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310409-Theory of computation

Komate AMPHAWAN. 310409-Theory of computation. DESCRIPTION-1. Studies concepts of: grammars, automata, languages, computability and complexity. DESCRIPTION-2. the relationship between automata and various classes of languages Turing machine and equivalent models of computation

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310409-Theory of computation

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  1. Komate AMPHAWAN 310409-Theory of computation

  2. DESCRIPTION-1 • Studies concepts of: • grammars, • automata, • languages, • computability and complexity

  3. DESCRIPTION-2 • the relationship between automata and various classes of languages • Turing machine and equivalent models of computation • the Chomsky hierarchy context-free grammar, push-down automata, • Pumping lemmas and variants, closure properties and decision properties; parsing algorithms.

  4. Computability • What is “Computation”? • What is Computable? • Problem of determining whether a mathematical statement is true or false.

  5. Evaluation and Grading • Midterm exam 40% • Final exam 40% • Quiz 10% • Class attendance 10%

  6. Text-1 • “Introduction to Languages and the Theory of Computation”—John C. Martin,4ed, 2010.

  7. Text-2 • “Introduction to the Theory of Computation”—Michael Sipser, 2ed, 2005.

  8. Main topic • We shall study different types of theoretical machines that are mathematical modelsfor actual physical processes.

  9. Machine Model-1

  10. Machine Model-2

  11. Machine Model-3

  12. Prerequisites • Set Theory • Relations and Functions „ Relations and Functions • Elementary Graph Theory • Boolean Logic • Proof Mechanisms

  13. Review… on basic mathematics

  14. Set

  15. Sets-1 • Aset is a collection of "things," called the elements or members of the set. • It is essential to have a criterion for determining, for any given thing, whether it is or is not a member of the given set. • This criterion is called the membership criterion of the set.

  16. Sets-2 • There are two common ways of indicating the members of a set: • List all the elements, e.g. {a, e, i, o, u} • Provide some sort of an algorithm or rule, such as a grammar

  17. Sets-3 • Notation: • To indicate that x is a member of set S, we write x  S • We denote the empty set (the set with no members) as { } or  • If every element of set A is also an element of set B, we say that A is a subset of B, and write A  B • If every element of set A is also an element of set B, but B also has some elements not contained in A, we say that A is a proper subset of B, and write A  B

  18. Operations on Sets • The union of sets A and B, written A  B, is a set that contains everything that is in A, or in B, or in both. • The intersection of sets A and B, written A  B, is a set that contains exactly those elements that are in both A and B. • The set difference of set A and set B, written A - B, is a set that contains everything that is in A but not in B.

  19. Additional terminology • The cardinality of a set A, written |A|, is the number of elements in a set A. • The powerset of a set Q, written 2Q, is the set of all subsets of Q. The notation suggests the fact that a set containing n elements has a powersetcontaining 2n elements. • Two sets are disjoint if they have no elements in common, that is, if A  B = .

  20. Relations and Functions

  21. Relations and Functions-1 • A relation on sets S and T is a set of ordered pairs (s, t), where • s  S (s is a member of S), • t    T, • S and T need not be different, • The set of all first elements (s) is the domain of the relation, and • The set of all second elements is therange of the relation.

  22. Relations and Functions-2 • A relation is a functioniff every element of S occurs once and only once as a first element of the relation. • A relation is a partial functioniff every element of S occurs at most once as an element of the relation.

  23. Graphs

  24. Graph-1 • A graph consists of two sets • A set V of vertices (or nodes), and • A set E of edges (or arcs). • An edge consists of a pair of vertices in V. If the edges are ordered, the graph is a digraph (a contraction of "directed graph").

  25. Graph-2 • A walk is a sequence of edges, where the finish vertex of each edge is the start vertex of the next edge. Example: (a, e), (e, i), (i, o), (o, u). • A path is a walk with no repeated edges. • A simple path is a path with no repeated vertices.

  26. Trees

  27. Trees-1 • A tree is a kind of digraph: • It has one distinguished vertex called the root; • There is exactly one path from the root to each vertex; and • The level of a vertex is the length of the path to it from the root.

  28. Trees-2 • Terminology: • if there is an edge from A to B, then A is the parent of B, and B is the child of A. • A leaf is a node with no children. • The height of a tree is the largest level number of any vertex.

  29. Proof techniques

  30. Importance • Because this is a formal subject, the textbook is full of proofs • Proofs are encapsulated understanding • You may be asked to learn a very few important proofs

  31. Proof by induction • Prove something about P1 (the basis) • Prove that if it is true for Pn, then it is true for Pn+1 (the inductive assumption) • Conclude that it is true for all P

  32. Proof by contradiction • also called reduction ad absurdum • Assume some fact P is false • Show that this leads to a contradiction • Conclude P must be true

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