Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V - PowerPoint PPT Presentation

Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai 23 February, 2007 Joint work with Nutan Limaye and Meena Mahajan Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 1 / 20

Boolean Circuits ( x 1 x 3 ∨ x 1 ¯ x 4 ∨ x 2 x 3 ∨ x 2 ¯ x 4 ∨ ¯ x 1 ¯ x 2 ¯ x 3 ) ∨ ∧ ∧ ∨ ∨ ∧ ¯ ¯ ¯ x 1 x 2 x 3 x 4 x 1 x 3 Definition Directed acyclic graph where nodes labeled with {∨ , ∧ , ¬ , 0 , 1 , x 1 , . . . , x n } . A node of out-degree zero, called output node of the circuit { x 1 , · · · , x n } are the inputs for the circuit, where x i ∈ { 0 , 1 } fan in(fan out) of a node is its in-degree(out-degree) depth - length of longest path from output node to input node width - maximum number of nodes at any particular level Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 2 / 20

NC 1 Definition Class of problems which can be decided by O (log n ) depth, poly size, constant fan-in circuits Examples: ◮ parity of n bits, ◮ sorting n numbers, ◮ evaluating a boolean formula Upper bound: NC 1 ⊆ DLOG Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 3 / 20

Classes Equivalent to NC 1 Bounded Width Branching Programs : BWBP Bounded Width Circuits : BWC Log Width Formula : LWF Branching Program over NFA : BP-NFA Branching Program over Visibly Pushdown Automata : BP-VPA Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 4 / 20

Branching Programs . . . s t ¯ x 2 x 1 x 2 ¯ x 1 x 2 ¯ x 2 Figure: Width-2 branching program for parity Definition ( BWBP ) Bounded Width Branching Programs (poly size). Theorem ( Barrington ’89 ) BWBP = NC 1 Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 5 / 20

Bounded Width Circuits Definition ( SC i ) SC i = DTIME - SPACE ( poly , log i n ) = CircuitSize , Width ( poly , log i n ) BWC = SC 0 (by definition) BWC ⊆ NC 1 (divide-and-conquer) BP =skew-circuits, hence BWBP ⊆ BWC Example ¯ x 1 x 2 ¯ x 2 1 x 1 ∨ ∧ ∨ ∧ ∨ x 2 ¯ x 2 x 1 ∧ ∧ ∧ x 2 ¯ x 2 x 1 ∨ ∧ ∨ ¯ x 2 ¯ x 2 Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 6 / 20

Log Width Formula Definition (Formula) A circuit where fan out is at most one (i.e a tree ) Definition (LWF) Logarithmic Width Formula. Theorem ( Istrail, Zivkovic ’94 ) NC 1 = LWF Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 7 / 20

BP - NFA Two way input tape One way input tape Program Input P Accept/Reject NFA · · · Projection · · · · · · If x i = 0 then a else b · · · Σ ∗ ∆ ∗ Theorem (Barrington ’89 ) BWBP = BP-NFA Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 8 / 20

Visibly Pushdown Automaton: VPA Definition (Mehlhorn ’80... Alur,Madhusudan’04 ) A Visibly Pushdown Automaton is a Pushdown automaton M , with input alphabet ∆ partitioned as (∆ push , ∆ pop , ∆ int ) M’s action is guided by the input alphabet Example Language { a n b n | n ≥ 0 } can be recognized by VPA s. Theorem ( follows from Dymond ’88 ) BP-VPA ⊆ NC 1 Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 9 / 20

Classes Equivalent to NC 1 NC 1 log depth, poly size circuits with fanin-2 BWBP bounded width branching programs BWC bounded width circuits log width formula LWF BP - NFA programs over NFA programs over VPA BP - VPA Are their arithmetic versions equivalent? Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 10 / 20

Arithmetization over N Circuits ◮ Replace ∧ with × and ∨ with +. ◮ Counting number of proving subtrees Branching Programs ◮ Counting the number of s - t paths in a branching program ◮ Counting the number of accepting paths in M , for BP - M ( M = VPA , NFA ) Arithmetic classes: # NC 1 , # BWBP , # LWF , # BP - NFA , # BP - VPA , # BWC Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 11 / 20

Theorem (Caussinus et al ’98 ,Istrail Zivkovic ’94 ) # BWBP = # BP-NFA ⊆ # NC 1 = # LWF Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 12 / 20

Our Results # BP - VPA = # BP - NFA # BWC = # SC 0 needs restrictions to become interesting We propose a restriction # sSC 0 # NC 1 # LWF # BWBP FL # BP - NFA # BP - V PA # sSC 0 Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 13 / 20

Infeasibility of # BWC A width two circuit can compute super exponential values 1 · · · ⊕ ⊗ ⊗ ⊗ ⊗ 2 2 n 1 poly degree ⇒ feasible values poly degree, poly size = SAC 1 = LogCFL NLOG ⊆ LogCFL ⊆ NC 2 Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 14 / 20

Degree of a circuit ∧ x 1 x 2 x 3 + x 1 x 4 x 1 .x 2 + x 1 ∨ x 3 + x 4 ∨ x 1 .x 2 ∧ x 1 x 3 x 4 x 2 x 1 Figure: A Degree-3 circuit Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 15 / 20

Definition sSC i = Poly degree, poly sized circuits of width O (log i n ) NC 1 = sSC 0 = SC 0 Observation: sSC 1 ? = SC 1 (= DLOG ) Open Question: Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 16 / 20

Understanding sSC 1 NC 1 ⊆ sSC 1 ⊆ DLOG Examine closure properties. Is sSC 1 closed under complementation ? Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 17 / 20

Understanding sSC 1 NC 1 ⊆ sSC 1 ⊆ DLOG Examine closure properties. Is sSC 1 closed under complementation ? – Naive negation blows up the degree. Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 17 / 20

Theorem co-sSC i ⊆ sSC 2 i Proof. Main ideas Inductive counting on layered circuits, as used by Borodin, Cook, Dymond, Ruzzo, Tompa ’89 to complement SAC 1 . Monotone branching programs for threshold, as used by Vinay ’96 to complement log width BP s. Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 18 / 20

Arithmetizing sSC - Overall picture # SAC 1 # NC 1 NC 2 FL # BWBP # sSC 0 # sSC 1 SC 2 # sSC i SC i +1 Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 19 / 20

Open questions sSC 1 ? = DLOG ? co - sSC i = sSC i , i > 0 # sSC 1 ? ⊆ FL What closure properties do # sSC i have ? We show that # sSC 0 is closed under div by a constant, decrement. Arithmetizing Circuits around NC 1 and L Raghavendra Rao B V Institute of Mathematical Sciences, Chennai () 23 February, 2007 20 / 20