1. The document discusses the theory of sequential machines and finite automata. It covers topics like sequential circuits, sequential machines, realization of sequential functions, reachable and observable states, and minimal realization.
2. The theory of automata originated from studies on abstract models of sequential circuits in 1956. A key paper on finite automata was published in 1959 and investigated relationships between inputs and outputs of state transition functions.
3. An alphabet is a finite set of symbols. A word is a finite string of zero or more symbols from the alphabet, where the same symbol can occur multiple times.
Lesson 13: Exponential and Logarithmic Functions (Section 041 handout)Matthew Leingang
This document summarizes sections 3.1-3.2 of a Calculus I course at New York University on exponential and logarithmic functions taught on October 20, 2010. It outlines definitions and properties of exponential functions, introduces the special number e and natural exponential function, and defines logarithmic functions. Announcements are made that the midterm exam is nearly graded and a WebAssign assignment is due the following week.
The document analyzes the analytic solution of Burger's equations using the variational iteration method. It begins by introducing the variational iteration method and how it can be used to solve differential equations. It then applies the method to obtain exact solutions for the (1+1), (1+2), and (1+3) dimensional Burger equations. Lengthy iterative solutions are presented for each case. The variational iteration method is shown to provide exact solutions to these Burger equations without requiring linearization.
Correlation of dts by er. sanyam s. saini me (reg) 2012-14Sanyam Singh
This document discusses correlation of discrete-time signals. It defines correlation as a measure of similarity between two data sequences. Correlation techniques are widely used in signal processing applications like radar target detection. Cross correlation compares two separate signals, while auto correlation compares a signal to itself. Properties of correlation include detecting signals in noise and recognizing patterns. Examples of cross correlation, auto correlation and correlation of periodic sequences are provided. The main application discussed is using correlation for radar target detection.
This document outlines key concepts in linear models and estimation that will be covered in the STA721 Linear Models course, including:
1) Linear regression models decompose observed data into fixed and random components.
2) Maximum likelihood estimation finds parameter values that maximize the likelihood function.
3) Linear restrictions on the mean vector μ define a subspace and equivalent parameterizations represent the same subspace.
4) Inference should be independent of the parameterization or coordinate system used to represent μ.
11.final paper -0047www.iiste.org call-for_paper-58Alexander Decker
This document discusses generating new Julia sets and Mandelbrot sets using the tangent function. It introduces using the tangent function of the form tan(zn) + c, where n ≥ 2, and applying Ishikawa iteration to generate new Relative Superior Mandelbrot sets and Relative Superior Julia sets. The results are entirely different from existing literature on transcendental functions. It describes using escape criteria for polynomials to generate the fractals and discusses the geometry of the Relative Superior Mandelbrot and Julia sets generated, which possess symmetry along the real axis.
This document is a master's thesis written in Chinese that investigates the existence and uniqueness of solutions to stochastic differential equations (SDEs) with Lévy noise and non-Lipschitz coefficients. It introduces Lévy processes and their properties, including the Lévy-Itô decomposition. It defines stochastic integration with respect to compensated Poisson processes and provides Itô's formula for Lévy diffusions. The thesis proves that if weak existence and pathwise uniqueness hold for an SDE with Lévy noise, then it has a unique strong solution. It establishes conditions on the coefficients that ensure infinite lifetime and pathwise uniqueness of the solution.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
The document is a lecture on derivatives of exponential and logarithmic functions. It begins with announcements about homework and an upcoming midterm. It then provides objectives and an outline for sections on exponential and logarithmic functions. The body of the document defines exponential functions, establishes conventions for exponents of all types, discusses properties of exponential functions, and graphs various exponential functions. It focuses on setting up the necessary foundations before discussing derivatives of these functions.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trig functions like arcsin, arccos, and arctan and discusses their domains, ranges, and relationships to the original trig functions. It also provides examples of evaluating inverse trig functions at specific values.
Lesson 13: Exponential and Logarithmic Functions (Section 041 handout)Matthew Leingang
This document summarizes sections 3.1-3.2 of a Calculus I course at New York University on exponential and logarithmic functions taught on October 20, 2010. It outlines definitions and properties of exponential functions, introduces the special number e and natural exponential function, and defines logarithmic functions. Announcements are made that the midterm exam is nearly graded and a WebAssign assignment is due the following week.
The document analyzes the analytic solution of Burger's equations using the variational iteration method. It begins by introducing the variational iteration method and how it can be used to solve differential equations. It then applies the method to obtain exact solutions for the (1+1), (1+2), and (1+3) dimensional Burger equations. Lengthy iterative solutions are presented for each case. The variational iteration method is shown to provide exact solutions to these Burger equations without requiring linearization.
Correlation of dts by er. sanyam s. saini me (reg) 2012-14Sanyam Singh
This document discusses correlation of discrete-time signals. It defines correlation as a measure of similarity between two data sequences. Correlation techniques are widely used in signal processing applications like radar target detection. Cross correlation compares two separate signals, while auto correlation compares a signal to itself. Properties of correlation include detecting signals in noise and recognizing patterns. Examples of cross correlation, auto correlation and correlation of periodic sequences are provided. The main application discussed is using correlation for radar target detection.
This document outlines key concepts in linear models and estimation that will be covered in the STA721 Linear Models course, including:
1) Linear regression models decompose observed data into fixed and random components.
2) Maximum likelihood estimation finds parameter values that maximize the likelihood function.
3) Linear restrictions on the mean vector μ define a subspace and equivalent parameterizations represent the same subspace.
4) Inference should be independent of the parameterization or coordinate system used to represent μ.
11.final paper -0047www.iiste.org call-for_paper-58Alexander Decker
This document discusses generating new Julia sets and Mandelbrot sets using the tangent function. It introduces using the tangent function of the form tan(zn) + c, where n ≥ 2, and applying Ishikawa iteration to generate new Relative Superior Mandelbrot sets and Relative Superior Julia sets. The results are entirely different from existing literature on transcendental functions. It describes using escape criteria for polynomials to generate the fractals and discusses the geometry of the Relative Superior Mandelbrot and Julia sets generated, which possess symmetry along the real axis.
This document is a master's thesis written in Chinese that investigates the existence and uniqueness of solutions to stochastic differential equations (SDEs) with Lévy noise and non-Lipschitz coefficients. It introduces Lévy processes and their properties, including the Lévy-Itô decomposition. It defines stochastic integration with respect to compensated Poisson processes and provides Itô's formula for Lévy diffusions. The thesis proves that if weak existence and pathwise uniqueness hold for an SDE with Lévy noise, then it has a unique strong solution. It establishes conditions on the coefficients that ensure infinite lifetime and pathwise uniqueness of the solution.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
The document is a lecture on derivatives of exponential and logarithmic functions. It begins with announcements about homework and an upcoming midterm. It then provides objectives and an outline for sections on exponential and logarithmic functions. The body of the document defines exponential functions, establishes conventions for exponents of all types, discusses properties of exponential functions, and graphs various exponential functions. It focuses on setting up the necessary foundations before discussing derivatives of these functions.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trig functions like arcsin, arccos, and arctan and discusses their domains, ranges, and relationships to the original trig functions. It also provides examples of evaluating inverse trig functions at specific values.
Lesson 12: Linear Approximation and Differentials (Section 41 slides)Matthew Leingang
The line tangent to a curve is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
This document discusses using linear approximations to estimate functions. It provides an example estimating sin(61°) using linear approximations about a=0 and a=60°. When approximating about a=0, the estimate is 1.06465. When approximating about a=60°, the estimate is 0.87475, which is closer to the actual value of sin(61°) according to a calculator check. The document teaches that the tangent line provides the best linear approximation near a point, and its equation can be used to estimate function values.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
- Dr. Harish Garg has a Ph.D. in applied mathematics from IIT Roorkee and works as an assistant professor teaching stochastic matrices.
- A stochastic matrix models a randomly changing system over time and can be represented by a transition matrix where each row sums to 1.
- Markov chains are a type of stochastic process where the next state only depends on the present state and not on past states. The transitions between states are represented using a transition probability matrix.
This document discusses adiabatic gate teleportation and its applications. It begins with an overview of joint work done by Dave Bacon of the University of Washington along with Steve Flammia, Alice Neels, and Andrew Landahl on this topic. The rest of the document discusses the history of classical computing using unreliable components, ideas from Kitaev and Freedman on topological quantum computing using anyons, and an open controversy around whether topological quantum computing is truly fault-tolerant.
Tensor Decomposition and its ApplicationsKeisuke OTAKI
This document discusses tensor factorizations and decompositions and their applications in data mining. It introduces tensors as multi-dimensional arrays and covers 2nd order tensors (matrices) and 3rd order tensors. It describes how tensor decompositions like the Tucker model and CANDECOMP/PARAFAC (CP) model can be used to decompose tensors into core elements to interpret data. It also discusses singular value decomposition (SVD) as a way to decompose matrices and reduce dimensions while approximating the original matrix.
This document provides an introduction to blind source separation and non-negative matrix factorization. It describes blind source separation as a method to estimate original signals from observed mixed signals. Non-negative matrix factorization is introduced as a constraint-based approach to solving blind source separation using non-negativity. The alternating least squares algorithm is described for solving the non-negative matrix factorization problem. Experiments applying these methods to artificial and real image data are presented and discussed.
11.homotopy perturbation and elzaki transform for solving nonlinear partial d...Alexander Decker
The document discusses using a combination of the homotopy perturbation method and Elzaki transform to solve nonlinear partial differential equations. It begins by introducing the homotopy perturbation method and Elzaki transform individually. It then presents the homotopy perturbation Elzaki transform method, which applies Elzaki transform to reformulate the problem before using homotopy perturbation method to obtain approximations of the solution as a series. Finally, it applies the new combined method to solve an example nonlinear partial differential equation.
Homotopy perturbation and elzaki transform for solving nonlinear partial diff...Alexander Decker
The document presents a combination of the homotopy perturbation method and Elzaki transform to solve nonlinear partial differential equations. The homotopy perturbation method is used to handle the nonlinear terms, while the Elzaki transform is applied to reformulate the equations in terms of transformed variables, obtaining a series solution via inverse transformation. The method is demonstrated to be effective for both homogeneous and non-homogeneous nonlinear partial differential equations. Key steps include using integration by parts to obtain Elzaki transforms of partial derivatives and defining a convex homotopy to reformulate the equations for the homotopy perturbation method.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (handout)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
A likelihood-free version of the stochastic approximation EM algorithm (SAEM)...Umberto Picchini
I show how to obtain approximate maximum likelihood inference for "complex" models having some latent (unobservable) component. With "complex" I mean models having a so-called intractable likelihood, where the latter is unavailable in closed for or is too difficult to approximate. I construct a version of SAEM (and EM-type algorithm) that makes it possible to conduct inference for complex models. Traditionally SAEM is implementable only for models that are fairly tractable analytically. By introducing the concept of synthetic likelihood, where information is captured by a series of user-defined summary statistics (as in approximate Bayesian computation), it is possible to automatize SAEM to run on any model having some latent-component.
Lesson 15: Exponential Growth and Decay (handout)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Parameter Estimation for Semiparametric Models with CMARS and Its ApplicationsSSA KPI
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 11.
More info at http://summerschool.ssa.org.ua
Lesson 13: Exponential and Logarithmic Functions (Section 021 handout)Matthew Leingang
This document contains lecture notes from a Calculus I class at New York University on October 21, 2010 covering sections 3.1-3.2 on exponential functions. The notes include announcements about an upcoming midterm exam and homework assignment. Statistics on the recent midterm exam are provided, showing the average, median, standard deviation, and what constitutes a "good" or "great" score. The objectives of sections 3.1-3.2 are outlined as understanding exponential functions, their properties, and laws of logarithms. The notes provide definitions and derivations of exponential functions for various exponent values.
This document discusses independent component analysis (ICA) for blind source separation. ICA is a method to estimate original signals from observed signals consisting of mixed original signals and noise. It introduces the ICA model and approach, including whitening, maximizing non-Gaussianity using kurtosis and negentropy, and fast ICA algorithms. The document provides examples applying ICA to separate images and discusses approaches to improve ICA, including using differential filtering. ICA is an important technique for blind source separation and independent component estimation from observed signals.
This 3-sentence summary provides the essential information about the course syllabus:
The syllabus outlines the topics and requirements for the Foundations of Computer Science course, including two exams, programming assignments, and a final exam that will cover topics in computability theory, formal languages, and automata. The course is divided into two units, the first covering unrestricted languages and grammars and the second focusing on restricted languages and grammars as well as logic. Students will be evaluated based on exams, programming assignments, and final exam scores.
Symbolic Computations in Conformal Geometric Algebra for Three Dimensional O...Yoshihiro Mizoguchi
This document discusses symbolic computations in conformal geometric algebra (CGA) for modeling three-dimensional origami folds. CGA provides an algebraic framework for representing geometric objects and transformations. The document introduces CGA, shows examples of visualizing geometric objects and operations in CGA, and demonstrates how origami folds can be formalized and visualized using CGA operations. Each origami fold is represented by an element of CGA, allowing the geometric properties of folds to be investigated using symbolic computations.
Theory of Computer Science - Post Correspondence ProblemKaran Thakkar
The document discusses recursive and recursively enumerable languages, and undecidability. It can be summarized as:
1. Recursive languages are accepted by Turing machines that halt on all inputs, while recursively enumerable languages are accepted by machines that may halt or loop.
2. Whether a Turing machine halts on a given input is undecidable, as is determining if a context-free grammar is ambiguous.
3. Rice's Theorem states that any non-trivial property of recursively enumerable languages is undecidable to determine. The Post Correspondence Problem is also undecidable in general.
The document discusses the Post Correspondence Problem (PCP) and shows that it is undecidable. It defines PCP as determining if there is a sequence of string pairs from two lists A and B that match up. It then defines the Modified PCP (MPCP) which requires the first pair to match. It shows how to reduce the Universal Language Problem to MPCP by mapping a Turing Machine and input to lists A and B, and then how to reduce MPCP to PCP. Finally, it discusses Rice's Theorem and how properties of recursively enumerable languages are undecidable.
Lesson 12: Linear Approximation and Differentials (Section 41 slides)Matthew Leingang
The line tangent to a curve is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
This document discusses using linear approximations to estimate functions. It provides an example estimating sin(61°) using linear approximations about a=0 and a=60°. When approximating about a=0, the estimate is 1.06465. When approximating about a=60°, the estimate is 0.87475, which is closer to the actual value of sin(61°) according to a calculator check. The document teaches that the tangent line provides the best linear approximation near a point, and its equation can be used to estimate function values.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
- Dr. Harish Garg has a Ph.D. in applied mathematics from IIT Roorkee and works as an assistant professor teaching stochastic matrices.
- A stochastic matrix models a randomly changing system over time and can be represented by a transition matrix where each row sums to 1.
- Markov chains are a type of stochastic process where the next state only depends on the present state and not on past states. The transitions between states are represented using a transition probability matrix.
This document discusses adiabatic gate teleportation and its applications. It begins with an overview of joint work done by Dave Bacon of the University of Washington along with Steve Flammia, Alice Neels, and Andrew Landahl on this topic. The rest of the document discusses the history of classical computing using unreliable components, ideas from Kitaev and Freedman on topological quantum computing using anyons, and an open controversy around whether topological quantum computing is truly fault-tolerant.
Tensor Decomposition and its ApplicationsKeisuke OTAKI
This document discusses tensor factorizations and decompositions and their applications in data mining. It introduces tensors as multi-dimensional arrays and covers 2nd order tensors (matrices) and 3rd order tensors. It describes how tensor decompositions like the Tucker model and CANDECOMP/PARAFAC (CP) model can be used to decompose tensors into core elements to interpret data. It also discusses singular value decomposition (SVD) as a way to decompose matrices and reduce dimensions while approximating the original matrix.
This document provides an introduction to blind source separation and non-negative matrix factorization. It describes blind source separation as a method to estimate original signals from observed mixed signals. Non-negative matrix factorization is introduced as a constraint-based approach to solving blind source separation using non-negativity. The alternating least squares algorithm is described for solving the non-negative matrix factorization problem. Experiments applying these methods to artificial and real image data are presented and discussed.
11.homotopy perturbation and elzaki transform for solving nonlinear partial d...Alexander Decker
The document discusses using a combination of the homotopy perturbation method and Elzaki transform to solve nonlinear partial differential equations. It begins by introducing the homotopy perturbation method and Elzaki transform individually. It then presents the homotopy perturbation Elzaki transform method, which applies Elzaki transform to reformulate the problem before using homotopy perturbation method to obtain approximations of the solution as a series. Finally, it applies the new combined method to solve an example nonlinear partial differential equation.
Homotopy perturbation and elzaki transform for solving nonlinear partial diff...Alexander Decker
The document presents a combination of the homotopy perturbation method and Elzaki transform to solve nonlinear partial differential equations. The homotopy perturbation method is used to handle the nonlinear terms, while the Elzaki transform is applied to reformulate the equations in terms of transformed variables, obtaining a series solution via inverse transformation. The method is demonstrated to be effective for both homogeneous and non-homogeneous nonlinear partial differential equations. Key steps include using integration by parts to obtain Elzaki transforms of partial derivatives and defining a convex homotopy to reformulate the equations for the homotopy perturbation method.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (handout)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
A likelihood-free version of the stochastic approximation EM algorithm (SAEM)...Umberto Picchini
I show how to obtain approximate maximum likelihood inference for "complex" models having some latent (unobservable) component. With "complex" I mean models having a so-called intractable likelihood, where the latter is unavailable in closed for or is too difficult to approximate. I construct a version of SAEM (and EM-type algorithm) that makes it possible to conduct inference for complex models. Traditionally SAEM is implementable only for models that are fairly tractable analytically. By introducing the concept of synthetic likelihood, where information is captured by a series of user-defined summary statistics (as in approximate Bayesian computation), it is possible to automatize SAEM to run on any model having some latent-component.
Lesson 15: Exponential Growth and Decay (handout)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Parameter Estimation for Semiparametric Models with CMARS and Its ApplicationsSSA KPI
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 11.
More info at http://summerschool.ssa.org.ua
Lesson 13: Exponential and Logarithmic Functions (Section 021 handout)Matthew Leingang
This document contains lecture notes from a Calculus I class at New York University on October 21, 2010 covering sections 3.1-3.2 on exponential functions. The notes include announcements about an upcoming midterm exam and homework assignment. Statistics on the recent midterm exam are provided, showing the average, median, standard deviation, and what constitutes a "good" or "great" score. The objectives of sections 3.1-3.2 are outlined as understanding exponential functions, their properties, and laws of logarithms. The notes provide definitions and derivations of exponential functions for various exponent values.
This document discusses independent component analysis (ICA) for blind source separation. ICA is a method to estimate original signals from observed signals consisting of mixed original signals and noise. It introduces the ICA model and approach, including whitening, maximizing non-Gaussianity using kurtosis and negentropy, and fast ICA algorithms. The document provides examples applying ICA to separate images and discusses approaches to improve ICA, including using differential filtering. ICA is an important technique for blind source separation and independent component estimation from observed signals.
This 3-sentence summary provides the essential information about the course syllabus:
The syllabus outlines the topics and requirements for the Foundations of Computer Science course, including two exams, programming assignments, and a final exam that will cover topics in computability theory, formal languages, and automata. The course is divided into two units, the first covering unrestricted languages and grammars and the second focusing on restricted languages and grammars as well as logic. Students will be evaluated based on exams, programming assignments, and final exam scores.
Symbolic Computations in Conformal Geometric Algebra for Three Dimensional O...Yoshihiro Mizoguchi
This document discusses symbolic computations in conformal geometric algebra (CGA) for modeling three-dimensional origami folds. CGA provides an algebraic framework for representing geometric objects and transformations. The document introduces CGA, shows examples of visualizing geometric objects and operations in CGA, and demonstrates how origami folds can be formalized and visualized using CGA operations. Each origami fold is represented by an element of CGA, allowing the geometric properties of folds to be investigated using symbolic computations.
Theory of Computer Science - Post Correspondence ProblemKaran Thakkar
The document discusses recursive and recursively enumerable languages, and undecidability. It can be summarized as:
1. Recursive languages are accepted by Turing machines that halt on all inputs, while recursively enumerable languages are accepted by machines that may halt or loop.
2. Whether a Turing machine halts on a given input is undecidable, as is determining if a context-free grammar is ambiguous.
3. Rice's Theorem states that any non-trivial property of recursively enumerable languages is undecidable to determine. The Post Correspondence Problem is also undecidable in general.
The document discusses the Post Correspondence Problem (PCP) and shows that it is undecidable. It defines PCP as determining if there is a sequence of string pairs from two lists A and B that match up. It then defines the Modified PCP (MPCP) which requires the first pair to match. It shows how to reduce the Universal Language Problem to MPCP by mapping a Turing Machine and input to lists A and B, and then how to reduce MPCP to PCP. Finally, it discusses Rice's Theorem and how properties of recursively enumerable languages are undecidable.
A generalized transition graph (GTG) is composed of states, an alphabet of input letters, and directed edges between states labeled with regular expressions. An example GTG has three states - a start state, middle state, and final state. The edges are labeled with (ba + a)*, allowing strings to transition from the start to middle state or directly to the final state, and b* allowing transition from the middle to final state. This GTG accepts all strings without a double b.
This document discusses finite automata and provides definitions and examples. It defines deterministic finite automata (DFA) and nondeterministic finite automata (NFA) and their components. It describes how strings are processed by DFAs using transition functions. Notations for finite automata like transition diagrams and tables are presented. The reasons for nondeterminism and how to convert NFAs to equivalent DFAs are summarized. Examples of finite automata design are provided.
The document discusses the theory of automata and formal languages including:
- Different types of automata like finite automata, pushdown automata, and Turing machines.
- Context-free grammars and properties of regular, context-free, and recursively enumerable languages.
- Operations on strings and languages like concatenation, Kleene closure, and positive closure.
- Proofs techniques like proof by induction and proof by contradiction.
This document discusses finite state machines and their applications. It describes three types of finite state automata: 1) without output, 2) with output on transition (Mealy machine), and 3) with output on state (Moore machine). It then provides examples of Mealy and Moore machines, including ones that invert input, divide by 2, and act as a vending machine.
The document discusses Mealy and Moore machine models. Mealy machines have an output function that depends on the present state and input, while Moore machines have an output function that depends only on the present state. The document provides examples of converting between Mealy and Moore machine representations.
1) The document discusses Turing machines and their properties such as having a finite set of states and read/write tape memory. The output depends only on the input and previous output based on definite transition rules.
2) Reducibility is introduced as a primary method for proving problems are computationally unsolvable by converting one problem into another problem such that solving the second solves the first.
3) Decidability and undecidability of languages are defined. Undecidable problems have no algorithm to determine membership regardless of whether a Turing machine halts or not on all inputs.
Finite-State Machine
The document discusses finite-state machines (FSM), which model sequential logic circuits. It describes two types of FSMs: Mealy and Moore machines. Mealy machines output depends on the present state and input, changing asynchronously with the clock. Moore machines' output depends only on the present state, changing synchronously with state changes and clock. The document provides an example of designing an FSM to output 0 if an even number of 1's have been received on the input, and 1 for odd. It shows solutions as both a Mealy and Moore machine using state transition tables and logic diagrams.
1. Automata theory is the study of abstract machines and the problems they are able to solve. It is closely related to formal language theory as automata are often classified by the formal languages they can recognize.
2. A finite automaton is an abstract machine that consists of a finite number of states. It reads an input string and based on its current state and the next input symbol, transitions to the next state according to its transition function. If it ends in an accepting state, the input is accepted.
3. Deterministic finite automata (DFAs) are a type of finite automaton where the transition function maps each state-symbol pair to a unique next state. DFAs can be represented
Turing Machines are a simple mathematical model of a general purpose computer invented by Alan Turing in 1936. A Turing Machine consists of an infinite tape divided into cells, a head that reads and writes symbols on the tape, a finite set of states, and transition rules determining the behavior of the machine. The machine operates by reading a symbol on the tape, updating the symbol according to its transition rules, moving the head left or right, and transitioning to a new state. Turing Machines can simulate any algorithm and are capable of performing any calculation that can be performed by any computing machine.
Markov analysis examines dependent random events where the likelihood of future events depends on past events. It models this using a transition matrix showing the probabilities of moving between states. The document discusses Markov analysis of accounts receivable to predict future payment categories. It defines states like paid, overdue 1-3 months, etc. and a transition matrix showing the probabilities of moving between states. Markov analysis can then predict future distributions of accounts among the states by multiplying the current distribution by the transition matrix repeatedly.
This document covers key topics in seismic data processing including complex numbers, vectors, matrices, determinants, eigenvalues, singular values, matrix inversion, series, Taylor series, Fourier series, delta functions, and Fourier integrals. It provides examples of using Taylor series to approximate nonlinear systems as linear systems and using Fourier series to approximate periodic functions. The importance of Fourier transforms for spectral analysis and various geophysical applications is also discussed.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
This document discusses three techniques for identifying linear systems: frequency chirp method, inverse filtering method, and coherence function method. The frequency chirp method uses a wideband excitation input like a frequency chirp and obtains the frequency response as the discrete Fourier transform of the system's output. The inverse filtering method uses pseudoinverse to find the impulse response. The coherence function method uses the coherence function and input-output cross-spectrum to estimate the system response directly and inversely. These three techniques were demonstrated on a 10th order Chebyshev filter. The inverse filtering method provided an identification closest to the actual system response compared to the other two techniques.
This document discusses stochastic models for site characterization. It describes several continuous models for generating random fields including the multivariate normal method, LU decomposition method, and turning bands method. The multivariate normal method models a random vector as having a multivariate normal distribution defined by a mean vector and covariance matrix. The LU decomposition method generates a random field with a given covariance structure by decomposing the covariance matrix into lower and upper triangular matrices. It provides numerical examples of applying the LU decomposition method to generate correlated random variables at two points.
The document discusses quantum computing concepts such as wave functions, bra-ket notation, identity matrices, Pauli matrices, Hermitian matrices, and unitary matrices. It provides examples of applying Pauli matrices to quantum states |0> and |1> and explains how identity matrices do not change these states. The key aspects covered are mathematical representations of quantum states and operations, as well as basic principles of quantum information and computing.
Fixed Point Theorm In Probabilistic Analysisiosrjce
Probabilistic operator theory is the branch of probabilistic analysis which is concerned with the study of
operator-valued random variables and their properties. The development of a theory of random operators is of
interest in its own right as a probabilistic generalization of (deterministic) operator theory and just as operator
theory is of fundamental importance in the study of operator equations, the development of probabilistic operator
theory is required for the study of various classes of random equations
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Relative superior mandelbrot and julia sets for integer and non integer valueseSAT Journals
Abstract
The fractals generated from the self-squared function,
2 zz c where z and c are complex quantities have been studied
extensively in the literature. This paper studies the transformation of the function , 2 n zz c n and analyzed the z plane and
c plane fractal images generated from the iteration of these functions using Ishikawa iteration for integer and non-integer values.
Also, we explored the drastic changes that occurred in the visual characteristics of the images from n = integer value to n = non
integer value.
Keywords: Complex dynamics,
Relative Superior Julia set, Relative Superior Mandelbrot set.
1) The document discusses dynamics modeling for robotic manipulators using the Denavit-Hartenberg representation and Lagrangian mechanics. It describes using the Euler-Lagrange method to derive equations of motion for robotic links by computing kinetic and potential energy terms.
2) As an example, dynamics equations are derived for a simple 1 degree-of-freedom robotic arm. Kinetic and potential energy expressions are written and the Lagrangian is computed to obtain the equation of motion.
3) State-space modeling basics are reviewed using the example of a damped spring-mass system, showing how to write the system dynamics as state-space matrices to evaluate responses like step response.
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...inventionjournals
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Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
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Chapter 5
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Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
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Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
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1. .
.
Theory of Relations (2)
Sequential Machines and Finite Automata
Course of Mathematics
Pusan National University
.
.. .
.
Yoshhiro Mizoguchi
Institute of Mathematics for Industry
Kyushu University, JAPAN
ym@imi.kyushu-u.ac.jp
November 3-4, 2011
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 1 / 34
2. Table of Contents
.
..
1 Sequential Machine
Preliminary
Reachable and Observable
Mimimal Realization
.
..
2 Finite Automata
Introduction to Theory of Automata
The Myhill-Nerode theorem
Minimal Realization
.
..
3 Applications of Relational Calculus to Theory of Automata
Nondeterministic Finite Automaton
Coproduct and Product Automaton
Reverse, Concatenate, Closure
Examples
.
..
4 References
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 2 / 34
3. Sequential Circuit (1)
1 2 0 1 0 2 0
0 1 0 0 1 1 0
dc S
c
Q
0 1 0 0 0 1 0
1 0 0 1 0 0 0
dc c
R
One of the elementary units of sequential circuits, the RS Flip-flop circuit
produces an output signal sequence according to the sequence of input
signals. Output signals are 1(on) or 0(o f f ) so we write the outputs as
Y = {0, 1}. There are two input signals S(et) and R(eset). We consider the
pair (SR) of S and R, and we denote 0 = (00), 1 = (01) and 2 = (10). So
we can consider the inputs as X = {0, 1, 2}.
The model of a sequential circuit consists of the state set Q = {a, b}, the
state transition function δ : Q × X → Q, and the output function
β : Q → Y.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 3 / 34
4. Sequential Circuit (2)
The table below is the value of δ and β. The figure is the state transition
diagram of the RS Flip-flop circuit.
q x δ(q, x)
a 0 a
a 1 a q β(q) 0or1 0or2
‡ ‡
a 2 b a 0 a/0 2E b/1
b 0 b b 1 i1
b 1 a
b 2 b
The labels of a vertex consists of a state and an output symbol. An edge
means a state transition and the label on an edge corresponds to the input
symbol.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 4 / 34
5. Sequential Circuit (3)
An sequential circuit can be considered as a function from an input word to
an output word. That is we can consider an sequential circuit as a function
f : X∗ → Y ∗ where X∗ is the set of words over X including an empty
string ε.
.
Problem .
..
Construct an algebraic model (Sequential Machine) of sequential
circuits using a state set, a state transition function and an output
function.
What kind of function from input words to output words is realizable
by a finite state sequential machine?
How to construct an efficient sequential machine with small number of
. states.
.. .
.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 5 / 34
6. Epi-Mono Factorization
A function F : X → Y from a set X to a set Y can be represented by a
composition F(x) = m(e(x)) (x ∈ X) of an injective function m : Z → Y
and an surjective function e : X → Z . The set Z is uniquely determined by
f up to isomorphism. That is Z F(X) = {F(x) ∈ Y | x ∈ X} and
Z X/ ∼ = {[x] | x ∈ X}, where the equivalent relation ∼ on X is defined
by [ x ∼ x′ ⇔ F(x) = F(x′ )] and [x] = {x′ ∈ X | x ∼ x′ } is a set of
equivalence class including x.
(Note)
A function e : X → Z is surjective if there exists an element x ∈ X satisfying
e(x) = z for any element z ∈ Z . A function m : Z → Y is injective if m(z1 ) m(z2 )
for any two elements z1 , z2 ∈ Z satisfying z1 z2 . We denote an injection by an
arrow with tail , and a surjective by an arrow with head .
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 6 / 34
7. Sequential Machine
.
Definition (Sequential Machine) .
..
A sequential mmachine is a sextuple M = (X, Q, δ, q0 , Y, β) where
X is the set of inputs,
Q is the set of states,
δ:Q×X →Q is the transition function,
q0 ∈ Q is the initial state,
Y is the set of outputs, and
. β:Q→Y is the output map.
.. .
.
(Note) This sequential machine (SM) is called Moore style SM. The Mealy style
SM is defined by an alternate output map λ : Q × X → Y insted of β. These two
model are equivalent. Mealy style does not have an output for the initial state, but
the rest of relations between inputs and outputs are mutually transformable.
We sometime define SM by pentad without an initial state.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 7 / 34
8. Run map and Response map
.
Definition (Run map) .
..
Let δ : Q × X → Q be a transition function. We define its run map to be
the unique map δ∗ : Q × X∗ → Q defined inductively by δ∗ (q, ε) = q, and
δ∗ (q, xw) = δ∗ (δ(q, x), w) ( q ∈ Q, x ∈ X, w ∈ X∗ ).
.
.. .
.
Let f : X∗ → Y be a function. We also define the function f∗ : X∗ → Y ∗
defined inductively by f∗ (ε) = f (ε), and f∗ (wx) = f∗ (w) f (wx) ( x ∈ X,
w ∈ X∗ ).
.
Definition (Response map) .
..
Let M = (X, Q, δ, q0 , Y, β) be a sequential machine. We define its
response map to be the map ( f M )∗ : X∗ → Y ∗ where f M : X∗ → Y is
defined by f M (w) = β(δ∗ (q0 , w)) (w ∈ X∗ ).
.
.. .
.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 8 / 34
9. Realization (1)
.
Definition .
..
Let t : X ∗ → Y ∗ be a function. If there exists a sequential machine M
such that t = ( f M )∗ , then M is called a realization of t .
.
.. .
.
We call t : X∗ → Y ∗ is realizable if there exisits a function f : X∗ → Y
such that t = f∗ .
.
Proposition .
..
A function t : X ∗ → Y ∗ is realizable if and only if for any w ∈ X ∗ , x ∈ X
there exists y ∈ Y such that t(wx) = t(w)y.
.
.. .
.
The condition is equivalent that the value t(wx) is depending on only w
and is not depending on x. We call a function t : X∗ → Y ∗ satisfying the
condition as a sequential function.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 9 / 34
10. Realization (2)
.
Proposition .
..
Let t : X∗ → Y ∗ be a sequential function. Then there exist a sequential
machine M which is a realization of t .
.
.. .
.
Let f : X∗ → Y be a function and t = f∗ . We introduce two kinds of
sequential machines which is a realization of t .
M I = (X, X∗ , δ I , ε, Y, f )
δ I (w, x) = wx (w ∈ X∗ , x ∈ X).
∗
MT = (X, Y X , δT , f, Y, βT )
∗
Y X is the set of all maps from X∗ to Y , that is { f | f : X∗ → Y},
δT ( f, x) : X∗ → Y is defined by δT ( f, x)(w) = f (xw) ( x ∈ X, w ∈ X∗ ),
∗
and βT ( f ) = f (ε) ( f ∈ Y X ).
We can verify easily f = f MI = f MT . We note that both M I and MT is not
a finite sequential machine. That is the state set is not a finite set.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 10 / 34
11. Finite Realization
.
Problem .
..
What kind of sequential function f : X ∗ → Y which have a finite state
realization?
.
.. .
.
The state sets of MT is infinite but the most of states are unreachable from
the initial state f . If the states reachable from f is finite, then we can
construct a finite sequential machine from MT . That is the condition to
∗
have a finite representation is that the set Z = {δ∗ ( f, w) ∈ Y X | w ∈ X∗ } is
T
finite.
∗
We define a function F : X∗ → Y X by F(w) = δ∗ ( f, w) (w ∈ X∗ ). F is
T
divided to the composition of a surjection and an injection. Since
Z = F(X∗ ), we have Z = X∗ / ∼ by the equivalence relation ∼ defined by
[w ∼ w′ ⇔ δ∗ ( f, w) = δ∗ ( f, w′ )]. We note δ∗ ( f, w)(z) = δ∗ ( f, w′ )(z) is
T T T T
f (wz) = f (w′ z) for any z ∈ X∗ .
If the number of the equivalence class is finite, then there exists a finite
realization.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 11 / 34
12. Reachable and Observable
Let M = (X, Q, δ, q0 , Y, β) be a sequential machine and its response map
∗
f M : X∗ → Y . We define F : X∗ → Y X by F(w) = δ∗ ( f M , w).
T ∗
F is divided into a composition of fe : X∗ → Q and f m : Q → Y X such
that
F(w) = f m( fe (w)),
where
fe (w) = δ∗ (q0 , w), and
f m(q)(w) = β(δ∗ (q, w)) (w ∈ X∗ , q ∈ Q).
If fe is a surjection then we call M as reachable.
If f m is an injection then we call M as obserbable or reduced.
We note that a reachable and obserbable sequential machine is
minimal representation of f M .
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 12 / 34
13. Minimal Realization
Let M = (X, Q, δ, q0 , Y, β) be a sequential machine. and F = f m ◦ fe .
∗
Assume fe : X∗ → Q is a surjection. If f m : Q → Y X is not an injection,
then we can construct a minimal realization using the epi-mono
factorization of f m.
The equivalence relation [ q ∼ q′ ⇔ f m(q) = f m(q′ )] on Q is that
f m(q)(w) = f m(q′ )(w) for any w ∈ X∗ . That is β(δ∗ (q, w)) = β(δ∗ (q′ , w)). If
Q is a finite set and |Q| = n, then it is sufficient to check the condition
β(δ∗ (q, w)) = β(δ∗ (q′ , w)) for finite number of words w with |w| n. So we
can check q ∼ q′ for any state q and q′ in finite steps, and we can
construct a minimal state sequential machine.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 13 / 34
14. Introduction to Theory of Automata
The first published study on automata [6] is the Automata studies in 1956
edited by C. E. Shanon who is famous as an originator of the information
theory and J. McCarthy who is a famous researcher of the fields artificial
intelligence.
At that time, they formalized an abstract model of a sequentil circuit and
investigated relationship between inputs and outputs analyzing a state
transition functions.
Once the notion of accept states is introduced, a machine is considered as
an acceptor and investigations of recognized language are started. This is
the origin of the theory of language and automata.
The first paper [4] about finite state automata is ’finite Automata and Their
Decision Problems’ by M. O. Rabin and D. Scott pubshed in 1959. They
were awarded an ACM Turing aword in 1976 for this research.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 14 / 34
15. Alphabet, Word, Concatenation
An alphabet is a finite, nonempty set. The elements of an plphabet are
feferred to as letters, or symbols. A word over an alphabet is a finite
string consisting of zero or more letters of the alphabet, in which the same
letter may occur several times. The string consisting of zero letters is
called the empty word, written ε. The length of a word w, denoted by |w|,
is the number of letters in w. Again by definition, |ε| = 0.
Let Σ be an alphabet. The set of all words over an alphabet Σ is denoted
by Σ∗ . The sets Σ∗ is infinite for any Σ. Algebraically speaking, Σ∗ is the
free monoid with the identity ε generated by Σ.
For words w1 and w2 , the juxtaposition w1 w2 is called the concatenation
of w1 and w2 . The empty word is an identity with respect to concatenation,
εw = wε = w holds for all word w.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 15 / 34
16. Kleene Closure
Subsets of Σ∗ are referred to as formal languages, or briefly, languages
over Σ.
The concatenation (or product) of two languages L1 and L2 is defined by
L1 L2 = {w1 w2 | w1 ∈ L1 , w2 ∈ L2 }.
The (Kleene) closure L∗ of a language L is defined to be the union of all
powers of L, that is L∗ = ∪∞ L n where L0 = {ε}, and L n = L n−1 · L
n=0
( n ≥ 1).
The closure of all words Σ is Σ∗ and there is no confusions defined as a
set of all words over Σ.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 16 / 34
17. Deterministic Automata (1)
.
Definition .
..
Deterministic Automata A determinisiic finite automaton is a pentad
M = (Σ, Q, δ, q0 , F) where
Σ is the alphabet,
Q is the finite set of states,
δ:Q×Σ→Q is the transition function,
q0 ∈ Q is the initial state, and
. F⊂Q is the set of accept states.
.. .
.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 17 / 34
18. Deterministic Automata (2)
s1 s2 · · · s n inputs
T E move head to right after a state transition
Controler( q)
Finite Automaton M
A finite automaton is illustrated as above figure. The input letters are on
input tape. The first state of controller is the initial state q0 . If the state is q
and input letter is s then the state is changed to δ(q, s). After changing the
state the head is moved to right. Repeating these procedures until the end
of an input word. If the head reached to the end of an input word, then
check the state is accept state or not.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 18 / 34
19. Recognized Language
Let δ : Q × Σ → Q be a state transition function. The function
δ∗ : Q × Σ∗ → Q is uniquely determined by δ∗ (q, ε) = q, and
δ∗ (q, wa) = δ(δ∗ (q, w), a) (w ∈ Σ∗ , a ∈ Σ).
.
Definition .
..
The recognized language L(M) ⊂ Σ∗ is defined by
L(M) = {w ∈ Σ∗ | δ∗ (q0 , w) ∈ F}.
L(M) is referred to as the language accepted by M.
.
.. .
.
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November 3-4, 2011 19 / 34
20. Example
.
Example .
..
Let M = (Q, Σ, δ, q0 , F) be a finite automaton where Q = {q0 , q1 , q2 },
F = {q1 }, q0 ∈ Q. The state transition function δ : Q × Σ → Q is defined
as follows
δ(q0 , a) = q1 , δ(q0 , b) = q2 ,
δ(q1 , a) = q2 , δ(q1 , b) = q0 ,
δ(q2 , a) = q2 , δ(q2 , b) = q2 .
For example, the word w = aba is acceptable δ∗ (q0 , aba)
= δ∗ (δ(q0 , a), ba) = δ∗ (q1 , ba) = δ∗ (δ(q1 , b), a) = δ∗ (q0 , a) = δ∗ (δ(q0 , a), ε)
= ∗
. δ (q1 , ε) = q1 ∈ F.
.. .
.
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November 3-4, 2011 20 / 34
21. State Transition Diagram
A finite automaton M is denoted by a following figure called state transition
diagram. Vertices are states and the initial state has an arrow without
label. According to the input letters follow arrows with same label with
input letter. The vertex corresponding to an accept state has double circle
and if the following the input letters ended at the vertices with double circle
then the input word is accepted.
a, b c
q0 ' b
q2 i
A
b q E
a
1
a
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November 3-4, 2011 21 / 34
22. The Myhill-Nerode theorem (1)
.
Definition .
..
An equivalence relation ∼ on Σ∗ is said to be right invariant if
”w1 ∼ w2 ⇒ w1 z ∼ w2 z (∀z ∈ Σ∗ )” for any w1 , w2 ∈ Σ∗ .
An equivalence relation ∼ is finite index if the number of equivalence
classes is finite. That is {[x] | x ∈ Σ∗ } is finite set where
. [x] = {x′ | x ∼ x′ }.
.. .
.
Let L Σ∗ . We define a relation w1 ∼ L w2 on Σ∗ by
w1 z ∈ L ⇔ w2 z ∈ L (∀z ∈ Σ∗ ).
The relation ∼ L is a right invariant equivalent relation.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 22 / 34
23. The Myhil-Nerode theorem (2)
.
Theorem (Nerode) .
..
Let L Σ∗ be a language on Σ. Then, the following three conditions (1),
(2) and (3) are equivalent.
(1) The set L is acpted by some finite automaton.
(2) L is the union of some of the equivalent classes of a right invariant
equivalence relation of finite index.
(3) The right invariant equivalence relation ∼ L induced by L is of finite
. index.
.. .
.
Let L be the language L in (3) and Q = {[w] | w ∈ Σ ∗ }, δ([w], a) = [wa],
q0 = [ε], F = {[w] | w ∈ L}. Then M = (Q, Σ, δ, q0 , F) is a finite automaton
and L = L(M). Further, if L = L(M′ ) for some finite automata
M′ = (Q′ , Σ, δ′ , q′ , F′ ) then |Q| ≤ |Q′ |. That is M is a minimal state
0
automaton with L = L(M).
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November 3-4, 2011 23 / 34
24. Minimal Realization
Let M = (X, Q, X, δ, q0 , Y, β) be a sequential machine and Y = {0, 1}.
There is an one-to-one coresspondence between an output map
β : Q → {0, 1} and a subset F = {q ∈ Q | β(q) = 1} of Q. That is a finite
automata is considered as a sequential machine with Y = {0, 1}.
A finite sequential machine is exactly a finite automaton. Further, a
function f : X∗ → Y is corresponde to a subset of X∗ that is a recognized
language.
In the previous section, we construct a minimal realization of f using an
equivalence relation w ∼ w′ defined by f (wz) = f (wz′ ) (∀z ∈ X∗ ). This is
the equivalence relation ∼ L induced by L.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 24 / 34
25. Example
.
Example (Minimalize) .
..
The automton M′ in right figure is the minimalized automaton of M in left
figure. We note p0 = {q0 , q2 }, p1 = {q1 , q3 }, and p2 = {q4 , q5 }.
.
.. .
.
35. b
j
$
q1 WB
$ X
a a
b B
a E
p2
a
M M′
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November 3-4, 2011 25 / 34
36. Nondeterministic Finite Automaton
Let Σ be an alphabet, I one point set {∗}.
.
Definition (Kawahara 1988[3]) .
..
A nondeterministic finite automaton (NFA) M = (Q, τ, δ a (a ∈ Σ), β) is a
quadruple where
Q is a finite set of states,
τ:I⇁Q is the inital state relation,
δa : Q ⇁ Q is the set of state-transition relations, and
. β:Q⇁I is the final state relation.
.. .
.
For an input string w = σ1 σ2 · · · σ n ∈ Σ∗ (σi ∈ Σ, 1 ≤ i ≤ n, 0 n), the iterative
state-transition relation δw : Q ⇁ Q is defined by δw = δσ1 δσ2 · · · δσn .
.
Definition .
..
The language accepted by a NFA M is defined by
. L(M) = {w ∈ Σ∗ | τδw β♯ = id I }.
.. .
.
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November 3-4, 2011 26 / 34
37. Coproduct Automaton
.
Definition .
..
Let M = (Q, τ, δ a (a ∈ Σ), β) and M ′ = (Q′ , τ′ , δ′ (a ∈ Σ), β′ ) be NFAs. The
a
coproduct automaton of M and M′ is defined by
M + M′ = (Q + Q′ , τ, δ a (a ∈ Σ), β) where τ = τ ⊥ τ′ , δ a = δ a + δ′ , and
ˆ ˆ ˆ ˆ ˆ
a
β=β⊥β
ˆ ′.
.
.. .
.
.
Proposition .
..
Let M + M′ be the coproduct automaton of M and M′ . Then
(a) δw = δw + δ′ for w ∈ Σ∗ .
ˆ
w
(b) τδw β♯ = τδw β♯ ⊔ τ′ δ′ (β′ )♯ .
ˆˆ ˆ w
′ ′
. (c) L(M + M ) = L(M) ∪ L(M ).
.. .
.
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November 3-4, 2011 27 / 34
38. Product Automatan
.
Definition .
..
Let M = (Q, τ, δ a (a ∈ Σ), β) and M ′ = (Q′ , τ′ , δ′ (a ∈ Σ), β′ ) be NFAs. The
a
product automaton of M and M ′ is defined by
M × M′ = (Q × Q′ , τ, δ a (a ∈ Σ), β) where τ = τ⊤τ′ , δ a = δ a × δ′ , and
ˆ ˆ ˆ ˆ ˆ
a
β = β⊤β
ˆ ′.
.
.. .
.
.
Proposition .
..
Let M + M′ be the coproduct automaton of M and M′ . Then
(a) δw = δw × δ′ for w ∈ Σ∗ .
ˆ
w
(b) τδw β♯ = τδw β♯ ⊓ τ′ δ′ (β′ )♯ .
ˆˆ ˆ w
′ ′
. (c) L(M × M ) = L(M) ∩ L(M ).
.. .
.
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November 3-4, 2011 28 / 34
39. Reverse Automaton
.
Definition .
..
Let M = (Q, τ, δ a (a ∈ Σ), β) be a NFA. The reverse automaton of M is
♯
defined by M R = (Q, τ R , δ R (a ∈ Σ), β R ) where τ R = β, δ R = δ a and
a a
. = τ.
βR
.. .
.
.
Proposition .
..
Let M R be the reverse automaton of M. Then
(a) δw = (δwR )♯ where w R is a reverse string of w ∈ Σ∗ .
R
(b) τ R δw (β R )♯ = (τδwR β♯ )♯ .
R
. (c) L(M ) = L(M) where L(M) = {w | w ∈ L(M)}.
R R R R
.. .
.
Y. Mizoguchi (Kyushu Univ.) Theory of Relations (2) Sequential Machines and Finite Automata
November 3-4, 2011 29 / 34
40. Concatenate Automatan
.
Definition .
..
Let M = (Q, τ, δ a (a ∈ Σ), β) and M ′ = (Q′ , τ′ , δ′ (a ∈ Σ), β′ ) be NFAs. The
a
concatenate automaton of M and M ′ is defined by
M · M′ = (Q + Q′ , τ, δ a (a ∈ Σ), β) where γ = β♯ τ′ , τ = τ(i ⊔ γ j),
ˆ ˆ ˆ ˆ
ˆ a = i♯ δ a i ⊔ i♯ γδ′ j ⊔ j♯ δ′ j, and β = β′ (γ♯ i ⊔ j).
δ ˆ
. a a
.. .
.
The function δw : Q ⇁ Q′ (w ∈ Σ∗ ) is uniquely determined by δε = 0QQ′
o o
and δwa = (δw γ ⊔ δw )δ′ .
o o
w
.
Proposition .
..
Let M · M′ be the concatenate automaton of M and M′ . Then
(a) δw = i♯ δw i ⊔ i♯ δw j ⊔ j♯ δ′ j.
ˆ o
w
(b) δw γ ⊔ δw ⊔ γδ′ = ⊔w=uv δu γδ′ .
o
w v
(c) τδw (β)♯ = ∪w=uv τδu γδ′ (β′ )♯ .
ˆˆ ˆ v
′ ′
. (d) L(M · M ) = L(M) · L(M ).
.. .
.
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November 3-4, 2011 30 / 34
41. Closure Automaton
.
Definition .
..
Let M = (Q, τ, δ a (a ∈ Σ), β) be a NFA. The coosure automaton of M + is
defined by M+ = (Q, τ, δ+ (a ∈ Σ), β) where γ = β♯ τ, and δ+ = (idQ ⊔ γ)δ a .
. a a
.. .
.
.
Proposition .
..
Let M+ be the closure automaton of M. Then
(a) δ+ = ⊔w=u1 ···u k ,u k
w ε,k0 δu1 γδu2 γ · · · γδu k for w ε.
(b) τδ+ β♯ = ⊔w=u1 ···u k ,k0 τδu1 γδu2 γ · · · γδu k β♯
w for w ε.
. (c) L(M+ ) = L(M)+ .
.. .
.
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November 3-4, 2011 31 / 34
42. Examples
.
Proposition .
..
Let Mϕ = (I, τ, δ a (a ∈ Σ), β) be a NFS where β = 0 II . Then
L(Mϕ ) = ϕ.
Let Mε = (I, τ, δ a (a ∈ Σ), β)) be a NFS where τ = id I , δ a = 0 II
( a ∈ Σ), and β = id I . Then L(Mε ) = {ε}.
Let σ ∈ Σ and Mσ = (I + I, δ a (a ∈ Σ), β) where τ = i, δσ = i♯ j,
. δ a = 0 I+I,I+I ( a σ), and β = j. Then L(Mσ ) = {σ}.
.. .
.
The Kleene closure automaton of M is defined by M ∗ = Mε + M+ . Then
we have
T(M∗ ) = L(Mε + M+ ) = T(Mε ) ∪ T(M+ ) = {ε} ∪ T(M)+ = T(M)∗ .
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43. Regular Language
.
Definition .
..
A language L ⊂ Σ∗ is a regular if there exists a NFA M such that
. = L(M).
L
.. .
.
.
Proposition .
..
Let L and L′ be regular languages. Then
L = ϕ, L = {ε} and L = {σ} (σ ∈ Σ) are regular languages.
Σ∗ − L, L R , L+ and L∗ are regular languages.
. L ∩ L′ , L ∪ L′ and L · L′ are regular languages.
.. .
.
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44. References
[1] M.A.Arbib and E.G.Manes: Machines in a cagegory, an expository
introduction, SIAM Review, 16(1974), 163-192.
[2] J.E.Hopcroft and J.D.Ullman: Formal Languages and Their Relation to
Automata (2nd. Ed.), Addison-Wesley(2001).
[3] Y. Kawahara, Applications of relational calculus to computer
mathematics, Bulletin of Informatics and Cybernetics, 23(1988),
67-78.
[4] M.O.Rabin and D.Scott: Finite Automata and Their Decision
Problems, IBM Journal, 3(1959), 114-125.
[5] A. Salomaa: Computation and Automata, Cambridge University
Press(1985).
[6] C.E.Shannon and J.Mac Carthy (eds.): Automata Studies, Princeton
University Press(1956).
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