A Superpolynomial Lower Bound for Clique Function Circuits with at - PowerPoint PPT Presentation

Main Results Hawk’s Eye View Extending Montone Lower Bounds to Limited Negations Monotone Lower bound for Clique function using Bottleneck counting Connecting the two worlds Thank You A Superpolynomial Lower Bound for Clique Function Circuits with at most 1 6 loglog n Negation Gates Kazuyuki Amano and Akira Maruoka Sajin Koroth Department of Computer Science IIT Madras Advanced Complexity Theory Course - Spring 2012 Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui

Main Results Hawk’s Eye View Extending Montone Lower Bounds to Limited Negations Monotone Lower bound for Clique function using Bottleneck counting Connecting the two worlds Thank You Outline Main Results 1 Hawk’s Eye View 2 Extending Montone Lower Bounds to Limited Negations 3 Monotone Lower bound for Clique function using Bottleneck counting 4 Connecting the two worlds 5 Thank You 6 Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui

Main Results Hawk’s Eye View Extending Montone Lower Bounds to Limited Negations Monotone Lower bound for Clique function using Bottleneck counting Connecting the two worlds Thank You 1 Main Result : C with at most 1 6 loglog n negations requires 1 2 gates for detecting cliques of size 1 5 ( log m ) ( log m ) 2 1 2 in a graph with m vertices ( log m ) 3 ( log m ) 2 A better monotone lower bound for clique function : (using 1 � � bottleneck counting) exp Ω m 3 for appropriate parameters Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui

Main Results Hawk’s Eye View Extending Montone Lower Bounds to Limited Negations Monotone Lower bound for Clique function using Bottleneck counting Connecting the two worlds Thank You Are we there yet ? ( NP �⊂ P/poly ) Thanks to Fischer Limited Negation lower bounds for circuits with at most log n negations would do The monotone lowerbounds could be misleading : Perfect matching inspite of being in P has a superpolynomial lowerbound for monotone circuits Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui

Main Results Hawk’s Eye View Extending Montone Lower Bounds to Limited Negations Monotone Lower bound for Clique function using Bottleneck counting Connecting the two worlds Thank You 1 From the limited-negation non-monotone circuit for clique obtain an “appropriate” family of monotone functions 2 Obtain lowerbounds for each function in the family (which approximates clique function) 3 Use the lowerbounds on each function in the family to extend the lowerbound to any such family 4 Transfer this lowerbound to the limited-negation circuit for clique. Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui

Main Results Hawk’s Eye View Extending Montone Lower Bounds to Limited Negations Monotone Lower bound for Clique function using Bottleneck counting Connecting the two worlds Thank You 1 From the limited-negation non-monotone circuit for clique obtain an “appropriate” family of monotone functions 2 Obtain lowerbounds for each function in the family (which approximates clique function) 3 Use the lowerbounds on each function in the family to extend the lowerbound to any such family 4 Transfer this lowerbound to the limited-negation circuit for clique. Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui

Main Results Hawk’s Eye View Extending Montone Lower Bounds to Limited Negations Monotone Lower bound for Clique function using Bottleneck counting Connecting the two worlds Thank You 1 From the limited-negation non-monotone circuit for clique obtain an “appropriate” family of monotone functions 2 Obtain lowerbounds for each function in the family (which approximates clique function) 3 Use the lowerbounds on each function in the family to extend the lowerbound to any such family 4 Transfer this lowerbound to the limited-negation circuit for clique. Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui

Main Results Hawk’s Eye View Extending Montone Lower Bounds to Limited Negations Monotone Lower bound for Clique function using Bottleneck counting Connecting the two worlds Thank You 1 From the limited-negation non-monotone circuit for clique obtain an “appropriate” family of monotone functions 2 Obtain lowerbounds for each function in the family (which approximates clique function) 3 Use the lowerbounds on each function in the family to extend the lowerbound to any such family 4 Transfer this lowerbound to the limited-negation circuit for clique. Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui

Main Results Hawk’s Eye View Obtaining Montone family from Limited Negations Circuit Extending Montone Lower Bounds to Limited Negations Characterizing the family for tight counting Monotone Lower bound for Clique function using Bottleneck counting Trasferring the Lowerbound from family to negation limited c Connecting the two worlds Thank You Outline Main Results 1 Hawk’s Eye View 2 Extending Montone Lower Bounds to Limited Negations 3 Monotone Lower bound for Clique function using Bottleneck counting 4 Connecting the two worlds 5 Thank You 6 Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui

Main Results Hawk’s Eye View Obtaining Montone family from Limited Negations Circuit Extending Montone Lower Bounds to Limited Negations Characterizing the family for tight counting Monotone Lower bound for Clique function using Bottleneck counting Trasferring the Lowerbound from family to negation limited c Connecting the two worlds Thank You Idea Use restrictions to eliminate negation gates by fixing their outputs Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui

Main Results Hawk’s Eye View Obtaining Montone family from Limited Negations Circuit Extending Montone Lower Bounds to Limited Negations Characterizing the family for tight counting Monotone Lower bound for Clique function using Bottleneck counting Trasferring the Lowerbound from family to negation limited c Connecting the two worlds Thank You Figure: Motone Circuit’s from Non-monotone one’s using restrictions Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui

Main Results Hawk’s Eye View Obtaining Montone family from Limited Negations Circuit Extending Montone Lower Bounds to Limited Negations Characterizing the family for tight counting Monotone Lower bound for Clique function using Bottleneck counting Trasferring the Lowerbound from family to negation limited c Connecting the two worlds Thank You Figure: Motone Circuit’s from Non-monotone one’s using restrictions Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui

Main Results Hawk’s Eye View Obtaining Montone family from Limited Negations Circuit Extending Montone Lower Bounds to Limited Negations Characterizing the family for tight counting Monotone Lower bound for Clique function using Bottleneck counting Trasferring the Lowerbound from family to negation limited c Connecting the two worlds Thank You Figure: Motone Circuit’s from Non-monotone one’s using restrictions Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui

Main Results Hawk’s Eye View Obtaining Montone family from Limited Negations Circuit Extending Montone Lower Bounds to Limited Negations Characterizing the family for tight counting Monotone Lower bound for Clique function using Bottleneck counting Trasferring the Lowerbound from family to negation limited c Connecting the two worlds Thank You Figure: Motone Circuit’s from Non-monotone one’s using restrictions Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui

Main Results Hawk’s Eye View Obtaining Montone family from Limited Negations Circuit Extending Montone Lower Bounds to Limited Negations Characterizing the family for tight counting Monotone Lower bound for Clique function using Bottleneck counting Trasferring the Lowerbound from family to negation limited c Connecting the two worlds Thank You Figure: Motone Circuit’s from Non-monotone one’s using restrictions Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui

Main Results Hawk’s Eye View Obtaining Montone family from Limited Negations Circuit Extending Montone Lower Bounds to Limited Negations Characterizing the family for tight counting Monotone Lower bound for Clique function using Bottleneck counting Trasferring the Lowerbound from family to negation limited c Connecting the two worlds Thank You Figure: Motone Circuit’s from Non-monotone one’s using restrictions Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui

Main Results Hawk’s Eye View Obtaining Montone family from Limited Negations Circuit Extending Montone Lower Bounds to Limited Negations Characterizing the family for tight counting Monotone Lower bound for Clique function using Bottleneck counting Trasferring the Lowerbound from family to negation limited c Connecting the two worlds Thank You Figure: Motone Circuit’s from Non-monotone one’s using restrictions Sajin Koroth A Superpolynomial Lower Bound for Clique Function Circui