csr2011 june15 11_30_sokolov

The complexity of inversion of explicit Goldreich’sfunction by DPLL algorithms

Dmitry Itsykson, Dmitry Sokolov

Steklov Institute of Mathematics at St. Petersburg,Academic University

CSR 2011, Saint-PetersburgJune 15

Sokolov D. 1/9 | Inversion of Goldreich’s function

Goldreich’s function

f : f0; 1gn ! f0; 1gn

G (X ;Y ;E ) is a bipartitegraph;8y 2 Y deg(y) = dd is a constant.

Goldreich’s conjecture:P is a random predicate;G is an expander;

then function f is a one-way.

f is computed by constant depth circuit;[Applebaum, Ishai, Kushilevitz 2006] If one-way functions existthen there is a one-way function that can be computed byconstant depth circuit.

f : f0; 1gn ! f0; 1gn

P(xi1; xi2; : : : ; xid )

f : f0; 1gn ! f0; 1gn

P(xi1; xi2; : : : ; xid )G (X ;Y ;E ) is a bipartitegraph;

8y 2 Y deg(y) = dd is a constant.

f : f0; 1gn ! f0; 1gn

P(xi1; xi2; : : : ; xid )G (X ;Y ;E ) is a bipartitegraph;8y 2 Y deg(y) = d

d is a constant.

f : f0; 1gn ! f0; 1gn

P(xi1; xi2; : : : ; xid )G (X ;Y ;E ) is a bipartitegraph;8y 2 Y deg(y) = dd is a constant.

f : f0; 1gn ! f0; 1gn

f is computed by constant depth circuit;

[Applebaum, Ishai, Kushilevitz 2006] If one-way functions existthen there is a one-way function that can be computed byconstant depth circuit.

f : f0; 1gn ! f0; 1gn

DPLL algorithms

xj := c1

xj := 1� c1

xi := c

xk := c2

xk := 1� c2

xi := 1� c

Heuristic A chooses a variable for splitting.Heuristic B chooses first value.Simplification rules:

unit clause elimination;pure literal rule.

DPLL algorithms

xj := c1

xj := 1� c1

xi := c

xk := c2

xk := 1� c2

xi := 1� c

DPLL algorithms

xj := c1

xj := 1� c1

xi := c

xk := c2

xk := 1� c2

xi := 1� c

DPLL algorithms

xj := c1

xj := 1� c1

xi := c

xk := c2

xk := 1� c2

xi := 1� c

DPLL algorithms

xj := c1

xj := 1� c1

xi := c

xk := c2

xk := 1� c2

xi := 1� c

DPLL algorithms

xj := c1

xj := 1� c1

xi := c

xk := c2

xk := 1� c2

xi := 1� c

Heuristic A chooses a variable for splitting.

Heuristic B chooses first value.Simplification rules:

DPLL algorithms

xj := c1

xj := 1� c1

xi := c

xk := c2

xk := 1� c2

xi := 1� c

Heuristic A chooses a variable for splitting.Heuristic B chooses first value.

Simplification rules:unit clause elimination;pure literal rule.

DPLL algorithms

xj := c1

xj := 1� c1

xi := c

xk := c2

xk := 1� c2

xi := 1� c

Lower bounds for DPLL algorithms

Unsatisfiable formulasExponential lower bounds for resolution refutations ofunsatisfiable formulas translate to backtracking algorithms.[Tseitin, 1968] ... [Pudlak, Implagliazzo, 2000].

Satisfiable formulasIf P = NP then there are no superpolynomial lower bounds forbacktracking algorithms since heuristic B may choose correctvalue.Inverting of functions corresponds to satisfiable formulas.[Nikolenko 2002], [Achilioptas, Beame, Molloy 2003-2004]exponential lower bound for specific backtracking algorithms.[Alekhnovich, Hirsch, Itsykson 2005] Exponential lower boundfor myopic and drunken algorithms.Exponential lower bound for inversion of Goldreich’s functionby myopic [J. Cook et al. 2009] and drunken [Itsykson 2010]algorithms.

Satisfiable formulasIf P = NP then there are no superpolynomial lower bounds forbacktracking algorithms since heuristic B may choose correctvalue.

Inverting of functions corresponds to satisfiable formulas.[Nikolenko 2002], [Achilioptas, Beame, Molloy 2003-2004]exponential lower bound for specific backtracking algorithms.[Alekhnovich, Hirsch, Itsykson 2005] Exponential lower boundfor myopic and drunken algorithms.Exponential lower bound for inversion of Goldreich’s functionby myopic [J. Cook et al. 2009] and drunken [Itsykson 2010]algorithms.

Satisfiable formulasIf P = NP then there are no superpolynomial lower bounds forbacktracking algorithms since heuristic B may choose correctvalue.Inverting of functions corresponds to satisfiable formulas.

[Nikolenko 2002], [Achilioptas, Beame, Molloy 2003-2004]exponential lower bound for specific backtracking algorithms.[Alekhnovich, Hirsch, Itsykson 2005] Exponential lower boundfor myopic and drunken algorithms.Exponential lower bound for inversion of Goldreich’s functionby myopic [J. Cook et al. 2009] and drunken [Itsykson 2010]algorithms.

Satisfiable formulasIf P = NP then there are no superpolynomial lower bounds forbacktracking algorithms since heuristic B may choose correctvalue.Inverting of functions corresponds to satisfiable formulas.[Nikolenko 2002], [Achilioptas, Beame, Molloy 2003-2004]exponential lower bound for specific backtracking algorithms.[Alekhnovich, Hirsch, Itsykson 2005] Exponential lower boundfor myopic and drunken algorithms.

Exponential lower bound for inversion of Goldreich’s functionby myopic [J. Cook et al. 2009] and drunken [Itsykson 2010]algorithms.

Drunken and myopic algorithms

DefinitionDrunken algorithms:

A: any;B: random 50 : 50.

DefinitionMyopic heuristic:

sees structure of the formula;doesn’t see negation signs;

requests negations in K = n1�� clause.

(x1 _ x3 _ x5)

(x2 _ x3)

(x2 _ x4 _ x5)

(x1 _ x4 _ x6)

(x1 _ x3 _ x5)

(x2 _ :x3)

(x2 _ x4 _ x5)

(x1 _ :x4 _ x6)

(x1 _ x3 _ x5)

(x2 _ x3)

(x2 _ x4 _ x5)

(x1 _ x4 _ x6)

(x1 _ x3 _ x5)

(x2 _ :x3)

(x2 _ x4 _ x5)

(x1 _ :x4 _ x6)

(x1 _ x3 _ x5)

(x2 _ x3)

(x2 _ x4 _ x5)

(x1 _ x4 _ x6)

(x1 _ x3 _ x5)

(x2 _ :x3)

(x2 _ x4 _ x5)

(x1 _ :x4 _ x6)

sees structure of the formula;

doesn’t see negation signs;

(x1 _ x3 _ x5)

(x2 _ x3)

(x2 _ x4 _ x5)

(x1 _ x4 _ x6)

(x1 _ x3 _ x5)

(x2 _ :x3)

(x2 _ x4 _ x5)

(x1 _ :x4 _ x6)

(x1 _ x3 _ x5)

(x2 _ x3)

(x2 _ x4 _ x5)

(x1 _ x4 _ x6)

(x1 _ x3 _ x5)

(x2 _ :x3)

(x2 _ x4 _ x5)

(x1 _ :x4 _ x6)

sees structure of the formula;doesn’t see negation signs;requests negations in K = n1�� clause.

(x1 _ x3 _ x5)

(x2 _ x3)

(x2 _ x4 _ x5)

(x1 _ x4 _ x6)

(x1 _ x3 _ x5)

(x2 _ :x3)

(x2 _ x4 _ x5)

(x1 _ :x4 _ x6)

(x1 _ x3 _ x5)

(x2 _ x3)

(x2 _ x4 _ x5)

(x1 _ x4 _ x6)

(x1 _ x3 _ x5)

(x2 _ :x3)

(x2 _ x4 _ x5)

(x1 _ :x4 _ x6)

(x1 _ x3 _ x5)

(x2 _ x3)

(x2 _ x4 _ x5)

(x1 _ x4 _ x6)

(x1 _ x3 _ x5)

(x2 _ :x3)

(x2 _ x4 _ x5)

(x1 _ :x4 _ x6)

(x1 _ x3 _ x5)

(x2 _ x3)

(x2 _ x4 _ x5)

(x1 _ x4 _ x6)

(x1 _ x3 _ x5)

(x2 _ :x3)

(x2 _ x4 _ x5)

(x1 _ :x4 _ x6)

Myopic algorithms

DefinitionMyopic algorithm:

A;B are myopic heuristics.

[Alekhnovich, Hirsch, Itsykson 2005]P is a linear predicate.G is a random graph.

[J. Cook, Etesami, Miller, Trevisan 2009]P = x1 + x2 + � � � + xd�2 + xd�1xd .In fact: P = x1 + x2 + � � � + xd�k + Q(xd�k+1; : : : ; xd).

Disadvantages:

G is a random graph.

K is a constant.

Too complicatedproof.

In our work:

G is based onexpander.

K = n1��.

“Simple” proof.

Myopic algorithms

Disadvantages:

K is a constant.

In our work:

K = n1��.

“Simple” proof.

Myopic algorithms

Disadvantages:

K is a constant.

In our work:

K = n1��.

“Simple” proof.

Myopic algorithms

[J. Cook, Etesami, Miller, Trevisan 2009]

P = x1 + x2 + � � � + xd�2 + xd�1xd .In fact: P = x1 + x2 + � � � + xd�k + Q(xd�k+1; : : : ; xd).

Disadvantages:

K is a constant.

In our work:

K = n1��.

“Simple” proof.

Myopic algorithms

Disadvantages:

K is a constant.

In our work:

K = n1��.

“Simple” proof.

Myopic algorithms

Disadvantages:

K is a constant.

In our work:

K = n1��.

“Simple” proof.

Myopic algorithms

Disadvantages:

K is a constant.

In our work:

K = n1��.

“Simple” proof.

Myopic algorithms

Disadvantages:

K is a constant.

In our work:

K = n1��.

“Simple” proof.

Myopic algorithms

Disadvantages:

K is a constant.

In our work:

K = n1��.

“Simple” proof.

Myopic algorithms

Disadvantages:

K is a constant.

In our work:

K = n1��.

“Simple” proof.

Myopic algorithms

Disadvantages:

K is a constant.

In our work:

K = n1��.

“Simple” proof.

Myopic algorithms

Disadvantages:

K is a constant.

In our work:

K = n1��.

“Simple” proof.

Our results

P(x1; : : : ; xd ) = x1 � x2 � : : :� xd�k �Q(xd�k+1; : : : ; xd ), Q is anarbitrary, k < d=4.

TheoremThere exists an explicit graph G such that every myopic or drunkenDPLL algorithm makes at least 2n(1) steps on “f (x) = f (a)” foralmost all a 2 f0; 1gn.

G is based on expander instead of random graph.For drunken algorithms the proof follows [Itsykson, CSR-2010]For myopic

we simplify previous proof andK = n1��

Our results

G is based on expander instead of random graph.

For drunken algorithms the proof follows [Itsykson, CSR-2010]For myopic

Our results

G is based on expander instead of random graph.For drunken algorithms the proof follows [Itsykson, CSR-2010]

For myopicwe simplify previous proof andK = n1��

Our results

Graph construction

P(x1; : : : ; xd ) = x1 � x2 � : : :� xd�k � Q(xd�k+1; : : : ; xd )

G is an expander;G + T has full rank. 8y 2 Y � T ; deg(y) = 1;

G + T is an expander.R contains nonlinear edges. jfx j x 2 X ; deg(x) 6= 0gj � n�.

G + T + R is an expander.Size of preimages no more than 2n� .

We can invert fG+T+R;P in time poly(n)2n� , but this is still much!

Graph construction

G is an expander;

G + T has full rank. 8y 2 Y � T ; deg(y) = 1;G + T is an expander.

R contains nonlinear edges. jfx j x 2 X ; deg(x) 6= 0gj � n�.G + T + R is an expander.Size of preimages no more than 2n� .

Graph construction

G is an expander;

G + T has full rank. 8y 2 Y � T ; deg(y) = 1;G + T is an expander.

Graph construction

G + T is an expander.

Graph construction

G + T + R is an expander.

Size of preimages no more than 2n� .

Graph construction

Plan of the proof

Lower bounds for unsatisfiable formulas.G is an expander.P is almost linear.Lower bounds for resolution proofs

With probability 1� 2�(n) after several steps current formulabecomes unsatisfiable.

G is an expander.f is almost bijection.Myopic algorithm can’t recognize different absolute terms.

Plan of the proof

Lower bounds for unsatisfiable formulas.

G is an expander.P is almost linear.Lower bounds for resolution proofs

Plan of the proof

csr2011 june15 11_30_sokolov

pure literal

x4 x6

unit clause

exponential

drunkendpll

x2 x3

constant depth

specific backtracking

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