Addition is exponentially harder than counting for shallow monotone circuits
Rocco Servedio, Columbia University Joint work with
Xi Chen, Columbia University Igor Oliveira, Columbia University / Charles University
Circuit complexity
A circuit with AND/OR/NOT gates: (de Morgan circuit) Size = # of wires. Depth = depth. (Unbounded fan-in.)
x1 ¬ x1 x2 ¬ x2
….........
xn ¬ xn
Can consider circuits with other types of gates as well instead of AND/OR: • Maj circuit: can use gates that computes the majority function • “weighted threshold” circuit: can use gates that compute “weighted threshold” functions, that output 1 iff
The goal of circuit complexity Goal: Strong lower bounds on size of unrestricted circuits computing explicit functions.
Holy grail: size lower bound
Unfortunately, we have no idea how to prove that an explicit function cannot be computed by unrestricted circuits of size 10n. So, we look at restricted circuits: • Constant-depth circuits • Monotone circuits (for monotone functions)
Monotone circuits • Monotone de Morgan circuits: No negations, only AND/OR gates • Monotone Maj circuits: No negations, only Maj gates (and constants 0,1)
Maj
• Monotone weighted threshold circuits: No “negative weights” allowed: each gate computes some function where each
What do we know about monotone circuits? • [Razborov85,Andreev85,AlonBoppana87,KarchmerWigderson88, Tardos88,RazWigderson92,HarnikRaz00,PitassiGoos14]: strong lower bounds on monotone AND/OR-circuit size and depth for explicit monotone functions • [AjtaiGurevich87]: Explicit monotone function in poly(n)-size AC0 which does not have poly(n)-size monotone-AC0 circuits • [AjtaiKomlosSzemeredi83,Valiant84]: has poly(n)-size monotone AND/OR formulas (tree circuits) And more.
But we don’t know everything… Monotone-TC0: Constant-depth, poly(n)-size circuits of monotone weighted threshold gates Monotone-NC1: poly(n)-size monotone AND/OR formulas (= poly(n)-size, log-depth fan-in 2 monotone formulas) (= poly(n)-size, log-depth fan-in 2 monotone circuits) • [Yao89]: Monotone-NC1 not contained in monotone-TC0. (AND/OR tree, or Rec-Maj-3) • Question 1: Is monotone-TC0 contained in monotone-NC1? ([BeimelWeinreb05]: “almost Yes”: monotone-TC0 is contained in nO(log n) size monotone-NC1.)
Something else we don’t know Recall [AjtaiGurevich87]: There is a monotone function in AC0 which does not have poly(n)-size monotone-AC0circuits. Question 2: Could there be a monotone function in AC0 which does not have poly(n)-size monotone circuits, period?
This talk will answer of these questions! (But it describes partial progress on both of them.)
Question 1 revisited Question 1: Can monotone-NC1 compute monotone-TC0? Possible approach, if you are an optimist: • Recall [AKS,Valiant]: monotone-NC1 can compute . • So if constant-depth monotone circuits of Maj gates can compute monotone-TC0, then monotone-NC1 circuits can compute monotone-TC0. (constant depth à only polynomial blowup in size when we compose.) Our first main result: This won’t work.
First result Define the function as follows: • The input is a list of –bit strings (which we view as binary #s ) • The output is 1 iff
Example: input is
For this input, have
First result is computed by a single monotone threshold gate:
Gates “can count”
Theorem 1: Any depth-d monotone Maj circuit computing must have size “Addition is exponentially harder than counting for shallow monotone circuits”
Discussion Theorem 1: Any depth-d monotone Maj circuit computing must have size • Theorem is tight in two ways:
• Any n-variable monotone weighted threshold function (such as ) can be computed by a poly-size depth-2 non-monotone Maj circuit [GoldmannHastadRazborov92, GoldmannKarpinski93, Hofmeister96, AmanoMaruoka05] • can be computed by a depth-d, sizemonotone Maj circuit
• Theorem answers question of [GK93,Hastad10,Hastad14] • [Hofmeister92] proved the d=2 case
Second question and result Question 2: Could there be a monotone function in AC0 which does not have poly(n)-size monotone circuits?
Theorem: [AjtaiGurevich87] There is an n-variable monotone function such that • has a poly(n)-size, constant-depth AND/OR/NOT circuit; but • Any depth-d AND/OR circuit for has size at least where
Second main result: Strengthening of Ajtai-Gurevich Theorem : There is an -variable monotone function such that • has a poly(n)-size, depth-3 AND/OR/NOT circuit; • Any depth-d monotone Maj-circuit computing has size at least
Compare with [AjtaiGurevich87]: • stronger size bound (exponential for every d) • works against stronger circuits (Maj gates, not just AND/OR) • very different proof (no switching lemma)
Proving Theorem 2 from Theorem 1 Theorem 1: Any depth-d monotone Maj circuit computing must have size Theorem 2: There is an -variable monotone function such that • has a poly(n)-size, depth-3 AND/OR/NOT circuit; • Any depth-d monotone Maj-circuit computing has size at least
The function is where
A glimpse at the proof of Theorem 1 Theorem 1: Any depth-d Maj circuit computing must have size Proof is a induction.
Inductively construct distributions that are “hard” for deeper and deeper circuits.
A glimpse at the proof of Theorem 1 Let Initial pair of distributions : both over . Every has . Every has Moreover, any depth-1 monotone Maj circuit of size satisfies
A glimpse at the proof of Theorem 1 Key lemma: Suppose any depthmonotone Maj circuit over of size satisfies
Then any depth- monotone Maj circuit over of size satisfies
where The case gives the theorem.
To construct over from over the proof actually needs three pairs of distributions: over over and over some other domain that’s intermediate between and Proof maintains very tight control over all three pairs of distributions.
What does this have to do with Fourier analysis?
Not sure. Here are some very brief notes on this.
“You might try using analysis of Boolean functions whenever you’re faced with a problem involving Boolean strings in which … the uniform probability distribution … play[s] a role.” - R. O’Donnell, Analysis of Boolean Functions
Kalai’s questions Kalai has asked several “uniform distribution” questions about related circuit classes: Question: Can monotone-TC0 approximate Rec-Maj-3? Question: Are monotone-TC0 functions “noise stable”? Question (“approximate Ajtai-Gurevich”) Are there monotone functions in AC0 that can’t be approximated by monotone-AC0? Question (“approximate Ajtai-Gurevich for TC0”) Are there monotone functions in TC0 that can’t be approximated by monotone-TC0?
A closing observation and question Theorem 1: Any depth-d monotone Maj circuit computing must have size Observation: “approximate” version of Theorem 1 does not hold already for depth-1 circuits of Maj gates, in a strong sense: • Any n-variable monotone weighted threshold function is 0.99approximated by a single Maj gate of size O(n) [DDFS12]. Question: what about “d>1” version of the above? Can constantdepth monotone-Maj circuits approximate constant-depth monotone weighted threshold functions?
Thank you! Thank you!