On the PPA -completeness of the Combinatorial Nullstellensatz and - PowerPoint PPT Presentation

On the PPA -completeness of the Combinatorial Nullstellensatz and the Chevalley-Warning Theorem Miklos Santha CNRS, Universit´ e Paris Diderot, France and Centre for Quantum Technologies, NUS, Singapore joint work with Alexander Belov G´ abor Ivanyos Youming Qiao Siyi Yang U. of Latvia, Riga SZTAKI, Budapest U. Tech., Sydney CQT, Singapore 1/44

Overview of the talk 1 The class PPA 2 CNSS and Chevalley-Warning Theorem 3 Arithmetic circuits and parse subcircuits 4 The problems PPA-Circuit Chevalley and PPA-Circuit CNSS 5 PPA -hardness and PPA -easiness 2/44

The class PPA 3/44

Functional NP ( FNP ) NP -search problems are defined by binary relations R ⊆ { 0 , 1 } ∗ × { 0 , 1 } ∗ such that – R ∈ P , – for some polynomial p ( n ) , R ( x , y ) = ⇒ | y | ≤ p ( | x | ). Search problem Π R Input: x Output: A solution y such that R ( x , y ) if there is any, or “failure” Π R is reducible to Π S if there exist polynomial time computable functions f and g such that, for every positive x , S ( f ( x ) , y ) = ⇒ R ( x , g ( x , y )) . 4/44

Total Functional NP ( TFNP ) [MP’91] An NP -search problem is total if for all x there exists a solution y . Facts: • If FNP ⊆ TFNP then NP = coNP . • If TFNP ⊆ P then P = NP ∩ coNP . TFNP is a semantic complexity class Syntactical subclasses of TFNP : • Polynomial Local Search PLS Examples: Local optima, pure equilibrium in potential games • Polynomial Pigeonhole Principle PPP Examples: Pigeonhole SubsetSum, Discrete Logarithm • Polynomial Parity Argument classes PPA , PPAD . 5/44

Polynomial Parity Argument [P’94] Parity Principle: In a graph the number of odd vertices is even. Definition: PPA is the set of total problems reducible to Leaf Leaf Input: ( z , M , ω ), where • z is a binary string • M is a polynomial TM that defines a graph G z = ( V z , E z ) • V z = { 0 , 1 } p ( | z | ) for some polynomial p • for v ∈ V z , M ( z , v ) is a set of at most two vertices • { v , v ′ } ∈ E z if v ′ ∈ M ( z , v ) and v ∈ M ( z , v ′ ) • ω ∈ V z is a degree one vertex, the standard leaf Output: A leaf different from ω . ω 6/44

PPA with edge recognition and pairing Graphs G z = ( V z , E z ) of unbounded degree can be defined by two polynomial time algorithms ǫ and φ : • Edge recognition: { v , v ′ } ∈ E z ⇔ ǫ ( v , v ′ ) = 1 • Pairing: For every vertex v , • if deg( v ) is even the function φ ( v , · ) is a pairing between the vertices adjacent to v . • if deg( v ) is odd then there exists exactly one neighbor w such that φ ( v , w ) = w , and on the remaining neighbors φ ( v , · ) is a pairing. Fact: A problem defined in terms of . . . ǫ and π is in PPA . Proof: Let G ′ z = ( V ′ z , E ′ z ) be defined as • V ′ z = E z • {{ v , w } , { v , w ′ }} ∈ E ′ z φ ( w ) = w ′ . if . . . 7/44

Examples of problems in PPA Few complete problems are known, all discretizations or combinatorial analogues of topological fixed point theorems: • 3 -D Sperner in some non-orientable space [G’01] • Locally 2 -D Sperner [FISV’06] • Sperner and Tucker on the M¨ obius band [DEFLQX16] • 2 -D Tucker in the Euclidean space [ABB’15] Many problems of various origins are in PPA : • Graph theory: Smith , Hamiltonian decomp. [P’94] • Combinatorics: Necklace splitting and Discrete ham sandwich [P’94] • Algebra: Explicit Chevalley [P’94] • Number theory: Square root and Factoring [J’16] 8/44

Combinatorial Nullstellensatz and Chevalley-Warning Theorem 9/44

Combinatorial Nullstellensatz Theorem [Alon’99]: Let F be a field, let d 1 , . . . , d n be non-negative integers, and let P ∈ F [ x 1 , . . . , x n ] be a polynomial. Suppose that • deg( P ) = � n i =1 d i , • the coefficient of x d 1 1 . . . x d n is non-zero. n Then for every subsets S 1 , . . . , S n of F with | S i | > d i , there exists ( s 1 , . . . s n ) ∈ S 1 × · · · × S n such that P ( s 1 , . . . , s n ) � = 0 . Consequences in algebra, graph theory, combinatorics, additive number theory ... 10/44

Chevalley-Warning Theorem Theorem [Chevalley’36, Warning’36]: Let F be a field of characteristic p , and let P 1 , . . . , P k ∈ F [ x 1 , . . . , x n ] be non-zero polynomials. If � k i =1 deg( P i ) < n , then the number of common zeros of P 1 , . . . , P k is divisible by p . In particular, if the polynomials have a common root, they also have another one. 11/44

The theorems over F 2 Definition A multilinear polynomial over F 2 is � � M ( x 1 , . . . , x n ) = where c T ∈ F 2 c T x i , T ⊆{ 1 ,..., n } i ∈ T Fact: For every P over F 2 , there exists a unique multilinear polynomial M P such that P and M P compute the same function. Definition: The multilinear degree of P is mdeg ( P ) = deg( M P ). Theorem [Combinatorial Nullstellensatz over F 2 ]: Let P be such that mdeg ( P ) = n . Then there exists a ∈ F n 2 such that P ( a ) = 1. Theorem[Chevalley-Warning over F 2 ]: Let P such that mdeg ( P ) < n , and let a ∈ F n 2 such that P ( a ) = 0. Then there exists b � = a such that P ( b ) = 0. Theorem: mdeg ( P ) < n ⇐ ⇒ the number of zeros is even 12/44

How to make them search problems? Theorem[P’94]: The following problem is in PPA . Explicit Chevalley Input: Explicitly given polynomials P 1 , . . . , P k over F 2 such that � k i =1 deg( P i ) < n , and a common root a ∈ F n 2 . Output: Another common root a ′ � = a . Remark: a is common root ⇔ P ( a ) = 0 where P = 1 + � k i =1 ( P i + 1) Could this be PPA -hard? Probably not. Two restrictions: • P is given by an arithmetic circuit of specific form • even the degree of P is less than n 13/44

Arithmetic circuits and parse subcircuits 14/44

Arithmetic circuits C is a labeled, directed, acyclic graph. × Labels = { + , ×} , + + G + = sum gates, G × = product gates. x 1 x 4 + Computational gates have indegree 2: x 2 x 3 left and right child Polynomial computed by C C ( x ) = ( x 1 + x 2 + x 3 ) × ( x 2 + x 3 + x 4 ) = x 1 x 2 + x 1 x 3 + x 1 x 4 + x 2 2 + x 2 x 4 + x 2 3 + x 3 x 4 15/44

Lagrange-circuits Circuits computing the Lagrange basis polynomials L a ( x ) L a ( x ) = 1 ⇐ ⇒ x = a × x 1 + + x 2 x 3 1 Lagrange-circuit L 100 16/44

Degrees in a circuit There are 3 types of degree 3 2 1 × × × 2 1 2 0 1 0 × + × + × + 1 0 0 x 1 x 1 x 1 x 1 x 1 x 1 + 1 + 1 + 1 x 2 x 2 x 2 x 2 x 2 x 2 Formal degree = 3 Polynomial degree = 2 Multilinear degree =1 x 2 = x 2 = 0 easy to compute ?? We are interested in the multilinear degree 17/44

Multilinear degree and monomials How can we certify mdeg ( C ( x )) = n ? ( x 1 + x 2 ) 2 n × What is the the complexity of . Mdeg = { C : mdeg ( C ( x )) = n } ? . . We wish Mdeg ∈ NP ( x 1 + x 2 ) 2 2 × A monomial m computed by C is maximal if mdeg ( m ) = n ( x 1 + x 2 ) 2 1 × Fact: mdeg ( C ( x )) = n ⇐ ⇒ odd number of maximal monomials ( x 1 + x 2 ) 2 0 + x 1 x 2 Difficulty: the number of monomials computed by C can be doubly exponential in the size of C We can certainly say that Mdeg ∈ ⊕ EXP 18/44

Monomials in arithmetic formulae Let F be an arithmetic formula Monomials are computed by parse subtrees defined by the marking of appropriate sum gates: S : G + → { ℓ, r , ∗} : × ℓ r + + x 2 x 2 x 1 1 19/44

Parse subcircuits C arithmetic circuit. A parse subcircuit is a partial marking S : G + → { ℓ, r , ∗} such that marked vertices = accessible vertices × × r ℓ ℓ r + + + + r ∗ x 1 + x 4 x 1 + x 4 x 2 x 3 x 2 x 3 computes x 2 computes x 1 x 4 3 20/44

Parse subcircuits witness monomials S ( C ) = set of parse subcircuits of C , m S ( x ) = monomial computed by parse sub circuit S Theorem: Let F be a field of characteristic 2. Then � C ( x ) = m S ( x ) . S ∈S ( C ) × × + + + + U W U ′ W ′ g g x 1 + x 4 x 1 + x 4 x 2 x 3 x 2 x 3 m U ′ m W ′ = x 2 x 3 m U m W = x 2 x 3 Corollary: Mdeg ∈ ⊕ P Proposition: Mdeg is ⊕ P -hard. 21/44

The problems PPA-Circuit Chevalley and PPA-Circuit CNSS 22/44

Towards PPA -circuits We would like to characterize PPA with arithmetic circuits Auxiliary circuits I and I ⋄ C : × · · · + + × · · · 1 C + · · · + · · · x 1 · · · x n y 1 · · · y n x 1 · · · x n 1 I ( x 1 , . . . , x n , y 1 , . . . , y n ) = � n i =1 ( x i + y i + 1) I ( x , y ) = 1 ⇐ ⇒ x = y I ⋄ C ( x ) = 1 ⇐ ⇒ C ( x ) = x 23/44

PPA-circuits Definition: A PPA -circuit is the PPA -composition C D , F of two n -variable, n -output arithmetic circuits D and F over F 2 + I 1 ⋄ D 1 ◦ F 1 I 2 ⋄ F 2 ◦ D 2 I 3 ⋄ D 3 ◦ D 4 I 4 ⋄ D 5 I 5 ⋄ F 3 ◦ F 4 I 5 ⋄ F 5 · · · · · · · · · · · · · · · · · · x 1 · · · x n C D , F PPA-Circuit Matching Lemma: If C is a PPA -circuit then in polynomial time a perfect matching µ can be computed between the maximal parse subcircuits of C . 24/44

PPA-Circuit Matching Lemma We want to define a polynomial time computable µ : perfect matching on the maximal parse subcircuits of C D , F + + + + C 1 C 2 C 3 I 1 ⋄ D 1 ◦ F 1 I 2 ⋄ F 2 ◦ D 2 I 3 ⋄ D 3 ◦ D 4 I 4 ⋄ D 5 I 5 ⋄ F 3 ◦ F 4 I 5 ⋄ F 5 · · · · · · · · · · · · · · · · · · x 1 · · · x n C D , F = C 1 + C 2 + C 3 µ is defined inside C 1 , inside C 2 and inside C 3 25/44