1 / 16

Circuit Complexity of Regular Languages

Circuit Complexity of Regular Languages. Michal Koucký (Institute of Mathemaics, AS Č R, Praha). Regular languages Introduced in 50’s. Many equivalent definitions: Languages recognized by finite automata. Languages described by regular expressions.

jescie-soto
Download Presentation

Circuit Complexity of Regular Languages

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

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

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

  4. Boolean circuits: AND OROR OR AND x1x2 x4 x7 → non-uniform model of time-bounded computation. Fundamental question: How large circuits does one need to compute specific Boolean functions, e.g., SAT?

  5. All regular languages are computable by logarithmic depth linear size circuits (NC1). → AND, OR of fan-in 2, NOT of fan-in 1. a b b a a … b b a a … b

  6. Some regular languages are computable by constant depth polynomial size circuits (AC0) → AND, OR of arbitrary fan-in, NOT of fan-in 1. • AC0µNC1 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 AC0 circuits. Eg.: PARITY = { w in {0,1}* that contain even number of ones}.

  7. Some more regular languages are computable by constant depth polynomial size circuits with additional MOD-q gates (ACC0) → AND, OR, MOD-q of arbitrary fan-in, NOT of fan-in 1. • AC0 ( ACC0µ NC1 Eg.: PARITY = { w in {0,1}* that contain even number of ones}. Big Open Problem: ACC0 =NC1 ? • [Barrington] – regular NC1 – complete languages.

  8. All regular languages are computable by linear size NC1-circuits. • Thm: • All regular languages in AC0 are computable by AC0-circuits of size O( n . gO(d)( n )). • All regular languages in ACC0 are computable by ACC0-circuits of size O( n . gO(d )( n)) if they are not NC1-complete and of size O(n1+ε) otherwise. g0( n ) = n/2 g 2( n) = log* n g 1( n) = log n g d ( n) = g d -1*( n )

  9. Corollary: To separate ACC0 from NC1 it suffices to show that a chosen NC1-complete regular language cannot be computed by ACC0 circuits of size, say, O( n3/2). E.g.: Word problem over S5.

  10. Thm [CFL]: If a regular language has a group-free syntactic monoid then it is computable by AC0-circuits of size O( n . g d ( n)). Note: LENGTH(2) is in AC0 but its syntactic monoid contains a group.

  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 √n … √n√n√n →circuits of size n1+ε

  12. Chandra-Fortune-Lipton procedure: If evaluating the prefix product of n monoid elements can be done by circuits of size O( n . g2d ( n)) [*] then it can be done by circuits of size O( n . g2d +1( n)). …

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

  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 ACC0-circuits with linear number of gates but not with linear number of wires (it requires Θ( n . g d ( n)) wires.)

  15. Question: Is U = (ac*bc*)* computable by AC0-circuits with linear number of gates? Yes → U separates AC0-circuits with linear number of wires from that with linear number of gates. No → Integer Addition does not have AC0-circuits with linear number of gates. • [CFL] Integer Addition has AC0-circuits with almost linear number of wires.

  16. Open problems • Is ACC0 equal to NC1 ? • Has Integer Addition AC0 circuits with linear number of gates? • Has (ac* bc*)* AC0 circuits with linear number of gates?

More Related