Spring 2021

Carnegie Mellon University

This advanced course reexamines traditional concepts of discrete mathematics (relations, functions, logic, graphs, algebra, automata) in the context of computation and algorithms, necessitating a strong background in discrete math.

In a nutshell, the main idea behind this course is that the development of the digital computer, together with the theory of computation, is also one of the most important developments in mathematics in the 20th century. Consequently, this course takes a fresh look at some of the standard concepts of discrete mathematics (relations, functions, logic, graphs, algebra, automata), with strong and consistent emphasis on computation and algorithms. Another key concern is knowledge transfer: we need to realign traditional mathematical concepts with our new computational universe and find ways to apply ideas from one realm to the other. We begin with a brief introduction to computability and computational complexity. Other topics may include: iteration, orbits and fixed points; order and equivalence relations; transition systems; groups and actions; finite fields, randomness; propositional logic and satisfiability testing.

To get a better impression, take a look at some of the homework problems that were used in previous versions of this course: CDM Assignments. A few pictures can be found here: CDM Gallery (challenge: figure out what these pictures mean, for as many as you can manage). A slightly dated paper outlining the course philosophy can be found at Jeric.

Needless to say, a solid background in discrete math is necessary for this course. You should feel very much at home with all the 15-251 material.

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AutomataClosureCoding TheoryComputation & Discrete MathDecidabilityDeterminizationEffective ResistanceFeedback Shift RegistersFinite FieldsFirst-order logicGroupsHypercomputationInfinite WordsIteration Iteration LogicMinimizationNondeterminismOrbitsPolya CountingPrimitive RecursionRegister MachinesZero Space