[PPT] - Provability of weak circuit lower bounds J an Pich Kurt G odel PowerPoint Presentation

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Proof complexity, Dagstuhl 1st February 2018

Provability of weak circuit lower bounds

J´ an Pich Kurt G¨

del Research Center

University of Vienna based on a joint work with Moritz M¨ uller

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Constructive proofs of circuit lower bounds

Known circuit lower bounds for f given explicitly: AC0, AC0[p], etc. very constructive: p-time algorithm often recognizing when f is hard a.k.a natural proofs

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Constructive proofs of circuit lower bounds

Known circuit lower bounds for f given explicitly: AC0, AC0[p], etc. very constructive: p-time algorithm often recognizing when f is hard a.k.a natural proofs Razborov-Rudich: Cryptography works → no natural proof of P=NP.

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Constructive proofs of circuit lower bounds

Known circuit lower bounds for f given explicitly: AC0, AC0[p], etc. very constructive: p-time algorithm often recognizing when f is hard a.k.a natural proofs Razborov-Rudich: Cryptography works → no natural proof of P=NP. Mathematical logic:

upper bounds: Prove all known circuit lower bounds in a constructive

mathematical theory, e.g. PV1 (p-time reasoning).

exhibit a structure of algorithms recognizing hard functions?

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Constructive proofs of circuit lower bounds

Known circuit lower bounds for f given explicitly: AC0, AC0[p], etc. very constructive: p-time algorithm often recognizing when f is hard a.k.a natural proofs Razborov-Rudich: Cryptography works → no natural proof of P=NP. Mathematical logic:

upper bounds: Prove all known circuit lower bounds in a constructive

mathematical theory, e.g. PV1 (p-time reasoning).

exhibit a structure of algorithms recognizing hard functions?
lower bounds: PV1 ⊢ strong circuit lower bounds?
stronger ’natural proofs’ barrier: P=NP consistent with PV1?
circuit lower bounds as hard tautologies witnessing NP=coNP?

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Bounded arithmetic and propositional logic

PV1: first-order theory formalizing p-time reasoning (Cook ’75) APC1: formalizes probabilistic p-time reasoning (Jeˇ r´ abek ’07) APC1:= PV1 + “∃f / ∈ SIZE(2ǫn)”

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Bounded arithmetic and propositional logic

PV1: first-order theory formalizing p-time reasoning (Cook ’75) APC1: formalizes probabilistic p-time reasoning (Jeˇ r´ abek ’07) APC1:= PV1 + “∃f / ∈ SIZE(2ǫn)” If PV1 ⊢ ∀xA(x) for a p-time predicate A, then tautologies expressing ∀xA(x) have p-size Extended Frege EF proofs If APC1 ⊢ ∀xA(x) for a p-time predicate A, then tautologies expressing ∀xA(x) have p-size WF proofs WF: EF + “∃f / ∈ SIZE(2ǫn)” (Jeˇ r´ abek ’04)

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How to express circuit lower bounds formally

First-order formulation: LB(f , nk): ’every circuit of size nk fails to compute function f ’

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How to express circuit lower bounds formally

First-order formulation: LB(f , nk): ’every circuit of size nk fails to compute function f ’ ∀n > n0 ∀ circuit C of size ≤ nk ∃ input y, |y| = n; C(y) = f (y) where n0, k are fixed constants

If f ∈ NP, then LB(f , nk) is Πp

3 (i.e. ∀∃∀ statement)

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How to express circuit lower bounds formally

First-order formulation: LB(f , nk): ’every circuit of size nk fails to compute function f ’ ∀n > n0 ∀ circuit C of size ≤ nk ∃ input y, |y| = n; C(y) = f (y) where n0, k are fixed constants

If f ∈ NP, then LB(f , nk) is Πp

3 (i.e. ∀∃∀ statement)

different scaling: LBtt(f , nk): ∀m, n > n0, |m| = 2n ∀ circuit C of size ≤ nk ∃ y, |y| = n; C(y) = f (y)

If f = SAT, then LBtt(f , nk) is coNP

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How to express circuit lower bounds formally

First-order formulation: LB(f , nk): ’every circuit of size nk fails to compute function f ’ ∀n > n0 ∀ circuit C of size ≤ nk ∃ input y, |y| = n; C(y) = f (y) where n0, k are fixed constants

If f ∈ NP, then LB(f , nk) is Πp

3 (i.e. ∀∃∀ statement)

different scaling: LBtt(f , nk): ∀m, n > n0, |m| = 2n ∀ circuit C of size ≤ nk ∃ y, |y| = n; C(y) = f (y)

If f = SAT, then LBtt(f , nk) is coNP

Easier to reason about LBtt(f , nk) than about LB(f , nk).

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Propositional formulation: tt(f , nk):

y∈{0,1}n

f (y) = C(y) (expresses LBtt(f , nk)) 2n bits f (y), poly(n) variables for circuit C of size nk, total size: 2O(n)

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Propositional formulation: tt(f , nk):

y∈{0,1}n

f (y) = C(y) (expresses LBtt(f , nk)) 2n bits f (y), poly(n) variables for circuit C of size nk, total size: 2O(n) Lipton-Young: f / ∈ SIZE(nO(1)) ⇒ ∃A ⊆ {0, 1}n of size poly(n) s.t.

y∈A f (y) = C(y)

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Propositional formulation: tt(f , nk):

y∈{0,1}n

f (y) = C(y) (expresses LBtt(f , nk)) 2n bits f (y), poly(n) variables for circuit C of size nk, total size: 2O(n) Lipton-Young: f / ∈ SIZE(nO(1)) ⇒ ∃A ⊆ {0, 1}n of size poly(n) s.t.

y∈A f (y) = C(y)

lbA(f , nk):

y∈A f (y) = C(y)

same meaning as tt(f , nk) but poly(n) size

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Propositional formulation: tt(f , nk):

y∈{0,1}n

f (y) = C(y) (expresses LBtt(f , nk)) 2n bits f (y), poly(n) variables for circuit C of size nk, total size: 2O(n) Lipton-Young: f / ∈ SIZE(nO(1)) ⇒ ∃A ⊆ {0, 1}n of size poly(n) s.t.

y∈A f (y) = C(y)

lbA(f , nk):

y∈A f (y) = C(y)

same meaning as tt(f , nk) but poly(n) size

lbw(f , nk): poly(n) size formula expressing LB(f , nk) with existential quantifiers witnessed feasibly by w

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Propositional formulation: tt(f , nk):

y∈{0,1}n

f (y) = C(y) (expresses LBtt(f , nk)) 2n bits f (y), poly(n) variables for circuit C of size nk, total size: 2O(n) Lipton-Young: f / ∈ SIZE(nO(1)) ⇒ ∃A ⊆ {0, 1}n of size poly(n) s.t.

y∈A f (y) = C(y)

lbA(f , nk):

y∈A f (y) = C(y)

same meaning as tt(f , nk) but poly(n) size

lbw(f , nk): poly(n) size formula expressing LB(f , nk) with existential quantifiers witnessed feasibly by w Possible witnessing w of LB(f , nk): a p-time algorithm with input: circuit C of size nk

utput: y s.t. C(y) = f (y)

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Propositional formulation: tt(f , nk):

y∈{0,1}n

f (y) = C(y) (expresses LBtt(f , nk)) 2n bits f (y), poly(n) variables for circuit C of size nk, total size: 2O(n) Lipton-Young: f / ∈ SIZE(nO(1)) ⇒ ∃A ⊆ {0, 1}n of size poly(n) s.t.

y∈A f (y) = C(y)

lbA(f , nk):

y∈A f (y) = C(y)

same meaning as tt(f , nk) but poly(n) size

lbw(f , nk): poly(n) size formula expressing LB(f , nk) with existential quantifiers witnessed feasibly by w

∃w follows e.g. from PV1 ⊢ LB(f , nk)

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Propositional formulation: tt(f , nk):

y∈{0,1}n

f (y) = C(y) (expresses LBtt(f , nk)) 2n bits f (y), poly(n) variables for circuit C of size nk, total size: 2O(n) Lipton-Young: f / ∈ SIZE(nO(1)) ⇒ ∃A ⊆ {0, 1}n of size poly(n) s.t.

y∈A f (y) = C(y)

lbA(f , nk):

y∈A f (y) = C(y)

same meaning as tt(f , nk) but poly(n) size

lbw(f , nk): poly(n) size formula expressing LB(f , nk) with existential quantifiers witnessed feasibly by w

∃w follows e.g. from PV1 ⊢ LB(f , nk)

Fact: If tt(f , nk) has no poly-size constant-depth Frege proofs, then lbA(f , nk) has no poly-size (full) Frege proofs.

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Previous results

Lower bounds: Razborov: S2

2(α) ⊢“LBtt(SAT, nk)” unless cryptography breaks

P.: VNC1 ⊢ LB(SAT, nk) unless SIZE(nk) ⊆approx “subexp NC1” Kraj´ ıˇ cek-Oliveira: ∀k ∃f ∈ P s.t. PV1 ⊢ f ∈ SIZE(cnk) Buss: PV1 ⊢ NP = coNP unless P=NP “folklore”: V 0 ⊢ SAT ∈ P/poly

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Previous results

Lower bounds: Razborov: S2

2(α) ⊢“LBtt(SAT, nk)” unless cryptography breaks

P.: VNC1 ⊢ LB(SAT, nk) unless SIZE(nk) ⊆approx “subexp NC1” Kraj´ ıˇ cek-Oliveira: ∀k ∃f ∈ P s.t. PV1 ⊢ f ∈ SIZE(cnk) Buss: PV1 ⊢ NP = coNP unless P=NP “folklore”: V 0 ⊢ SAT ∈ P/poly Razborov-Kraj´ ıˇ cek: Propositional systems with feasible interpolation property have no p-size proofs of tt(f , nk) unless cryptography breaks. Raz: Resolution has no p-size proofs of tt(f , nk) (unconditionally). Razborov: Res(ǫ log n) does not have p-size proofs of tt(f , nω(1)).

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Previous results

Lower bounds: Razborov: S2

2(α) ⊢“LBtt(SAT, nk)” unless cryptography breaks

P.: VNC1 ⊢ LB(SAT, nk) unless SIZE(nk) ⊆approx “subexp NC1” Kraj´ ıˇ cek-Oliveira: ∀k ∃f ∈ P s.t. PV1 ⊢ f ∈ SIZE(cnk) Buss: PV1 ⊢ NP = coNP unless P=NP “folklore”: V 0 ⊢ SAT ∈ P/poly Razborov-Kraj´ ıˇ cek: Propositional systems with feasible interpolation property have no p-size proofs of tt(f , nk) unless cryptography breaks. Raz: Resolution has no p-size proofs of tt(f , nk) (unconditionally). Razborov: Res(ǫ log n) does not have p-size proofs of tt(f , nω(1)). tt(f , nk) considered as cadidate hard tautologies for EF.

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Upper bounds: Razborov: PV1 ⊢ LBtt(PARITY , AC0(nk))

AC0(nk): constant depth circuits of size nk

PV1 ⊢ LBtt(MODq, AC0[p](nk)) for p, q distinct primes

AC0[p](nk): AC0(nk) with modp gates

PV1 ⊢ LBtt(CLI, mSIZE(nk))

mSIZE(nk): monotone circuits of size nk

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Upper bounds: Razborov: PV1 ⊢ LBtt(PARITY , AC0(nk))

AC0(nk): constant depth circuits of size nk

PV1 ⊢ LBtt(MODq, AC0[p](nk)) for p, q distinct primes

AC0[p](nk): AC0(nk) with modp gates

PV1 ⊢ LBtt(CLI, mSIZE(nk))

mSIZE(nk): monotone circuits of size nk

Corollary: p-size EF proofs of the corresponding tt(f , nk) formulas In fact: Razborov’s formalization are below PV1 resp. EF

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Upper bounds: Razborov: PV1 ⊢ LBtt(PARITY , AC0(nk))

AC0(nk): constant depth circuits of size nk

PV1 ⊢ LBtt(MODq, AC0[p](nk)) for p, q distinct primes

AC0[p](nk): AC0(nk) with modp gates

PV1 ⊢ LBtt(CLI, mSIZE(nk))

mSIZE(nk): monotone circuits of size nk

Corollary: p-size EF proofs of the corresponding tt(f , nk) formulas In fact: Razborov’s formalization are below PV1 resp. EF Kraj´ ıˇ cek: APC1 ⊢ LB(PARITY , AC0(nk))

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Complexity theory formalizable in PV1 and APC1

Theory Theorem PV1 Cook-Levin’s theorem the PCP theorem Hardness amplification . . . APC1 AC0 lower bounds AC0[p] lower bounds (with 2logO(1) n ∈ Log) Monotone circuit lower bounds Nisan-Wigderson’s derandomization Impagliazzo-Wigderson’s derandomization Goldreich-Levin’s theorem Natural proofs barrier . . .

Table: A list of formalizations.

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New results

APC1 ⊢ LB(PARITY , AC0(nk)) APC1 ⊢ LB(MODq, AC0[p](nk)) for p, q distinct primes APC1 ⊢ LB(CLI, mSIZE(nk))

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New results

APC1 ⊢ LB(PARITY , AC0(nk))

standard proof using Jeˇ

r´ abek’s machinery of approximate counting

Pr[A] > p, for A ⊆ 2n, witnessed by a p-time surjection s : A → p2n
size of each set approximated by sampling poly(n) elements

e.g. there are poly(n) restrictions ρ1, . . . , ρt, t ≤ poly(n) s.t. each nk-size d-depth circuit is collapsed by some ρ ∈ {ρ1, . . . , ρt} (ρ leaving many variables unassigned) APC1 ⊢ LB(MODq, AC0[p](nk)) for p, q distinct primes APC1 ⊢ LB(CLI, mSIZE(nk))

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New results

APC1 ⊢ LB(PARITY , AC0(nk))

standard proof using Jeˇ

r´ abek’s machinery of approximate counting

Pr[A] > p, for A ⊆ 2n, witnessed by a p-time surjection s : A → p2n
size of each set approximated by sampling poly(n) elements

e.g. there are poly(n) restrictions ρ1, . . . , ρt, t ≤ poly(n) s.t. each nk-size d-depth circuit is collapsed by some ρ ∈ {ρ1, . . . , ρt} (ρ leaving many variables unassigned) APC1 ⊢ LB(MODq, AC0[p](nk)) for p, q distinct primes

standard proof infeasible: counts all 22n functions with n inputs
we scale the argument: count only functions with logO(1) n inputs
∃m, |m| = 2logO(1) n needed

APC1 ⊢ LB(CLI, mSIZE(nk))

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New results

APC1 ⊢ LB(PARITY , AC0(nk))

standard proof using Jeˇ

r´ abek’s machinery of approximate counting

Pr[A] > p, for A ⊆ 2n, witnessed by a p-time surjection s : A → p2n
size of each set approximated by sampling poly(n) elements

e.g. there are poly(n) restrictions ρ1, . . . , ρt, t ≤ poly(n) s.t. each nk-size d-depth circuit is collapsed by some ρ ∈ {ρ1, . . . , ρt} (ρ leaving many variables unassigned) APC1 ⊢ LB(MODq, AC0[p](nk)) for p, q distinct primes

standard proof infeasible: counts all 22n functions with n inputs
we scale the argument: count only functions with logO(1) n inputs
∃m, |m| = 2logO(1) n needed

APC1 ⊢ LB(CLI, mSIZE(nk))

standard proof with p-time surjections witnessing probabilities

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New results

APC1 ⊢ LB(PARITY , AC0(nk)) APC1 ⊢ LB(MODq, AC0[p](nk)) for p, q distinct primes APC1 ⊢ LB(CLI, mSIZE(nk)) Corollary: p-size EF proofs of the corresponding lbw formulas from the assumption that “∃g, tt(g, 2ǫn)”.

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New results

APC1 ⊢ LB(PARITY , AC0(nk)) APC1 ⊢ LB(MODq, AC0[p](nk)) for p, q distinct primes APC1 ⊢ LB(CLI, mSIZE(nk)) Corollary: p-size EF proofs of the corresponding lbw formulas from the assumption that “∃g, tt(g, 2ǫn)”. Problem: APC1 ⊢ LB(f , nk) ⇒ ∃ efficient witnessing w of LB(f , nk) but w is probabilistic resp. w depends on a hard function g so unconditional WF proofs of lbw(f , nk) do not follow directly

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New results

APC1 ⊢ LB(PARITY , AC0(nk)) APC1 ⊢ LB(MODq, AC0[p](nk)) for p, q distinct primes APC1 ⊢ LB(CLI, mSIZE(nk)) Corollary: p-size EF proofs of the corresponding lbw formulas from the assumption that “∃g, tt(g, 2ǫn)”. Problem: APC1 ⊢ LB(f , nk) ⇒ ∃ efficient witnessing w of LB(f , nk) but w is probabilistic resp. w depends on a hard function g so unconditional WF proofs of lbw(f , nk) do not follow directly

Possible solution (the road not taken): Derandomize the probabilistic

witnessing of AC0, AC0[p] and monotone circuit lower bounds in APC1.

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New results

APC1 ⊢ LB(PARITY , AC0(nk)) APC1 ⊢ LB(MODq, AC0[p](nk)) for p, q distinct primes APC1 ⊢ LB(CLI, mSIZE(nk)) Corollary: p-size EF proofs of the corresponding lbw formulas from the assumption that “∃g, tt(g, 2ǫn)”. Problem: APC1 ⊢ LB(f , nk) ⇒ ∃ efficient witnessing w of LB(f , nk) but w is probabilistic resp. w depends on a hard function g so unconditional WF proofs of lbw(f , nk) do not follow directly

Possible solution (the road not taken): Derandomize the probabilistic

witnessing of AC0, AC0[p] and monotone circuit lower bounds in APC1. To get WF proofs of lbA(f , AC0[p](nk)) formulas (unconditionally) we give a succinct naturalization of Razborov-Smolensky’s AC0[p] lower bound.

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Naturalization / automatizability

want a p-time algorithm which given lbA(f , nk) finds its proof if it exists
i.e. succinct natural proof

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Naturalization / automatizability

want a p-time algorithm which given lbA(f , nk) finds its proof if it exists
i.e. succinct natural proof

Learning: - given bits f (x1), . . . , f (xk) for k n-bit tuples x1, . . . , xk

want to predict f (xk+1) on a new input xk+1 ∈ {0, 1}n
minimal circuit C computing f on x1, . . . , xk has to determine f (xk+1)
say that the size of the minimal circuit C is s

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Naturalization / automatizability

want a p-time algorithm which given lbA(f , nk) finds its proof if it exists
i.e. succinct natural proof

Learning: - given bits f (x1), . . . , f (xk) for k n-bit tuples x1, . . . , xk

want to predict f (xk+1) on a new input xk+1 ∈ {0, 1}n
minimal circuit C computing f on x1, . . . , xk has to determine f (xk+1)
say that the size of the minimal circuit C is s

To predict f (xk+1) prove an s-size circuit lower bound (for ǫ ∈ {0, 1})

i=1,...,k

C(xi) = f (xi) ∨ C(xk+1) = ǫ

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Naturalization / automatizability

want a p-time algorithm which given lbA(f , nk) finds its proof if it exists
i.e. succinct natural proof

Learning: - given bits f (x1), . . . , f (xk) for k n-bit tuples x1, . . . , xk

want to predict f (xk+1) on a new input xk+1 ∈ {0, 1}n
minimal circuit C computing f on x1, . . . , xk has to determine f (xk+1)
say that the size of the minimal circuit C is s

To predict f (xk+1) prove an s-size circuit lower bound (for ǫ ∈ {0, 1})

i=1,...,k

C(xi) = f (xi) ∨ C(xk+1) = ǫ A more sophisticated connection between circuit lower bounds and learning algorithms recently demonstrated by Carmosino et al.

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Naturalization / automatizability

want a p-time algorithm which given lbA(f , nk) finds its proof if it exists
i.e. succinct natural proof

Learning: - given bits f (x1), . . . , f (xk) for k n-bit tuples x1, . . . , xk

want to predict f (xk+1) on a new input xk+1 ∈ {0, 1}n
minimal circuit C computing f on x1, . . . , xk has to determine f (xk+1)
say that the size of the minimal circuit C is s

To predict f (xk+1) prove an s-size circuit lower bound (for ǫ ∈ {0, 1})

i=1,...,k

C(xi) = f (xi) ∨ C(xk+1) = ǫ A more sophisticated connection between circuit lower bounds and learning algorithms recently demonstrated by Carmosino et al. Theorem: quasipolynomial-time algorithm generating WF proofs of lbA(f , AC0[p](nk)) for many functions f .

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Problems

Derandomize known circuit lower bounds, i.e. prove them inside PV1.

1st step: Derandomize witnessing of known circuit lower bounds.

Prove APC1 ⊢ LB(MODq, AC0[p](nk)) without ∃m, |m| = 2logO(1) n.
V 0 ⊢ LB(PARITY , AC0(nk))?

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Problems

Derandomize known circuit lower bounds, i.e. prove them inside PV1.

1st step: Derandomize witnessing of known circuit lower bounds.

Prove APC1 ⊢ LB(MODq, AC0[p](nk)) without ∃m, |m| = 2logO(1) n.
V 0 ⊢ LB(PARITY , AC0(nk))?

Thank You for Your Attention

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