Introduction Previous Results Our Model Read-Once Polynomials Our Results Complete Derandomization of Identity Testing of Read-Once Formulas Daniel Minahan Ilya Volkovich University of Michigan Presented by: Ramprasad Saptharishi
Introduction Previous Results Our Model Read-Once Polynomials Our Results Outline Introduction 1
Introduction Previous Results Our Model Read-Once Polynomials Our Results Outline Introduction 1 Previous Results 2
Introduction Previous Results Our Model Read-Once Polynomials Our Results Outline Introduction 1 Previous Results 2 Our Model 3
Introduction Previous Results Our Model Read-Once Polynomials Our Results Outline Introduction 1 Previous Results 2 Our Model 3 Read-Once Polynomials 4
Introduction Previous Results Our Model Read-Once Polynomials Our Results Outline Introduction 1 Previous Results 2 Our Model 3 Read-Once Polynomials 4 Our Results 5
Introduction Previous Results Our Model Read-Once Polynomials Our Results Problem Definition Definition (Polynomial Identity Testing (PIT)) Given a succinct representation of a polynomial P in n variables over a field F , decide efficiently if P is identically 0.
Introduction Previous Results Our Model Read-Once Polynomials Our Results Problem Definition Definition (Polynomial Identity Testing (PIT)) Given a succinct representation of a polynomial P in n variables over a field F , decide efficiently if P is identically 0. This requires a few things to be specified.
Introduction Previous Results Our Model Read-Once Polynomials Our Results Problem Definition Definition (Polynomial Identity Testing (PIT)) Given a succinct representation of a polynomial P in n variables over a field F , decide efficiently if P is identically 0. This requires a few things to be specified. Method of representation
Introduction Previous Results Our Model Read-Once Polynomials Our Results Problem Definition Definition (Polynomial Identity Testing (PIT)) Given a succinct representation of a polynomial P in n variables over a field F , decide efficiently if P is identically 0. This requires a few things to be specified. Method of representation Efficiency
Introduction Previous Results Our Model Read-Once Polynomials Our Results Problem Definition Definition (Polynomial Identity Testing (PIT)) Given a succinct representation of a polynomial P in n variables over a field F , decide efficiently if P is identically 0. This requires a few things to be specified. Method of representation Efficiency Allowed actions
Introduction Previous Results Our Model Read-Once Polynomials Our Results Succinct Representation
Introduction Previous Results Our Model Read-Once Polynomials Our Results Succinct Representation Definition An arithmetic formula is a directed tree from variables ( x 1 , . . . , x n ) to an output. Each leaf is assigned a variable or field element and each internal node, or gate, is assigned an operation + or × . Idea: Model small computational devices Figure: Example of an Arithmetic Formula for ( x 1 + 1)(2 x 2 ) + x 2 + x 3
Introduction Previous Results Our Model Read-Once Polynomials Our Results Black-Box Setting We cannot see the formula.
Introduction Previous Results Our Model Read-Once Polynomials Our Results Black-Box Setting We cannot see the formula. We can only send inputs to the formula and receive an output.
Introduction Previous Results Our Model Read-Once Polynomials Our Results Black-Box Setting We cannot see the formula. We can only send inputs to the formula and receive an output.
Introduction Previous Results Our Model Read-Once Polynomials Our Results Black-Box Setting We cannot see the formula. We can only send inputs to the formula and receive an output.
Introduction Previous Results Our Model Read-Once Polynomials Our Results Black-Box Setting We cannot see the formula. We can only send inputs to the formula and receive an output. We are allowed to query the formula on a small extension field.
Introduction Previous Results Our Model Read-Once Polynomials Our Results Black-Box Setting We cannot see the formula. We can only send inputs to the formula and receive an output. We are allowed to query the formula on a small extension field. Definition (Hitting Set) Let C ⊆ F [ x 1 , . . . , x n ]. A set H ⊆ F n is called a hitting set for C provided that for any P ∈ C with P �≡ 0, ∃ ¯ a ∈ H with P (¯ a ) � = 0.
Introduction Previous Results Our Model Read-Once Polynomials Our Results Black-Box Setting We cannot see the formula. We can only send inputs to the formula and receive an output. We are allowed to query the formula on a small extension field. Definition (Hitting Set) Let C ⊆ F [ x 1 , . . . , x n ]. A set H ⊆ F n is called a hitting set for C provided that for any P ∈ C with P �≡ 0, ∃ ¯ a ∈ H with P (¯ a ) � = 0. H gives an algorithm that runs in time O ( |H| ).
Introduction Previous Results Our Model Read-Once Polynomials Our Results Black-Box Setting We cannot see the formula. We can only send inputs to the formula and receive an output. We are allowed to query the formula on a small extension field. Definition (Hitting Set) Let C ⊆ F [ x 1 , . . . , x n ]. A set H ⊆ F n is called a hitting set for C provided that for any P ∈ C with P �≡ 0, ∃ ¯ a ∈ H with P (¯ a ) � = 0. H gives an algorithm that runs in time O ( |H| ). Goal: find a small hitting set for polynomials computed by small formulas.
Introduction Previous Results Our Model Read-Once Polynomials Our Results Applications Various application in Complexity Theory and Algorithm Design:
Introduction Previous Results Our Model Read-Once Polynomials Our Results Applications Various application in Complexity Theory and Algorithm Design: Fibonacci Identity: ( a 2 + b 2 )( c 2 + d 2 ) = ( ac − bd ) 2 + ( ad + bc ) 2
Introduction Previous Results Our Model Read-Once Polynomials Our Results Applications Various application in Complexity Theory and Algorithm Design: Fibonacci Identity: ( a 2 + b 2 )( c 2 + d 2 ) = ( ac − bd ) 2 + ( ad + bc ) 2 Graph Perfect Matching [MVV87]
Introduction Previous Results Our Model Read-Once Polynomials Our Results Applications Various application in Complexity Theory and Algorithm Design: Fibonacci Identity: ( a 2 + b 2 )( c 2 + d 2 ) = ( ac − bd ) 2 + ( ad + bc ) 2 Graph Perfect Matching [MVV87] Primality Testing [AKS04]
Introduction Previous Results Our Model Read-Once Polynomials Our Results Applications Various application in Complexity Theory and Algorithm Design: Fibonacci Identity: ( a 2 + b 2 )( c 2 + d 2 ) = ( ac − bd ) 2 + ( ad + bc ) 2 Graph Perfect Matching [MVV87] Primality Testing [AKS04] IP = PSPACE [Sha90]
Introduction Previous Results Our Model Read-Once Polynomials Our Results Applications Various application in Complexity Theory and Algorithm Design: Fibonacci Identity: ( a 2 + b 2 )( c 2 + d 2 ) = ( ac − bd ) 2 + ( ad + bc ) 2 Graph Perfect Matching [MVV87] Primality Testing [AKS04] IP = PSPACE [Sha90] PCP Theorem [AS98]
Introduction Previous Results Our Model Read-Once Polynomials Our Results Applications Various application in Complexity Theory and Algorithm Design: Fibonacci Identity: ( a 2 + b 2 )( c 2 + d 2 ) = ( ac − bd ) 2 + ( ad + bc ) 2 Graph Perfect Matching [MVV87] Primality Testing [AKS04] IP = PSPACE [Sha90] PCP Theorem [AS98] Geometric Complexity Theory [Mul12]
Introduction Previous Results Our Model Read-Once Polynomials Our Results Applications Various application in Complexity Theory and Algorithm Design: Fibonacci Identity: ( a 2 + b 2 )( c 2 + d 2 ) = ( ac − bd ) 2 + ( ad + bc ) 2 Graph Perfect Matching [MVV87] Primality Testing [AKS04] IP = PSPACE [Sha90] PCP Theorem [AS98] Geometric Complexity Theory [Mul12] ...
Introduction Previous Results Our Model Read-Once Polynomials Our Results Existing Algorithms Suppose we can generate a random element of a field F . Then we have a general PIT algorithm.
Introduction Previous Results Our Model Read-Once Polynomials Our Results Existing Algorithms Suppose we can generate a random element of a field F . Then we have a general PIT algorithm. Lemma (Schwartz-Zippel) Let P ∈ F [ x 1 , . . . , x n ] with total degree bounded by some d ∈ N . d Pick some finite S ⊆ F . Then Pr ¯ x ∈ S n [ P (¯ x ) = 0] ≤ | S | .
Introduction Previous Results Our Model Read-Once Polynomials Our Results Existing Algorithms Suppose we can generate a random element of a field F . Then we have a general PIT algorithm. Lemma (Schwartz-Zippel) Let P ∈ F [ x 1 , . . . , x n ] with total degree bounded by some d ∈ N . d Pick some finite S ⊆ F . Then Pr ¯ x ∈ S n [ P (¯ x ) = 0] ≤ | S | . This does not provide a deterministic algorithm but suggests that one might exist.
Introduction Previous Results Our Model Read-Once Polynomials Our Results Hardness Results [KI04]: Deterministic PIT = ⇒ super-polynomial circuit lower bounds: Boolean for NEXP or arithmetic for Permanent.
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