1. Finite monoids, regular languages, circuit complexity and logic Pascal Tesson Laval University, Quebec City, Canada

2. P.TessonDagstuhl Seminar on Circuits, Logic and Games 2 Fact: Circuit complexity is difficult...... so it’s important to develop numerous angles of attack on the central questions. Logic has been very helpful. (e.g. logical characterizations of AC 0, ACC 0.) Algebra has been helpful: –Smolensky’s lower bounds. –Programs over finite monoids (Barrington, Straubing, Thérien).

3. P.TessonDagstuhl Seminar on Circuits, Logic and Games 3 Quick circuit reminder NC 1 : languages recognized by family of AND/OR circuits with fan-in 2 and depth O(log n). AC 0 : languages recognized by family of AND/OR circuits with arbitrary fan-in, depth O(1) and poly-size. ACC 0 : languages recognized by family of AND/OR/MOD q circuits with arbitrary fan-in, depth O(1) and poly-size. CC 0 : languages recognized by family of MOD q circuits with arbitrary fan-in, depth O(1) and poly-size.

4. P.TessonDagstuhl Seminar on Circuits, Logic and Games 4 Regular languages and finite monoids In this talk... Finite monoids to understand regular languages. Finite monoids to characterize boolean circuit complexity classes. Finite monoids to understand the circuit complexity of regular languages. Circuit complexity First order logic over words

5. P.TessonDagstuhl Seminar on Circuits, Logic and Games 5 Outline I. Finite automata and finite monoids II. Programs over finite monoids III. Some tools for lower bounds IV. Circuit complexity of regular languages V.Conclusion

6. P.TessonDagstuhl Seminar on Circuits, Logic and Games 6 Finite monoids and automata Monoid: set + binary associative operation + identity element. Examples –  * is a monoid under concatenation. Empty word  is the identity element. –For every k, the set T k of functions t:[k]  [k] forms a monoid under composition. –For a finite automaton A, each finite word induces a transformation on the set of states. The transition monoid of A is the submonoid of T |A|.

7. P.TessonDagstuhl Seminar on Circuits, Logic and Games 7 Finite monoids = finite automata A finite monoid can be conveniently represented by a finite automaton –states: elements of the monoid. –alphabet: elements of the monoid. –transitions:  (m,n) = m  n Finite monoid  finite automata: just two points of view.

8. P.TessonDagstuhl Seminar on Circuits, Logic and Games 8 Def’n: A language L   * is recognized by the finite monoid M if there exists homomorphism  :  *  M and a subset F  M such that  -1 (F) = L. Theorem: L is regular iff it is recognized by some finite monoid M. Def’n: The syntactic monoid of L is the transition monoid of its minimal automaton.

9. P.TessonDagstuhl Seminar on Circuits, Logic and Games 9 Algebraic automata theory Why use this algebraic point of view? Important classes of regular languages can be characterized by the algebraic properties of their syntactic monoids. Often gives algorithms to test membership in a given class of regular languages.

10. P.TessonDagstuhl Seminar on Circuits, Logic and Games 10 Star-free languages A regular language is star-free if it can be defined by a regular expression using , ;, concatenation and boolean operations but without the Kleene star. Example: L = (ab)* is a star-free language since L = ; c b Å a ; c Å ( ; c aa ; c ) c Å ( ; c bb ; c ) c But how does one decide whether a given regular language is star-free?

11. P.TessonDagstuhl Seminar on Circuits, Logic and Games 11 Theorem: [Schützenberger] L is star-free iff its syntactic monoid is aperiodic, i.e. contains no non-trivial subgroup. It can also be shown that M is aperiodic iff there exists n s.t. x n+1 = x n for all x 2 M.

12. P.TessonDagstuhl Seminar on Circuits, Logic and Games 12 Logic over words View finite words over  as a linearly ordered  -colored structure. We construct first-order sentences using the following atomic formulas: 1.for each a 2  a unary predicate Q a w ² Q a xiff w x = a 2.x < y (with the obvious semantics) We can also augment the logic with modular counting quantifiers. 9 i mod p x  (x)(there exist i modulo p x’s s.t.  holds)

13. P.TessonDagstuhl Seminar on Circuits, Logic and Games 13  x  y Q a x  ((y < x)  Q b y) defines the language b * a  * 2.To define (ab)* we can use the sentence: 8 x 8 y (Q b x ! [ 9 z (z<x)]) Æ (Q a x ! [ 9 z (x < z)]) Æ [((x  y) Æ Q a x Æ Q a y) Ç ((x  y) Æ Q b x Æ Q b y)] ! 9 z [(x <z<y) Ç (y <z<x)]

14. P.TessonDagstuhl Seminar on Circuits, Logic and Games 14 Theorem: [McNaughton-Papert] L is star-free iff it is definable in FO[<] iff L’s syntactic monoid is aperiodic. Theorem: L is definable in FO+MOD[<] iff its syntactic monoid is solvable. Theorem: L is definable in MOD[<] iff its syntactic monoid is a solvable group.

15. P.TessonDagstuhl Seminar on Circuits, Logic and Games 15 Outline I. Finite automata and finite monoids II. Programs over finite monoids III. Some tools for lower bounds IV. Circuit complexity of regular languages V.Conclusion

16. P.TessonDagstuhl Seminar on Circuits, Logic and Games 16 From homomorphisms to programs a1a1 a2a2...a n-1 anan m1m1 m2m2...m n-1 mnmn homomorphism  Multiply in M result is m =  (input) accept if m  F

17. P.TessonDagstuhl Seminar on Circuits, Logic and Games 17 From homomorphisms to programs a1a1 a2a2...a n-1 anan m1m1 m2m2 m3m3...m s-1 msms program over M: each output element depends on a single input position Multiply in M result is m =  (input) accept if m  F

18. P.TessonDagstuhl Seminar on Circuits, Logic and Games 18 Programs over monoids An n-input program  over M of length s is a sequence of instructions I 1 I 2... I s where each instruction is a pair: I j = (f j, k j ) where f j :   M and 1  k j  n. The output of  on input w = a 1... a n is the monoid element  w) = f 1 (a k 1 )  f 2 (a k 2 ) ...  f s (a k s ) A language L   n is recognized by a program  over M if there exists F  M such that w  L iff  (w)  F.

19. P.TessonDagstuhl Seminar on Circuits, Logic and Games 19 Programs over monoids To recognize subsets of  * we use a family {  n } n  0 where each  n processes inputs of length n. The length of such a family is then a function of n. Often, we require that {  n } n ¸ 0 is s.t. the n th program is constructible within some resource bound. ) uniformity restrictions. (not our problem today!)

20. P.TessonDagstuhl Seminar on Circuits, Logic and Games 20 Bounded-width branching programs Programs over monoids: originally a point of view on bounded width branching programs. x1x1 x7x7 x5x5 x1x1 On input x 1 x 2... x n : the red arrows are followed when the queried bit is 0, the blue arrows are followed when the bit is 1. For example if x 1 = 0, x 7 = 1.

21. P.TessonDagstuhl Seminar on Circuits, Logic and Games 21 Barrington’s theorem Theorem: [Barrington] A language lies in NC 1 iff it can be recognized by a polynomial length family of programs over a finite monoid. In fact, any simple non-Abelian group will do. Recall: the commutator of two group elements [g,h] = g -1 h -1 gh. If G is simple and non-Abelian, then [G,G] = G.

22. P.TessonDagstuhl Seminar on Circuits, Logic and Games 22 Barrington’s theorem (proof) Easy direction: show that L recognized by a poly- length program over a finite monoid ) L 2 NC 1. (more on this later) Hard direction: suppose L 2 NC 1 and G simple non- Abelian. Show by induction on depth d: if C is an AND/OR circuit with binary fan-in and g 2 G, there is a program  C,g of length O(4 d ) whose output is g if C evaluates to 1 and 1 G otherwise.

23. P.TessonDagstuhl Seminar on Circuits, Logic and Games 23 Barrington theorem’s proof (cont’d) Suppose for simplicity that g = [g 1,g 2 ] and assume the ouput of C is AND(C 1,C 2 ). By induction, exists  C 1,g 1 and  C 2,g 2 of length O(4 d-1 ). Define  C,g = [  C 1,g 1,  C 2,g 2 ] If C 1 = 0 then  C 1,g 1 outputs 1 G and so [1 G,  C 2,g 2 ] =  C 2,g 2 1  C 2,g 2 = 1 G. If C 1 and C 2 both evaluate to 1, the program  C,g outputs [g 1,g 2 ] = g.

24. P.TessonDagstuhl Seminar on Circuits, Logic and Games 24 Alternative point of view: if L is a regular language whose syntactic monoid is non-solvable then L is NC 1 complete under non-uniform projections.

25. P.TessonDagstuhl Seminar on Circuits, Logic and Games 25 Algebraic characterizations in NC 1 Theorem: K 2 AC 0 iff K is recognized by a poly-length program over a finite aperiodic monoid. K 2 CC 0 iff K is recognized by a poly-length program over a finite solvable group. K 2 ACC 0 iff K is recognized by a poly-length program over a finite solvable monoid.

26. P.TessonDagstuhl Seminar on Circuits, Logic and Games 26 Contrast with: Theorem: K 2 AC 0 iff K is definable in FO[Arb], i.e. FO extended with arbitrary numerical predicates. K 2 CC 0 iff K is definable in MOD[Arb]. K 2 ACC 0 iff K is definable in FO+MOD[Arb]

27. P.TessonDagstuhl Seminar on Circuits, Logic and Games 27 So what? Looks nice but how is that useful? 1.Provides some insight into the power and limitations of these circuit classes. Since AC 0 corresponds to aperiodic monoids, it’s natural to see PARITY as the canonical example of a language that these circuits can’t compute. 2.Separation conjectures can be reformulated algebraically. Showing CC 0  ACC 0 translates into show AND cannot be computed by a poly-length program over a solvable group. 3.Roadmap: start with very simple classes of solvable groups and work your way up. 4.More generally, finer grain in the analysis: the power of the program depends both on length and on structure of underlying monoid.

28. P.TessonDagstuhl Seminar on Circuits, Logic and Games 28 Some results Theorem: [Barrington-Straubing] If L has a neutral letter and is recognized by a program of length o(n log log n) over some finite monoid M then L can be recognized via morphism by a direct product of M and its reverse M r.

29. P.TessonDagstuhl Seminar on Circuits, Logic and Games 29 Theorem [Barrington-Straubing-Thérien] Any program over a group G such that [G,G] is a p-group requires exponential length to compute AND.

30. P.TessonDagstuhl Seminar on Circuits, Logic and Games 30 Outline I. Finite automata and finite monoids II. Programs over finite monoids III. One useful tool for lower bounds IV. Circuit complexity of regular languages V.Conclusion

31. P.TessonDagstuhl Seminar on Circuits, Logic and Games 31 Communication complexity k players collaborate to determine if an input string w = a 1 a 2... a n belongs to a given language L. The player j sees each a i except those such that i  j (mod k). a 1 a 4 a 7...a 2 a 5 a 8...a 3 a 6 a 9... The k-party communication complexity of L is the least amount of bits that the parties need to exchange in the worst-case to determine if their input belongs to L. One can similarly define the communication complexity of a monoid M as the complexity of evaluating the product m 1 m 2... m n in M.

32. P.TessonDagstuhl Seminar on Circuits, Logic and Games 32 A general framework for lower bounds Theorem: If M has communication complexity O(f), then any language recognized by a program of length s over M has communication complexity at most O(f(s)). The same holds for various variants of the communication complexity model. This gives an algebraic point of view on these lower bound techniques and exposes their limits.

33. P.TessonDagstuhl Seminar on Circuits, Logic and Games 33 Theorem: [T., Thérien] In the two-party model, a monoid M has communication complexity:  (1) iff M is commutative  (log n) iff M is non commutative but every subgroup of M is abelian and M satisfies (xy) n (yx) n (xy) n = (xy) n for some n. (we denote this class as DO Å H(Ab) )  (n) otherwise.

34. P.TessonDagstuhl Seminar on Circuits, Logic and Games 34 Outline I. Finite automata and finite monoids II. Programs over finite monoids III. One useful tool for lower bounds IV. Circuit complexity of regular languages V.Conclusion

35. P.TessonDagstuhl Seminar on Circuits, Logic and Games 35 The classical result PARITY  AC 0 shows that understanding the circuit complexity of regular languages is central to progress in the field. By the algebraic characterizations of AC 0, CC 0 and ACC 0 we already have good tools for a first, rough classification. How precise can we be about the circuit complexity of regular languages?

36. P.TessonDagstuhl Seminar on Circuits, Logic and Games 36 Theorem: [Koucký, Pudlák, Thérien] A regular language (with a neutral letter) can be computed by an ACC 0 circuit using O(n) wires iff its syntactic monoid lies in DO Å H(Ab). Theorem: [Chandra, Fortune, Lipton] Any regular language in ACC 0 can be computed by a circuit using only O(ng -1 (n)) wires for any primitive recursive g.

37. P.TessonDagstuhl Seminar on Circuits, Logic and Games 37 Some ideas of the proof The upper bound relies on the combinatorial characterization of the regular languages with syntactic monoids in this class. For the lower bound: deep results about superconcentrators are needed to show that if L is regular with 2-party communication complexity  (n) then L requires a superlinear number of wires.

38. P.TessonDagstuhl Seminar on Circuits, Logic and Games 38 Open problem Question: What regular languages can be computed by an AC 0, CC 0 or ACC 0 circuit using only linearly many gates? Hint: These correspond exactly to FO 2 [Arb], MOD 2 [Arb] and FO+MOD 2 [Arb]. (see [Koucký,Lautemann,Thérien]) Conjectured answers: It is believed that the answer will have some algebraic form. In particular: show that AC 0 circuits for (ab)* require a superlinear number of gates in the presence of a neutral letter.

39. P.TessonDagstuhl Seminar on Circuits, Logic and Games 39 Conclusion Open problems suggested by the algebraic point of view: –Show that AND cannot be computed by poly-length program over S 4. –Show that AND cannot be computed by poly-length program over a super-solvable group.