Nullstellensatz is Equivalent to Sum-of-Squares, over Algebraic Circuits

1. Nullstellensatz is Equivalent to Sum-of-Squares, over Algebraic Circuits Edward Hirsch Iddo Tzameret St. Petersburg Royal Holloway, University of London

2. The Systems

3. Nullstellensatz (NS) , over the reals. By Hilbert’s Nullstellensatz the fi‘s have no 0-1 solution iff there exists and s.t. • Complexity of NS Degree: Maximal degree of a monomial in a summand Size: total number of monomials in each summand

4. Sum-of-Squares (SoS) ,over the reals. By Positivestellensatz the fi‘s have no 0-1 solution iff there exists , ti’s and s.t. • Complexityof such refutation is Degree: Maximal degree of a monomial in a summandSize: total number of monomials in each summand

5. Cone-Proof-System (CPS) , over reals. By (the actual) Positivestellensatz these equations and inequalities have no 0-1 solution iff there exists s.t. • Complexityof such refutation is Degree: Maximal degree of a monomial in a summandSize: total number of monomials in refutation

6. Known Separations • Seen on Hirsch’s talk: • E.g., NS cannot efficiently refute PHP, while SoS can.

7. What about Algebraic Circuit Size? output + ˟ ˟ ˟ X X -2 Y

8. Nullstellensatz over Circuits NScirc IPS = NScirc (Grochow-Pitassi ’14; Forbes et al. ’16) , over the reals. By Hilbert’s Nullstellensatz the fi‘s have no 0-1 solution iff there exists and s.t. Complexity: total algebraic circuit size of ,

9. Sum-of-Squares over Circuits (SoScirc) ,over the reals. By Positivestellensatz the fi‘s have no 0-1 solution iff there exists and s.t. Complexity: total algebraic circuit size of , ,

10. Cone-Proof-System over Circuits (CPScirc) By (the actual) Positivestellensatz these equations and inequalities have no 0-1 solution iff there exists and s.t. • Complexity: total algebraic circuit sizeof summands

11. Our Result Thm: Over ℤ, and for polynomial-degree proofs (IPS=p) NScirc =p SoScirc =p CPScirc . Cor: IPS admits short proofs of the Reflection Principle: IPS ⊢ For every integer inputs Y to C, written in binary, C(Y) outputs 0.

12. Proof • Thm: Over ℤ, and for polynomial-degree proofs • (IPS=p) NScirc =p SoScirc=p CPScirc . • Proof: • Let be an SoScirc proof • Hence, • has a proof in NScirc (every zero polynomial can be proved in NScirc).

13. Proof (cntd.) • “Reason” in NScirc (can simulate EF; though with exponential degree (Pit97; GP14)): • We’ve assumedfi=0, i in [m], and the Boolean axioms equal 0. • We’re left with: • Remains to refute in NScirc .

14. Proof (cntd.) • Refuting in NScirc . • Derive from this equation an algebraic circuit representing the Booleancircuit computing 0 where inputsare integers in binary: output ith-bit is ith bit of Carry-Save-Adder of the left and right inputs to the gate. + ⊢NScirc ˟ ith-bit is ith bit of Two’s Complement notation of -2. ˟ • ⊢NScirc -2 ˟ X 1st-bit is X; other bits are 0. Crucially uses Boolean Axioms • ⊢NScirc X Y

15. Let k := min number of bits needed to encode a number computed in the circuit for 0-1 inputs, in Two’s Complement. BIT definition (cntd.) Define to be the ith bit of algebraic circuit F: Base: Case 1:. Then and for i=1,..,k-1. Case 2: . Then, is ith bit of in Two’s Complement. Induction Step: Given a circuit G, we denote by the vector of individual bits Case 1: . . Then, Case 2: . . Then,

16. Proof (cntd.) • Given kBoolean values ,

17. Proof (finished) Lemma: Let Fbe an algebraic circuit of size sand syntactic-degree d. Then there is a Nullstellensatz proof of of algebraic circuit-size poly(s,d). • Corollary: • And RHS can be refuted, by construction of ADD/PROD: “1 + square is nonzero”. QED

18. Conclusions • We showed that when measuring algebraic circuit-size NS (i.e., IPS) is equivalent to semi-algebraic systems such as SoS (PS) and the Cone Proof System. • We showed that IPS can prove the reflection principle for polynomial identities(open in GP’14).

19. Thank you for Listening!