CDM
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Work on a textbook for CDM is currently in progress.
Below is a draft of some chapters of the book; use with care and let
me know about bugs and typos.
There is also an extensive collection of homework problems, some of which
will appear in the actual assignments this term.
Lastly, there slides from lectures for the last few incarnations of the course. Since the material changes every year there is some overlap between lectures from different semesters. Note that not all of this material is part of the current version (Fall 08). Still, all of it is relevant and you might find it useful to take a look. Some of the notebooks require the Automata package, see Computation. |
| Topic | Comment | |
| Foundations | Computability, Logic, Sets. | Ch1-5 |
| Assignments | Collection of assignments. | HW |
| Computation | ||
| Turing Machines | Universality and undecidability. | 6up pdf | (Primitive) Recursive Functions | Closure properties, bounded search, coding, Ackermann function. | 6up pdf |
| LOOP Programs | Comparison to prec functions, levels 0, 1, 2. | 6up pdf |
| Completeness | Turing, many-one, one-one reductions, degrees. | 6up pdf |
| Arithmetical Hierarchy | Projections and complements, definability theory. | 6up pdf |
| Deterministic Complexity | Tractable problems, deterministic complexity classes. | 6up pdf |
| Nondeterministic Complexity | NP, Sigma-1-definition, examples. | 6up pdf |
| Hardness | Polynomial time reductions, hard and complete problems. | 6up pdf |
| Kolmogorov Complexity | Program size complexity, incompressibility, Omega. | 6up pdf |
| Physics and Computation | Landauer, reversible computation, entropy, physical computation. | 6up pdf |
| LOOP programs | LoopProg.nb | |
| Finite State Machines | ||
| Deterministic Machines | Scan algorithms, accessibility and minimality. | 6up pdf |
| Nondeterministic Machines | Deterministic simulation, pattern matching. | 6up pdf |
| Minimization | Quotients and covers. | 6up pdf |
| Minimization algorithms. | 6up pdf | |
| Pattern Matching | Matching strings. | 6up pdf |
| General pattern matching. | 6up pdf | |
| Logic | ||
| Formal Systems | Motivation, axioms, derivations, Frege's system. | 6up pdf |
| Propositional Logic | Boolean functions, circuits, propositional logic. | 6up pdf |
| Algorithms for propositional logic, DPLL. | 6up pdf | |
| Equational Logic | Equational reasoning, Birkhoff's theorem, Boolean algebras. | 6up pdf |
| Predicate Logic I | Predicate calculus, structures, interpretations. | 6up pdf |
| Predicate Logic II | Semantics, completeness, compactness, incompleteness. | 6up pdf |
| Model Checking | Verification, Kripke structures, CTL, LTL. | 6up pdf |
| Puzzles, propositional logic, normal forms. | LogicHW.nb | |
| Sets | ||
| Set Theory | Sets, Cantor, Frege, Zermelo-Fraenkel | 6up pdf |
| Three Theorems | Cardinality, three theorems. | 6up pdf |
| Ordinals | Ordinals, well-orderings, transfinite induction. | 6up pdf |
| Sets, implementation, Cartesian product. | SetsHW.nb | |
| Relations and Functions | ||
| Relations | Cartesian products, relations, properties. | 6up pdf |
| Algorithms for Relations | Closures, equivalence relations. | 6up pdf |
| Functions | Functions, functionals, iteration. | 6up pdf |
| Hash functions, universal hashing, perfect hashing. | ||
| Relations, Boolean matrices, visual representation. | Relations.nb | |
| Functions, functionals, iteration. | Functions.nb | |
| Relations, functions, closure. | RelFuncHW.nb | |
| Induction | ||
| Induction | IND, LEP, well-orders, recursive datatypes, recursive functions | 6up pdf |
| Iteration | Orbits and fixed points, Collatz, Ducci, Floyd's trick, Knaster-Tarski | 6up pdf |
| Induction | Induction.nb | |
| Iteration | Collatz.nb | |
| Recursion | Recursion.nb | |
| Induction, iteration, recursion. | IteRecInd.nb | |
| Stack turtle. | Turtle.nb | |
| Combinatorics | ||
| Basic Combinatorics | Combinations, permutations, binomials, occupancy, inclusion/exclusion. | 6up pdf |
| Fibonacci | Recurrence equations, Fibonacci numbers. | 6up pdf |
| Moebius | Necklaces, bracelets, Lyndon words, Moebius inversion. | 6up pdf |
| Actions and Counting | Group actions, Burnside, Polya-Redfield. | 6up pdf |
| Enumerations | Semigroup actions, enumeration, ranking, unranking. | 6up pdf |
| Randomness | Pseudo-randomness, physics, practical RNGs. | 6up pdf |
| Shortest paths, Dijkstra, Floyd-Warshall, Bellman-Ford. | ||
| Strongly connected components, biconnected components. | ||
| Permutations | Permutations.nb | |
| Fibonacci | Fibonacci.nb | |
| Binomials | Binomials.nb | |
| Enumerating functions (injective, increasing, bijective). | EnumFunctions.nb | |
| Various exercises. | CombinatoricsHW.nb | |
| Algebra | Modular Arithmetic | Euclidean algorithm, Chinese remainder theorem, RSA. | 6up pdf |
| Groups | Subgroups, Lagrange's theorem, Cayley's theorem. | 6up pdf |
| Polynomials | Evaluation, interpolation, roots. | 6up pdf |
| Finite Fields | Irreducible polynomials, primitive elements. | 6up pdf |
| FF Applications | GPS, Sigma Automata. | 6up pdf |
| Cryptography | Permutation codes, one-time pads, RSA. | 6up pdf |
| Primality testing, Pratt, Solovay-Strassen. | ||
| Divisibility | Divisibility.nb | |
| Modular arithmetic. | ModArithHW.nb | |
| Cellular Automata and FSRs | ||
| Feedback Shift Register | GPS, Fibonacci, Galois feedback shift registers. | 6up pdf |
| Cellular Automata | One- and two-dimensional cellular automata, orbits. | 6up pdf |
| CA and FSMs | CA and regular languages. | 6up pdf |
| CA and Algebra | Patterns and generating functions. | 6up pdf |
| CA and Computation | Computation on CA, universal CA. | 6up pdf |
| Reversibility | Reversible CA, Hedlund, decision algorithms. | 6up pdf |
| Feedback shift registers | FeedbackSR.nb | |
| Miscellany | ||
| Ehrenfeucht-Mycielsky | Non-learning, disjunctiveness, balance. | 6up pdf |
| Sigma Automata | Additive cellular automata, Fibonacci polynomials. | 6up pdf |
| All-Ones Problem, binary Fibonacci polynomials. | FibonacciP.nb | |