Models of Computation: 2008-2009
Information
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Degrees |
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Term |
Trinity Term 2009 |
Overview
This course introduces the classical mathematical models used to analyse computation, including finite state automata, grammars, and Turing Machines.A computer scientist should be able to distinguish between what can be computed and what cannot. This distinction can only be made with a good scientific model of computers and computation. This course introduces the powerful idea of using a mathematical model to analyse computation.
This course describes a number of different models of computation which were proposed and analysed over the past century. Many of these models were found to be equivalent, in the sense that they allow exactly the same computations to be carried out. Other models were shown to be less powerful, but simpler to implement, and so useful for some purposes.
Learning outcomes
At the end of the course students will be able to:
- Describe in detail what is meant by a finite state automaton, and a Turing machine, and calculate the behaviour of simple examples of these devices.
- Design machines of these types to carry out simple computational tasks.
- Describe a computational problem in terms of a language, and specify simple languages using a grammar or regular expression.
- Reason about the capabilities of standard machines, and demonstrate that they have limitations.
Synopsis
Part 1: Finite-state machines and regular languages
- General introduction. The FSM model. DFA's. Transition diagrams. Examples.
- NFA. The subset construction. Reduction of NFA to DFA.
- Regular Languages. Closure properties. Constructions on machines.
- Regular expressions. Some regular algebra. Algebraic laws.
- Compiling regular expressions into NFA's, and vice versa.
- Regular grammars. Equivalence to regular languages.
- Limits to FSM's. The Pumping Lemma.
- Context-free Languages and Pushdown Automata. The Chomsky hierarchy.
- Turing's Analysis of computation. The Turing machine model: FSM's plus memory (the "tape").
- The intuitive notion of computability. Decision problems. Encoding problems as sets of strings.
- Expressive power of Turing machines. Church's thesis.
- Undecidable problems. Diagonalization. The halting problem.
- Survey of decidable and undecidable problems. A glimpse beyond: complexity, P=NP?
Syllabus
The finite-state machine. DFA and NFA and their equivalence. Regular languages and regular expressions. Regular grammars. The pumping lemma.The Turing machine. Church's Thesis. Decision problems and undecidability. The halting problem. Chomsky grammars.
Reading list
Primary Text- M Sipser, Introduction to the Theory of Computation, PWS Publishing Company, January 1997.
- J E Hopcroft, R Motwani and J D Ullman, Introduction to Automata Theory, Languages and Computation,Addison-Wesley, Second edition, 2001.
- G. Boolos and R. Jeffrey, Computability and Logic, Cambridge University Press, 3rd Edition, 1989.