1. Circuit Complexity of Regular Languages Michal Koucký (Institute of Mathemaics, AS ČR, Praha)

2. 2 Regular languages Introduced in 50’s. Many equivalent definitions: Languages recognized by finite automata. Languages recognized by finite automata. Languages described by regular expressions. Languages described by regular expressions. Languages corresponding to word problems over finite monoids. Languages corresponding to word problems over finite monoids.

3. 3 Why regular languages today? Used in practice, formal verification of systems… Used in practice, formal verification of systems… Provide insight into computation and circuit complexity. Provide insight into computation and circuit complexity. Provide understanding of elementary functions such as Integer Addition. Provide understanding of elementary functions such as Integer Addition. ( )

4. 4 Boolean circuits: AND AND OR OR OR OR OR OR AND AND x 1 x 2 x 4 x 7 x 1 x 2 x 4 x 7 → non-uniform model of time-bounded computation. Fundamental question: How large circuits does one need to compute specific Boolean functions, e.g., SAT?

5. 5 All regular languages are computable by logarithmic depth linear size circuits (NC 1 ). All regular languages are computable by logarithmic depth linear size circuits (NC 1 ). → AND, OR of fan-in 2, NOT of fan-in 1. b a a … b b a a … b … b a a b a a a b a b b

6. 6 Some regular languages are computable by constant depth polynomial size circuits (AC 0 ) Some regular languages are computable by constant depth polynomial size circuits (AC 0 ) → AND, OR of arbitrary fan-in, NOT of fan-in 1. AC 0 µ NC 1 AC 0 µ NC 1 Eg.: TH-2 = { w in {0,1}* that contain at least 2 ones} LENGTH(2) = { w in {0,1}* of even length} [FSS’84]: Not all regular languages are computable by AC 0 circuits. [FSS’84]: Not all regular languages are computable by AC 0 circuits. Eg.: PARITY = { w in {0,1}* that contain even number of ones}.

7. 7 Some more regular languages are computable by constant depth polynomial size circuits with additional MOD-q gates (ACC 0 ) Some more regular languages are computable by constant depth polynomial size circuits with additional MOD-q gates (ACC 0 ) → AND, OR, MOD-q of arbitrary fan-in, NOT of fan-in 1. AC 0 ( ACC 0 µ NC 1 AC 0 ( ACC 0 µ NC 1 Eg.: PARITY = { w in {0,1}* that contain even number of ones}. Big Open Problem: ACC 0 = NC 1 ? [Barrington] – regular NC 1 – complete languages. [Barrington] – regular NC 1 – complete languages.

8. 8 All regular languages are computable by linear size NC 1 -circuits. All regular languages are computable by linear size NC 1 -circuits. Thm: Thm: All regular languages in AC 0 are computable by AC 0 -circuits of size O( n. g O(d ) ( n )). All regular languages in AC 0 are computable by AC 0 -circuits of size O( n. g O(d ) ( n )). All regular languages in ACC 0 are computable by ACC 0 -circuits of size O( n. g O(d ) ( n )) if they are not NC 1 -complete and of size O(n 1+ε ) otherwise. All regular languages in ACC 0 are computable by ACC 0 -circuits of size O( n. g O(d ) ( n )) if they are not NC 1 -complete and of size O(n 1+ε ) otherwise. g 0 ( n ) = n/2 g 2 ( n ) = log* n g 1 ( n ) = log n g d ( n ) = g d -1 *( n )

9. 9 Corollary: To separate ACC 0 from NC 1 it suffices to show that a chosen NC 1 -complete regular language cannot be computed by ACC 0 circuits of size, say, O( n 3/2 ). Corollary: To separate ACC 0 from NC 1 it suffices to show that a chosen NC 1 -complete regular language cannot be computed by ACC 0 circuits of size, say, O( n 3/2 ). E.g.: Word problem over S 5.

10. 10 Thm [CFL]: If a regular language has a group- free syntactic monoid then it is computable by AC 0 -circuits of size O( n. g d ( n )). Thm [CFL]: If a regular language has a group- free syntactic monoid then it is computable by AC 0 -circuits of size O( n. g d ( n )). Note: LENGTH(2) is in AC 0 but its syntactic monoid contains a group.

11. 11 Proof: (ideas) If evaluating the product of n monoid elements can be done by circuits of size n k then it can be done by circuits of size n (1 + k)/2 If evaluating the product of n monoid elements can be done by circuits of size n k then it can be done by circuits of size n (1 + k)/2 … √n √n √n √n √n √n √n√n√n√n → circuits of size n 1+ε

12. 12 Chandra-Fortune-Lipton procedure: Chandra-Fortune-Lipton procedure: If evaluating the prefix product of n monoid elements can be done by circuits of size O( n. g 2 d ( n )) [*] then it can be done by circuits of size O( n. g 2 d +1 ( n )). …

13. 13 Last ingredient: description of regular languages by regular expressions [S,T]. Last ingredient: description of regular languages by regular expressions [S,T].

14. 14 → all regular languages are computable by their respective circuits of almost linear size. (size measured by wires and/or gates.) Question: Is it possible that all regular languages are computable by their respective circuits of linear size? Thm [KPT]: Language U = (ac*bc*)* is computable by ACC 0 -circuits with linear number of gates but not with linear number of wires (it requires Θ( n. g d ( n )) wires.)

15. 15 Question: Is U = (ac*bc*)* computable by AC 0 - circuits with linear number of gates? Yes → U separates AC 0 -circuits with linear number of wires from that with linear number of gates. No → Integer Addition does not have AC 0 - circuits with linear number of gates. [CFL] Integer Addition has AC 0 -circuits with almost linear number of wires. [CFL] Integer Addition has AC 0 -circuits with almost linear number of wires.

16. 16 Open problems Is ACC 0 equal to NC 1 ? Is ACC 0 equal to NC 1 ? Has Integer Addition AC 0 circuits with linear number of gates? Has Integer Addition AC 0 circuits with linear number of gates? Has (ac* bc*)* AC 0 circuits with linear number of gates? Has (ac* bc*)* AC 0 circuits with linear number of gates?