Computation and Complexity

MATH63012 Computation and Complexity 15 Semester 2 Postgraduate Taught Level 6 Nora Szakacs Unit coordinator Department of Mathematics 1 FHEQ level (Framework for Higher Education Qualifications) ' Masters/Integrated Masters P4 ' European Credit Transfer & Accumulation System Rating : 7.5 Quite a lot of the mathematics you have studied so far involves using algorithms to solve computational problems. For example, you have probably used Euclid's algorithm to solve the problem of finding the greatest common divisor of two integers. In this course, we abstract a level further, and study the properties of problems and algorithms themselves. The kind of questions we ask are "is there an algorithm to solve EVERY problem?" and "what problems can be solved by an EFFICIENT algorithm?". Compared with most of mathematics, this area is in its infancy, and many important things remain unknown. The course will take you to the point where you understand the statement of one of the most important open questions in mathematics and computer science: the "P vs NP" problem, for which the Clay Mathematics Foundation is offering a $1,000,000 prize. And who knows, perhaps one day you will be the one to solve it! Quite a lot of the mathematics you have studied so far involves using algorithms to solve computational problems. For example, you have probably used Euclid's algorithm to solve the problem of finding the greatest common divisor of two integers. In this course, we abstract a level further, and study the properties of problems and algorithms themselves. The kind of questions we ask are "is there an algorithm to solve EVERY problem?" and "what problems can be solved by an EFFICIENT algorithm?". Compared with most of mathematics, this area is in its infancy, and many important things remain unknown. The course will take you to the point where you understand the statement of one of the most important open questions in mathematics and computer science: the "P vs NP" problem, for which the Clay Mathematics Foundation is offering a $1,000,000 prize. And who knows, perhaps one day you will be the one to solve it! The course unit aims<ul><li>to introduce the main model of computation currently being employed in the theory of computation, Turing machines;</li><li>to introduce the key parameters quantifying computational complexity (deterministic, non-deterministic, time, space) and the relationships between them.</li></ul> <p style="-webkit-text-stroke-width:0px;background-color:rgb(255, 255, 255);color:rgb(81, 81, 81);font-family:Arial, sans-serif;font-size:12px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:-webkit-left;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:normal;widows:2;word-spacing:0px;">On successful completion of the course, students will be able to:<ul><li style="-webkit-text-stroke-width:0px;background-color:rgb(255, 255, 255);color:rgb(81, 81, 81);font-family:Arial, sans-serif;font-size:12px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:-webkit-left;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:normal;widows:2;word-spacing:0px;">Define finite state automata and regular languages, explain the relationship between these, construct and analyse simple examples;</li><li style="-webkit-text-stroke-width:0px;background-color:rgb(255, 255, 255);color:rgb(81, 81, 81);font-family:Arial, sans-serif;font-size:12px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:-webkit-left;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:normal;widows:2;word-spacing:0px;">define the syntactic monoid of a language, discuss its relevance, and calculate it in simple examples; </li><li style="-webkit-text-stroke-width:0px;background-color:rgb(255, 255, 255);color:rgb(81, 81, 81);font-family:Arial, sans-serif;font-size:12px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:-webkit-left;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:normal;widows:2;word-spacing:0px;">define Turing machines, discuss their capabilities and limitations, and construct and analyse simple examples;</li><li style="-webkit-text-stroke-width:0px;background-color:rgb(255, 255, 255);color:rgb(81, 81, 81);font-family:Arial, sans-serif;font-size:12px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:-webkit-left;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:normal;widows:2;word-spacing:0px;">define the key concepts of computation and complexity (including the main computability and complexity classes), discuss the relationships between them, and prove simple seen and unseen facts about them;</li><li style="-webkit-text-stroke-width:0px;background-color:rgb(255, 255, 255);color:rgb(81, 81, 81);font-family:Arial, sans-serif;font-size:12px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:-webkit-left;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:normal;widows:2;word-spacing:0px;">recall and analyse a range of computational problems and their properties;</li><li style="-webkit-text-stroke-width:0px;background-color:rgb(255, 255, 255);color:rgb(81, 81, 81);font-family:Arial, sans-serif;font-size:12px;font-style:normal;font-variant-caps:normal;font-variant-ligatures:normal;font-weight:400;letter-spacing:normal;orphans:2;text-align:-webkit-left;text-decoration-color:initial;text-decoration-style:initial;text-decoration-thickness:initial;text-indent:0px;text-transform:none;white-space:normal;widows:2;word-spacing:0px;">state, apply and 0. INTRODUCTION: outline introduction to computability and complexity; course practicalities. 1. FINITE STATE AUTOMATA: alphabets and languages; finite state automata as models of computation; regular languages; Kleene’s theorem; semigroups and syntactic monoids; Schützenberger’s theorem. 2. COMPUTABILITY: problems and solutions; Turing machines; recursiveness and the Church-Turing Thesis; multitape machines; coding machines and non-recursive languages; universal computation; non-determinism. 3. COMPUTATIONAL COMPLEXITY: time and space; linear speed up and space reduction; complexity classes; lower bounds and crossing arguments; space and time hierarchy theorems; tractability and P vs NP; polynomial time reduction. 4. COMPLETENESS: NP-completeness; SAT and the Cook-Levin Theorem; NP-completeness by reduction; further examples of NP-complete languages. 1 Written exam 100%  Tutorials will provide an opportunity for students’ work to be discussed and provide feedback on their understanding. MATH63012 Computation and Complexity Anti-requisite Compulsory Pre-Requisites: There are no explicit prerequisites; however, this is a heavily proof-based course and experience with proof-based mathematics is essential – eg MATH21120 or MATH20101 etc.Students are not permitted to take, for credit, MATH43012 in an undergraduate programme and then MATH63012 in a postgraduate programme at the University of Manchester, as the courses are identical. N Printed notes will be supplied and you should not need to refer to any books. But if you would like an alternative viewpoint, the following texts cover most of the course material:<ul><li>Bovet and Crescenzi, Introduction to the Theory of Complexity, 1994,</li><li>Papadimitriou, Computational Complexity, 1994,</li><li>Sipser, Introduction to the Theory of Computation (second edition), 2006.</li></ul> Lectures 11 Tutorials 11 0 128