Circuit Lower-bounds Lecture 24 Weak circuits are indeed weak 1 - - PowerPoint PPT Presentation

Circuit Lower-bounds

Lecture 24 Weak circuits are indeed weak

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Circuit Lower-bounds

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Circuit Lower-bounds

Today:

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Circuit Lower-bounds

Today: PARITY ∉ AC0

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Circuit Lower-bounds

Today: PARITY ∉ AC0 Two different proofs! (Latter generalizes to ACC0)

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Circuit Lower-bounds

Today: PARITY ∉ AC0 Two different proofs! (Latter generalizes to ACC0) CLIQUE cannot be decided by poly-sized monotone circuits

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Circuit Lower-bounds

Today: PARITY ∉ AC0 Two different proofs! (Latter generalizes to ACC0) CLIQUE cannot be decided by poly-sized monotone circuits (Only sketches/partial proofs. See textbook or lecture- notes from linked courses)

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PARITY ∉ AC0

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PARITY ∉ AC0

Recall AC0

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PARITY ∉ AC0

Recall AC0 Poly size, constant depth (unbounded fan-in)

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PARITY ∉ AC0

Recall AC0 Poly size, constant depth (unbounded fan-in) Today, non-uniform AC0

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PARITY ∉ AC0

Recall AC0 Poly size, constant depth (unbounded fan-in) Today, non-uniform AC0 How powerful can AC0 be?

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PARITY ∉ AC0

Recall AC0 Poly size, constant depth (unbounded fan-in) Today, non-uniform AC0 How powerful can AC0 be? Recall PARITY

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PARITY ∉ AC0

Recall AC0 Poly size, constant depth (unbounded fan-in) Today, non-uniform AC0 How powerful can AC0 be? Recall PARITY How shallow can a poly-sized circuit family for PARITY be?

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A Switching Argument

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A Switching Argument

Suppose constant depth (say ≤ d, d being minimal) circuits for PARITY

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A Switching Argument

Suppose constant depth (say ≤ d, d being minimal) circuits for PARITY Plan for contradiction: Show depth d-1 circuits for every input size n: start from depth d circuit for a larger n’, and construct one for the smaller n.

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A Switching Argument

Suppose constant depth (say ≤ d, d being minimal) circuits for PARITY Plan for contradiction: Show depth d-1 circuits for every input size n: start from depth d circuit for a larger n’, and construct one for the smaller n. By “restricting” to n inputs

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A Switching Argument

Suppose constant depth (say ≤ d, d being minimal) circuits for PARITY Plan for contradiction: Show depth d-1 circuits for every input size n: start from depth d circuit for a larger n’, and construct one for the smaller n. By “restricting” to n inputs And showing how to rewrite with depth d-1, staying poly sized

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AND-OR trees

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AND-OR trees

Any function can be written as depth 2 AND-OR tree or an OR-AND tree

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AND-OR trees

Any function can be written as depth 2 AND-OR tree or an OR-AND tree But exponential size

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AND-OR trees

Any function can be written as depth 2 AND-OR tree or an OR-AND tree But exponential size Any circuit can be rewritten as an AND-OR tree (each leaf has a literal, possibly shared with other leaves)

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AND-OR trees

Any function can be written as depth 2 AND-OR tree or an OR-AND tree But exponential size Any circuit can be rewritten as an AND-OR tree (each leaf has a literal, possibly shared with other leaves) If polynomial size and constant depth (AC0), stays so

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Switching

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Switching

In an AND-OR tree, if bottom two levels can be replaced by equivalent two levels with switched AND-OR order, and polynomial size

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Switching

In an AND-OR tree, if bottom two levels can be replaced by equivalent two levels with switched AND-OR order, and polynomial size

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Switching

In an AND-OR tree, if bottom two levels can be replaced by equivalent two levels with switched AND-OR order, and polynomial size

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Switching

In an AND-OR tree, if bottom two levels can be replaced by equivalent two levels with switched AND-OR order, and polynomial size

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Switching

In an AND-OR tree, if bottom two levels can be replaced by equivalent two levels with switched AND-OR order, and polynomial size A depth d AC0 circuit changes into depth d-1

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Switching

In an AND-OR tree, if bottom two levels can be replaced by equivalent two levels with switched AND-OR order, and polynomial size A depth d AC0 circuit changes into depth d-1 When is switching possible?

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Switching

In an AND-OR tree, if bottom two levels can be replaced by equivalent two levels with switched AND-OR order, and polynomial size A depth d AC0 circuit changes into depth d-1 When is switching possible?

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Switching

In an AND-OR tree, if bottom two levels can be replaced by equivalent two levels with switched AND-OR order, and polynomial size A depth d AC0 circuit changes into depth d-1 When is switching possible?

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Switching

In an AND-OR tree, if bottom two levels can be replaced by equivalent two levels with switched AND-OR order, and polynomial size A depth d AC0 circuit changes into depth d-1 When is switching possible? Distributivity

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Switching

In an AND-OR tree, if bottom two levels can be replaced by equivalent two levels with switched AND-OR order, and polynomial size A depth d AC0 circuit changes into depth d-1 When is switching possible? Distributivity a b

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Switching

In an AND-OR tree, if bottom two levels can be replaced by equivalent two levels with switched AND-OR order, and polynomial size A depth d AC0 circuit changes into depth d-1 When is switching possible? Distributivity a b ba a

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Switching

In an AND-OR tree, if bottom two levels can be replaced by equivalent two levels with switched AND-OR order, and polynomial size A depth d AC0 circuit changes into depth d-1 When is switching possible? Distributivity But may increase size to exponential a b ba a

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Switching Lemma

a b ba a a b

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Switching Lemma

A modified function has a switched circuit a b ba a a b

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Switching Lemma

A modified function has a switched circuit Size stays polynomial even after switching a b ba a a b

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Switching Lemma

A modified function has a switched circuit Size stays polynomial even after switching Computes same function as before but with most variables already set to specific values (a “restriction” of the original function) a b ba a a b

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Switching Lemma

A modified function has a switched circuit Size stays polynomial even after switching Computes same function as before but with most variables already set to specific values (a “restriction” of the original function) a b ba a c’ c a b

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Switching Lemma

A modified function has a switched circuit Size stays polynomial even after switching Computes same function as before but with most variables already set to specific values (a “restriction” of the original function) a b ba a c’ c a b c’’ c’

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Switching Lemma

A modified function has a switched circuit Size stays polynomial even after switching Computes same function as before but with most variables already set to specific values (a “restriction” of the original function) Still function of many variables a b ba a c’ c a b c’’ c’

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Switching Lemma

A modified function has a switched circuit Size stays polynomial even after switching Computes same function as before but with most variables already set to specific values (a “restriction” of the original function) Still function of many variables How to find such a modified function a b ba a c’ c a b c’’ c’

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Switching Lemma

A modified function has a switched circuit Size stays polynomial even after switching Computes same function as before but with most variables already set to specific values (a “restriction” of the original function) Still function of many variables How to find such a modified function Random restriction a b ba a c’ c a b c’’ c’

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Switching Lemma

A modified function has a switched circuit Size stays polynomial even after switching Computes same function as before but with most variables already set to specific values (a “restriction” of the original function) Still function of many variables How to find such a modified function Random restriction a b ba a c’ c

fix bits with

• prob. 1-n-2/3

a b c’’ c’

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Switching Lemma

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Switching Lemma

Random restriction. With positive probability:

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Switching Lemma

Random restriction. With positive probability: can switch bottom levels, staying poly sized

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Switching Lemma

Random restriction. With positive probability: can switch bottom levels, staying poly sized with high probability for each node above the leaf level (switching lemma); then union bound

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Switching Lemma

Random restriction. With positive probability: can switch bottom levels, staying poly sized with high probability for each node above the leaf level (switching lemma); then union bound computes PARITY for n2/3 variables (Chernoff)

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Switching Lemma

Random restriction. With positive probability: can switch bottom levels, staying poly sized with high probability for each node above the leaf level (switching lemma); then union bound computes PARITY for n2/3 variables (Chernoff) Depth d-1, poly-sized circuit family for PARITY

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Switching Lemma

Random restriction. With positive probability: can switch bottom levels, staying poly sized with high probability for each node above the leaf level (switching lemma); then union bound computes PARITY for n2/3 variables (Chernoff) Depth d-1, poly-sized circuit family for PARITY Contradiction: started with minimal depth!

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Razborov-Smolensky

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Razborov-Smolensky

An alternate proof that PARITY ∉ AC0

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Razborov-Smolensky

An alternate proof that PARITY ∉ AC0 Generalizes to ACC0(p) for odd primes p

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Razborov-Smolensky

An alternate proof that PARITY ∉ AC0 Generalizes to ACC0(p) for odd primes p Plan:

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Razborov-Smolensky

An alternate proof that PARITY ∉ AC0 Generalizes to ACC0(p) for odd primes p Plan: Given a circuit C, can find a polynomial s.t.

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Razborov-Smolensky

An alternate proof that PARITY ∉ AC0 Generalizes to ACC0(p) for odd primes p Plan: Given a circuit C, can find a polynomial s.t. Polynomial has “low degree”

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Razborov-Smolensky

An alternate proof that PARITY ∉ AC0 Generalizes to ACC0(p) for odd primes p Plan: Given a circuit C, can find a polynomial s.t. Polynomial has “low degree” Polynomial agrees with C on most inputs

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Razborov-Smolensky

An alternate proof that PARITY ∉ AC0 Generalizes to ACC0(p) for odd primes p Plan: Given a circuit C, can find a polynomial s.t. Polynomial has “low degree” Polynomial agrees with C on most inputs Show that no low degree polynomial can agree with PARITY on that many inputs

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Polynomial from Circuit

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Polynomial from Circuit

Assume circuit has OR, NOT gates

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Polynomial from Circuit

Assume circuit has OR, NOT gates Replace gates by polynomials (over some field), and compose together into one big polynomial

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Polynomial from Circuit

Assume circuit has OR, NOT gates Replace gates by polynomials (over some field), and compose together into one big polynomial If we do this faithfully, degree will be large

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Polynomial from Circuit

Assume circuit has OR, NOT gates Replace gates by polynomials (over some field), and compose together into one big polynomial If we do this faithfully, degree will be large Large enough to evaluate PARITY

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Polynomial from Circuit

Assume circuit has OR, NOT gates Replace gates by polynomials (over some field), and compose together into one big polynomial If we do this faithfully, degree will be large Large enough to evaluate PARITY So allow polynomials which err on some inputs

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Polynomial from Circuit

Assume circuit has OR, NOT gates Replace gates by polynomials (over some field), and compose together into one big polynomial If we do this faithfully, degree will be large Large enough to evaluate PARITY So allow polynomials which err on some inputs At each gate will pick polynomial from a distribution

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Polynomial from Circuit

Assume circuit has OR, NOT gates Replace gates by polynomials (over some field), and compose together into one big polynomial If we do this faithfully, degree will be large Large enough to evaluate PARITY So allow polynomials which err on some inputs At each gate will pick polynomial from a distribution Composed polynomial will be good with prob. > 0

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Polynomials for OR, NOT and PARITY

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Polynomials for OR, NOT and PARITY

Want that PARITY is complex (high degree) while OR, NOT are simple (low degree)

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Polynomials for OR, NOT and PARITY

Want that PARITY is complex (high degree) while OR, NOT are simple (low degree) If over GF(2), PARITY is just sum (degree 1)!

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Polynomials for OR, NOT and PARITY

Want that PARITY is complex (high degree) while OR, NOT are simple (low degree) If over GF(2), PARITY is just sum (degree 1)! We will work over GF(q), q>2

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Polynomials for OR, NOT and PARITY

Want that PARITY is complex (high degree) while OR, NOT are simple (low degree) If over GF(2), PARITY is just sum (degree 1)! We will work over GF(q), q>2 PARITY = [1-(1-2x1)(1-2x2)....(1-2xn)]/2 (if 2≠0)

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Polynomials for OR, NOT and PARITY

Want that PARITY is complex (high degree) while OR, NOT are simple (low degree) If over GF(2), PARITY is just sum (degree 1)! We will work over GF(q), q>2 PARITY = [1-(1-2x1)(1-2x2)....(1-2xn)]/2 (if 2≠0) NOT = 1-x.

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Polynomials for OR, NOT and PARITY

Want that PARITY is complex (high degree) while OR, NOT are simple (low degree) If over GF(2), PARITY is just sum (degree 1)! We will work over GF(q), q>2 PARITY = [1-(1-2x1)(1-2x2)....(1-2xn)]/2 (if 2≠0) NOT = 1-x. OR = 1- (1-x1)...(1-xn)

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Polynomials for OR, NOT and PARITY

Want that PARITY is complex (high degree) while OR, NOT are simple (low degree) If over GF(2), PARITY is just sum (degree 1)! We will work over GF(q), q>2 PARITY = [1-(1-2x1)(1-2x2)....(1-2xn)]/2 (if 2≠0) NOT = 1-x. OR = 1- (1-x1)...(1-xn) But high degree! Need OR to be simple!

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Approximate Polynomials

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Approximate Polynomials

Consider (random) polynomial p(x1,...,xn) = (a1x1 +...+ anxn)q-1 where ai are picked at random from the field

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Approximate Polynomials

Consider (random) polynomial p(x1,...,xn) = (a1x1 +...+ anxn)q-1 where ai are picked at random from the field If OR(x1,...,xn)=0 then p(x1,...,xn)=0

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Approximate Polynomials

Consider (random) polynomial p(x1,...,xn) = (a1x1 +...+ anxn)q-1 where ai are picked at random from the field If OR(x1,...,xn)=0 then p(x1,...,xn)=0 If OR(x1,...,xn)=1

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Approximate Polynomials

Consider (random) polynomial p(x1,...,xn) = (a1x1 +...+ anxn)q-1 where ai are picked at random from the field If OR(x1,...,xn)=0 then p(x1,...,xn)=0 If OR(x1,...,xn)=1 Pra[ a1x1 +...+ anxn = 0 ] ≤ 1/q (why?)

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Approximate Polynomials

Consider (random) polynomial p(x1,...,xn) = (a1x1 +...+ anxn)q-1 where ai are picked at random from the field If OR(x1,...,xn)=0 then p(x1,...,xn)=0 If OR(x1,...,xn)=1 Pra[ a1x1 +...+ anxn = 0 ] ≤ 1/q (why?) Recall in GF(q), uq-1 = 1 unless u=0 (since non-0 elements form a group of order q-1 under multiplication)

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Approximate Polynomials

Consider (random) polynomial p(x1,...,xn) = (a1x1 +...+ anxn)q-1 where ai are picked at random from the field If OR(x1,...,xn)=0 then p(x1,...,xn)=0 If OR(x1,...,xn)=1 Pra[ a1x1 +...+ anxn = 0 ] ≤ 1/q (why?) Recall in GF(q), uq-1 = 1 unless u=0 (since non-0 elements form a group of order q-1 under multiplication) i.e. Pra[ (a1x1 +...+ anxn)q-1 = 1 ] ≥ 1-1/q

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Approximate Polynomials

Consider (random) polynomial p(x1,...,xn) = (a1x1 +...+ anxn)q-1 where ai are picked at random from the field If OR(x1,...,xn)=0 then p(x1,...,xn)=0 If OR(x1,...,xn)=1 Pra[ a1x1 +...+ anxn = 0 ] ≤ 1/q (why?) Recall in GF(q), uq-1 = 1 unless u=0 (since non-0 elements form a group of order q-1 under multiplication) i.e. Pra[ (a1x1 +...+ anxn)q-1 = 1 ] ≥ 1-1/q Can boost probability by doing (exact) OR t times: deg < qt

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Approximate Polynomials

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Approximate Polynomials

OR: a random polynomial of degree O(log 1/ε), such that it is correct with prob. > 1-ε

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Approximate Polynomials

OR: a random polynomial of degree O(log 1/ε), such that it is correct with prob. > 1-ε Composing gate-polynomials into circuit-polynomial

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Approximate Polynomials

OR: a random polynomial of degree O(log 1/ε), such that it is correct with prob. > 1-ε Composing gate-polynomials into circuit-polynomial Substitute child polynomials as variables

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Approximate Polynomials

OR: a random polynomial of degree O(log 1/ε), such that it is correct with prob. > 1-ε Composing gate-polynomials into circuit-polynomial Substitute child polynomials as variables Degree multiplies: depth d circuit gives deg O(log 1/ε)d

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Approximate Polynomials

OR: a random polynomial of degree O(log 1/ε), such that it is correct with prob. > 1-ε Composing gate-polynomials into circuit-polynomial Substitute child polynomials as variables Degree multiplies: depth d circuit gives deg O(log 1/ε)d Error adds (by union bound): size s circuit gives error < sε

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Approximate Polynomials

OR: a random polynomial of degree O(log 1/ε), such that it is correct with prob. > 1-ε Composing gate-polynomials into circuit-polynomial Substitute child polynomials as variables Degree multiplies: depth d circuit gives deg O(log 1/ε)d Error adds (by union bound): size s circuit gives error < sε Using ε=1/(4s), degree O(log s)d polynomial, correct w.p. > 3/4

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Approximate Polynomials

OR: a random polynomial of degree O(log 1/ε), such that it is correct with prob. > 1-ε Composing gate-polynomials into circuit-polynomial Substitute child polynomials as variables Degree multiplies: depth d circuit gives deg O(log 1/ε)d Error adds (by union bound): size s circuit gives error < sε Using ε=1/(4s), degree O(log s)d polynomial, correct w.p. > 3/4 One polynomial, correct on > 3/4 fraction of inputs (why?)

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How about PARITY?

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How about PARITY?

Can PARITY also be approximated (i.e., calculated for some large input set S) by a low-degree polynomial?

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How about PARITY?

Can PARITY also be approximated (i.e., calculated for some large input set S) by a low-degree polynomial? PARITY is essentially Πi=1 to n xi, for inputs from {+1,-1}n

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How about PARITY?

Can PARITY also be approximated (i.e., calculated for some large input set S) by a low-degree polynomial? PARITY is essentially Πi=1 to n xi, for inputs from {+1,-1}n If can calculate Πi=1 to n xi (for S ⊆ {+1,-1}n) using degree D, then can calculate (for S) any polynomial using degree D+n/2 polynomial (why?)

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How about PARITY?

Can PARITY also be approximated (i.e., calculated for some large input set S) by a low-degree polynomial? PARITY is essentially Πi=1 to n xi, for inputs from {+1,-1}n If can calculate Πi=1 to n xi (for S ⊆ {+1,-1}n) using degree D, then can calculate (for S) any polynomial using degree D+n/2 polynomial (why?) But if S large, too many polynomials, distinct for S

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How about PARITY?

Can PARITY also be approximated (i.e., calculated for some large input set S) by a low-degree polynomial? PARITY is essentially Πi=1 to n xi, for inputs from {+1,-1}n If can calculate Πi=1 to n xi (for S ⊆ {+1,-1}n) using degree D, then can calculate (for S) any polynomial using degree D+n/2 polynomial (why?) But if S large, too many polynomials, distinct for S Need D = Ω(√n) to have enough degree D+n/2 polys.

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PARITY ∉ ACC(q)0

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PARITY ∉ ACC(q)0

Given depth d, size s circuit C, there is a polynomial of degree O(log(s))d which agrees with C on 3/4 of inputs

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PARITY ∉ ACC(q)0

Given depth d, size s circuit C, there is a polynomial of degree O(log(s))d which agrees with C on 3/4 of inputs Using approximate OR polynomials

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PARITY ∉ ACC(q)0

Given depth d, size s circuit C, there is a polynomial of degree O(log(s))d which agrees with C on 3/4 of inputs Using approximate OR polynomials Even if circuit has Modq (boolean) gates: (x1+...+xn)q-1

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PARITY ∉ ACC(q)0

Given depth d, size s circuit C, there is a polynomial of degree O(log(s))d which agrees with C on 3/4 of inputs Using approximate OR polynomials Even if circuit has Modq (boolean) gates: (x1+...+xn)q-1 PARITY needs degree Ω(√n) polynomial for approximation

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PARITY ∉ ACC(q)0

Given depth d, size s circuit C, there is a polynomial of degree O(log(s))d which agrees with C on 3/4 of inputs Using approximate OR polynomials Even if circuit has Modq (boolean) gates: (x1+...+xn)q-1 PARITY needs degree Ω(√n) polynomial for approximation log(s) = Ω(√n)1/d or s = 2Ω(n)^(1/2d) : i.e., if depth is constant then size not poly (in fact exponential)

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Monotone Circuits

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Monotone Circuits

Another restricted class for which strong lower-bounds are known

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Monotone Circuits

Another restricted class for which strong lower-bounds are known Monotone circuits: no NOT gate (and no neg. literal)

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Monotone Circuits

Another restricted class for which strong lower-bounds are known Monotone circuits: no NOT gate (and no neg. literal) For monotonic functions f

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Monotone Circuits

Another restricted class for which strong lower-bounds are known Monotone circuits: no NOT gate (and no neg. literal) For monotonic functions f To show that f has no poly-sized monotone circuit family

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Monotone Circuits

Another restricted class for which strong lower-bounds are known Monotone circuits: no NOT gate (and no neg. literal) For monotonic functions f To show that f has no poly-sized monotone circuit family Still possible that f may have a more efficient non- monotone circuit family (or even be in P)

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CLIQUE

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CLIQUE

CLIQUEn,k does not have poly-sized monotone circuits

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CLIQUE

CLIQUEn,k does not have poly-sized monotone circuits A way to turn a circuit into an approximately correct circuit, gate by gate (AND/OR gate → “approximation gate”)

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CLIQUE

CLIQUEn,k does not have poly-sized monotone circuits A way to turn a circuit into an approximately correct circuit, gate by gate (AND/OR gate → “approximation gate”) Will consider effect of this change on some Yes examples and some No examples

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CLIQUE

CLIQUEn,k does not have poly-sized monotone circuits A way to turn a circuit into an approximately correct circuit, gate by gate (AND/OR gate → “approximation gate”) Will consider effect of this change on some Yes examples and some No examples Converting each gate to approximation makes only a few extra examples go wrong

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CLIQUE

CLIQUEn,k does not have poly-sized monotone circuits A way to turn a circuit into an approximately correct circuit, gate by gate (AND/OR gate → “approximation gate”) Will consider effect of this change on some Yes examples and some No examples Converting each gate to approximation makes only a few extra examples go wrong A circuit with only approximation gates errs on a large number of the examples

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CLIQUE

CLIQUEn,k does not have poly-sized monotone circuits A way to turn a circuit into an approximately correct circuit, gate by gate (AND/OR gate → “approximation gate”) Will consider effect of this change on some Yes examples and some No examples Converting each gate to approximation makes only a few extra examples go wrong A circuit with only approximation gates errs on a large number of the examples Original circuit must have been large

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Proof Sketch

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Input sets

Proof Sketch

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Input sets Yes set: graphs with no edges except a single k-clique. No set: complete (k-1)-partite graphs

Proof Sketch

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Input sets Yes set: graphs with no edges except a single k-clique. No set: complete (k-1)-partite graphs Since monotone circuit, we can label each gate with a set of subgraphs which will make the gate’ s output 1

Proof Sketch

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Input sets Yes set: graphs with no edges except a single k-clique. No set: complete (k-1)-partite graphs Since monotone circuit, we can label each gate with a set of subgraphs which will make the gate’ s output 1 Input gates: edges

Proof Sketch

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Input sets Yes set: graphs with no edges except a single k-clique. No set: complete (k-1)-partite graphs Since monotone circuit, we can label each gate with a set of subgraphs which will make the gate’ s output 1 Input gates: edges OR: take union of the two sets of input-subsets

Proof Sketch

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Input sets Yes set: graphs with no edges except a single k-clique. No set: complete (k-1)-partite graphs Since monotone circuit, we can label each gate with a set of subgraphs which will make the gate’ s output 1 Input gates: edges OR: take union of the two sets of input-subsets AND: take set of pair-wise unions of input-subsets

Proof Sketch

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Proof Sketch

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Approximation gate: output wire labeled by a sample of M cliques of at most t vertices. Value 1 if at least one of those M cliques is present in the input

Proof Sketch

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Approximation gate: output wire labeled by a sample of M cliques of at most t vertices. Value 1 if at least one of those M cliques is present in the input Input gates: edges

Proof Sketch

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Approximation gate: output wire labeled by a sample of M cliques of at most t vertices. Value 1 if at least one of those M cliques is present in the input Input gates: edges OR: take union of the two sets of subsets, and “prune” to M subsets

Proof Sketch

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Approximation gate: output wire labeled by a sample of M cliques of at most t vertices. Value 1 if at least one of those M cliques is present in the input Input gates: edges OR: take union of the two sets of subsets, and “prune” to M subsets AND: take set of pair-wise unions of subsets which are at most t vertices, and “prune” to M subsets

Proof Sketch

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Approximation gate: output wire labeled by a sample of M cliques of at most t vertices. Value 1 if at least one of those M cliques is present in the input Input gates: edges OR: take union of the two sets of subsets, and “prune” to M subsets AND: take set of pair-wise unions of subsets which are at most t vertices, and “prune” to M subsets Pruning uses “sunflower lemma”: find a sunflower and replace petals by core

Proof Sketch

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Proof Sketch

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Proof Sketch

Converting each gate to approximation makes only a few more examples go wrong

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Proof Sketch

Converting each gate to approximation makes only a few more examples go wrong Bounding new false positives among No sets and false negatives among Yes sets introduced by pruning

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Proof Sketch

Converting each gate to approximation makes only a few more examples go wrong Bounding new false positives among No sets and false negatives among Yes sets introduced by pruning A circuit with only approximation gates errs on a large number of the examples

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Proof Sketch

Converting each gate to approximation makes only a few more examples go wrong Bounding new false positives among No sets and false negatives among Yes sets introduced by pruning A circuit with only approximation gates errs on a large number of the examples If output identically “No” then errs on entire Yes set. Else, output wire’ s label has some subset X, |X| ≤ t = O(√k), and then a constant fraction of No-examples get accepted

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