Toda’s theorem in arithmetic and proof complexity
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 2012
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Toda’s theorem in arithmetic and proof complexity
Introduction
Major problem in propositional proof complexity: lower bounds
(ideally, exponential) on bounded depth proofs with mod 2 gates:
⊕(φ1 , . . . , φn ) = an odd number of φi have value 1.
Bounded depth Frege with mod 2 gates = AC0 [2]-Frege.
Related problem in bounded arithmetic: “interesting” independence
result for T2 (⊕)(α), bounded arithmetic with a parity quantifier.
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Toda’s theorem in arithmetic and proof complexity
Introduction
Major problem in propositional proof complexity: lower bounds
(ideally, exponential) on bounded depth proofs with mod 2 gates:
⊕(φ1 , . . . , φn ) = an odd number of φi have value 1.
Bounded depth Frege with mod 2 gates = AC0 [2]-Frege.
Related problem in bounded arithmetic: “interesting” independence
result for T2 (⊕)(α), bounded arithmetic with a parity quantifier.
More modest aim: better understanding of AC0 [2]-Frege and
T2 (⊕)(α) (and analogues for prime p ̸= 2).
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Toda’s theorem in arithmetic and proof complexity
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 AC0 [2]
circuits to a very simple form, with quasipolynomial increase in size.
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Toda’s theorem in arithmetic and proof complexity
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 AC0 [2]
circuits to a very simple form, with quasipolynomial increase in size.
Can something similar be done for AC0 [2]-Frege proofs?
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Toda’s theorem in arithmetic and proof complexity
Collapsing AC0 [2]-Frege?
Maciel-Pitassi 1998:
simulation of AC0 [2]-Frege by proofs of simple form,
but the simulating system has exact counting (threshold) gates.
New development since then (Jeřábek 2004-9):
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 AC0 [2]-Frege proofs?
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Toda’s theorem in arithmetic and proof complexity
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.
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Toda’s theorem in arithmetic and proof complexity
Bounded arithmetic: a very quick review
Σbi — class of arithmetic formulas corresponding
to Σpi .
∪
T2i — induction for Σbi formulas. T2 = i T2i .
PV — induction for polytime properties (“right notion” of T20 ).
For new predicate α (oracle), Σbi (α) and T2i (α) can be defined.
Paris-Wilkie translation: translates arithmetic formulas (with α)
into families of propositional formulas, and proofs in T2i (α)
into uniform families of fixed-depth quasipolynomial
size proofs.
∧ ∨
(atoms in α
variables, quantifiers
/ , etc.)
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Toda’s theorem in arithmetic and proof complexity
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 a2 suffices.)
APC1 = PV + sWPHP(FP). APC2 = T21 + sWPHP(FPNP ).
APC1 is contained in T22 . It can approximate the size
of polytime set X ⊆ 2n up to 1/poly(n) fraction of 2n .
APC2 can do the same for X ∈ PNP , while for X ∈ NP
it finds surjections witnessing m X m + m/polylog(m).
It is contained in T23 .
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Toda’s theorem in arithmetic and proof complexity
Bounded arithmetic with a parity quantifier
Two ways of adding the new quantifier:
I
T2 (⊕): add ⊕x < y to the usual language,
induction available for all bounded formulas.
I
T2i,⊕P : allow ⊕x < y only in front of polytime formulas.
T2i,⊕P has Σib,⊕P induction.
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Toda’s theorem in arithmetic and proof complexity
Toda’s Theorem in APC⊕P
2
L ∈ BP · ⊕P if for some polytime functions u(x), f (x, r),
x ∈ L → Prr<u(x) [f (x, r) ∈
/ ⊕SAT] < 1/4,
x∈
/ L → Prr<u(x) [f (x, r) ∈ ⊕SAT] < 1/4,
where probabilities stated using approximate counting.
Theorem
Every Σb,⊕P
formula can be assigned a BP · ⊕P
∞
representation which is provably correct in APC⊕P
2 .
As a consequence, T2 (⊕) is conservative over APC⊕P
2 .
The theorem smoothly relativizes to a new oracle α.
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Toda’s theorem in arithmetic and proof complexity
Toda in APC⊕P
2 : 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,
2k S 2k+1 .
This seems to need APC⊕P
2 .
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Toda’s theorem in arithmetic and proof complexity
Back to propositional proofs
The proof system PCKi : lines are cedents of ∧/∨ formulas of depth i
with literals replaced by “low-degree” polynomials over F2
(“low” = logarithmic in the proof size).
Intended meaning of the PCK1 line “f1 , f2 ∧ f3 ∧ f4 ” is:
“f1 is 0 or f2 , f3 , f4 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 PCKi is a subsystem of AC0 [2]-Frege.)
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Toda’s theorem in arithmetic and proof complexity
Propositional proofs: rules
ψ, ¬ψ
Γ, ψi , where i ∈ I ∨
∨
( )
Γ, i∈I ψi
Γ (weakening)
Γ, ∆
Γ, f
(·)
Γ, fg
Axiom
Γ, ψi , all i ∈ I ∧
∧
( )
Γ, i∈I ψi
Γ, ¬ψ
Γ, ψ
Γ
(cut)
Γ, f
Γ, g
(+)
Γ, f + g
(¬ is DeMorgan negation, and ¬f is 1 + f .)
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Toda’s theorem in arithmetic and proof complexity
Propositional proofs: correspondence
An arithmetic formula A(x, α) has propositional translations JAKn ,
with variables for bits of α, meaning “A(n, α) holds”.
Theorem
A
provable in
Σb,⊕P
(α)
i
T2i,⊕P (α)
Σb,⊕P
i+1 (α)
T2i,⊕P (α)
Σb,⊕P
(α)
1
T21,⊕P (α)
Σb,⊕P
(α)
1
PV⊕P (α)
J¬AK have qpoly size refutations in
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Toda’s theorem in arithmetic and proof complexity
Propositional proofs: correspondence
An arithmetic formula A(x, α) has propositional translations JAKn ,
with variables for bits of α, meaning “A(n, α) holds”.
Theorem
A
provable in
J¬AK have qpoly size refutations in
Σb,⊕P
(α)
i
T2i,⊕P (α)
PCKi−2 , treelike PCKi−1
Σb,⊕P
i+1 (α)
T2i,⊕P (α)
treelike PCKi−1
Σb,⊕P
(α)
1
T21,⊕P (α)
Σb,⊕P
(α)
1
PV⊕P (α)
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Toda’s theorem in arithmetic and proof complexity
Propositional proofs: correspondence
An arithmetic formula A(x, α) has propositional translations JAKn ,
with variables for bits of α, meaning “A(n, α) holds”.
Theorem
A
provable in
J¬AK have qpoly size refutations in
Σb,⊕P
(α)
i
T2i,⊕P (α)
PCKi−2 , treelike PCKi−1
Σb,⊕P
i+1 (α)
T2i,⊕P (α)
treelike PCKi−1
Σb,⊕P
(α)
1
T21,⊕P (α)
polylog degree Polynomial Calculus
Σb,⊕P
(α)
1
PV⊕P (α)
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Toda’s theorem in arithmetic and proof complexity
Propositional proofs: correspondence
An arithmetic formula A(x, α) has propositional translations JAKn ,
with variables for bits of α, meaning “A(n, α) holds”.
Theorem
A
provable in
J¬AK have qpoly size refutations in
Σb,⊕P
(α)
i
T2i,⊕P (α)
PCKi−2 , treelike PCKi−1
Σb,⊕P
i+1 (α)
T2i,⊕P (α)
treelike PCKi−1
Σb,⊕P
(α)
1
T21,⊕P (α)
polylog degree Polynomial Calculus
Σb,⊕P
(α)
1
PV⊕P (α)
polylog degree Nullstellensatz
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Toda’s theorem in arithmetic and proof complexity
Propositional proofs: collapse
Corollary
∨∧∨
For proofs of simple enough formulas (
⊕“small ∧”),
0
AC [2]-Frege is quasipolynomially simulated by PCK1 .
Proof.
I
by conservativity, T23 (α) proves reflection for AC0 [2]-Frege:
every provable formula is true.
I
so PCK1 refutes J¬ReflectionK.
I
by substituting bits of an actual AC0 [2]-Frege proof of φ,
we get a PCK1 refutation of J“φ is false”K.
I
modulo cosmetic changes, that is a refutation of ¬φ.
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Toda’s theorem in arithmetic and proof complexity
Propositional proofs: collapse (cont’d)
Corollary
∨∧∨
For proofs of simple enough formulas (
⊕“small ∧”),
0
AC [2]-Frege is quasipolynomially simulated by PCK1 .
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Toda’s theorem in arithmetic and proof complexity
Propositional proofs: collapse (cont’d)
Corollary
∨∧∨
For proofs of simple enough formulas (
⊕“small ∧”),
0
AC [2]-Frege is quasipolynomially simulated by PCK1 .
3,⊕P
Using conservativity over APC⊕P
, we get:
2 instead of T2
Corollary
∨∧
For proofs of simple enough formulas (
⊕“small ∧”),
0
AC [2]-Frege is quasipolynomially simulated by treelike PCK0
extended by axioms corresponding to sWPHP(FPNP (⊕P)).
(Using partial conservativity of sWPHP over so-called “retraction
WPHP”, one could even replace treelike PCK0 by polylog degree
Polynomial Calculus, but the extra axioms become less natural.)
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Toda’s theorem in arithmetic and proof complexity
The picture right now
APC⊕P
2 (α)
treelike PCK1 + sWPHP
T21,⊕P (α)
APC⊕P
1 (α)
polylog degree PC, treelike PCK1
polylog degree NS + sWPHP
PV⊕P
2 (α)
polylog degree NS
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Toda’s theorem in arithmetic and proof complexity
The picture right now
APC⊕P
2 (α)
treelike PCK1 + sWPHP
T21,⊕P (α)
APC⊕P
1 (α)
polylog degree PC, treelike PCK1
polylog degree NS + sWPHP
PV⊕P
2 (α)
polylog degree NS
The pigeonhole PHPn+1
n (α) is independent from:
I
I
T21,⊕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 T21,⊕P (α) + sWPHP(FPNP (α)),
but sWPHP for functions involving ⊕P seems very difficult to deal
with.
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Toda’s theorem in arithmetic and proof complexity
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 < t2 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...
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Toda’s theorem in arithmetic and proof complexity
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 < t2 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 φ̃ to
φ. This has polylog many new variables, and for any assignment to
the old variables, almost all assignments to the new variables make φ̃
true. But φ̃ joined with ¬φ is refutable in low degree Nullstellensatz!
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