This document summarizes an algorithm proposed by the author for solving NP-complete problems in polynomial time. The algorithm, called the Dual Expression Algorithm (DEA), works by constructing a path through the clauses of a Boolean formula in conjunctive normal form (CNF). It does this by starting with a random clause and either adding another clause or removing a clause while maintaining satisfiability. The author argues this algorithm could solve important problems in automated theorem proving and fulfill Hilbert's program. An appendix provides an example of DEA on a simple formula and discusses modeling its dynamics as a Markov chain.
An alternative scheme for approximating a periodic functionijscmcj
Fourier series is generally used in applied mathematics and non-linear mechanics to solve the problems containing periodic functions. The aim of this paper is to present a new scheme through involving the Taylor’s series after some process modification for all such problems. It will save the long evaluation process involve in computing the different constants of Fourier series. The result obtained by the present
method has been compared with the result from the Fourier series expansion w.r.t. the exact solution and
found to be more satisfactory.
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...mathsjournal
The following document presents some novel numerical methods valid for one and several variables, which
using the fractional derivative, allow us to find solutions for some nonlinear systems in the complex space using
real initial conditions. The origin of these methods is the fractional Newton-Raphson method, but unlike the
latter, the orders proposed here for the fractional derivatives are functions. In the first method, a function is
used to guarantee an order of convergence (at least) quadratic, and in the other, a function is used to avoid the
discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible
that the method has at most an order of convergence (at least) linear.
The presentation outlines an approach for invariant-free clausal temporal resolution. It introduces temporal logic and its role in modeling dynamic systems. The temporal logic PLTL is described, as well as existing techniques for clausal resolution and clausal normal forms. The presentation proposes an invariant-free approach to temporal resolution and discusses ongoing and future work.
Oscillation Criteria for First Order Nonlinear Neutral Delay Difference Equat...inventionjournals
1) The document discusses oscillation criteria for first order nonlinear neutral delay difference equations with variable coefficients of the form x(n+1) - a(n)f(x(n-k)) = 0, where a(n) are positive sequences and establishes sufficient conditions for all solutions to oscillate.
2) Five theorems are presented that give oscillation criteria based on relationships between the sequences a(n) and b(n) in the equation.
3) Examples are given that satisfy the conditions of three of the theorems, confirming that all solutions of the example equations oscillate.
The document discusses Boolean satisfiability (SAT) problems and whether they exhibit genuine phase transitions. It summarizes that while 2-SAT has a proven discontinuous phase transition, the conjectured transition for 3-SAT at α ≈ 4.2 has not been proven. A toy model is presented showing that 3-SAT may not display a real phase transition but only a threshold phenomenon induced by statistics. The model supports investigating quantitative parameters like number of solutions instead of just existence of a solution. The document questions whether k-SAT problems truly exhibit phase transitions or if usage of the term needs clarification.
This document discusses inference in first-order logic and various proof strategies. It begins by describing a general proof procedure that uses binary resolution and represents proofs as trees. It then discusses different proof strategies like unit preference, set of support strategy, input resolution, linear resolution, and SLD-resolution. SLD-resolution is described as a sound and complete proof procedure for definite clauses. The document also introduces the concepts of non-monotonic reasoning and default reasoning, describing both non-monotonic logic and default logic as approaches to modeling this type of reasoning.
Formal Languages and Automata Theory unit 5Srimatre K
This document summarizes key concepts from Unit 5, including types of Turing machines, undecidability, recursively enumerable languages, Post's correspondence problem, and counter machines. It defines undecidable problems as those with no algorithm to solve them. Examples of undecidable problems include the halting problem and determining if a Turing machine accepts a language that is not recursively enumerable. Post's correspondence problem and its modified version are presented with examples. Recursively enumerable languages are defined and properties like concatenation, Kleene closure, union, and intersection are described. Counter machines are defined as having states, input alphabet, start/final states, and transitions that allow incrementing, decrementing, and checking if a
The document provides an overview of first-order logic (FOL) including its syntax, semantics, and inference rules. It defines the basic components of FOL such as terms, atomic formulas, literals, clauses, and formulas. It also explains substitutions, unification, semantics, and provides an example of representing a block world in FOL. The goal is for students to understand FOL as a knowledge representation language and be able to apply inference rules and implement automated theorem provers.
An alternative scheme for approximating a periodic functionijscmcj
Fourier series is generally used in applied mathematics and non-linear mechanics to solve the problems containing periodic functions. The aim of this paper is to present a new scheme through involving the Taylor’s series after some process modification for all such problems. It will save the long evaluation process involve in computing the different constants of Fourier series. The result obtained by the present
method has been compared with the result from the Fourier series expansion w.r.t. the exact solution and
found to be more satisfactory.
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...mathsjournal
The following document presents some novel numerical methods valid for one and several variables, which
using the fractional derivative, allow us to find solutions for some nonlinear systems in the complex space using
real initial conditions. The origin of these methods is the fractional Newton-Raphson method, but unlike the
latter, the orders proposed here for the fractional derivatives are functions. In the first method, a function is
used to guarantee an order of convergence (at least) quadratic, and in the other, a function is used to avoid the
discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible
that the method has at most an order of convergence (at least) linear.
The presentation outlines an approach for invariant-free clausal temporal resolution. It introduces temporal logic and its role in modeling dynamic systems. The temporal logic PLTL is described, as well as existing techniques for clausal resolution and clausal normal forms. The presentation proposes an invariant-free approach to temporal resolution and discusses ongoing and future work.
Oscillation Criteria for First Order Nonlinear Neutral Delay Difference Equat...inventionjournals
1) The document discusses oscillation criteria for first order nonlinear neutral delay difference equations with variable coefficients of the form x(n+1) - a(n)f(x(n-k)) = 0, where a(n) are positive sequences and establishes sufficient conditions for all solutions to oscillate.
2) Five theorems are presented that give oscillation criteria based on relationships between the sequences a(n) and b(n) in the equation.
3) Examples are given that satisfy the conditions of three of the theorems, confirming that all solutions of the example equations oscillate.
The document discusses Boolean satisfiability (SAT) problems and whether they exhibit genuine phase transitions. It summarizes that while 2-SAT has a proven discontinuous phase transition, the conjectured transition for 3-SAT at α ≈ 4.2 has not been proven. A toy model is presented showing that 3-SAT may not display a real phase transition but only a threshold phenomenon induced by statistics. The model supports investigating quantitative parameters like number of solutions instead of just existence of a solution. The document questions whether k-SAT problems truly exhibit phase transitions or if usage of the term needs clarification.
This document discusses inference in first-order logic and various proof strategies. It begins by describing a general proof procedure that uses binary resolution and represents proofs as trees. It then discusses different proof strategies like unit preference, set of support strategy, input resolution, linear resolution, and SLD-resolution. SLD-resolution is described as a sound and complete proof procedure for definite clauses. The document also introduces the concepts of non-monotonic reasoning and default reasoning, describing both non-monotonic logic and default logic as approaches to modeling this type of reasoning.
Formal Languages and Automata Theory unit 5Srimatre K
This document summarizes key concepts from Unit 5, including types of Turing machines, undecidability, recursively enumerable languages, Post's correspondence problem, and counter machines. It defines undecidable problems as those with no algorithm to solve them. Examples of undecidable problems include the halting problem and determining if a Turing machine accepts a language that is not recursively enumerable. Post's correspondence problem and its modified version are presented with examples. Recursively enumerable languages are defined and properties like concatenation, Kleene closure, union, and intersection are described. Counter machines are defined as having states, input alphabet, start/final states, and transitions that allow incrementing, decrementing, and checking if a
The document provides an overview of first-order logic (FOL) including its syntax, semantics, and inference rules. It defines the basic components of FOL such as terms, atomic formulas, literals, clauses, and formulas. It also explains substitutions, unification, semantics, and provides an example of representing a block world in FOL. The goal is for students to understand FOL as a knowledge representation language and be able to apply inference rules and implement automated theorem provers.
SSOS group of companies is dedicated to provide facility management, HR solutions. We provide lots of services which facilitate you in many ways. We provide several types of best and reliable facilities like housekeeping services, pantry services, support half services, pest control services, carpet & Sofa shampoo services, Façade/Glass cleaning Service, Gardening Services, Security services, Electro-Mechanical services, HVAC (operation & Management), Plumbing services etc.
Stand-out leaders don't hire. Yes they hire to fill a job but they recruit to bring in the right "human essence" to their organization to ensure its long term success. Discover these competencies in people and gather them around you...
Govinda Prasad is a UI/Front-End Developer with over 2 years of experience developing websites using HTML, CSS, JavaScript, AngularJS, jQuery, and Adobe tools. He has worked on projects for clients like Procter & Gamble and Mindtree developing front-ends in Sitecore and responsive designs using Bootstrap. Govinda holds a B.E. in Information Science and Technology from V.T.U. Belgaum and is proficient in English, Kannada, and Hindi.
New College Nottingham provides English language courses and teacher development programs. Veriko Michitashvili participated in a program from January to February 2012 that included sessions on contemporary teaching methodology, learner autonomy, using instructional technology, and materials analysis. The program also included visits to several British cities and towns. It promoted project-based learning and fostering learner autonomy through various classroom activities. The document discussed using multimedia resources like videos and the benefits of computer-assisted language learning.
Kumar Rishabh has over 5 years of experience in business development and currently works as a Senior Business Manager at Hacker-German Made Pvt. Ltd. He has expertise in operations, corporate tie-ups, vendor management, business development through marketing and branding activities. Previously, he worked as a Business Development Manager at Zest Agro Pvt. Ltd. and a Business Manager at Lodhi Sports Pvt. Ltd. Rishabh holds an MBA in Retail Management and is a Microsoft Certified Systems Engineer.
The International Medical City (IMC) project proposes:
1) Establishing a dedicated healthcare cluster in Belo Horizonte, Brazil that incorporates hospitals, clinics, research centers and medical schools to improve healthcare delivery and innovation.
2) Adopting a "patient centric" approach called P4 Medicine that shifts the focus from disease to wellness using personalized, predictive, preventive and participative care.
3) Creating an integrated healthcare intelligence system using patient data and advanced technologies to enable preventive care, disease prediction and participation of individuals in their own health management.
Shibuthankachan is seeking a challenging role in sales, business development, client relationship management, or a similar position, preferably in the banking or financial sector. He has over 9 years of experience in roles like business development, sales, client relationship management, and collections. Currently he works as a Regional Head of Sales at ICICI Bank, where he is responsible for meeting sales targets, managing dealer partnerships, and other duties. He has a history of consistently achieving sales goals and has received appreciation letters and cash incentives for his performance.
This document provides an agenda and overview of topics to be covered in an architecture course, including finishing a discussion of Vienna and moving to the Arts and Crafts movement in the UK. It summarizes key figures of the time period like Josef Hoffman, Adolf Loos and Gustav Klimt in Vienna as well as William Morris and the founding of the Arts and Crafts movement in response to the ill effects of industrialization. It also highlights some of Loos' major works like the Looshaus in Vienna and the influential Villa Müller in Prague, as well as key Arts and Crafts designers like Charles Rennie Mackintosh and buildings such as the Hill House outside Glasgow.
Mulbery disorders A Lecture By Allah Dad Khan To FFS Trainee Mr.Allah Dad Khan
Mulberry trees can experience physiological disorders caused by environmental stressors like low soil moisture and high temperatures. One such disorder is summer scorch, where the leaves develop chlorotic or yellow margins due to these stress conditions. Proper irrigation and mulching can help prevent summer scorch and other physiological issues for mulberry trees during hot and dry periods.
This document discusses choosing a university degree and career path. It presents arguments for both following your passion and studying something you love, versus choosing a more practical degree with better career prospects. The document also discusses how college degrees are becoming less valuable as more people obtain them, leading to credential inflation and underemployment of graduates. Choosing a career path is a significant decision that will impact future job satisfaction and financial security. The document advocates considering both personal interests and realistic career opportunities when selecting a field of study.
The document promotes an environmentally friendly freight and logistics company called Pinks Go Green. It emphasizes using non-GMO and sustainable products and working to preserve clean air, water, and healthy food. The company aims to utilize its knowledge of logistics to provide excellent customer service while helping to preserve natural resources and the environment. It works with carriers that have high safety ratings and promotes fair trade from field to delivery.
The document discusses several abused drugs including heroin, methamphetamine, LSD, ecstasy, inhalants, cocaine, prescription drugs, marijuana, tobacco, and alcohol. It provides brief descriptions of the effects of each drug, such as the euphoric rush from heroin, the energy and partying effects of meth, the unpredictable hallucinations of LSD, and feelings of closeness from ecstasy. However, it also notes that nearly two-thirds of ecstasy pills contain other dangerous substances. The document concludes by stating that over half of Americans drink alcohol, and about a quarter participate in binge drinking, making alcohol the most commonly abused substance in the US.
This document discusses sampling and sample size in statistics. It defines key terms like population, sample, sampling unit, sampling frame, and sampling schemes. It explains that sampling allows researchers to generalize results from a subset of the population. The main advantages of sampling are that it is less costly, takes less time, and can provide more accurate results than studying the entire population. The document also discusses different sampling methods like simple random sampling, systematic random sampling, stratified random sampling, and cluster sampling. It notes that sample size depends on several factors and must result in a truly representative sample with small errors.
Novel analysis of transition probabilities in randomized k sat algorithmijfcstjournal
This document summarizes a research paper that proposes a new analysis of transition probabilities in randomized k-SAT algorithms. Specifically:
- It shows the probability of correctly flipping a literal in 2-SAT and 3-SAT approaches 2/3 and 4/7 respectively, using Karnaugh maps to analyze all possible variable combinations.
- It extends this analysis to general k-SAT, showing the transition probability of the Markov chain in randomized k-SAT algorithms approaches 0.5.
- Using this result, it determines the probability and complexity of finding a satisfying assignment for randomized k-SAT, showing values within a polynomial factor of (0.9272)^n and (1.0785)^n for satisf
This document discusses various methods for finding the roots of equations, including bracketing methods like bisection and false position, open methods like fixed point iteration and Newton-Raphson, and the secant method. It provides formulas and explanations of how each method works to successively approximate a root through iterative calculations. Examples are given of applying the methods to solve engineering problems involving equations of state.
This document provides an introduction and overview to Unit 13 of the course MST209 Mathematical methods and models. Unit 13 focuses on modelling systems using non-linear differential equations. It discusses two main examples - modeling the interaction between predator and prey populations using the Lotka-Volterra equations, and modeling the motion of a pendulum using differential equations. The unit emphasizes qualitative analysis and interpretation of solutions rather than explicit solutions. It introduces concepts like equilibrium solutions and linearizing near equilibria to understand behavior. Sections 1 and 2 develop these ideas for the Lotka-Volterra population model, while Section 3 applies similar techniques to the pendulum motion model.
This document summarizes an algorithm for directly solving 3-SAT instances using packed state stochastic processes, without first reducing the problem to 3-RSS. The algorithm, called PSSP-1, represents variables as having true, false, or "packed" states. When choosing an unsatisfied clause, it prioritizes literals in packed or minority states for flipping. Experimental results show PSSP-1 can solve most SATLIB benchmarks within polynomial time, though a proposed "Worst Case 4" 3-SAT instance seems intractable for packed computation algorithms. The paper also compares PSSP-1 to modern versions of Schöning's algorithm that incorporate variable selection probabilities.
The binary search algorithm allows for faster searching of an ordered array compared to linear search. It works by first examining the middle element of the array and eliminating half of the elements from further search based on whether the target value is less than or greater than the middle element. This process continues, halving the search space on each iteration, allowing binary search to have a time complexity of O(log n).
This document discusses reductions between various NP-complete and NP-hard problems such as Boolean satisfiability, 3-SAT, maximum clique, and others. It introduces several new NP-hard problems and reductions between problems. Specifically, it shows reductions from Boolean satisfiability to rules states satisfiability (RSS), from RSS to 3-RSS, from 3-SAT and 3-RSS to each other, from a type of timetabling problem to 3-RSS, from k-SAT to k-maximum hyper clique, from maximum clique to 3-maximum hyper clique, from maximum clique to Max True 2-SAT, and from Max 2-SAT to a new problem called Max Var 2-
SSOS group of companies is dedicated to provide facility management, HR solutions. We provide lots of services which facilitate you in many ways. We provide several types of best and reliable facilities like housekeeping services, pantry services, support half services, pest control services, carpet & Sofa shampoo services, Façade/Glass cleaning Service, Gardening Services, Security services, Electro-Mechanical services, HVAC (operation & Management), Plumbing services etc.
Stand-out leaders don't hire. Yes they hire to fill a job but they recruit to bring in the right "human essence" to their organization to ensure its long term success. Discover these competencies in people and gather them around you...
Govinda Prasad is a UI/Front-End Developer with over 2 years of experience developing websites using HTML, CSS, JavaScript, AngularJS, jQuery, and Adobe tools. He has worked on projects for clients like Procter & Gamble and Mindtree developing front-ends in Sitecore and responsive designs using Bootstrap. Govinda holds a B.E. in Information Science and Technology from V.T.U. Belgaum and is proficient in English, Kannada, and Hindi.
New College Nottingham provides English language courses and teacher development programs. Veriko Michitashvili participated in a program from January to February 2012 that included sessions on contemporary teaching methodology, learner autonomy, using instructional technology, and materials analysis. The program also included visits to several British cities and towns. It promoted project-based learning and fostering learner autonomy through various classroom activities. The document discussed using multimedia resources like videos and the benefits of computer-assisted language learning.
Kumar Rishabh has over 5 years of experience in business development and currently works as a Senior Business Manager at Hacker-German Made Pvt. Ltd. He has expertise in operations, corporate tie-ups, vendor management, business development through marketing and branding activities. Previously, he worked as a Business Development Manager at Zest Agro Pvt. Ltd. and a Business Manager at Lodhi Sports Pvt. Ltd. Rishabh holds an MBA in Retail Management and is a Microsoft Certified Systems Engineer.
The International Medical City (IMC) project proposes:
1) Establishing a dedicated healthcare cluster in Belo Horizonte, Brazil that incorporates hospitals, clinics, research centers and medical schools to improve healthcare delivery and innovation.
2) Adopting a "patient centric" approach called P4 Medicine that shifts the focus from disease to wellness using personalized, predictive, preventive and participative care.
3) Creating an integrated healthcare intelligence system using patient data and advanced technologies to enable preventive care, disease prediction and participation of individuals in their own health management.
Shibuthankachan is seeking a challenging role in sales, business development, client relationship management, or a similar position, preferably in the banking or financial sector. He has over 9 years of experience in roles like business development, sales, client relationship management, and collections. Currently he works as a Regional Head of Sales at ICICI Bank, where he is responsible for meeting sales targets, managing dealer partnerships, and other duties. He has a history of consistently achieving sales goals and has received appreciation letters and cash incentives for his performance.
This document provides an agenda and overview of topics to be covered in an architecture course, including finishing a discussion of Vienna and moving to the Arts and Crafts movement in the UK. It summarizes key figures of the time period like Josef Hoffman, Adolf Loos and Gustav Klimt in Vienna as well as William Morris and the founding of the Arts and Crafts movement in response to the ill effects of industrialization. It also highlights some of Loos' major works like the Looshaus in Vienna and the influential Villa Müller in Prague, as well as key Arts and Crafts designers like Charles Rennie Mackintosh and buildings such as the Hill House outside Glasgow.
Mulbery disorders A Lecture By Allah Dad Khan To FFS Trainee Mr.Allah Dad Khan
Mulberry trees can experience physiological disorders caused by environmental stressors like low soil moisture and high temperatures. One such disorder is summer scorch, where the leaves develop chlorotic or yellow margins due to these stress conditions. Proper irrigation and mulching can help prevent summer scorch and other physiological issues for mulberry trees during hot and dry periods.
This document discusses choosing a university degree and career path. It presents arguments for both following your passion and studying something you love, versus choosing a more practical degree with better career prospects. The document also discusses how college degrees are becoming less valuable as more people obtain them, leading to credential inflation and underemployment of graduates. Choosing a career path is a significant decision that will impact future job satisfaction and financial security. The document advocates considering both personal interests and realistic career opportunities when selecting a field of study.
The document promotes an environmentally friendly freight and logistics company called Pinks Go Green. It emphasizes using non-GMO and sustainable products and working to preserve clean air, water, and healthy food. The company aims to utilize its knowledge of logistics to provide excellent customer service while helping to preserve natural resources and the environment. It works with carriers that have high safety ratings and promotes fair trade from field to delivery.
The document discusses several abused drugs including heroin, methamphetamine, LSD, ecstasy, inhalants, cocaine, prescription drugs, marijuana, tobacco, and alcohol. It provides brief descriptions of the effects of each drug, such as the euphoric rush from heroin, the energy and partying effects of meth, the unpredictable hallucinations of LSD, and feelings of closeness from ecstasy. However, it also notes that nearly two-thirds of ecstasy pills contain other dangerous substances. The document concludes by stating that over half of Americans drink alcohol, and about a quarter participate in binge drinking, making alcohol the most commonly abused substance in the US.
This document discusses sampling and sample size in statistics. It defines key terms like population, sample, sampling unit, sampling frame, and sampling schemes. It explains that sampling allows researchers to generalize results from a subset of the population. The main advantages of sampling are that it is less costly, takes less time, and can provide more accurate results than studying the entire population. The document also discusses different sampling methods like simple random sampling, systematic random sampling, stratified random sampling, and cluster sampling. It notes that sample size depends on several factors and must result in a truly representative sample with small errors.
Novel analysis of transition probabilities in randomized k sat algorithmijfcstjournal
This document summarizes a research paper that proposes a new analysis of transition probabilities in randomized k-SAT algorithms. Specifically:
- It shows the probability of correctly flipping a literal in 2-SAT and 3-SAT approaches 2/3 and 4/7 respectively, using Karnaugh maps to analyze all possible variable combinations.
- It extends this analysis to general k-SAT, showing the transition probability of the Markov chain in randomized k-SAT algorithms approaches 0.5.
- Using this result, it determines the probability and complexity of finding a satisfying assignment for randomized k-SAT, showing values within a polynomial factor of (0.9272)^n and (1.0785)^n for satisf
This document discusses various methods for finding the roots of equations, including bracketing methods like bisection and false position, open methods like fixed point iteration and Newton-Raphson, and the secant method. It provides formulas and explanations of how each method works to successively approximate a root through iterative calculations. Examples are given of applying the methods to solve engineering problems involving equations of state.
This document provides an introduction and overview to Unit 13 of the course MST209 Mathematical methods and models. Unit 13 focuses on modelling systems using non-linear differential equations. It discusses two main examples - modeling the interaction between predator and prey populations using the Lotka-Volterra equations, and modeling the motion of a pendulum using differential equations. The unit emphasizes qualitative analysis and interpretation of solutions rather than explicit solutions. It introduces concepts like equilibrium solutions and linearizing near equilibria to understand behavior. Sections 1 and 2 develop these ideas for the Lotka-Volterra population model, while Section 3 applies similar techniques to the pendulum motion model.
This document summarizes an algorithm for directly solving 3-SAT instances using packed state stochastic processes, without first reducing the problem to 3-RSS. The algorithm, called PSSP-1, represents variables as having true, false, or "packed" states. When choosing an unsatisfied clause, it prioritizes literals in packed or minority states for flipping. Experimental results show PSSP-1 can solve most SATLIB benchmarks within polynomial time, though a proposed "Worst Case 4" 3-SAT instance seems intractable for packed computation algorithms. The paper also compares PSSP-1 to modern versions of Schöning's algorithm that incorporate variable selection probabilities.
The binary search algorithm allows for faster searching of an ordered array compared to linear search. It works by first examining the middle element of the array and eliminating half of the elements from further search based on whether the target value is less than or greater than the middle element. This process continues, halving the search space on each iteration, allowing binary search to have a time complexity of O(log n).
This document discusses reductions between various NP-complete and NP-hard problems such as Boolean satisfiability, 3-SAT, maximum clique, and others. It introduces several new NP-hard problems and reductions between problems. Specifically, it shows reductions from Boolean satisfiability to rules states satisfiability (RSS), from RSS to 3-RSS, from 3-SAT and 3-RSS to each other, from a type of timetabling problem to 3-RSS, from k-SAT to k-maximum hyper clique, from maximum clique to 3-maximum hyper clique, from maximum clique to Max True 2-SAT, and from Max 2-SAT to a new problem called Max Var 2-
This document provides an overview of mathematical patterns and sequences. It defines key terms like sequence, term, recursive formula, and explicit formula. Examples demonstrate how to find sequence terms using recursive and explicit rules, write rules for linear and quadratic patterns, and calculate terms of fractal sequences generated by repeated iterations. The document also illustrates sequence patterns with examples like the Fibonacci sequence, flushing water in a toilet, and the Cantor set fractal.
Dr Marcel Remon, Professor of Statistics, Fundamentals of Mathematics and Probability at Namur University, presented an overview of his research as part of the SMART Seminar Series on 31st August 2017.
More information: http://www.uoweis.co/event/a-polynomial-algorithm-to-solve-hard-np-3-cnf-sat-problems/
Keep updated with future events: http://www.uoweis.co/tag/smart-infrastructure/
This document summarizes the topological structure of ternary residue curve maps, which describe the dynamics of ternary distillation processes. It introduces differential equations that model ternary distillation and place a meaningful structure on ternary phase diagrams. By recognizing this structure is subject to the Poincaré-Hopf index theorem, the authors obtained a topological relationship between azeotropes and pure components in ternary mixtures. This relationship provides useful information about ternary mixture distillation behavior and predicts situations where ternary azeotropes cannot occur.
This document discusses numerical methods for solving partial differential equations (PDEs). It begins by classifying PDEs as parabolic, elliptic, or hyperbolic based on their coefficients. It then introduces finite difference methods, which approximate PDE solutions on a grid by replacing derivatives with finite differences. In particular, it describes the forward time centered space (FTCS) scheme for solving the 1D heat equation numerically and analyzing its stability using von Neumann analysis.
Newton's method is an iterative method for finding roots of functions. It requires calculating the derivative of the function. The method may fail to converge if the derivative does not exist at the root, is discontinuous at the root, or if the root has a multiplicity greater than one. It also may not converge if the starting point is too far from the root or if a stationary point is encountered. When it does converge, the rate is quadratic close to a simple root where the derivative is non-zero, but may be linear or fail to converge in other cases.
A Markov model assumes that the current state captures all relevant information for predicting the future. It can be used for language modeling by assigning probabilities to word sequences. Google's PageRank algorithm ranks web pages based on the principle that more authoritative pages, as determined by other pages linking to them, should rank higher. It models the probability of being on a page as a stationary distribution of a Markov chain defined by the link structure of the web.
This document summarizes Newton's method, an iterative process for finding approximations of the zeroes of a function. It works by using tangent lines to get better approximations with each iteration. The method starts with an initial guess x1 and calculates successive approximations x2, x3, etc. by finding the x-intercept of the tangent line at the previous point. If the approximations converge to a limit, Newton's method has found a zero of the function. The document provides examples of functions where Newton's method does and does not converge.
Invariant Manifolds, Passage through Resonance, Stability and a Computer Assi...Diego Tognola
1) The document is a dissertation submitted to ETH Zurich that studies invariant manifolds, passage through resonance, stability, and applies these concepts to a synchronous motor model.
2) It first develops theory for a general Hamiltonian system coupled to a linear system by weak periodic perturbations, showing the persistence of invariant manifolds. It then uses averaging techniques to analyze global dynamics, assuming a finite number of resonances.
3) It represents the reduced system in a way suitable for stability analysis, covering both non-degenerate and degenerate cases.
4) The second part applies these methods to explicitly model a miniature synchronous motor, analytically deriving approximations and numerically simulating and confirming the dynamics, showing approach
This document summarizes an algorithm called Connection Sweeping with Synthesis (CSS) for solving 2-SAT problems and discusses attempts to extend CSS to 3-SAT problems. CSS models a 2-SAT problem as a multipartite graph and uses a ternary operator to sweep connections, performing synthesis after each scan. The algorithm is proven correct and runs in O(n^3) time, faster than resolution methods. While CSS was extended to 3-SAT, the author proves these extensions are incorrect, matching prior results showing resolution is exponential for 3-SAT.
This document summarizes recent convergence results for the fuzzy c-means clustering algorithm (FCM). It discusses both numerical convergence, referring to how well the algorithm attains the minima of an objective function, and stochastic convergence, referring to how accurately the minima represent the actual cluster structure in data. For numerical convergence, the document outlines global and local convergence theorems, showing FCM converges to minima or saddle points globally and linearly to local minima. For stochastic convergence, it discusses a consistency result showing the minima accurately represent cluster structure under certain statistical assumptions.
A Probabilistic Attack On NP-Complete ProblemsBrittany Allen
This document discusses reformulating NP-complete problems in terms of continuous mathematics using probability theory. Specifically, it considers the 3-SAT NP-complete problem and introduces new probability variables to represent bit assignments. A cost function is constructed as a sum of clause satisfaction probabilities. Key properties of the cost function are that it is harmonic over subsets of variables and its Hessian has zero diagonal entries. The cost function is always positive inside the problem's domain and achieves its min/max on the boundary. The spectrum of cost function values on vertices corresponds to number of unsatisfied clauses. Overall, the approach reformulates 3-SAT in terms of a harmonic cost function to manipulate solutions without examining them individually.
This document discusses recursion versus iteration in Lisp and Common Lisp. It notes that the original Lisp language was purely functional and used recursion to solve problems since it lacked local variables and iteration. While recursion is more elegant and leads to less code in some cases, it is also harder to understand, debug, and less efficient due to function call overhead. However, recursion is necessary for some problems like tree and graph traversals. Common Lisp makes recursion easier through its runtime stack and debugger. Several examples of recursive functions are provided, including ones for lists, trees, searching, and factorials. Tail recursion is discussed as a way to make recursion more efficient.
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...mathsjournal
The following document presents some novel numerical methods valid for one and several variables, which
using the fractional derivative, allow us to find solutions for some nonlinear systems in the complex space using
real initial conditions. The origin of these methods is the fractional Newton-Raphson method, but unlike the
latter, the orders proposed here for the fractional derivatives are functions. In the first method, a function is
used to guarantee an order of convergence (at least) quadratic, and in the other, a function is used to avoid the
discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible
that the method has at most an order of convergence (at least) linear
It covers knowledge representation techniques using propositional and predicate logic. It also discusses about the knowledge inference using resolution refutation process, rule based system and bayesian network.
1. Researchjournali’s Journal of Computer Science
Vol. 1 | No. 4 August | 2014 ISSN 2349-5391
1
Cristian Dumitrescu
BSc. in Mathematics, Freelance Mathematician,
119 Young St., Ap. 11, Kitchener, Ontario N2H 4Z3,
Canada
An Efficient
Probabilistic
Algorithm For An
Important NP
Complete Problem In
Mathematics
2. Researchjournali’s Journal of Computer Science
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ABSTRACT
In this article I describe an efficient, randomized algorithm (section 3) that I think solves the 3- SAT problem
(known to be NP complete) with high probability in polynomial time, and a bit of the history of the problem
under consideration. In the last section I present an interesting application, based on an idea that belongs to
Godel. The appendix contains a Markov chain model for the dynamics of the algorithm.
Keywords: The Satisfiability Problem, Markov chains, random walk with absorbing barriers
1. INTRODUCTION
In this article I propose an algorithm that has the potential of changing the way we do mathematics (based on
an idea that belongs to Godel), and also has many applications in many fields of activity. I build on the work
of Professor Uwe Schoning, but my algorithm is more efficient. Ultimately, my algorithm is inspired by
nature, since it can be related to gene acquisition/deletion and speciation in the greater theory of evolution.
Section 1. Useful notions that are used for the analysis of the algorithm
A Boolean expression is said to be in conjunctive normal form (CNF) if it is of the form
, and each Ei, called a clause (or conjunct), is of the form
, where each ij is a literal, either x or x, for some variable x.
A Boolean expression is said to be in disjunctive normal form (DNF) if it is of the form
, and each Fj, called a clause (or disjunct), is of the form
, where each jk is a literal, either y or y, for some variable y.
A Boolean expression in CNF form is called satisfiable if there is some assignment of 0’s and 1’s to the
variables that gives the expression the value 1.
The satisfiability problem is to determine, given a Boolean expression, whether it is satisfiable.
An expression is said to be 3 - CNF if each clause has exactly three distinct literals.
Theorem 1 (See reference [1]). L3SAT, the satisfiability problem for 3 - CNF expressions, is NP - complete
The Hamming distance dH(x, y) between two vectors x, y is the number of components in which they differ. It
is known that the Hamming distance dH(x, y) satisfies the conditions for a metric. Related to the theory of
symmetric random walks (in one dimension), we have the following theorem.
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Theorem 2 (See reference [2]). Limit theorem for first passages. For fixed t, the probability that the first
passage through r occurs before epoch tends to ,
as , where N is the normal distribution function. We note that when , then P tends to 1.
Section 2. The description of Schoning’s algorithm
Input: a formula in 3-CNF with n variables.
Guess an initial assignment for the n variables, uniform at random.
Repeat 3n times:
If the formula is satisfied by the actual assignment: stop and accept.
Let C be some clause not being satisfied by the actual assignment.
Pick one of the 3 literals in the clause at random, and flip its value in the current assignment.
Schoning proves (see reference [3]) that the complexity of k-SAT (with this algorithm) is within a polynomial
factor of . This means that this algorithm does not have direct practical value, since the
expected time needed to hit a solution grows exponentially with the number of variables.
Section 3. The presentation of the DEA class of algorithms
3.1 THE PROPOSED DUAL EXPRESSION ALGORITHM (DEA2)
Basically, given a 3-CNF expression, we want to construct a 2-CNF path through all the 3-clauses from the 3-
CNF expression. In the following, I will explain what I mean by a 2-CNF path though the 3-CNF expression.
Each 3-clause in the original 3-CNF expression has the form , where each represents a variable
or the negation of one of the original variables that appear in the given 3-CNF expression. For example, the 3-
clause contains all the 2 – clauses , , and .
By the active 2-CNF chain, I mean a conjunction of 2 – clauses so that each 2 – clause is chosen from a
corresponding 3 – clause from the original 3-CNF expression. In the current state (during the execution of the
algorithm), the length of the active 2-CNF chain could be equal to the number of clauses or less than the
number of clauses in the original 3-CNF expression.
The function Test2CNF will check whether the conjunction of all the 2 – clauses from the active 2-CNF chain
(in the current state) is satisfiable or not. We note that we do not work with the original variables, the function
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Test2CNF is just a 2-CNF tester/solver, it will not look for a satisfiable assignment in the original variables
unless all the 3 – clauses from the original 3-CNF expression are satisfied.
We also notice that 2-SAT can be solved in linear time. The reason why I call it DEA2 (dual expression
algorithm) has to do (intuitively) with the fact that the 3 – clauses from the original 3-CNF expression, for
example, can also be written as , and I look at the 2-
clauses as some sort of new variables (but the way you look at it does not affect the understanding of the
algorithm, but this is the way I thought about it when I invented the algorithm).
We also assume that we order the clauses in the 3-CNF expression such that when we talk about the next
unsatisfied clause (for example), or the first clause satisfying a certain condition, we know which clause we
are talking about. When we choose a clause at random, it does not matter.
Here is the algorithm, the dual expression algorithm (DEA2):
Input: a formula in 3-CNF with n variables .
Choose a 2 – clause from the first 3 – clause (of the given 3-CNf expression) and add it to the active 2-CNF
chain.
Repeat A(n) times (where A(n) is a polynomial discussed later):
Call the routine Test2CNF for the conjunction of all the 2 – clauses that are currently part of the active 2-
CNF chain.
If Test2CNF finds that the active 2-CNF chain is satisfied (I will sometimes say that the chain is consistent),
and if all the 3 – clauses from the original 3-CNF expression are satisfied, then the original 3-CNF
expression is satisfied, and we finished, return “expression satisfiable”.
If Test2CNF finds that the active 2-CNF chain is satisfied (I will sometimes say that the chain is consistent),
but not all the 3 – clauses from the original 3-CNF expression are satisfied (or not considered yet), then
choose a random unsatisfied 3 – clause, and choose a random 2 – clause from it (from the three possible) and
add this 2 – clause to the active 2-CNF chain.
If Test2CNF finds that the active 2-CNF chain is not satisfied (I will sometimes say that the chain is
inconsistent), then discard a 2 – clause (chosen at random) from the active 2-CNF chain.
Repeat cycle.
If a solution has not been found yet, return “expression unsatisfiable”.
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In a slightly different version of this algorithm, if Test2CNF finds consistency, then we look at all 2- clauses
(there are three of them) from an unsatisfied 3 - clause such that the 3 – clause can be satisfied while
maintaining consistency of the active 2-CNF chain (we try to force consistency).
The modified version of the DEA2 algorithm can be written as follows:
Input: a formula in 3-CNF with n variables .
Choose a 2 – clause from the first 3 – clause (of the given 3-CNF expression) and add it to the active 2-CNF
chain.
Repeat A(n) times (where A(n) is a polynomial discussed later):
Call the routine Test2CNF for the conjunction of all the 2 – clauses that are currently part of the active 2-
CNF chain.
If Test2CNF finds that the active 2-CNF chain is satisfied (I will sometimes say that the chain is consistent),
and if all the 3 – clauses from the original 3-CNF expression are satisfied, then the original 3-CNF
expression is satisfied, and we finished, return “expression satisfiable”.
If Test2CNF finds that the active 2-CNF chain is satisfied (I will sometimes say that the chain is consistent),
but not all the 3 – clauses from the original 3-CNF expression are satisfied (or not considered yet), then
choose at random an unsatisfied 3 – clause, and choose another 2 – clause from it (from the three possible, in
order) and add this 2 – clause to the active 2-CNF chain only if the 2-CNF chain stays consistent, otherwise
do not add any 2 – clause to the active 2-CNF chain. Basically, we add a 2- clause to the active 2-CNF chain
only if we are sure that the chain stays consistent.
If Test2CNF finds that the active 2-CNF chain is not satisfied (I will sometimes say that the chain is
inconsistent), then discard a 2 – clause (chosen at random) from the active 2-CNF chain.
Repeat cycle.
If a solution has not been found yet, return “expression unsatisfiable”.
3.2 THE PROPOSED DUAL EXPRESSION ALGORITHM (SIMPLIFIED VERSION, DEA1)
We will consider now a simplified version of the algorithm, a version that is more intuitive and easy to
understand. We will call a literal, a variable or its negation. For example, or are literals. We also note
that the negation of the literal is , and the negation of the literal is . Given a 3-CNF expression, by
a 1-CNF path through all the clauses of the 3-CNF expression we mean a conjunction of literals, one from
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each clause, such that once a literal appears in the 1-CNF path, its negation does not appear in the
conjunction.
By the active 1-CNF chain, I mean a conjunction of literals so that each literal is chosen from a corresponding
3 – clause from the original 3-CNF expression. In the current state (during the execution of the algorithm), the
length of the active 1-CNF chain could be equal to the number of clauses or less than the number of clauses in
the original 3-CNF expression.
The function Test1CNF will check whether the conjunction of all literals from the active 1-CNF chain (in the
current state) is satisfiable or not. We note that in this case we work with the original variables. The function
Test1CNF basically checks that if one literal appears in the active 1-CNF chain, then its negation does not
appear in it. Here is the algorithm, the dual expression algorithm (DEA1):
Input: a formula in 3-CNF with n variables .
Choose a literal from the first 3 – clause (of the given 3-CNf expression) and add it to the active 1-CNF
chain.
Repeat A(n) times (where A(n) is a polynomial discussed later):
Call the routine Test1CNF for the conjunction of all the literals that are currently part of the active 1-CNF
chain.
If Test1CNF finds that the active 1-CNF chain is satisfied (I will sometimes say that the chain is consistent),
and if all the 3 – clauses from the original 3-CNF expression are satisfied, then the original 3-CNF
expression is satisfied, and we finished, return “expression satisfiable”.
If Test1CNF finds that the active 1-CNF chain is satisfied (I will sometimes say that the chain is consistent),
but not all the 3 – clauses from the original 3-CNF expression are satisfied (or not considered yet), then
choose a random unsatisfied 3 – clause, and choose a random literal from it (from the three possible) and add
this literal to the active 1-CNF chain.
If Test1CNF finds that the active 1-CNF chain is not satisfied (I will sometimes say that the chain is
inconsistent), then discard a literal (chosen at random) from the active 1-CNF chain.
Repeat cycle.
If a solution has not been found yet, return “expression unsatisfiable”.
In a slightly different version of this algorithm, if Test1CNF finds consistency, then we look at all the literals
(there are three of them) from an unsatisfied 3 - clause such that the 3 – clause can be satisfied while
maintaining consistency of the active 1-CNF chain (we try to force consistency). The modified version of the
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DEA1 algorithm can be written similarly to the modified version of the DEA2 algorithm (the terminology
emphasizes this analogy).
Section 4. Godel’s letter to von Newmann.
For general implications, related to efficiently solving NP – complete problems, see [4]. An interesting
application is related to the problem of automated theorem proving using an efficient algorithm for NP –
complete problems.
We know that we can solve the following problem in polynomial time:
Given two well formed formulas α and β, in a given axiomatic system (like ZFC), is β a ZFC – proof of α?
Therefore, the following problem is in NP (it can be easily proved):
Given a formula α, and a number n, is there a ZFC – proof of size at most n for α?
Any efficient solution for NP-complete problems would make automated theorem proving a reality. We can
have an automated system that would tell us (with probability as close to 1 as we want) that no solution to a
given problem exists, that can be written in (for example) less than 10000 pages, or hit upon (find) such a
proof.
In a letter in 1956, Godel asked John von Newmann whether there was a general method to find proofs of size
n, using time that increases only as n or . If such a method existed, Godel argued that this “ would have
consequences of the greatest magnitude. That is to say, it would clearly indicate that the mental effort of the
mathematician in the case of yes or no questions could be completely replaced by machines. One would
indeed have to simply select an n so large that, if the machine yields no result, there would then also be no
reason to think further about the problem. “.
This is not just a problem of optimization, or applied mathematics. I think that this problem should be the
focus of attention for the core of the mathematicians, a problem the solution of which could transform
mathematics and fulfill (to some extent) Hilbert’s dream, by following an idea that belongs to Godel. Another
interesting path is to consider quantum algorithms, quantum random walks, in particular, but we will not go
into this issue here.
CONCLUSIONS
For all practical purposes, we can assume that P = NP, even if the conjecture P NP might be true, if we
exclude randomized algorithms. This article can be considered a review article, but the ideas expressed in
section 3, the DEA algorithm are original though.
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APPENDIX
In this appendix, we will give one example of the DEA2 algorithm dynamics for a very simple 3 – CNF
expression, and we will present an approximate Markov chain model for the DEA2 algorithm (a similar
analysis can be made for DEA1).
A1. Example
We will give an example of the dynamics of the DEA2 algorithm, in a very simple case. We consider the 3 –
CNF expression:
.
We choose the 2-clause from the first 3-clause of the 3-CNF expression and we add it to the active
2-CNF chain. The active 2-CNF chain will be, at this moment:
We call the function Test2CNF. The active 2-CNF chain (at this moment) is satisfiable (I will also say
consistent). Assume that we choose the 2-clause, from the second 3-clause of the 3-CNF
expression E, and we add it to the active 2-CNF chain C. We have then:
We call the function Test2CNF. The active 2-CNF chain (at this moment) is satsifiable (consistent). Assume
that we choose the 2-clause from the third 3-clause of the 3-CNF expression E, and we add it to
the active 2-CNF chain C. We have then:
We call the function Test2CNF. The active 2-CNF chain (at this moment) is satisfiable. Assume that we
choose the 2-clause from the fourth 3-clause of the 3-CNF expression E, and we add it to the
active 2-CNF chain C. We have then:
We call the function Test2CNF. This time the active 2-CNF chain is not satisfiable (not consistent). Now we
have to discard a 2-clause from the active 2-CNF chain. Assume that we discard the 2-clause ,
taken from the third 3-clause of the 3-CNF expression. We have then:
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We call the function Test2CNF. The active 2-CNF chain is satisfiable. Assume that we choose the 2-clause
from the third 3-clause of the 3-CNF expression (because at this moment this is the clause that
is not satisfied yet), and we add it to the active 2-CNF chain. We have then:
We call the function Test2CNF. The active 2-CNF chain is satisfiable. We also notice that all the 3-clauses
from the 3-CNF expression E are satisfied. As a consequence, the original 3-CNF expression E is satisfiable,
and we can finally ask Test2CNF to give us a solution in terms of the original variables ,
for example.
A2. The Markov chain approximate model (for DEA2)
We will call a correct 3 - clause, a clause from where a correct 2 - clause has been chosen in the current active
2-CNF chain. A 3 - clause from which an incorrect 2 - clause has been chosen will be called an incorrect
clause. We will consider the pairs (i, j), where i is the number of correct clauses from which the 2 - clauses
have been chosen in the current active 2-CNF chain, and j is the number of incorrect clauses from which the 2
- clauses have been chosen in the current active 2-CNF chain. In this case, will be the length of the
current active 2-CNF chain, and in general we have , where N is the number of clauses in our 3 –
CNF expression. We note that when we talk about correct and incorrect clauses, the assumption is that a
solution does exist, otherwise this terminology is not meaningful.
We assume that in the current state of the Markov chain, Test2CNF has found consistency, then (following
the algorithm) we choose another 2 - clause from an unsatisfied 3 - clause, and we add it to the current active
2-CNF chain. We have the following transition probabilities for the Markov chain:
. We choose the correct 2- clause from the unsatisfied 3 – clause with probability .
. We choose the incorrect 2 - clause from the unsatisfied 3 - clause with probability .
If in the current state of the Markov chain, Test2CNF has found inconsistency, then (following the algorithm)
we discard a (random) 2 - clause from the current active 2-CNF chain. We have the following transition
probabilities for the Markov chain:
We discard a 2 - clause from a correct 3 - clause with probability .
. We discard a 2- clause from an incorrect 3 - clause with probability .
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We assume that the Markov chain hits consistency with probability α, and it hits inconsistency with
probability β, where . We need to properly define α and β (and prove that they exist). This problem
is not addressed in the current version of this article. At the moment, we can assume that we consider all the
runs of the algorithm in which we have T calls to the function Test2CNF, and in of them the function
finds that the active chain is consistent, and in of them the function finds that the active chain is
inconsistent (how probable that is, this question is not answered here). The approximate model will be a
Markov chain with transition probabilities:
.
.
.
.
Since we know that , that means that the Markov chain dynamics takes place on a 2
– dimensional grid inside the triangle OAB, where O(0,0), A(N, 0), and B(0, N). When the state representing
our current state moves on the grid inside the triangle OAB (one unit up, down, left or right), we consider the
projection on the i – axis. The motion of this projection will not be governed by a Markov chain, but we can
couple it with the dynamics of a Markov chain in such a manner that if this Markov chain (a birth and death
chain, in fact) is expected to reach the point A(N,0) in linear time (linear in N), then the projection mentioned
above will also be expected to reach A(N,0) in linear time. This birth and death chain will have the transition
probabilities (we do not need the value of p(i,i)):
.
.
.
We note that this birth and death chain will have the transition probability towards A(N,0) equal to the
corresponding transition probability of the projection mentioned above, and the transition probability away
from A(N,0) will be greater or equal to the corresponding transition probability of the projection. As a
consequence, if this chain is expected to reach A(N,0) in linear time, then the projection will also reach
A(N,0) in expected linear time.
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Let t(i, i+1) be the random time it takes for the Markov chain to go from i to i+1. An analysis based on
conditioning on the first transition and iteration leads us to the relation:
This formula is valid for any birth and death chain. We note that , and if , then , and
the chain will hit A(N, 0) in linear time in N. Starting from anywhere inside the triangle OAB (following a
distribution that can be easily calculated), and staying inside the triangle, the Markov chain will reach the
point A(N, 0) in expected linear time in N. The condition , together with the equation
will lead us to the conclusion that is a sufficient condition, so that our Markov chain will hit A(N,0)
in expected linear time in N. If our Markov chain (the approximation considered here) hits inconsistency less
than 40% of the time, then it will find a solution in expected linear time in N, the number of clauses in the 3-
CNF expression. We note that in the second version of the DEA algorithm (considered above), where we
avoid inconsistency as much as possible, the probability of hitting inconsistency could easily be less than
40%. of the time. Since N is at most cubic in n (the number of variables ), we see that our algorithm is at
most biquadratic in n (taking into consideration the time needed for each call of Test2CNF), the number of
variables (but in most cases much smaller time is sufficient). In fact, the expected time to hit a solution (if
it exists) has the order of magnitude (up to a constant), where n is the number of variables and N is
the number of clauses in the 3 – CNF expression. We can the take A(n) from section 3 of the form ,
where C is a constant. In the most interesting cases, when is around 4.5, the DEA2 algorithm is quadratic.
The power of the DEA algorithm is in the fact that a current active 2-CNF chain is connected to a whole class
of assignments for the variables, in other words, we work with many assignments in parallel, at every
step of the algorithm.
REFERENCES
[1] J. E. Hopcroft, J. D. Ullman, “ Introduction to Automata Theory, Langiages, and Computation “, Addison - Wesley Publishing
Company, 1979.
[2] W. Feller, “ An Introduction to Probability Theory and Its Applications “, John Wiley & Sons, 1968.
[3] Uwe Schoning, “ A probabilistic Algorithm for k-SAT and Constraint Satifaction Problems“, Research Supported by the ESPRIT
Basic Research, 1991.
[4] L. Fortnow, “ The Golden Ticket, P, NP, and The Search For The Impossible “, Princeton University Press, 2013.