Toda s theorem in bounded arithmetic with parity quantifiers and bounded depth proof systems with parity gates

1 / 17 Toda s theorem in bounded arithmetic with parity quantifiers and bounded depth proof systems with parity gates Leszek Kołodziejczyk University of Warsaw/UCSD (joint work with Sam Buss and Konrad Zdanowski) Logical Approaches to Barriers in Complexity II Cambridge, March 01

/ 17 Introduction Major problem in propositional proof complexity: lower bounds (ideally, exponential) on bounded depth proofs with mod gates: (φ 1,..., φ n ) = an odd number of φ i have value 1. Bounded depth Frege with mod gates = AC 0 []-Frege. Related problem in bounded arithmetic: interesting independence result for T ( )(α), bounded arithmetic with a parity quantifier.

/ 17 Introduction Major problem in propositional proof complexity: lower bounds (ideally, exponential) on bounded depth proofs with mod gates: (φ 1,..., φ n ) = an odd number of φ i have value 1. Bounded depth Frege with mod gates = AC 0 []-Frege. Related problem in bounded arithmetic: interesting independence result for T ( )(α), bounded arithmetic with a parity quantifier. More modest aim: better understanding of AC 0 []-Frege and T ( )(α) (and analogues for prime p ).

3 / 17 Toda s Theorem (A version of) Toda s Theorem: PH( ), the polynomial hierarchy with a parity quantifier, collapses to BP P. Observation: the relativized version of this can be seen as a collapse of AC 0 [] circuits to a very simple form, with quasipolynomial increase in size.

3 / 17 Toda s Theorem (A version of) Toda s Theorem: PH( ), the polynomial hierarchy with a parity quantifier, collapses to BP P. Observation: the relativized version of this can be seen as a collapse of AC 0 [] circuits to a very simple form, with quasipolynomial increase in size. Can something similar be done for AC 0 []-Frege proofs?

4 / 17 Collapsing AC 0 []-Frege? Maciel-Pitassi 1998: simulation of AC 0 []-Frege by proofs of simple form, but the simulating system has exact counting (threshold) gates. New development since then (Jeřábek 004-9): bounded arithmetic has reasonable notions of approximate cardinality and probabilistic complexity classes. So: why not try to prove Toda in bounded arithmetic with parity quantifiers, and see what that says about AC 0 []-Frege proofs?

5 / 17 Plan for rest of talk There won t be any really interesting proofs in this talk. There won t even be too many pictures/diagrams. So, the above is offered as a form of compensation.

6 / 17 Bounded arithmetic: a very quick review Σ b i class of arithmetic formulas corresponding to Σ p i. T i induction for Σb i formulas. T = i Ti. PV induction for polytime properties ( right notion of T 0 ). For new predicate α (oracle), Σ b i (α) and Ti (α) can be defined. Paris-Wilkie translation: translates arithmetic formulas (with α) into families of propositional formulas, and proofs in T i (α) into uniform families of fixed-depth quasipolynomial size proofs. (atoms in α variables, quantifiers /, etc.)

7 / 17 Approximate counting in bounded arithmetic swphp(γ) surjective WPHP for function class Γ: no function f Γ is surjection a a(1 + 1/(log a)). (in many contexts, ruling out a a suffices.) APC 1 = PV + swphp(fp). APC = T 1 + swphp(fpnp ). APC 1 is contained in T. It can approximate the size of polytime set X n up to 1/poly(n) fraction of n. APC can do the same for X P NP, while for X NP it finds surjections witnessing m X m + m/polylog(m). It is contained in T 3.

8 / 17 Bounded arithmetic with a parity quantifier Two ways of adding the new quantifier: T ( ): add x < y to the usual language, induction available for all bounded formulas. T i, P : allow x < y only in front of polytime formulas. T i, P has i induction.

9 / 17 Toda s Theorem in APC P L BP P if for some polytime functions u(x), f (x, r), x L Pr r<u(x) [f (x, r) / SAT] < 1/4, x / L Pr r<u(x) [f (x, r) SAT] < 1/4, where probabilities stated using approximate counting. Theorem Every formula can be assigned a BP P representation which is provably correct in APC P. As a consequence, T ( ) is conservative over APC P. The theorem smoothly relativizes to a new oracle α.

10 / 17 Toda in APC P : comments on proof Essentially a formalization of the textbook proof. Induction on formula complexity, some technicalities involved. The base case uses a version of the Valiant-Vazirani Theorem: SAT is probabilistically reducible to Unique-SAT. One point in the proof of V-V: given propositional formula φ, if S is the set of satisfying assignments for φ, then for some k, This seems to need APC P. k S k+1.

11 / 17 Back to propositional proofs The proof system PCK i : lines are cedents of / formulas of depth i with literals replaced by low-degree polynomials over F ( low = logarithmic in the proof size). Intended meaning of the PCK 1 line f 1, f f 3 f 4 is: f 1 is 0 or f, f 3, f 4 all are. So, constant 1 plays the role of. (Btw, low-degree polynomials s of small conjunctions, it s just that algebraic rules are sometimes less clumsy than boolean. So PCK i is a subsystem of AC 0 []-Frege.)

1 / 17 Propositional proofs: rules ψ, ψ Axiom Γ, ψ i, where i I Γ, i I ψ i ( ) Γ, ψ i, all i I Γ, i I ψ i ( ) Γ Γ, (weakening) Γ, ψ Γ, ψ (cut) Γ Γ, f Γ, fg ( ) Γ, f Γ, g (+) Γ, f + g ( is DeMorgan negation, and f is 1 + f.)

Propositional proofs: correspondence An arithmetic formula A(x, α) has propositional translations A n, with variables for bits of α, meaning A(n, α) holds. Theorem A provable in A have qpoly size refutations in i (α) T i, P (α) i+1 (α) Ti, P (α) 1 (α) T 1, P (α) 1 (α) PV P (α) 13 / 17

Propositional proofs: correspondence An arithmetic formula A(x, α) has propositional translations A n, with variables for bits of α, meaning A(n, α) holds. Theorem A provable in A have qpoly size refutations in i (α) T i, P (α) PCK i, treelike PCK i 1 i+1 (α) Ti, P (α) treelike PCK i 1 1 (α) T 1, P (α) 1 (α) PV P (α) 13 / 17

Propositional proofs: correspondence An arithmetic formula A(x, α) has propositional translations A n, with variables for bits of α, meaning A(n, α) holds. Theorem A provable in A have qpoly size refutations in i (α) T i, P (α) PCK i, treelike PCK i 1 i+1 (α) Ti, P (α) treelike PCK i 1 1 (α) T 1, P (α) polylog degree Polynomial Calculus 1 (α) PV P (α) 13 / 17

Propositional proofs: correspondence An arithmetic formula A(x, α) has propositional translations A n, with variables for bits of α, meaning A(n, α) holds. Theorem A provable in A have qpoly size refutations in i (α) T i, P (α) PCK i, treelike PCK i 1 i+1 (α) Ti, P (α) treelike PCK i 1 1 (α) T 1, P (α) polylog degree Polynomial Calculus 1 (α) PV P (α) polylog degree Nullstellensatz 13 / 17

14 / 17 Propositional proofs: collapse Corollary For proofs of simple enough formulas ( small ), AC 0 []-Frege is quasipolynomially simulated by PCK 1. Proof. by conservativity, T 3 (α) proves reflection for AC0 []-Frege: every provable formula is true. so PCK 1 refutes Reflection. by substituting bits of an actual AC 0 []-Frege proof of φ, we get a PCK 1 refutation of φ is false. modulo cosmetic changes, that is a refutation of φ.

15 / 17 Propositional proofs: collapse (cont d) Corollary For proofs of simple enough formulas ( small ), AC 0 []-Frege is quasipolynomially simulated by PCK 1.

15 / 17 Propositional proofs: collapse (cont d) Corollary For proofs of simple enough formulas ( small ), AC 0 []-Frege is quasipolynomially simulated by PCK 1. Using conservativity over APC P instead of T 3, P, we get: Corollary For proofs of simple enough formulas ( small ), AC 0 []-Frege is quasipolynomially simulated by treelike PCK 0 extended by axioms corresponding to swphp(fp NP ( P)). (Using partial conservativity of swphp over so-called retraction WPHP, one could even replace treelike PCK 0 by polylog degree Polynomial Calculus, but the extra axioms become less natural.)

16 / 17 The picture right now T 1, P (α) polylog degree PC, treelike PCK 1 APC P (α) treelike PCK 1 + swphp PV P (α) polylog degree NS APC P 1 (α) polylog degree NS + swphp

The picture right now T 1, P (α) polylog degree PC, treelike PCK 1 APC P (α) treelike PCK 1 + swphp PV P (α) polylog degree NS APC P 1 (α) polylog degree NS + swphp The pigeonhole PHP n+1 n (α) is independent from: T 1, P (α), by known Polynomial Calculus lower bounds, PV P (α) + swphp(fp(α)), by combining Nullstellensatz lower bounds with switching lemma techniques. Seems within reach to extend this to T 1, P (α) + swphp(fp NP (α)), but swphp for functions involving P seems very difficult to deal with. 16 / 17

17 / 17 Problems with an approach to lower bounds Let φ be a small formula. Want to show: φ has no refutations in low degree Nullstellensatz + swphp(fp( P)) axioms. The swphp axiom says c < t not in F([0, t)) ; has polylog many new variables for bits of c. For any assignment to the φ variables, almost all assignments to the new variables make the axiom true. So maybe...

17 / 17 Problems with an approach to lower bounds Let φ be a small formula. Want to show: φ has no refutations in low degree Nullstellensatz + swphp(fp( P)) axioms. The swphp axiom says c < t not in F([0, t)) ; has polylog many new variables for bits of c. For any assignment to the φ variables, almost all assignments to the new variables make the axiom true. So maybe... However: take a suitably constructed low degree approximation φ to φ. This has polylog many new variables, and for any assignment to the old variables, almost all assignments to the new variables make φ true. But φ joined with φ is refutable in low degree Nullstellensatz!