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

2 Clemens Lautemann

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

4 Computational complexity of regular languages Algebraic properties of regular lang’s  computational complexity of these lang’s [B, BT, Sz, TT, KPT, …] Algebraic properties of regular lang’s  computational complexity of these lang’s [B, BT, Sz, TT, KPT, …] A*(ac*a)A* 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?

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

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

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

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

9 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 [McNaughton]: f is expressible by a first order formula using only “<“ predicate iff f is a star-free regular language in AC 0

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

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

12 Non-expressibility of functions in first order logic Non-expressibility of functions in first order logic  impossibility to compute these functions by AC 0 circuits. So far: Impossibility to compute 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.

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

14 f is expressible by a first order formula of quantifier depth k using structures S 1,., S 2,., … 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 S n, u and S n, 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 S n,.  strings u, w, f ( u )=0 and f ( w )=1 such that Duplicator has a winning strategy on S n, u and S n, w.

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

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

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

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