Monotone Circuit can implement an algorithm to run Non-Deterministic Polynomial time complexity (NP) problem in Polynomial time complexity (P). I developed a method to implement all algorithms without "Not" operations. Using this information, I manage to prove that NP is not equal to P. Kung Fu Computer Science, Geometric complexity theory
Beyond Shannon, Sipser and Razborov; Solve Clique Problem like an Electronic ...Sing Kuang Tan
Convert any Boolean algebra into monotone circuit and use that to prove that NP is not equal to P as monotone circuit cannot solve Clique problem in Polynomial time complexity. NP vs P is a Millennium Prize problem. Kung Fu Computer Science, Geometric complexity theory
Brief explanation of NP vs P. Prove Np not equal P using Markov Random Field ...Sing Kuang Tan
In this paper, we proved that Non-deterministic Polynomial time complexity (NP) is not equal to Polynomial time complexity (P). We developed the Boolean algebra that will infer the solution of two variables of a Non-deterministic Polynomial computation time Markov Random Field. We showed that no matter how we simplified the Boolean algebra, it can never run in Polynomial computation time (NP not equal to P). We also developed proof that all Polynomial computation time multi-layer Boolean algebra can be transformed to another Polynomial computation time multi-layer Boolean algebra where there are only 'Not' operations in the first layer. So in the process of simplifying the Boolean algebra, we only need to consider factorization operations that only assumes only 'Not' operations in the first layer. We also developed Polynomial computation time Boolean algebra for Markov Random Field Chain and 2sat problem represented in Markov Random Field form to give examples of Polynomial computation time Markov Random Field. Kung Fu Computer Science, Geometric complexity theory
A method for solving quadratic programming problems having linearly factoriz...IJMER
A new method namely, objective separable method based on simplex method is proposed for
finding an optimal solution to a quadratic programming problem in which the objective function can be
factorized into two linear functions. The solution procedure of the proposed method is illustrated with the
numerical example.
Quadratic Programming : KKT conditions with inequality constraintsMrinmoy Majumder
In the case of Quadratic Programming for optimization, the objective function is a quadratic function. One of the techniques for solving quadratic optimization problems is KKT Conditions which is explained with an example in this tutorial.
Beyond Shannon, Sipser and Razborov; Solve Clique Problem like an Electronic ...Sing Kuang Tan
Convert any Boolean algebra into monotone circuit and use that to prove that NP is not equal to P as monotone circuit cannot solve Clique problem in Polynomial time complexity. NP vs P is a Millennium Prize problem. Kung Fu Computer Science, Geometric complexity theory
Brief explanation of NP vs P. Prove Np not equal P using Markov Random Field ...Sing Kuang Tan
In this paper, we proved that Non-deterministic Polynomial time complexity (NP) is not equal to Polynomial time complexity (P). We developed the Boolean algebra that will infer the solution of two variables of a Non-deterministic Polynomial computation time Markov Random Field. We showed that no matter how we simplified the Boolean algebra, it can never run in Polynomial computation time (NP not equal to P). We also developed proof that all Polynomial computation time multi-layer Boolean algebra can be transformed to another Polynomial computation time multi-layer Boolean algebra where there are only 'Not' operations in the first layer. So in the process of simplifying the Boolean algebra, we only need to consider factorization operations that only assumes only 'Not' operations in the first layer. We also developed Polynomial computation time Boolean algebra for Markov Random Field Chain and 2sat problem represented in Markov Random Field form to give examples of Polynomial computation time Markov Random Field. Kung Fu Computer Science, Geometric complexity theory
A method for solving quadratic programming problems having linearly factoriz...IJMER
A new method namely, objective separable method based on simplex method is proposed for
finding an optimal solution to a quadratic programming problem in which the objective function can be
factorized into two linear functions. The solution procedure of the proposed method is illustrated with the
numerical example.
Quadratic Programming : KKT conditions with inequality constraintsMrinmoy Majumder
In the case of Quadratic Programming for optimization, the objective function is a quadratic function. One of the techniques for solving quadratic optimization problems is KKT Conditions which is explained with an example in this tutorial.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Analysis and design of algorithms part 4Deepak John
Complexity Theory - Introduction. P and NP. NP-Complete problems. Approximation algorithms. Bin packing, Graph coloring. Traveling salesperson Problem.
Digital Signals and Systems (May – 2016) [Question Paper | IDOL: Revised Course]Mumbai B.Sc.IT Study
Data Warehousing (April – 2016) [Question Paper | CBSGS: 75:25 Pattern]
april - 2017, april - 2016, april - 2015, april - 2014, april - 2013, october - 2017, october - 2016, october - 2015, october - 2014, may - 2016, may - 2017, december - 2017, 75:25 pattern, 60:40 pattern, revised course, old course, mumbai bscit study, mumbai university, bscit semester vi, bscit question paper, old question paper, previous year question paper, semester vi question paper, question paper, CBSGS, IDOL, kamal t, internet technology, digital signals and systems, data warehousing, ipr and cyber laws, project management, geographic information system
I have used "Exact Exponential Algorithms" book by 'Fedor V. Fomin and Dieter Kratsch' to prepare this presentation. It will be helpful in understanding split and list technique in breaking and solving hard problems in exponential time.
Digital Signals and Systems (April – 2015) [Question Paper | IDOL: Revised Co...Mumbai B.Sc.IT Study
Data Warehousing (April – 2016) [Question Paper | CBSGS: 75:25 Pattern]
april - 2017, april - 2016, april - 2015, april - 2014, april - 2013, october - 2017, october - 2016, october - 2015, october - 2014, may - 2016, may - 2017, december - 2017, 75:25 pattern, 60:40 pattern, revised course, old course, mumbai bscit study, mumbai university, bscit semester vi, bscit question paper, old question paper, previous year question paper, semester vi question paper, question paper, CBSGS, IDOL, kamal t, internet technology, digital signals and systems, data warehousing, ipr and cyber laws, project management, geographic information system
Digital Signals and System (May – 2016) [Revised Syllabus | Question Paper]Mumbai B.Sc.IT Study
mumbai bscit study, kamal t, mumbai university, old question paper, previous year question paper, bscit question paper, bscit semester vi, internet technology, april - 2015, 75:25 Pattern, 60:40 Pattern, revised syllabus, old syllabus, cbsgc, question paper, may - 2016, april - 2017, april - 2014, april - 2013, may – 2016, october – 2016, digital signals and system
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
"SSumM: Sparse Summarization of Massive Graphs", KDD 2020KyuhanLee4
A presentation slides of Kyuhan Lee, Hyeonsoo Jo, Jihoon Ko, Sungsu Lim, Kijung Shin, "SSumM: Sparse Summarization of Massive Graphs", KDD 2020.
Given a graph G and the desired size k in bits, how can we summarize G within k bits, while minimizing the information loss?
Large-scale graphs have become omnipresent, posing considerable computational challenges. Analyzing such large graphs can be fast and easy if they are compressed sufficiently to fit in main memory or even cache. Graph summarization, which yields a coarse-grained summary graph with merged nodes, stands out with several advantages among graph compression techniques. Thus, a number of algorithms have been developed for obtaining a concise summary graph with little information loss or equivalently small reconstruction error. However, the existing methods focus solely on reducing the number of nodes, and they often yield dense summary graphs, failing to achieve better compression rates. Moreover, due to their limited scalability, they can be applied only to moderate-size graphs.
In this work, we propose SSumM, a scalable and effective graph-summarization algorithm that yields a sparse summary graph. SSumM not only merges nodes together but also sparsifies the summary graph, and the two strategies are carefully balanced based on the minimum description length principle. Compared with state-of-the-art competitors, SSumM is (a) Concise: yields up to 11.2X smaller summary graphs with similar reconstruction error, (b) Accurate: achieves up to 4.2X smaller reconstruction error with similarly concise outputs, and (c) Scalable: summarizes 26X larger graphs while exhibiting linear scalability. We validate these advantages through extensive experiments on 10 real-world graphs.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Analysis and design of algorithms part 4Deepak John
Complexity Theory - Introduction. P and NP. NP-Complete problems. Approximation algorithms. Bin packing, Graph coloring. Traveling salesperson Problem.
Digital Signals and Systems (May – 2016) [Question Paper | IDOL: Revised Course]Mumbai B.Sc.IT Study
Data Warehousing (April – 2016) [Question Paper | CBSGS: 75:25 Pattern]
april - 2017, april - 2016, april - 2015, april - 2014, april - 2013, october - 2017, october - 2016, october - 2015, october - 2014, may - 2016, may - 2017, december - 2017, 75:25 pattern, 60:40 pattern, revised course, old course, mumbai bscit study, mumbai university, bscit semester vi, bscit question paper, old question paper, previous year question paper, semester vi question paper, question paper, CBSGS, IDOL, kamal t, internet technology, digital signals and systems, data warehousing, ipr and cyber laws, project management, geographic information system
I have used "Exact Exponential Algorithms" book by 'Fedor V. Fomin and Dieter Kratsch' to prepare this presentation. It will be helpful in understanding split and list technique in breaking and solving hard problems in exponential time.
Digital Signals and Systems (April – 2015) [Question Paper | IDOL: Revised Co...Mumbai B.Sc.IT Study
Data Warehousing (April – 2016) [Question Paper | CBSGS: 75:25 Pattern]
april - 2017, april - 2016, april - 2015, april - 2014, april - 2013, october - 2017, october - 2016, october - 2015, october - 2014, may - 2016, may - 2017, december - 2017, 75:25 pattern, 60:40 pattern, revised course, old course, mumbai bscit study, mumbai university, bscit semester vi, bscit question paper, old question paper, previous year question paper, semester vi question paper, question paper, CBSGS, IDOL, kamal t, internet technology, digital signals and systems, data warehousing, ipr and cyber laws, project management, geographic information system
Digital Signals and System (May – 2016) [Revised Syllabus | Question Paper]Mumbai B.Sc.IT Study
mumbai bscit study, kamal t, mumbai university, old question paper, previous year question paper, bscit question paper, bscit semester vi, internet technology, april - 2015, 75:25 Pattern, 60:40 Pattern, revised syllabus, old syllabus, cbsgc, question paper, may - 2016, april - 2017, april - 2014, april - 2013, may – 2016, october – 2016, digital signals and system
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
"SSumM: Sparse Summarization of Massive Graphs", KDD 2020KyuhanLee4
A presentation slides of Kyuhan Lee, Hyeonsoo Jo, Jihoon Ko, Sungsu Lim, Kijung Shin, "SSumM: Sparse Summarization of Massive Graphs", KDD 2020.
Given a graph G and the desired size k in bits, how can we summarize G within k bits, while minimizing the information loss?
Large-scale graphs have become omnipresent, posing considerable computational challenges. Analyzing such large graphs can be fast and easy if they are compressed sufficiently to fit in main memory or even cache. Graph summarization, which yields a coarse-grained summary graph with merged nodes, stands out with several advantages among graph compression techniques. Thus, a number of algorithms have been developed for obtaining a concise summary graph with little information loss or equivalently small reconstruction error. However, the existing methods focus solely on reducing the number of nodes, and they often yield dense summary graphs, failing to achieve better compression rates. Moreover, due to their limited scalability, they can be applied only to moderate-size graphs.
In this work, we propose SSumM, a scalable and effective graph-summarization algorithm that yields a sparse summary graph. SSumM not only merges nodes together but also sparsifies the summary graph, and the two strategies are carefully balanced based on the minimum description length principle. Compared with state-of-the-art competitors, SSumM is (a) Concise: yields up to 11.2X smaller summary graphs with similar reconstruction error, (b) Accurate: achieves up to 4.2X smaller reconstruction error with similarly concise outputs, and (c) Scalable: summarizes 26X larger graphs while exhibiting linear scalability. We validate these advantages through extensive experiments on 10 real-world graphs.
Selection of the optimal parameters for machine learning tasks is challenging. Some results may be bad not because the data is noisy or the used learning algorithm is weak, but due to the bad selection of the parameters values. This presentation gives a brief introduction about evolutionary algorithms (EAs) and describes genetic algorithm (GA) which is one of the simplest random-based EAs. A step-by-step example is given in addition to its implementation in Python 3.5.
---------------------------------
Read more about GA:
Yu, Xinjie, and Mitsuo Gen. Introduction to evolutionary algorithms. Springer Science & Business Media, 2010.
https://www.kdnuggets.com/2018/03/introduction-optimization-with-genetic-algorithm.html
https://www.linkedin.com/pulse/introduction-optimization-genetic-algorithm-ahmed-gad
A brief study material for teaching mathematics of back propagation.
The notations are based on
https://www.cs.huji.ac.il/~shais/UnderstandingMachineLearning/understanding-machine-learning-theory-algorithms.pdf
APL is an interesting beast, with an immense level of expressiveness locked away in a syntax that instantly scares away all but the initiated.
In this talk, we'll take a gentle introduction to APL and demonstrate its power to implement complex algorithms in very little code. We'll compare examples with equivalents in C# to illustrate why APL is much better suited as a "thinking tool" for programmers.
Using my BSnet deep learnin network, each neuron is designed not to overfit. It achieves this by concatenating the positive and negative inputs so that it becomes more separable in high dimension space. This allows it to be used for general purpose classification problems such as MNIST dataset to recognize handwriting number digits. BSnet is based on the principles of Boolean algebra and monotone circuit. Using the same principles, I also design BSautonet autoencoder, that can be used to denoise image, learn embeddings and unsupervised learning.
This powerpoint gives a technique to approximate (relaxation) discrete Markov Random Field (MRF) using convex programming. This approximated MRF can be used to approximate NP problem. This also proves that NP is not equal P because the MRF convex programming and the approximate MRF convex programming are not the same with removal of some product terms.
kung fu Computer Science, Geometric complexity theory
NP vs P Proof using Deterministic Finite AutomataSing Kuang Tan
Prove that Clique problem is NP and cannot be reduced to P because the Deterministic Finite Automata of the Clique problem has exponential number of states. We can use the same concept to prove that NP is not equal to P using Turing Machine. We figured out a way to unify Mathematics. This proof is for those Theoretical Computing guys who do not know Boolean algebra but know Turing Machine. Kung fu computer science, Geometric complexity theory
Use Inductive or Deductive Logic to solve NP vs P?Sing Kuang Tan
Use Inductive or Deductive Logic to solve NP vs P? I use circuit complexity and deductive logic to solve NP vs P. Kung fu computer science, Geometric complexity theory
Simplify a Clique Problem Boolean algebra by factorization. Show that Clique Problem is Non-Deterministic Polynomial Time (NP) and cannot be simplified to Polynomial Time (P). Kung Fu Computer Science, Geometric complexity theory
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Event Management System Vb Net Project Report.pdfKamal Acharya
In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
My project named “Event Management System” is software that store and maintained all events coordinated in college. It also helpful to print related reports. My project will help to record the events coordinated by faculties with their Name, Event subject, date & details in an efficient & effective ways.
In my system we have to make a system by which a user can record all events coordinated by a particular faculty. In our proposed system some more featured are added which differs it from the existing system such as security.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
AKS UNIVERSITY Satna Final Year Project By OM Hardaha.pdf
A Weird Soviet Method to Partially Solve the Perebor Problem
1. A Weird Soviet Method to
Partially Solve the Perebor
Problem
How is this method related to my NP vs P solution
Sing Kuang Tan
singkuangtan@gmail.com
17 August 2021
2. Perebor Problems
• Perebor problems are difficult problems which current methods can
only solve it by brute force algorithm
• https://drdoane.com/the-perebor-problem/
• Perebor problems are the NP (Non-Deterministic Polynomial time
complexity) problem
• Can we convert NP into P (Polynomial time complexity) problem?
• My answer is NO.
• My solution to NP vs P problem is NP≠P
Link to my paper
https://www.slideshare.net/SingKuangTan/brief-np-vspexplain-249524831
Prove Np not equal P using Markov Random Field and Boolean Algebra
Simplification
https://vixra.org/abs/2105.0181
3. Alexander Razborov (from Soviet Union)
• He developed the method to partially solve Perebor Problem.
• Bio: https://en.wikipedia.org/wiki/Alexander_Razborov
• In 1985, Razborov has proved that Clique problem of a graph requires a Non-
Deterministic Polynomial time complexity (NP) algorithm if the algorithm is implemented
using Boolean algebra consist of only “And” and “Or” operations
• As there is no “Not” operation, so it only partially proves that NP≠P
• Later I will show how I circumvent the problem of no “Not” operation
• https://blog.computationalcomplexity.org/2005/09/circuit-complexity-and-p-versus-np.html
• https://www.cs.utexas.edu/~danama/courses/adv-comp/scribe5.pdf
• A. A. Razborov, Lower bounds on the monotone complexity of some Boolean functions, Dokl.
Akad. Nauk. SSSR, 281(4):798-801, 1985
• At that time, Soviet is one of the country leading in the problem on NP vs P
• Soviet Song “Our Army” LOL
• https://www.youtube.com/watch?v=5yYQIa-4rLE
• Enable subtitles
4. Clique Problem
• https://en.wikipedia.org/wiki/Clique_problem
4-clique in this 7 vertices graph can
be found using brute force
“Or” operation
“And” operation
This digital circuit is used to detect whether a 3-clique exists
in this 4 vertices graph
This circuit has only “And” and “Or” operations (In literature,
this is called a monotone circuit)
5. Cellular Automata
• What is Cellular Automata?
• https://mathworld.wolfram.com/CellularAutomaton.ht
ml
• https://natureofcode.com/book/chapter-7-cellular-
automata/
• https://towardsdatascience.com/simple-but-stunning-
animated-cellular-automata-in-python-c912e0c156a9
• https://towardsdatascience.com/algorithmic-beauty-an-
introduction-to-cellular-automata-f53179b3cf8f
• Animation
• https://www.youtube.com/watch?v=3MJ8deSCOCE
6. “Not” operations are unnecessary
• I will show using “Not” operations are unnecessary for
implementation of any algorithms using Boolean algebra
• I will use Cellular Automata as an Universal Computing model
• A Polynomial time (P) Cellular Automata can be implemented without any
“Not” operation
First layer is
the input of an
algorithm
Last layer is the
output of an
algorithm
Cellular Automata
can implement any
algorithm
Different algorithm can be
implemented by changing the
rule
7. Cellular Automata can Emulate any Computer
Algorithm
An algorithm implemented
on a modern computer
Can also be implemented using
Cellular Automata
A Polynomial time complexity (P)
algorithm remains Polynomial time (P)
after implemented on Cellular
Automata
A Non-Deterministic Polynomial time
complexity (NP) algorithm remains
Non-Deterministic Polynomial time
(NP) after implemented on Cellular
Automata
9. Simple Cellular Automata Example
Each intermediate output is
dependent on input
Each output is dependent on
intermediate outputs
• This looks like a convolutional
neural network
• Neural Network can be
represented by multi-layer
Boolean algebra
• Multi-layer Boolean algebra
simplification gives us insight on
how to train a neural network
10. Simply Cellular Automata Example
a1,1 a1,2 a1,3 a1,4 a1,5
a2,2 a2,3 a2,4
a3,3
I give a variable name to each
cell of the Cellular Automata
a1,1
Boolean
Algebra or
Circuit
𝑎3,3
a1,2
a1,3
a1,4
a1,5
11. Simply Cellular Automata Example
ai,j ai,j+1 Ai,j+2
ai+1,j+1
𝑎𝑖+1,𝑗+1 = 𝑎𝑖,𝑗 𝑎𝑖,𝑗+1 𝑎𝑖,𝑗+2 +
𝑎𝑖,𝑗𝑎𝑖,𝑗+1 𝑎𝑖,𝑗+2+ 𝑎𝑖,𝑗 𝑎𝑖,𝑗+1 𝑎𝑖,𝑗+2 +
𝑎𝑖,𝑗 𝑎𝑖,𝑗+1 𝑎𝑖,𝑗+2
Bar above ai,j means “Not” operation on ai,j
Addition means “Or” operation
Multiplication means “And” operation
The rule can be implemented
using the Boolean algebra below
14. Simply Cellular Automata Example
a1,1 a1,2 a1,3 a1,4 a1,5
a2,2 a2,3 a2,4
a3,3
Find a3,3 in terms of a1,1, a1,2, a1,3, a1,4, a1,5 is
a3,3=a1,1,0a1,2,0a1,3,0a1,4,0a1,5,1
+ a1,1,0a1,2,0a1,3,0a1,4,1a1,5,0
+ a1,1,0a1,2,0a1,3,0a1,4,1a1,5,1
+ a1,1,0a1,2,1a1,3,1a1,4,0a1,5,1
+ a1,1,0a1,2,1a1,3,1a1,4,1a1,5,0
+ a1,1,0a1,2,1a1,3,1a1,4,1a1,5,1
+ a1,1,1a1,2,0a1,3,0a1,4,0a1,5,0
+ a1,1,1a1,2,0a1,3,1a1,4,0a1,5,0
+ a1,1,1a1,2,0a1,3,1a1,4,0a1,5,1
+ a1,1,1a1,2,0a1,3,1a1,4,1a1,5,0
+ a1,1,1a1,2,0a1,3,1a1,4,1a1,5,1
+ a1,1,1a1,2,1a1,3,0a1,4,0a1,5,0
+ a1,1,1a1,2,1a1,3,0a1,4,0a1,5,1
+ a1,1,1a1,2,1a1,3,0a1,4,1a1,5,0
+ a1,1,1a1,2,1a1,3,0a1,4,1a1,5,1
+ a1,1,1a1,2,1a1,3,1a1,4,0a1,5,0
Use the
representation below
to simplify the
equations,
𝑎𝑖,𝑗,0 = 𝑎𝑖,𝑗
𝑎𝑖,𝑗,1 = 𝑎𝑖,𝑗
15. Simply Cellular Automata Example
a1,1,0
Use the
representation below
to simplify the
equations,
𝑎𝑖,𝑗,0 = 𝑎𝑖,𝑗
𝑎𝑖,𝑗,1 = 𝑎𝑖,𝑗
a1,1,1 a1,2,0 a1,2,1 a1,3,0 a1,3,1 a1,4,0 a1,4,1 a1,5,0 a1,5,1
a2,2,0 a2,2,1 a2,3,0 a2,3,1 a2,4,0 a2,4,1
A3,3,0 A3,3,1
Represent the Cellular Automata by
Boolean algebra,
𝑎2,2,0 = 𝑎1,1,1𝑎1,2,1𝑎1,3,1 +
𝑎1,1,1𝑎1,2,1𝑎1,3,0+ 𝑎1,1,1𝑎1,2,0𝑎1,3,1
+ 𝑎1,1,0𝑎1,2,0𝑎1,3,0
𝑎2,2,1 = 𝑎1,1,1𝑎1,2,0𝑎1,3,0 +
𝑎1,1,0𝑎1,2,1𝑎1,3,1+ 𝑎1,1,0𝑎1,2,1𝑎1,3,0
+ 𝑎1,1,0𝑎1,2,0𝑎1,3,1
𝑎2,3,0 = 𝑎1,2,1𝑎1,3,1𝑎1,4,1 +
𝑎1,2,1𝑎1,3,1𝑎1,4,0+ 𝑎1,2,1𝑎1,3,0𝑎1,4,1
+ 𝑎1,2,0𝑎1,3,0𝑎1,4,0
𝑎2,3,1 = 𝑎1,2,1𝑎1,3,0𝑎1,4,0 +
𝑎1,2,0𝑎1,3,1𝑎1,4,1+ 𝑎1,2,0𝑎1,3,1𝑎1,4,0
+ 𝑎1,2,0𝑎1,3,0𝑎1,4,1
𝑎2,4,0 = 𝑎1,3,1𝑎1,4,1𝑎1,5,1 +
𝑎1,3,1𝑎1,4,1𝑎1,5,0+ 𝑎1,3,1𝑎1,4,0𝑎1,5,1
+ 𝑎1,3,0𝑎1,4,0𝑎1,5,0
𝑎2,4,1 = 𝑎1,3,1𝑎1,4,0𝑎1,5,0 +
𝑎1,3,0𝑎1,4,1𝑎1,5,1+ 𝑎1,3,0𝑎1,4,1𝑎1,5,0
+ 𝑎1,3,0𝑎1,4,0𝑎1,5,1
𝑎3,3,0 = 𝑎2,2,1𝑎2,3,1𝑎2,4,1 +
𝑎2,2,1𝑎2,3,1𝑎2,4,0+ 𝑎2,2,1𝑎2,3,0𝑎2,4,1
+ 𝑎2,2,0𝑎2,3,0𝑎2,4,0
𝑎3,3,1 = 𝑎2,2,1𝑎2,3,0𝑎2,4,0 +
𝑎2,2,0𝑎2,3,1𝑎2,4,1+ 𝑎2,2,0𝑎2,3,1𝑎2,4,0
+ 𝑎2,2,0𝑎2,3,0𝑎2,4,1
The use of double
variables remove the
need of “Not” operation
in the Boolean algebra
16. Simply Cellular Automata Example
a1,1,0
Find a3,3,1 in terms of a1,1,0, a1,2,0, a1,3,0, a1,4,0, a1,5,0, a1,1,1,
a1,2,1, a1,3,1, a1,4,1, a1,5,1 is
a3,3,1=a1,1,0a1,2,0a1,3,0a1,4,0a1,5,1
+ a1,1,0a1,2,0a1,3,0a1,4,1a1,5,0
+ a1,1,0a1,2,0a1,3,0a1,4,1a1,5,1
+ a1,1,0a1,2,1a1,3,1a1,4,0a1,5,1
+ a1,1,0a1,2,1a1,3,1a1,4,1a1,5,0
+ a1,1,0a1,2,1a1,3,1a1,4,1a1,5,1
+ a1,1,1a1,2,0a1,3,0a1,4,0a1,5,0
+ a1,1,1a1,2,0a1,3,1a1,4,0a1,5,0
+ a1,1,1a1,2,0a1,3,1a1,4,0a1,5,1
+ a1,1,1a1,2,0a1,3,1a1,4,1a1,5,0
+ a1,1,1a1,2,0a1,3,1a1,4,1a1,5,1
+ a1,1,1a1,2,1a1,3,0a1,4,0a1,5,0
+ a1,1,1a1,2,1a1,3,0a1,4,0a1,5,1
+ a1,1,1a1,2,1a1,3,0a1,4,1a1,5,0
+ a1,1,1a1,2,1a1,3,0a1,4,1a1,5,1
+ a1,1,1a1,2,1a1,3,1a1,4,0a1,5,0
Use the
representation below
to simplify the
equations,
𝑎𝑖,𝑗,0 = 𝑎𝑖,𝑗
𝑎𝑖,𝑗,1 = 𝑎𝑖,𝑗
a1,1,1 a1,2,0 a1,2,1 a1,3,0 a1,3,1 a1,4,0 a1,4,1 a1,5,0 a1,5,1
a2,2,0 a2,2,1 a2,3,0 a2,3,1 a2,4,0 a2,4,1
A3,3,0 A3,3,1
The use of double
variables remove the
need of “Not” operation
in the Boolean algebra
17. Other Models without “Not” operation
• Creating a Boolean algebra without “Not” operation may seem weird
in the first place
• But there is no “Not” operation in
• Fault tree analysis
• AND OR graph (for solving e.g. TIC-TAC-TOE)
• Bayesian network or Markov Random Field
18. Expansion of the Boolean Algebra
a3,3=a1,1,0a1,2,0a1,3,0a1,4,0a1,5,1
+ a1,1,0a1,2,0a1,3,0a1,4,1a1,5,0
+ a1,1,0a1,2,0a1,3,0a1,4,1a1,5,1
+ a1,1,0a1,2,1a1,3,1a1,4,0a1,5,1
+ a1,1,0a1,2,1a1,3,1a1,4,1a1,5,0
+ a1,1,0a1,2,1a1,3,1a1,4,1a1,5,1
+ a1,1,1a1,2,0a1,3,0a1,4,0a1,5,0
+ a1,1,1a1,2,0a1,3,1a1,4,0a1,5,0
+ a1,1,1a1,2,0a1,3,1a1,4,0a1,5,1
+ a1,1,1a1,2,0a1,3,1a1,4,1a1,5,0
+ a1,1,1a1,2,0a1,3,1a1,4,1a1,5,1
+ a1,1,1a1,2,1a1,3,0a1,4,0a1,5,0
+ a1,1,1a1,2,1a1,3,0a1,4,0a1,5,1
+ a1,1,1a1,2,1a1,3,0a1,4,1a1,5,0
+ a1,1,1a1,2,1a1,3,0a1,4,1a1,5,1
+ a1,1,1a1,2,1a1,3,1a1,4,0a1,5,0
Polynomial Time Complexity Boolean algebra
Non-Deterministic Polynomial Time
Complexity Boolean algebra
The Boolean algebra can
be expanded
𝑎2,2,0 = 𝑎1,1,1𝑎1,2,1𝑎1,3,1 + 𝑎1,1,1𝑎1,2,1𝑎1,3,0+
𝑎1,1,1𝑎1,2,0𝑎1,3,1 +𝑎1,1,0𝑎1,2,0𝑎1,3,0
𝑎2,2,1 = 𝑎1,1,1𝑎1,2,0𝑎1,3,0 + 𝑎1,1,0𝑎1,2,1𝑎1,3,1+
𝑎1,1,0𝑎1,2,1𝑎1,3,0 +𝑎1,1,0𝑎1,2,0𝑎1,3,1
𝑎2,3,0 = 𝑎1,2,1𝑎1,3,1𝑎1,4,1 + 𝑎1,2,1𝑎1,3,1𝑎1,4,0+
𝑎1,2,1𝑎1,3,0𝑎1,4,1 +𝑎1,2,0𝑎1,3,0𝑎1,4,0
𝑎2,3,1 = 𝑎1,2,1𝑎1,3,0𝑎1,4,0 + 𝑎1,2,0𝑎1,3,1𝑎1,4,1+
𝑎1,2,0𝑎1,3,1𝑎1,4,0 +𝑎1,2,0𝑎1,3,0𝑎1,4,1
𝑎2,4,0 = 𝑎1,3,1𝑎1,4,1𝑎1,5,1 + 𝑎1,3,1𝑎1,4,1𝑎1,5,0+
𝑎1,3,1𝑎1,4,0𝑎1,5,1 +𝑎1,3,0𝑎1,4,0𝑎1,5,0
𝑎2,4,1 = 𝑎1,3,1𝑎1,4,0𝑎1,5,0 + 𝑎1,3,0𝑎1,4,1𝑎1,5,1+
𝑎1,3,0𝑎1,4,1𝑎1,5,0 +𝑎1,3,0𝑎1,4,0𝑎1,5,1
𝑎3,3,0 = 𝑎2,2,1𝑎2,3,1𝑎2,4,1 + 𝑎2,2,1𝑎2,3,1𝑎2,4,0+
𝑎2,2,1𝑎2,3,0𝑎2,4,1 +𝑎2,2,0𝑎2,3,0𝑎2,4,0
𝑎3,3,1 = 𝑎2,2,1𝑎2,3,0𝑎2,4,0 + 𝑎2,2,0𝑎2,3,1𝑎2,4,1+
𝑎2,2,0𝑎2,3,1𝑎2,4,0 +𝑎2,2,0𝑎2,3,0𝑎2,4,1
• Note that all Boolean
algebras do not have
“Not” operation
• Expansion is simply
multiply throughout
19. Factorization of the Boolean Algebra
a3,3
=a1,1,0a1,2,0a1,3,0a1,4,0a1,5,1+a1,1,0a1,2,0a1,3,0a1,4,1a1,5,0+
a1,1,0a1,2,0a1,3,0a1,4,1a1,5,1+ a1,1,0a1,2,1a1,3,1a1,4,0a1,5,1+
a1,1,0a1,2,1a1,3,1a1,4,1a1,5,0+ a1,1,0a1,2,1a1,3,1a1,4,1a1,5,1+
a1,1,1a1,2,0a1,3,0a1,4,0a1,5,0+ a1,1,1a1,2,0a1,3,1a1,4,0a1,5,0+
a1,1,1a1,2,0a1,3,1a1,4,0a1,5,1+ a1,1,1a1,2,0a1,3,1a1,4,1a1,5,0+
a1,1,1a1,2,0a1,3,1a1,4,1a1,5,1+ a1,1,1a1,2,1a1,3,0a1,4,0a1,5,0+
a1,1,1a1,2,1a1,3,0a1,4,0a1,5,1+ a1,1,1a1,2,1a1,3,0a1,4,1a1,5,0+
a1,1,1a1,2,1a1,3,0a1,4,1a1,5,1+ a1,1,1a1,2,1a1,3,1a1,4,0a1,5,0
Polynomial Time Complexity Boolean algebra
Non-Deterministic Polynomial Time
Complexity Boolean algebra
The Boolean algebra can
be factorized into multi-
layer Boolean algebra that
is more efficient if a more
efficient Boolean algebra
exists
Naïve problem definition that solves a
problem using brute force
𝑎2,2,0 = 𝑎1,1,1𝑎1,2,1𝑎1,3,1 + 𝑎1,1,1𝑎1,2,1𝑎1,3,0+
𝑎1,1,1𝑎1,2,0𝑎1,3,1 +𝑎1,1,0𝑎1,2,0𝑎1,3,0
𝑎2,2,1 = 𝑎1,1,1𝑎1,2,0𝑎1,3,0 + 𝑎1,1,0𝑎1,2,1𝑎1,3,1+
𝑎1,1,0𝑎1,2,1𝑎1,3,0 +𝑎1,1,0𝑎1,2,0𝑎1,3,1
𝑎2,3,0 = 𝑎1,2,1𝑎1,3,1𝑎1,4,1 + 𝑎1,2,1𝑎1,3,1𝑎1,4,0+
𝑎1,2,1𝑎1,3,0𝑎1,4,1 +𝑎1,2,0𝑎1,3,0𝑎1,4,0
𝑎2,3,1 = 𝑎1,2,1𝑎1,3,0𝑎1,4,0 + 𝑎1,2,0𝑎1,3,1𝑎1,4,1+
𝑎1,2,0𝑎1,3,1𝑎1,4,0 +𝑎1,2,0𝑎1,3,0𝑎1,4,1
𝑎2,4,0 = 𝑎1,3,1𝑎1,4,1𝑎1,5,1 + 𝑎1,3,1𝑎1,4,1𝑎1,5,0+
𝑎1,3,1𝑎1,4,0𝑎1,5,1 +𝑎1,3,0𝑎1,4,0𝑎1,5,0
𝑎2,4,1 = 𝑎1,3,1𝑎1,4,0𝑎1,5,0 + 𝑎1,3,0𝑎1,4,1𝑎1,5,1+
𝑎1,3,0𝑎1,4,1𝑎1,5,0 +𝑎1,3,0𝑎1,4,0𝑎1,5,1
𝑎3,3,0 = 𝑎2,2,1𝑎2,3,1𝑎2,4,1 + 𝑎2,2,1𝑎2,3,1𝑎2,4,0+
𝑎2,2,1𝑎2,3,0𝑎2,4,1 +𝑎2,2,0𝑎2,3,0𝑎2,4,0
𝑎3,3,1 = 𝑎2,2,1𝑎2,3,0𝑎2,4,0 + 𝑎2,2,0𝑎2,3,1𝑎2,4,1+
𝑎2,2,0𝑎2,3,1𝑎2,4,0 +𝑎2,2,0𝑎2,3,0𝑎2,4,1
• Simplification is simply
factoring as there is no
“Not” operation
• We can find optimal time
efficient Boolean algebra
by factoring
20. Example of an Non-Deterministic problem (NP)
expressed in Markov Random Field representation
a1 a2
a3
a4
𝑓 𝑎1, 𝑎2, 𝑎3, 𝑎4 =
𝑎1,𝑎2,𝑎3,𝑎4
ℎ(𝑎1, 𝑎2)ℎ(𝑎1, 𝑎3)ℎ(𝑎1, 𝑎4)ℎ(𝑎2, 𝑎3)ℎ(𝑎2, 𝑎4)ℎ(𝑎3, 𝑎4)
Example of an NP problem expressed as a Graphical model (or
Markov Random Field)
Each variable ai can take finite values, e.g. 𝑎𝑖 ∈ {0,1,2,3}
ℎ 𝑎𝑖, 𝑎𝑗 = 0 𝑜𝑟 1
If the NP problem has a solution, 𝑓 𝑎1, 𝑎2, 𝑎3, 𝑎4 >0
Addition is “Or” operation
Multiplication is “And” Operation
21. Boolean Algebra of NP Problem
• Boolean algebra
• 𝑓 𝑎1, 𝑎2, 𝑎3, 𝑎4 =
𝑎1,𝑎2,𝑎3,𝑎4
ℎ(𝑎1, 𝑎2)ℎ(𝑎1, 𝑎3)ℎ(𝑎1, 𝑎4)ℎ(𝑎2, 𝑎3)ℎ(𝑎2, 𝑎4)ℎ(𝑎3, 𝑎4)
• Each h term, h(ai,aj), is a Boolean input variable taking an input
0 or 1
• Addition is “Or” operation
• Multiplication is “And” operation
• We can try to simplify this Boolean algebra by factorization
h(a1=0,a2=0)
h(a1=1,a2=0)
h(a1=0,a2=1)
h(a1=0,a2=2)
…
h(a1=0,a3=0)
h(a1=1,a3=0)
h(a1=0,a3=1)
h(a1=0,a3=2)
…
h(a3=0,a4=0)
h(a3=1,a4=0)
h(a3=0,a4=1)
h(a3=0,a4=2)
…
Boolean
Algebra or
Circuit
⋮
⋮
22. “Not” Operations on h() Terms can be
Replaced
• Do not have to consider ℎ(𝑎𝑖 = 𝑣𝑖, 𝑎𝑗 = 𝑣𝑗) because “Not” operation on
input can be replaced by
• 𝑣𝑘,𝑣𝑙 (𝑣𝑖,𝑣𝑗) ℎ(𝑎𝑖 = 𝑣𝑘, 𝑎𝑗 = 𝑣𝑙)
• which means sum of all h(ai,aj) terms not including h(ai=vi,aj=vj)
• After replacing the ℎ(𝑎𝑖 = 𝑣𝑖, 𝑎𝑗 = 𝑣𝑗) term, the Boolean algebra remains
Polynomial Time complexity if it is originally Polynomial time
• Because it is replaced by at most 4*4-1=15 terms, where n=4 is the number of variables
• Assume that the number of values a variable can take is 4
• E.g. ℎ(𝑎𝑖 = 0, 𝑎𝑗 = 0) can be replaced by h(ai=0,aj=1)+h(ai=0,aj=2)
+h(ai=0,aj=3)+h(ai=1,aj=0)+h(ai=1,aj=1)+h(ai=1,aj=2)
+h(ai=1,aj=3)+h(ai=2,aj=0)+h(ai=2,aj=1)+h(ai=2,aj=2)+h(ai=2,aj=3)+h(ai=3,aj=0)
+h(ai=3,aj=1)+h(ai=3,aj=2)+h(ai=3,aj=3)
23. NP problem cannot be simplified into P
𝑓 𝑎1, 𝑎2, 𝑎3, 𝑎4 =
𝑎1,𝑎2,𝑎3,𝑎4
ℎ(𝑎1, 𝑎2)ℎ(𝑎1, 𝑎3)ℎ(𝑎1, 𝑎4)ℎ(𝑎2, 𝑎3)ℎ(𝑎2, 𝑎4)ℎ(𝑎3, 𝑎4)
𝑓 𝑎1, 𝑎2, 𝑎3, 𝑎4 =
𝑎1,𝑎2
ℎ(𝑎1, 𝑎2)
𝑎3
ℎ(𝑎1, 𝑎3)ℎ(𝑎2, 𝑎3)
𝑎4
ℎ(𝑎1, 𝑎4) ℎ(𝑎2, 𝑎4)ℎ(𝑎3, 𝑎4)
One possible factorization is shown below. There are many possible different factorizations
An NP (Non-Deterministic Polynomial) problem cannot be simplified into Polynomial time by
factorization
Because of these 3 terms, it takes NP time to evaluate
a1 a2
a3
a4
Actual proof is much more complex than this
For details, please read my paper (end of slides)
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