Circuit Lower Bounds via EhrenfeuchtFrass Games Michal Kouck

Circuit Lower Bounds via Ehrenfeucht-Fraïssé Games Michal Koucký Joint work with: Clemens Lautemann, Sebastian Poloczek, Denis Thérien

Clemens Lautemann 2

Circuit complexity of Boolean functions n Relationship among circuit classes: AC 0 ACC 0 TC 0 NC 1 n Circuit complexity of concrete functions: e. g. , INTEGER ADDITION - (n g O (d )(n )) wires - O (n g O (d )(n )) gates 3

Computational complexity of regular languages n n Algebraic properties of regular lang’s computational complexity of these lang’s [B, BT, Sz, TT, KPT, …] A*(ac*a)A* - (n g O (d )(n )) wires - O (n g O (d )(n )) gates Question: Does a linear number of gates suffices to compute the above language? 4

Possible tools to answer these questions → descriptive complexity – characterization of complexity classes in terms of logic. → possibility to use tools from logic. 5

Our results: n Logic characterization of languages computable by linear size AC 0 circuits. (→ Lin-AC 0 = FO 2[arb] ) n Arguments using Ehrenfeucht-Fraïssé games of nonexpressibility of certain functions in first order logic. (→ PARITY is not in AC 0 ) AC 0 circuits … constant-depth circuits consisting of polynomially many , , gates. 6

First order structure 0 0 1 universe U = {1, …, n } n numerical predicates – relations R 1, …, Rm n input predicate ( i ) is true iff w i = 1 n 7

Representing a Boolean function f : {0, 1}* {0, 1} n First order formula x y z ( P(x, y ) ( R(x, z ) ( z )) ) n Sequence of first order structures S 1, . , S 2, . , S 3, . , … For all i, w : Si, w has universe {1, …n } Si, . have the same numerical predicates → f ( w )=1 iff Si, w 8

Thm [Immerman]: f is expressible by a first order formula iff f is in AC 0 Thm [BIS]: f is expressible by a first order formula using only “BIT“ predicate iff f is in uniform AC 0 Thm [Mc. Naughton]: f is expressible by a first order formula using only “<“ predicate iff f is a star-free regular language in AC 0 9

Thm: f is expressible by a first order formula using only two variables iff f is a in linear size AC 0 10

Example: n Function “at least two input bits are set to one”: x y ( x < y ( x ) ( y ) ) n “at least three input bits are set to one” x ( ( x ) y ( ( y ) x < y ( x ( x ) y < x ))) 11

n Non-expressibility of functions in first order logic n impossibility to compute these functions by AC 0 circuits. So far: Impossibility to compute functions by AC 0 circuits non-expressibility of functions in first order logic. Thm: PARITY is not expressible in first order logic. Cor: PARITY is not in AC 0. 12

Ehrenfeucht-Fraïssé games: 0 1 1 0 0 0 Spoiler : wants to point out a difference Duplicator : wants to show that structures are isomorphic 13

n f is expressible by a first order formula of quantifier depth k using structures S 1, . , S 2, . , … Spoiler has a winning strategy in k-round EF game on Sn , u and Sn , w for any u , w s. t. f ( u )=0 and f ( w )=1. To prove non-expressibility Want: For n large enough and any choice of numerical predicates for structure Sn , . strings u , w , f ( u )=0 and f ( w )=1 such that Duplicator has a winning strategy on Sn , u and Sn , w. 14

Duplicator has a winning strategy 1 localy isomorphic type) 1 structures (elt’s of same game Claim: enough to assign 0/1 to only part of the universe. 15

Proof overview n Induction on number of pebbles n Switching lemma 16

Conclusions n Lin-AC 0 formulas with two variables n Non-expressibility of functions using Ehrenfeucht-Fraïssé games n Cons: n Not as simple (as we hoped for) n Too powerful n Pros: n Could be tuned up for e. g. uniform lower-bounds n Could be possibly simpler 17

Open problems n Simple proof of non-expressibility n Is integer ADDITION in AC 0 with linear number of gates? n Is A*(ac*a)A* in AC 0 with linear number of gates? 18