Circuit Lower Circuit Lower Bounds via Bounds via

Ehrenfeucht-Ehrenfeucht-FraïssFraïsséé Games Games

MichalMichal Koucký Koucký

Joint work with: Clemens LautemannJoint work with: Clemens Lautemann, , Sebastian PoloczekSebastian Poloczek, Denis Thérien, Denis Thérien

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ClemensClemens Lautemann Lautemann

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Circuit complexity of Boolean functionsCircuit complexity of Boolean functions

Relationship among circuit classes:Relationship among circuit classes:

ACAC00 ACC ACC00 TC TC00 NC NC11

Circuit complexity of concrete Circuit complexity of concrete functions:functions:

e.g., INTEGER ADDITIONe.g., INTEGER ADDITION

- - ((n gn g O O((d d ))((n n )) wires)) wires

- - OO((n gn g O O((d d ))((n n )) gates)) gates

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Computational complexity of regular languagesComputational complexity of regular languages

Algebraic properties of regular lang’s Algebraic properties of regular lang’s computational complexity of these lang’s [B, computational complexity of these lang’s [B, BT, Sz, TT, KPT, …]BT, Sz, TT, KPT, …]

AA*(*(acac**aa))AA**

- - ((n gn g O O((d d ))((n n )) wires)) wires

- - OO((n gn g O O((d d ))((n n )) gates)) gates

Question:Question: Does a linear number of gates Does a linear number of gates suffices to compute the above language? suffices to compute the above language?

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Possible tools to answer these Possible tools to answer these questionsquestions

→→ descriptive complexity – descriptive complexity – characterization of complexity characterization of complexity classes in terms of logic.classes in terms of logic.

→→ possibility to use tools from possibility to use tools from logic.logic.

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Our results:Our results:

Logic characterization of languages Logic characterization of languages computable by linear size ACcomputable by linear size AC00 circuits. circuits.

((→→ Lin-AC Lin-AC0 0 = FO= FO22[arb] )[arb] )

Arguments using Ehrenfeucht-FraïssArguments using Ehrenfeucht-Fraïsséé games games of non-expressibility of certain functions in of non-expressibility of certain functions in first order logic. first order logic.

((→→ PARITY is not in AC PARITY is not in AC0 0 ))

ACAC00 circuits … circuits … constant-depth circuits constant-depth circuits consisting of consisting of polynomially polynomially many many , , , , gates. gates.

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First order structureFirst order structure

universe universe UU = {1, …, = {1, …, nn }} numerical numerical predicates – relations predicates – relations RR11, …, , …,

RRmm

input input predicate predicate ( ( ii ) is true iff ) is true iff wwii = = 11

00

00

11

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Representing a Boolean function Representing a Boolean function f f : : {0,1}*{0,1}*{0,1}{0,1}

First order formula First order formula

xx yy zz ( P( ( P(xx, , yy) ) ( R( ( R(xx, , zz )) ( ( zz )) )) ))

Sequence of first order structuresSequence of first order structures

SS1,.1,. , , SS2,.2,. , , SS3,.3,. , … , …

For all For all ii, , ww : : SSii, , ww has universe {1,… has universe {1,…n n }}

SSii, , . . have the same numerical have the same numerical predicatespredicates

→→ ff ( ( ww )=1)=1 iff iff SSii, , ww

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ThmThm [Immerman]: [Immerman]: ff is expressible by a first order formula is expressible by a first order formula

iffiffff is in AC is in AC00

ThmThm [BIS]: [BIS]: ff is expressible by a first order formula using only is expressible by a first order formula using only

“BIT“ predicate“BIT“ predicateiffiff

f f is in uniform AC is in uniform AC00

ThmThm [McNaughton]: [McNaughton]: ff is expressible by a first order formula using only is expressible by a first order formula using only

“<“ predicate“<“ predicateiffiff

ff is a star-free regular language in AC is a star-free regular language in AC00

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Thm:Thm:

ff is expressible by a first order formula is expressible by a first order formula using only two variablesusing only two variables

iffiff

f f is a in linear size AC is a in linear size AC00

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Example:Example:

Function “at least two input bits are Function “at least two input bits are set to one”:set to one”:

xx yy ( ( xx < < yy ( ( xx ) ) ( ( yy ) ) ))

““at least three input bits are set to at least three input bits are set to one”one”

xx (( ( ( xx ) ) yy ( ( ( ( yy ) ) xx < < yy ( ( xx ( ( xx ) ) yy < < xx ))))))

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Non-expressibility of functions in first Non-expressibility of functions in first order logicorder logic

impossibility to compute these impossibility to compute these functions by functions by ACAC00 circuits. circuits.

So far:So far: Impossibility to compute functions Impossibility to compute functions by ACby AC00 circuits circuits

non-expressibility of functions non-expressibility of functions in first order in first order logic.logic.

Thm:Thm: PARITY is not expressible in first PARITY is not expressible in first order logic.order logic.

Cor:Cor: PARITY is not in AC PARITY is not in AC00..

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Ehrenfeucht-FraïssEhrenfeucht-Fraïsséé games: games:

Spoiler Spoiler : wants to point out a difference: wants to point out a difference

DuplicatorDuplicator : wants to show that structures : wants to show that structures are isomorphicare isomorphic

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1100

00

0000

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ff is expressible by a first order formula of is expressible by a first order formula of quantifier depth quantifier depth kk using structures using structures SS1,.1,. , , SS2,.2,. , … , …

Spoiler has a winning strategy in Spoiler has a winning strategy in kk-round EF -round EF game on game on SSnn, , uu and and SSnn, , ww for any for any uu, , ww s.t. s.t. f f ( ( u u )=0 )=0 and and f f ( ( w w )=1.)=1.

To prove non-expressibilityTo prove non-expressibility

Want:Want: For For nn large enough and any choice of large enough and any choice of numerical predicates for structure numerical predicates for structure SSnn, ., .

strings strings uu, , ww , , f f ( ( u u )=0 and )=0 and f f ( ( w w )=1 such )=1 such that that Duplicator has a winning strategy on Duplicator has a winning strategy on SSnn, , uu and and SSnn, , ww . .

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Duplicator has a winning strategyDuplicator has a winning strategy

localy isomorphiclocaly isomorphic structures (elt’s of structures (elt’s of same same game typegame type))

Claim: Claim: enough to assign 0/1 to only enough to assign 0/1 to only part of the part of the universe.universe.

11 11

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Proof overviewProof overview

Induction on number of pebblesInduction on number of pebbles

Switching lemmaSwitching lemma

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ConclusionsConclusions Lin-ACLin-AC00 formulas with two variables formulas with two variables Non-expressibility of functions using Non-expressibility of functions using

Ehrenfeucht-FraïssEhrenfeucht-Fraïsséé games games Cons:Cons:

Not as simple (as we hoped for)Not as simple (as we hoped for) Too powerfulToo powerful

Pros:Pros: Could be tuned up for e.g. uniform lower-Could be tuned up for e.g. uniform lower-

boundsbounds Could be possibly simplerCould be possibly simpler

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Open problemsOpen problems Simple proof of non-Simple proof of non-

expressibilityexpressibility

Is integer ADDITION in ACIs integer ADDITION in AC00 with with linear number of gates?linear number of gates?

Is Is AA*(*(acac**aa))AA** in AC in AC00 with linear with linear number of gates?number of gates?

polyPtt/ polyP

tt/ polyP

tt/ polyP

tt/ polyP

tt/ polyP

tt/ polyP

tt/ polyP

tt/