Introduction To Computation Theory

[Theo Pontbriand]

Thistextbook connects three vibrant areas at the interface between economics and computer science: algorithmic game theory, computational social choice, and fair division. It thus offers an interdisciplinary treatment of collective decision making from an economic and computational perspective. Part I introduces to algorithmic game theory, focusing on both noncooperative and cooperative game theory. Part II introduces to computational social choice, focusing on both preference aggregation (voting) and judgment aggregation. Part III introduces to fair division, focusing on the division of both a single divisible resource ("cake-cutting") and multiple indivisible and unshareable resources ("multiagent resource allocation"). In all these parts, much weight is given to the algorithmic and complexity-theoretic aspects of problems arising in these areas, and the interconnections between the three parts are of central interest.

Jrg Rothe is Professor and the Vicechair of Department of Computer Science at Heinrich-Heine-Universitt Dsseldorf (Germany). Previously he was a visiting Assistant Professor at the University of Rochester (USA) and a visiting Professor at Stanford University (USA). His research focuses on computational complexity, computational social choice, collective decision making, algorithmic game theory, fair division, argumentation theory, algorithmics and cryptology.

Topics: Economic Theory/Quantitative Economics/Mathematical Methods, Microeconomics, Algorithm Analysis and Problem Complexity, Philosophy of Mathematics, Complex Systems, Political Economy/Economic Systems

CS 3306 - Introduction to Theory of ComputationCredits: 3 Class: 3 Lab: 0

Prerequisite(s): Grade of C or better in CS 1410 and MATH 2305 .

An introduction to the modern theory of computing. Topics selected from abstract algebra, finite automata, regular expressions, regular languages, pushdown automata, context-free languages, and Turing machines. The capabilities and limitations of abstract computing devices are investigated from a theoretical perspective.

The question "Can machines think?" is one that has fascinated people for a long time (see here for a non-technical perspective on this question by E.B. White). This is pretty close to the question "Can machines learn?", which has been studied from different points of view by many researchers in computer science.

This course will give an introduction to some of the central topics in computational learning theory, a field which approaches the above question from a theoretical computer science perspective. We will study well-defined mathematical and computational models of learning in which it is possible to give precise and rigorous analyses of learning problems and learning algorithms. A big focus of the course will be the computational efficiency of learning in these models. We'll develop computationally efficient algorithms for certain learning problems, and will see why efficient algorithms are not likely to exist for other problems.

This is a preliminary list of core topics. Other topics may be covered depending on how the semester progresses. Most topics will take several lectures. For more information, click on the "Lectures" tab above.

Computational complexity theory is a mathematical research area in which the goal is to quantify the resources required to solve computational problems. It is concerned with algorithms, which are computational methods for solving problems.

Now to get to the point. A dimer arrangement with $k$ dimers has a weight which we write as $x^k$, where $x$ is a variable representing the contribution of an individual dimer. The dimer partition function is the sum of the weights of all dimer arrangements. So for this graph, the dimer partition function is $f(x) = 1 + 5x + 3x^2$. That is because there is $1$ dimer arrangement with no dimers and this arrangement has weight $x^0=1$. There are $5$ dimer arrangements with a single dimer, and these each have weight $x^1 =x$. Finally, there are $3$ dimer arrangements with $2$ dimers, and these each have weight $x^2$.

Approximating the dimer partition function is just one example of the sort of computational problem that we study. By studying these, our real objective is to understand computation, and in particular to understand how varying parameters ultimately determines which computational problems are feasibly solvable, and which, provably, are not.

I remember, back when I was working on my computer science degree, studying about Turing machines and the Church-Turing Thesis in my Intro to Computational Theory class. Back then I thought it was a big waste of time. I just wanted to program computers and I could care less about this long dead Turing-guy (or this Church-guy) nor his stupid theoretical machines.

Turing Machines are funny little theoretical machines that have a read/write head and a (hypothetical) paper tape that it can read from or write to. Based on what the Turing Machine reads it puts the Turing Machine into an action state that performs some sort of combination of tasks consisting of either moving the read/write head forward or backward, reading from a new position on the tape, or writing so a new position on the tape. Try to visualize what a Turing Machine looks like via this drawing:

One thing of interest is that a Turing Machine is, despite surface appearances, actually somewhat similar to a modern computer. In a modern computer the Central Processing Unit (CPU) is equivalent to the read/write head of a Turing Machine. The memory chips (RAM or ROM) is very similar to the cells of the long paper tape that the turning machine can read from or write to. So modern computers are (roughly) equivalent to a Turing Machine.

[1] The Turing Principle. So named by Roger Penrose, who does not believe in it (at least not in current form). It was developed into the Turing-Deutsch principle by David Deutsch, who does believe in it, at least in his form of it.) (See article in Wikipedia for more details)

Setting aside the religious aspect for now, it seems that the human brain might blow the Church-Turing theory out of the water. It indicates that there is very likely a more powerful computational machine than a Turing out there: a biological one.

On the religious front, we could see our brains as such machines with infinite potential and the ability to self-modify. Machines that somehow have the ability to interface with spiritual matter: to bridge the gap between spirit and physicality. Machines that are capable of being manipulated by another force capable of reason and deduction, which would indicate that there is, in fact, a machine even greater than it is.

The one thing I really like about Deutsch is that he is so dang hopeful. He is really, if you will, coming up with a sort of Theology of Nature that goes so far beyond the incredibly lame stuff that most scientists try to cite for their reasons for finding important morality in nature.

The idea of something that is truly non-computational is at once refreshing and scary. It proves there is more than the world of science can detect which is refreshing, but places it utterly outside the bounds of our comprehension also, which is kind of scary. So I always have mixed feelings no matter which way the wind ends up blowing on this.

I guess my comment was more aimed at undermining the idea that folk psychology needs or ever could have a mathematical basis. I see no reason for why math or computation should be the base for FP rather than the other way around since the latter came first and would certainly be the structure around which the former was be built.

So the question of something being non-computational would be more one of a limit to our understanding of math. Just like even miracles and magic can be explained with science. Any lack of ability to explain through science is a lack in the science, not a lack in the capacity to explain.

Thanks for the further explanation, Jeff G. Where do you live? We so need to sit down for a couple of hours and have a few beers with undergrowth and chat and you can explain all your ideas to me better. ?

Comprehensive introduction to the neural network models currently under intensive study for computational applications. It also provides coverage of neural network applications in a variety of problems of both theoretical and practical interest.

1. Examples of Vector Spaces. 1.1. First Vector Space: Tuples. 1.2. Dot Product. 1.3. Application: Geometry. 1.4. Second Vector Space: Matrices. 1.5. Matrix Multiplication. 2. Matrices and Linear Systems. 2.1. Systems of Linear Equations. 2.2. Gaussian Elimination. 2.3. Application: Markov Chains. 2.4. Application: The Simplex Method. 2.5. Elementary Matrices and Matrix Equivalence. 2.6. Inverse of a Matrix. 2.7. Application: The Simplex Method Revisited. 2.8. Homogeneous/Nonhomogeneous Systems and Rank. 2.9. Determinant. 2.10. Applications of the Determinant. 2.11. Application: Lu Factorization. 3. Vector Spaces. 3.1. Definition and Examples. 3.2. Subspace. 3.3. Linear Independence. 3.4. Span. 3.5. Basis and Dimension. 3.6. Subspaces Associated with a Matrix. 3.7. Application: Dimension Theorems. 4. Linear Transformations. 4.1. Definition and Examples. 4.2. Kernel and Image. 4.3. Matrix Representation. 4.4. Inverse and Isomorphism. 4.5. Similarity of Matrices. 4.6. Eigenvalues and Diagonalization. 4.7. Axiomatic Determinant. 4.8. Quotient Vector Space. 4.9. Dual Vector Space. 5. Inner Product Spaces. 5.1. Definition, Examples and Properties. 5.2. Orthogonal and Orthonormal. 5.3. Orthogonal Matrices. 5.4. Application: QR Factorization. 5.5. Schur Triangularization Theorem. 5.6. Orthogonal Projections and Best Approximation. 5.7. Real Symmetric Matrices. 5.8. Singular Value Decomposition. 5.9. Application: Least Squares Optimization. 6. Applications in Data Analytics. 6.1. Introduction. 6.2. Direction of Maximal Spread. 6.3. Principal Component Analysis. 6.4. Dimensionality Reduction. 6.5. Mahalanobis Distance. 6.6. Data Sphering. 6.7. Fisher Linear Discriminant Function. 6.8. Linear Discriminant Functions in Feature Space. 6.9. Minimal Square Error Linear Discriminant Function. 7. Quadratic Forms. 7.1. Introduction to Quadratic Forms. 7.2. Principal Minor Criterion. 7.3. Eigenvalue Criterion. 7.4. Application: Unconstrained Nonlinear Optimization. 7.5. General Quadratic Forms. Appendix A. Regular Matrices. Appendix B. Rotations and Reflections in Two Dimensions. Appendix C. Answers to Selected Exercises.

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