Introduction and Applications of Discrete Mathematics
- 1. Bart Selman CS2800 1 Introduction to Discrete Mathematics
- 2. Bart Selman CS2800 2 Discrete vs. Continuous Mathematics Continuous Mathematics It considers objects that vary continuously; Example: analog wristwatch (separate hour, minute, and second hands). From an analog watch perspective, between 1 :25 p.m. and 1 :26 p.m. there are infinitely many possible different times as the second hand moves around the watch face. Real-number system --- core of continuous mathematics; Continuous mathematics --- models and tools for analyzing real-world phenomena that change smoothly over time. (Differential equations etc.)
- 3. Bart Selman CS2800 3 Discrete vs. Continuous Mathematics Discrete Mathematics It considers objects that vary in a discrete way. Example: digital wristwatch. On a digital watch, there are only finitely many possible different times between 1 :25 P.M. and 1:27 P.M. A digital watch does not show split seconds: - no time between 1 :25:03 and 1 :25:04. The watch moves from one time to the next. Integers --- core of discrete mathematics Discrete mathematics --- models and tools for analyzing real-world phenomena that change discretely over time and therefore ideal for studying computer science – computers are digital! (numbers as finite bit strings; data structures, all discrete! Historical aside: earliest computers were analogue.)
- 4. Bart Selman CS2800 4 Examples of the uses of Discrete Mathematics
- 5. E.g. between you and the Bank of America. Bart Selman CS2800 5 Number Theory: RSA and Public-key Cryptography Alice and Bob have never met but they would like to exchange a message. Eve would like to eavesdrop. They could come up with a good encryption algorithm and exchange the encryption key – but how to do it without Eve getting it? (If Eve gets it, all security is lost.) CS folks found the solution: public key encryption. Quite remarkable that that is feasible.
- 6. Bart Selman CS2800 6 Graphs and Networks •Many problems can be represented by a graphical network representation. •Examples: – Distribution problems – Routing problems – Maximum flow problems – Designing computer / phone / road networks – Equipment replacement – And of course the Internet Aside: finding the right problem representation is one of the key issues.
- 7. Networks are pervasive 7 New Science of Networks Sub-Category Graph No Threshold NYS Electric Power Grid (Thorp,Strogatz,Watts) Cybercommunities (Automatically discovered) Kleinberg et al Network of computer scientists ReferralWeb System (Kautz and Selman) Neural network of the nematode worm C- elegans (Strogatz, Watts) Utility Patent network 1972-1999 (3 Million patents) Gomes,Hopcroft,Lesser,Selman
- 8. 8 Scheduling of Final Exams How can the final exams at Kent State be scheduled so that no student has two exams at the same time? (Note: it is not obvious that this has anything to do with graphs!) Graph: A vertex correspond to a course. An edge between two vertices denotes that there is at least one common student in the courses they represent. Each time slot for a final exam is represented by a different color. 1 7 2 6 3 5 4
- 9. 9 Probability and Chance Importance of concepts from probability is rapidly increasing in CS: • Randomized algorithms (e.g. primality testing; randomized search algorithms, such as simulated annealing, Google’s PageRank, “just” a random walk on the web!) In computation, having a few random bits really helps! • Machine Learning / Data Mining: Find statistical regularities in large amounts of data. (e.g. Naïve Bayes alg.) • Natural language understanding: dealing with the ambiguity of language (words have multiple meanings, sentences have multiple parsings --- key: find the most likely (i.e., most probable) coherent interpretation of a sentence (the “holy grail” of NLU).
- 10. 10 Probability: Bayesian Reasoning Bayesian networks provide a means of expressing joint probability over many interrelated hypotheses and therefore reason about them. Bayesian networks have been successfully applied in diverse fields such as medical diagnosis, image recognition, language understanding, search algorithms, and many others. Bayes Rule Example of Query: what is the most likely diagnosis for the infection given all the symptoms? “18th-century theory is new force in computing” CNET ’07
- 11. Goals of the Discrete Mathematics Course Introduce students to a range of mathematical tools from discrete mathematics that are key in computer science Mathematical Sophistication How to write statements rigorously How to read and write theorems, lemmas, etc. How to write rigorous proofs Areas we will cover: Actually, only practice works! Logic and proofs Set Theory Induction and Recursion Counting and combinatorics Probability theory Number Theory (if time permits) Practice works! Note: Learning to do proofs from watching the slides is like trying to learn to play tennis from watching it on TV! So, do the exercises! Aside: We’re not after the shortest or most elegant proofs; verbose but rigorous is just fine!
- 12. Bart Selman CS2800 12 The End Now that you have completed the introduction to Discrete Mathematics, please explore the next topic on Sets