arithmetizing classes around nc1 and l

8/10/2019 Arithmetizing classes around NC1 and L

1/89

IntroductionClasses Computationally Equivalent to NC1

Arithmetic VersionsBounded width Arithmetic circuits

Open questions

Arithmetizing Circuits around NC1 an L

Raghavendra Rao B VJoint work with

Nutan Limaye and Meena Mahajan

February 6, 2007

Raghavendra Rao B V Joint work with Nutan Limaye and Meena Mahajan

http://find/


2/89



Open questions

Boolean Circuits

Definitions

DefinitionBoolean Circuit

Directed acyclic graph where nodes labelled with{, , , 0, 1, x1, . . . , xn}.


http://find/


3/89



Open questions

Boolean Circuits

Definitions



A node of out-degree zero, calledoutputnode of the circuit


http://find/


4/89



Open questions

Boolean Circuits

Definitions




{x1, , xn} are the inputs for the circuit, where xi {0, 1}

Raghavendra Rao B V Joint work with Nutan Limaye and Meena MahajanArithmetizing Circuits around NC1 an L

I d i
http://find/


5/89



Open questions

Boolean Circuits

Definitions





fan in(fan out) of a node is its in-degree(out-degree)


I t d ti
http://find/


6/89



Open questions

Boolean Circuits

Definitions





fan in(fan out) of a node is its in-degree(out-degree) depth- length of longest path from output node to input node


Introduction
http://find/


7/89



Open questions

Boolean Circuits

Definitions





fan in(fan out) of a node is its in-degree(out-degree) depth- length of longest path from output node to input node

width- maximum number of nodes at any particular level


Introduction
http://find/


8/89



Open questions

Boolean Circuits

NC1

Class of problems which can be decided by O(log n) depth,poly size, constant fan-in circuits


Introduction
http://find/


9/89



Open questions

Boolean Circuits

NC1

Class of problems which can be decided by O(log n) depth,poly size, constant fan-in circuits

Examples: parity ofn bits, sorting.


Introduction
http://find/


10/89



Open questions

Boolean Circuits

. . ... . . . .

x1 x2 x3 x4 xnxn1xn2xn3

O(logn)

Figure: An NC1 circuit for parityofn bits.Raghavendra Rao B V Joint work with Nutan Limaye and Meena MahajanArithmetizing Circuits around NC1 an L

IntroductionB hi P
http://find/


11/89



Open questions

Branching ProgramsBounded Width CircuitsPrograms over automataVisibly pushdown automata-VPA

Classes Equivalent to NC1

Bounded Width Branching Programs - BWBP


IntroductionBranching Programs
http://find/


12/89

Classes Computationally Equivalent to NC1


Open questions




Bounded Width Circuits - BWC


Introduction1 Branching Programs
http://find/


13/89



Open questions





Log width formula - LWF


Introduction1 Branching Programs
http://find/


14/89



Open questions






Branching Program over NFA - BP-NFA


IntroductionCl C i ll E i l NC1

Branching Programs
http://find/http://goback/


15/89



Open questions






Branching Program over NFA - BP-NFA

Branching Program over Visibly Pushdown Automata -BP-VPA


IntroductionCl C t ti ll E i l t t NC1

Branching Programs
http://find/


16/89



Open questions

a c g og a sBounded Width CircuitsPrograms over automataVisibly pushdown automata-VPA

Branching Programs

Definition

Layered acyclic digraph P, edges labelled with literals,{x1, , xn, x1, , xn, 0, 1}



Branching Programs
http://find/


17/89



Open questions

g gBounded Width CircuitsPrograms over automataVisibly pushdown automata-VPA

Branching Programs

Definition


Two special nodes swith zero in degree and twith zero outdegree



Branching Programs
http://find/


18/89

Classes Computationally Equivalent to NCArithmetic Versions

Bounded width Arithmetic circuitsOpen questions

Bounded Width CircuitsPrograms over automataVisibly pushdown automata-VPA

Branching Programs

Definition


Two special nodes swith zero in degree and twith zero outdegree

Acceptance of input = Existence of an s t path.



Branching Programs
http://find/


19/89




BWBP

DefinitionBWBP - Languages accepted bypoly-sizebranching programs ofconstant width



Branching ProgramsB d d Wid h Ci i
http://find/http://goback/


20/89




BWBP

DefinitionBWBP - Languages accepted bypoly-sizebranching programs ofconstant width

Example

. . .Odd

Even

x1

x1

x2

x2

x2

x2

Figure: Width-2 branching program for parity



Branching ProgramsB d d Width Ci it
http://find/


21/89

p y qArithmetic Versions



Theorem (Barrington 89)BWBP=NC1



Branching ProgramsBounded Width Circuits
http://find/


22/89

p y qArithmetic Versions



Bounded width circuits

DefinitionBWC (SC0 ) -Accepted byconstantwidth,poly-sizecircuits



http://find/


23/89


Open questions


Bounded width circuits

DefinitionBWC (SC0 ) -Accepted byconstantwidth,poly-sizecircuits

Example1x1

x2

x2

x1

x2

x2

x1

x1

x2

x2

x2

x2



http://find/


24/89


Open questions


Theorem (Barrington 89)BWBP=BWC



A i h i V i

http://find/


25/89


Open questions


Formula

Definition

Boolean Formula: Boolean circuits, where out-degree(fan-out) ofevery node is at most one.



A ith ti V i

http://find/


26/89


Open questions

Programs over automataVisibly pushdown automata-VPA

Formula

Definition

Boolean Formula: Boolean circuits, where out-degree(fan-out) ofevery node is at most one.LWF : Accepted by poly-size, logarithmicwidth formula.



Arithmetic Versions

http://find/


27/89


Open questions


Formula

Definition

Boolean Formula: Boolean circuits, where out-degree(fan-out) ofevery node is at most one.LWF : Accepted by poly-size, logarithmicwidth formula.

Theorem (Istrail, Zivkovic )

NC1 =LWF



Arithmetic Versions

http://find/


28/89


Open questions


BP-NFA

ProgramIn

put

P

Projection

Two way input tape

NFA

One way input tape

Accept/Reject

Ifxi= 0 then aelse b



Arithmetic Versions

http://find/


29/89


Open questions


BP-NFA

ProgramIn

put

P

Projection

Two way input tape

NFA

One way input tape

Accept/Reject

Ifxi= 0 then aelse b

Theorem (Barrinton, CMTV )

BWBP=BP-NFA



Arithmetic Versions

Branching ProgramsBounded Width CircuitsP
http://find/


30/89


Open questions


VPA

Definition (Alur,Madhusudan )

VPA is a

A Pushdown automata M, with input alphabet partitionedas (c, r, int)



Arithmetic Versions

Branching ProgramsBounded Width CircuitsP g ms t m t
http://find/


31/89



VPA


VPA is a


Ms action is guided by the input alphabet



Arithmetic Versions

Branching ProgramsBounded Width CircuitsPrograms over automata
http://find/


32/89



VPA


VPA is a



Letters from c force Mto perform onlypush



Arithmetic Versions

http://find/


33/89



VPA


VPA is a




r forcepop



Arithmetic Versions

http://find/


34/89



VPA


VPA is a




r forcepop int force internaloperations



Arithmetic VersionsB d d idth A ith ti i it

http://find/


35/89



BP-VPA

P

Accept/Reject

.

.

. .

.

.Progr

amI

nput

Projection

VPA



Arithmetic VersionsBo nded width Arithmetic circ its

http://find/


36/89


gVisibly pushdown automata-VPA

BP-VPA

P

Accept/Reject

.

.

. .

.

.Progr

amI

nput

Projection

VPA

Theorem (Dymond)

BP-VPANC1




http://find/


37/89


gVisibly pushdown automata-VPA


NC1

BWBP

BWC

LWF

BP-NFA

BP-VPA



http://find/


38/89


Arithmetization over N

Replace withand with +, NOTgates are allowed onlyat input.



http://find/


39/89



Replace withand with +, NOTgates are allowed onlyat input.

Counting number of proving subtrees

Example

x1 x2 x3 x1

x4 x3



http://find/


40/89


0 0 1 1

1 1

0 0 1 1

1 1

x1= 0, x2= 0, x3= 1, x4= 1

Figure: Two different proving subtrees, withx1 = 0, x2 = 0, x3 = 1, x4 = 1



http://find/


41/89

Open questions


Circuits Replace with and with +. Counting number of proving subtrees

Branching Programs Counting the number ofs-tpaths in a branching program



http://find/


42/89

Open questions

Definition

#NC1 : Functions f :{0, 1}n N, computed by NC1 circuits over{, +}



http://find/


43/89

Open questions

Definition


#BWBP : Functions f :{0, 1}n

N, computed as number ofs-tpaths in a branching program




O i
http://find/


44/89

Open questions

Definition




#LWF : Computed by log width, poly size formula over {, +}




O ti
http://find/


45/89

Open questions

Definition





BP-NFA : Counting number of accepting paths ofNFA




Open questions
http://find/


46/89

Open questions

Definition





BP-NFA : Counting number of accepting paths ofNFA

BP-VPA : Counting number of accepting paths ofVPA




Open questions
http://find/


47/89

Open questions

Theorem (Caussinus et al,IZ )

#BWBP= #BP-NFA #NC1 = #LWF




Open questions
http://find/


48/89

Open questions

Theorem#BP-VPA= #BP-NFA= #BWBP




Open questions
http://find/


49/89

Open questions


Proof

#BP-NFA#BP-VPA, straightforward.

M= (Q, , , , QF) be a VPA.




Open questions
http://find/


50/89

p q


Proof


M= (Q, , , , QF) be a VPA. let y=y1 yn be the input for M




Open questions
http://find/


51/89

p q


Proof



Construct a branching program P, with n+ 2 layers




Open questions
http://find/


52/89


Proof




0th layer consists of a single node(q0, $, 0), marked as s




Open questions
http://find/


53/89


Proof





ith

layer has vertices labelled (q, X, i), for qQ and X ,except for i= 0, n+ 1.




Open questions
http://find/


54/89


Proof





ith

layer has vertices labelled (q, X, i), for qQ and X ,except for i= 0, n+ 1.

Layer n+ 1 contains t=(qf, $, n+ 1)




Open questions
http://find/


55/89

Case 1 Ifaint and p(q, yi), then edge(q, A, i)(p, A, i+ 1), for all A




Open questions
http://find/


56/89

Case 1 Ifaint and p(q, yi), then edge(q, A, i)(p, A, i+ 1), for all A

Case 2 Ifacand (p, B) (q, yi), then edge(q, A, i)(p, B, i+ 1)




Open questions
http://find/


57/89

Case 3 : yir

Consider p(q, A, yi)


Introduction



Open questions
http://find/


58/89

Case 3 : yir

Consider p(q, A, yi)

BA

BBA

StackContent

Time

B pushed

Figure: Stack contents between,B is pushed and A is popped


Introduction

Classes Computationally Equivalent to NC1Arithmetic Versions

http://find/


59/89

Case 2 : yir Consider p(q, A, yi)

Using Dymonds result, Bcan be computed in NC1, hence in

BWBP(deterministic)


Introduction


http://find/


60/89



BWBP(deterministic) Substitute this deterministic branching program P in between

nodes (q, A, i) and (p, B, i+ 1), and do necessary staggering


Introduction


http://find/


61/89



BWBP(deterministic) Substitute this deterministic branching program P in between

nodes (q, A, i) and (p, B, i+ 1), and do necessary staggering

Accepting paths ofMare preserved in the new branchingprogram.


Introduction



Polynomial degree restriction
http://find/


62/89

Infeasibility of #BWC

A width two circuit can compute super exponential values

22n

1

1


Introduction



http://find/


63/89

x1 x2

x1 x3 x4

x1.x2

x1x2x3 + x1x4

x3 + x4x1.x2 + x1

Figure: Degree of a circuit,

http://find/


64/89

Introduction





65/89

Poly degree circuits are studied in the literature.

Poly degree, poly size = log depth, poly size,unbounded ORfanin called SAC1


Introduction



http://find/


66/89



SAC1 =LogCFL


Introduction



Open questions

http://find/


67/89



SAC1 =LogCFL

Degree restriction, bounds the maximum value that can becomputed


Introduction



Open questions

http://find/


68/89

sSCi : Poly degree, poly sized circuits of width O(logi n)


Introduction



Open questions

http://find/


69/89

sSCi : Poly degree, poly sized circuits of width O(logi n)

ObservationBPs are skew circuits which have polynomial degreehence BP-width O(logi)sSCi


Introduction



Open questions

http://find/


70/89

LogCFL

sSC

NL

OR fanin polylog

OR fanin-2

NC1 =BWBP

sSC0

BP width(log)

OR fanin log

sSC1

OR fanin logi

BPwidth(logi)

sSCi


Introduction



Open questions

http://find/


71/89

Theoremco sSCi sSC2i

Proof.Main ideas

Inductive counting for circuits, that ofBCDRT instead ofIS

Monotone branching programs for threshold(Vinay)


Introduction



Open questions

http://find/


72/89

Proof. C, be the given circuit of width w, length l, to construct C.


Introduction



Open questions

http://find/


73/89


Assume alternate ANDs and ORs.

One level ofANDs and ORs =one layer

Layer i: For every gate g, add gates cc(g|k) for 0 kw


Introduction



Open questions

http://find/


74/89




Layer i: For every gate g, add gates cc(g|k) for 0 kw cc(g|k)(x): conditional complement ofg, assuming exactly

kat previous layer evaluate to 1


Introduction



Open questions

http://find/


75/89






Add gates, count(i, k), 0 kw


Introduction



Open questions

http://find/


76/89






Add gates, count(i, k), 0 kw

count(i, k) = 1 iff exactly kgates evaluate to 1 at layer i

Raghavendra Rao B V Joint work with Nutan Limaye and Meena MahajanArithmetizing Circuits around NC1

an L

Introduction



Open questions

http://find/


77/89

Output gate: at level l

d[cc(g|d) count(l 1, d)]


an L

Introduction



Open questions

http://find/


79/89

count(i, k)

Thk(f) Thk(f)

f=all the gates at leveli f=complements of all thegates at level i

Figure: Computing count(c, k)


an L

Introduction



Open questions

http://find/


80/89

Thwk(f) =

dThwk(cc(f|d)) count(i 1, d)

d d = w

Thwk(cc(f|d)) count(i 1, d)

d = 0

Figure: Computing count(c, k)


an L

Introduction



Open questions


Open question
http://find/


81/89

Open question

DLOG sSC1?


an L

Introduction



Open questions


#sSC i
http://find/


82/89

#sSC

Definition#sSCi : The class of functions f :{0, 1}n N, computed bypoly-size, poly-degree arithmetic circuits with width O(logi n)


an L

Introduction

Classes Computationally Equivalent to NC

1


Open questions


Arithmetic classes arround NC 1
http://find/


83/89

Arithmetic classes arround NC

#NC1

#LWF


an L Introduction


1


Open questions


Arithmetic classes arround NC 1
http://find/


84/89

Arithmetic classes arround NC

#NC1

#LWF

#BWBP

#BP-NFA

#BP-VPA


an L Introduction


1


Open questions


Arithmetic classes arround NC1
http://find/


85/89

#NC1

#LWF

#BWBP

#BP-NFA

#BP-VPA

#sSC0


an L Introduction


1


Open questions


3 potential equivalent classes?
http://find/


86/89

#BWBP

#sSC0

#NC1

#sSC1

FL

SC2

#SAC1 NC2


an L Introduction


1


Open questions

Open questions
http://find/


87/89

Is co-sSCi

=sSCi

, i >0


an L Introduction


1


Open questions

Open questions
http://find/


88/89

Is co-sSCi

=sSCi

, i >0 #sSCi FL?


an L Introduction


1


Open questions

Open questions
http://find/


89/89

Is co-sSCi

=sSCi

, i >0 #sSCi FL?

Closure properties of #sSCi-div, decrement


an L
http://find/

arithmetizing classes around nc1 and l

Documents