results of [GKST15]. ◮ This work: BIWA for UPT circuits, extends hitting sets of

[AGKS15,GKST15,GKS15].

Coefficient span [RS05,FS13]

. . . . . . u1 ui uw s t f1 fi fw

Preserve nonzeroness of an arbitrary linear combination of fis. Mf =    ← f1 → ← . . . → ← fw →    ∈ F[y]k ≡ Fk[y]

Coefficient span [RS05,FS13]

. . . . . . u1 ui uw s t f1 fi fw

Preserve nonzeroness of an arbitrary linear combination of fis. Mf =    ← f1 → ← . . . → ← fw →    ∈ F[y]k ≡ Fk[y] Consider φ : y → F[t1, . . . , tk] (k ∼ log n) Such a φ sends columns of Mf to nO(k) columns.

Coefficient span [RS05,FS13]

. . . . . . u1 ui uw s t f1 fi fw

Preserve nonzeroness of an arbitrary linear combination of fis. Mf =    ← f1 → ← . . . → ← fw →    ∈ F[y]k ≡ Fk[y] Consider φ : y → F[t1, . . . , tk] (k ∼ log n) Such a φ sends columns of Mf to nO(k) columns.

[FS13] A φ that preserves colSpan(Mf ) suffices.

ColSpan(Mf ) = CoeffSpan(f1, . . . , fw)

Coefficient span [RS05,FS13]

. . . . . . u1 ui uw s t f1 fi fw

[FS13] A φ that preserves colSpan(Mf ) suffices.

Let (wt1, . . . , wtk) : y → [N]k and φwt be such that φwt : yi → twt1(yi)

1

· · · twtk(yi)

k

.

[AGKS15] If wt is a basis isolating weight assignment (BIWA) for

Mf , then φwt will preserve CoeffSpan. How do we construct a BIWA?

Basis Isolation [AGKS15]

. . . . . . . . . . . . . . . . . . v1 vk vw s1 si sw t1 tj tw gi,ks hk,js

Mf Mg Mh    ← f1,1 → ← . . . → ← fw,w →       ← g1,1 → ← . . . → ← gw,w →       ← h1,1 → ← . . . → ← hw,w →    Define Vf , Vg, Vh, where V∗ = rowSpan(M∗).

Basis Isolation [AGKS15]

. . . . . . . . . . . . . . . . . . v1 vk vw s1 si sw t1 tj tw gi,ks hk,js

Mf Mg Mh    ← f1,1 → ← . . . → ← fw,w →       ← g1,1 → ← . . . → ← gw,w →       ← h1,1 → ← . . . → ← hw,w →    Define Vf , Vg, Vh, where V∗ = rowSpan(M∗). fi,j =

• k∈[w]

gi,khk,j ∈ Vf ⊆ Vg ⊗ Vh

Basis Isolation [AGKS15]

. . . . . . . . . . . . . . . . . . v1 vk vw s1 si sw t1 tj tw gi,ks hk,js

Mf Mg Mh    ← f1,1 → ← . . . → ← fw,w →       ← g1,1 → ← . . . → ← gw,w →       ← h1,1 → ← . . . → ← hw,w →    Define Vf , Vg, Vh, where V∗ = rowSpan(M∗). fi,j =

• k∈[w]

gi,khk,j ∈ Vf ⊆ Vg ⊗ Vh

Basis Isolation [AGKS15]

. . . . . . . . . . . . . . . . . . v1 vk vw s1 si sw t1 tj tw gi,ks hk,js

Define Vf , Vg, Vh, where V∗ = rowSpan(M∗). fi,j =

• k∈[w]

gi,khk,j ∈ Vf ⊆ Vg ⊗ Vh BIWA [AGKS15]: If wt : y → [N]k is a BIWA for Vg and Vh, then poly(n) time construction for wt′ : y → [N] such that (wt, wt′) : y → [N]k+1 is a BIWA for Vf .

So far...

Abstract view of [AGKS15]

◮ Each layer segment with w2 edges naturally yields a vector

space.

◮ Space Vf resulting from paths across consecutive layers

(Vg, Vh) satisfies Vf ⊆ Vg ⊗ Vh.

◮ BIWA for Vg and Vh can be extended to a BIWA for Vf by

adding an extra coordinate, in poly(n) time.

So far...

Abstract view of [AGKS15]

◮ Each layer segment with w2 edges naturally yields a vector

space.

◮ Space Vf resulting from paths across consecutive layers

(Vg, Vh) satisfies Vf ⊆ Vg ⊗ Vh.

◮ BIWA for Vg and Vh can be extended to a BIWA for Vf by

adding an extra coordinate, in poly(n) time. Properties of UPT circuits

◮ All parse trees have the same shape, each gate ∼ node. ◮ Analogous notion of width for nodes. ◮ All product gates are position disjoint.

Extending AGKS to UPT ckts

+ τ + · · · · · · + × × · · · × τ ′ + · · · + τL + · · · + τR + × + + τ τ ′ τL τR

Extending AGKS to UPT ckts

+ τ f1, . . . , fw + · · · · · · + × × · · · × τ ′ p1,1 p1,2 pw,w + · · · + τL g1, . . . , gw + · · · + τR h1, . . . , hw

pi,j = gi × hj fk ∈ {gi × hj : (i, j) ∈ [w]2}

Extending AGKS to UPT ckts

+ τ f1, . . . , fw + · · · · · · + × × · · · × τ ′ p1,1 p1,2 pw,w + · · · + τL g1, . . . , gw + · · · + τR h1, . . . , hw

pi,j = gi × hj fk ∈ {gi × hj : (i, j) ∈ [w]2} Vτ ≡    ← f1 → . . . ← fw →    Vτ ′, VτL, VτR.

Extending AGKS to UPT ckts

+ τ f1, . . . , fw + · · · · · · + × × · · · × τ ′ p1,1 p1,2 pw,w + · · · + τL g1, . . . , gw + · · · + τR h1, . . . , hw

pi,j = gi × hj fk ∈ {gi × hj : (i, j) ∈ [w]2} Vτ ≡    ← f1 → . . . ← fw →    Vτ ′, VτL, VτR. Vτ ⊆ Vτ ′ Vτ ′ = VτL ⊗ VτR Vτ ⊆ VτL ⊗ VτR

BIWA for UPT circuits

+ × + + VτL VτR Vτ

Lemma [AGKS15] If (wt1, . . . , wtk) is a BIWA for both VτL and VτR, then in poly(n) time we can find wtk+1 such that (wt1, . . . , wtk+1) is a BIWA for all Vτ.

BIWA for UPT circuits

+ × + + VτL VτR Vτ

Lemma [AGKS15] If (wt1, . . . , wtk) is a BIWA for both VτL and VτR, then in poly(n) time we can find wtk+1 such that (wt1, . . . , wtk+1) is a BIWA for all Vτ. BIWA for Vroot with at most as many coordinates as depth(C).

BIWA for UPT circuits

+ × + + VτL VτR Vτ

Lemma [AGKS15] If (wt1, . . . , wtk) is a BIWA for both VτL and VτR, then in poly(n) time we can find wtk+1 such that (wt1, . . . , wtk+1) is a BIWA for all Vτ. BIWA for Vroot with at most as many coordinates as depth(C). Depth Reduction by shuffling For every UPT C of degree d, an equivalent UPT σ(C) of depth O(log d) exists.

Concluding remarks

Not covered:

◮ Extending hitting sets for sum of c ROABPs [GKST15] and

constant width ROABPs [GKS16] to UPT circuits.

◮ Exponential lower bound against UPT circuits under shufflings

for the moving pallindrome defined in [LMP16].

◮ Quasipolynomial (tight) separation between ABPs and UPT

circuits under shufflings, extension of [HY16].

Concluding remarks

Not covered:

◮ Extending hitting sets for sum of c ROABPs [GKST15] and

constant width ROABPs [GKS16] to UPT circuits.

◮ Exponential lower bound against UPT circuits under shufflings

for the moving pallindrome defined in [LMP16].

◮ Quasipolynomial (tight) separation between ABPs and UPT

circuits under shufflings, extension of [HY16]. Question: ABPs are UPT circuits with skew trees. Can we construct hitting sets for skew circuits?

Concluding remarks

Not covered:

◮ Extending hitting sets for sum of c ROABPs [GKST15] and

constant width ROABPs [GKS16] to UPT circuits.

◮ Exponential lower bound against UPT circuits under shufflings

for the moving pallindrome defined in [LMP16].

◮ Quasipolynomial (tight) separation between ABPs and UPT

circuits under shufflings, extension of [HY16]. Question: ABPs are UPT circuits with skew trees. Can we construct hitting sets for skew circuits?

Thank you.