1. Matrix PI-algebras and Lower Bounds on Arithmetic Proofs (work in progress) Iddo Tzameret Joint work with Fu Li Tsinghua University

2. In short
Motivation: provide algebraic technique that may imply polynomial (or even super-polynomial) lower bounds on very strong arithmetic or even propositional proof systems (Extended Frege).
Note: No conditional (even non-explicit) lower bounds on strong propositional proofs is known (unless we condition on NP!=coNP)
Our results: generalize to commutativity axioms of high degrees prior lower bounds by Hrubeš (2011).
Still in progress: the connections to proof systems.

3. The Algebraic Problem

4. The algebraic problem
Let F<X> be the ring of noncommutative polynomials over variables x1,x2,…
i.e., every polynomial is a formal sum of noncommutative monomials with coefficients from the field F.
E.g., the commutator [x1,x2]:= x1x2 – x2x1 is not the zero polynomial.

5. The algebraic problem
Let A be a (not necessarily commutative, but associative) F -algebra.
E.g.: the dxd matrix algebra Matd( ).
An identity of A is a noncommutative polynomial f(x1,..,xn) in F<X>,, where for all vectors a from An, f(a)=0.
E.g.: x1x2 – x2x1 is an identity of
Mat1( F ) (but not of Matd( ) if d>1)

6. for some polynomials Qi’s, ti’s and Pi’s in F<X .
Consider the set of identities over Matd[F].
Kemer ‘87: Identities of Matd[F] can be generated (in the two-sided ideal) by substitution instances of a finite set G of polynomials g1…gc in F<X
I.e., every identity f in F<X> over Matd(F) can be written as:
for some polynomials Qi’s, ti’s and Pi’s in F<X .

7. Example for d=1 case (Mat1[F]):
All identities of Mat1[F] can be generated by substitution instances of a single polynomial:
the commutator [x,y]=xy-yx :
f is an identity of Mat1[F] iff
f in <[x_i,x_j]: i neq j in N
(all ideals are two sided ideals).

8. Complexity measure: how many substitution instances of generators are needed to generate an identity for Matd[F] ?
The case of d=1:
Let Q1(f) be the minimal number of substitution instances of commutators [x,y] needed to generate identities of Mat1[F].
i.e., min k such that f in I<[t1,t’1],…,[tk,t’k] , for some ti’s in F<X>.
Example: Q1(x1x3-x3x1+x2x3-x3x2)=?
=1 since: (x1+x2)x3-x3(x1+x2)=x1x3-x3x1+x2x3-x3x2

9. Upper bound on Q1(f): O(n2).
We have a matching non-explicit lower bound:
Thm (Hrubeš 2011): There exists a (non-explicit) polynomial identity of Mat1[F], f in F<X>, where Q1(f)=Ω(n2).
Proof follows by a counting argument and some algebraic properties of the commutators.

10. The case of Matd[F] for d>1
For d>1, what are the identities of Matd[F] ?
not all cases are known precisely; dates back to Amitzur and Levitski 1950.
We’ll see some cases are known, and some are conjectured.

11. Matd[F] for d=2
Thm (Drenski 1981): For d=2 and char(F)=0, all identities of Matd(F) are generated by
s2d formulas and the hall formulas
where
and h(x1,x2):=[[x1,x2]2,x1]
Note: assume from now that char(F)=0.
Every identity f in F<X> over Mat2(F) can be written as:
For some polynomials Qi’s, Pi’s and sequences of polynomials gi’s in F<X>.

12. Mat2[F]
Define QMat2[F](f) as the min number of substitution instances needed to generate an identity in Mat2[F].
Upper bound on QMat2(f) ?
It must be O(n4): every identity f with n variables can be generated by all s2d(xi1,xi2,xi3,xi4), for all 𝒏 𝟒 choices x vars, and the hall identities h(xi1,xi2), for all 𝒏 𝟐 variables.
We prove a corresponding lower bound:
Thm: There exists (non-explicit) f such that f is identity of Mat2(F) and QMat2(f)=Ω(n4).
We generalize this to any degree d….

13. Generalization
Leron (1973): All multilinear polynomial identities of degree 2d+1 of Matd(F) are generated by the s2d identities.
for some polynomials gi’s.
We show:
Thm: For any d, there exists a degree 2d+1 (non-explicit) polynomial identity f of Matd(F) with n variables, such that Qs2d(f)=Ω(n2d).

14. Generalization
The hard polynomials f in thm have a special form. We also have:
Proposition: Let f be of the above form. Then, showing an explicit such f with Qs2d(f)=Ω(n2d) implies giving an explicit order 2d+1 tensor A:[n]2d+1F with tensor-rank Ω(n2d).

15. The lower bound proof

16. What are the hard identities f ?
We call it the s-formulas:
where
For some n fixed fi’s:

17. We’ll only show that to generate n fi’s: f1,…,fn by s2d polynomials we need Ω(n2d) many generators:
Q(f1,…,fn)= Ω(n2d)
(To combine them into
we need more work.)

18. By counting (total # of n-tuples of f1,…,fn) vs.
(total # of n-tuples f1,…,fn we can generate with q s2d generators)
(We show that q=Ω(n2d).)
Lemma: For any d and polynomials p1,…,pn:

19. Thus we can assume w.l.o.g. that the substitutions in the generators’ variables are linear forms:

20. Recall:
So, total # of possible n-tuples of fi’s:
(for each i=1,..n choose which of the
cj’s in fi are 1).

21. total # of n-tuples f1,…,fn we can generate with q s2d-generators:
choose 2d x q linear forms x choose q field elements for coefficients of linear combination:
We get:
implying:
Q.E.D.
Assume field is finite. The other case can also be handled.

22. Lemma Lemma: For any d and polynomials p1,…,pn:
deg > d monomials in pi not counted in LHS
Property of s2d(x1,..,x2d): assigning a constant to a variable makes it 0.
Thus:
Degree 0 monomial in pi doesn’t contribute to LHS;
Degree >1 monomial in pi can contibute to LHS only if it multiplies a constant in some pj, j≠j. Hence, we get 0 again.

23. Motivation: What we aim at

24. A fundamental hardness question in complexity:
Prove super-polynomial lower bounds on propositional proofs!
Propositional proofs:
Standard logical proofs: a sequence of Boolean formulas;
Every formula: an axiom (a tautology) or derived from previous lines by a simple deduction rule; 6’
Motivations: understand the circuit based Frege hierarchy; separations, possible separations, understand better analogy with circuit complexity, if any.
--Note: the rules have to be slightly accommodated when speaking about circuits and not formulas: the similarity rule
- Don’t require poly-size for NC2-Frege proofs

25. Until you derive the desired tautology…
Propositional proofs
Start from some axioms,
Aᴠ¬A, A(AᴠB), etc.
and successively apply inference rules to derive new formulas or “lines”
FᴠX FᴠY
Fᴠ(XΛY)
A AB B
Until you derive the desired tautology…

26. Probably. But nobody knows how to prove it.
Is there a family of tautologies {fi} with no polynomial size propositional proofs?
Probably. But nobody knows how to prove it.
Only Ω(n) lower bound on number of lines is known!
Consider stronger proofs: every line is a circuit (we can make this formal with additional simple rules). (AKA Extended Frege)
Again, only Ω(n) lower bound on number of lines.

27. We would hope to give a framework where:
The ideal goal: prove polynomial (or even super-polynomial) lower bounds on circuit-based propositional proofs.
A bit less ideal goal: Conditional lower bounds on circuit-based propositional proofs (from algebraic assumptions)
Closer goal: Conditional lower bounds on a fragment of propositional circuit-based proofs (from algebraic assumptions): i.e., lower bounds on arithmetic proofs

28. if F and G are identical when the circuits are unwinded into trees
Arithmetic proofs
Establish commutative polynomial identities
Proof-lines: equations between algebraic circuits
Axioms: polynomial-ring axioms
Rules: Transitivity of “=“; +,x introduction, etc.
Circuit-axiom: F=G,
if F and G are identical when the circuits are unwinded into trees
15’

29. field identities axiom
Arithmetic proofs
field identities axiom
2∙3=6
x=x
reflexivity axiom
product rule
commutativity axiom
15’
3x∙2=2∙3x
2∙3x=6x
transitivity
3x∙2=6x

30. Arithmetic proofs (over GF(2))
By Rekchow’s Thm: Over GF(2) (and over the rationals) arithmetic proofs are also propositional proofs of the translated tautology:
Translate every arithmetic circuit to Boolean circuit:
Note difference: Arithmetic proofs
cannot prove all propositional tautologies;
cannot use every possible proof-lines (only translations of identities between algebraic circuits)
Propositional
proofs
Arithmetic proofs (over GF(2))

31. What’s the connection?
For the case d=1: recall Q1(f) is the minimal number of substitution instances of commutators [x,y] needed to generate an identity.
Observation (Hrubes ’11): The minimal arithmetic proof of f=g ≥ Q1( ), where is the noncommutative poly computed by circuit f.
Proof: By induction on number of lines in the arithmetic proof.

32. What’s the connection? What about the case d>1?
We can’t have the same idea, since identity f=g of Matd[F] for d>1, will only need O(n2) commutativity axioms !

33. Translating matrix identities
But any matrix identity of Matd[F] corresponds to a set of d2 polynomial identities in F[X].
Example: If AB=1 is an “identity” in Mat2[F], then it corresponds to the 4 identities 𝑨𝑩=𝟏 𝒅 induced by:
𝑨 𝟏𝟏 𝑨 𝟏𝟐 𝑨 𝟐𝟏 𝑨 𝟐𝟐 𝑩 𝟏𝟏 𝑩 𝟏𝟐 𝑩 𝟐𝟏 𝑩 𝟐𝟐 = 𝟏 𝟎 𝟎 𝟏
i.e., A11B11+A12B21=1 ….
Conjecture: The minimal size of an arithmetic proof of 𝒇=𝒈 𝒅 is at least 𝑸 𝐌𝐚𝐭 𝒅 𝑭 (𝒇−𝒈).

34. Conjecture
In other words: proving matrix identities of Matd[F] entry-wise cannot be faster than “proving” them using substitution instances of the generating sets of Matd[F].
Define proof system PMatd: consider arithmetic proofs and replace the commutativity axiom gf=fg with the (finite) set of generators of Matd[F].
We get a proof system sound & complete for all F=G that are identities over Matd[F].

35. Conjecture So the conjecture boils down to:
let f,g be two circuits where = be a true identity of Matd[F].
Size s arithmetic proofs of 𝒇=𝒈 𝒅 
size s PMatd proof of f=g

36. Conclusions so far
We considered an algebraic question not directly connected to proof complexity—seems to yield lower bounds on proofs.
We show that the same reasoning of Hrubes gives not just Ω(n2) bounds, but Ω(nd) bounds for any d.
We can hope for even super-polynomial lower bounds (the lower bounds increase exponentially in d, but increases polynomially in d and |f|,|g|).
𝒇=𝒈 𝒅

37. Thank you !

38. What’s the connection?
Observation (Hrubes ’11): The minimal arithmetic proof of f=g >= Q1(\hat f-\hat g), where \hat f is the noncommutative poly computed by circuit f.
Proof: By induction on number of lines t in proof.
Base: t=1. f=g is an axiom.
If f=g not the commutativity axiom, say h+0=h, then \hat (h+0)-\hat h =0\in F<X>. Hence Q1(0)=0.
Otherwise, f=g is the axiom uv=vu, for u,v circuits, and so Q1(uv-vu)=1.

39. Complexity measure: how many substitution instances of generators are needed to generate an identity for Matd[F] ?
The case of d=1:
Let Q1(f) be the minimal number of substitution instances of commutators [x,y] needed to generate identities of Mat1[F].
i.e., min k such that f in I<[t1,t’1],…,[tk,t’k]>, for some t’s in F<X>.
Example: Q1(sum_{i,j\in n} xixj ) = 1
sum_{i,j\in n} xixj = (x1+…+xn)*(x1+…+xn)