CSCI-2400 Models of Computation

1. CSCI-2400 Models of Computation

2. Computation memory CPU

3. temporary memory input memory CPU output memory Program memory

4. Example: temporary memory input memory CPU output memory Program memory compute compute

5. temporary memory input memory CPU output memory Program memory compute compute

6. temporary memory input memory CPU output memory Program memory compute compute

7. temporary memory input memory CPU Program memory output memory compute compute

8. Automaton temporary memory Automaton input memory CPU output memory Program memory

9. Different Kinds of Automata • Automata are distinguished by the temporary memory • Finite Automata: no temporary memory • Pushdown Automata: stack • Turing Machines: random access memory

10. Finite Automaton temporary memory input memory Finite Automaton output memory Vending Machines (small computing power)

11. Pushdown Automaton Stack Push, Pop input memory Pushdown Automaton output memory Programming Languages (medium computing power)

12. Turing Machine Random Access Memory input memory Turing Machine output memory Algorithms (highest computing power)

13. Power of Automata Finite Automata Pushdown Automata Turing Machine

14. We will show later in class • How to build compilers for programming languages • Some computational problems cannot be solved • Some problems are hard to solve

15. Mathematical Preliminaries

16. Mathematical Preliminaries • Sets • Functions • Relations • Graphs • Proof Techniques

17. SETS A set is a collection of elements We write

18. Set Representations • C = { a, b, c, d, e, f, g, h, i, j, k } • C = { a, b, …, k } • S = { 2, 4, 6, … } • S = { j : j > 0, and j = 2k for some k>0 } • S = { j : j is nonnegative and even } finite set infinite set

19. U A 6 8 2 3 1 7 4 5 9 10 A = { 1, 2, 3, 4, 5 } • Universal Set: All possible elements • U = { 1 , … , 10 }

20. B A • Set Operations • A = { 1, 2, 3 } B = { 2, 3, 4, 5} • Union • A U B = { 1, 2, 3, 4, 5 } • Intersection • A B = { 2, 3 } • Difference • A - B = { 1 } • B - A = { 4, 5 } U A-B

21. Complement • Universal set = {1, …, 7} • A = { 1, 2, 3 } A = { 4, 5, 6, 7} 4 A A 6 3 1 2 5 7 A = A

22. { even integers } = { odd integers } Integers 1 odd 0 5 even 6 2 4 3 7

23. DeMorgan’s Laws A U B = A B U A B = A U B U

24. Empty, Null Set: = { } S U = S S = S - = S - S = U = Universal Set

25. U A B U A B Subset A = { 1, 2, 3} B = { 1, 2, 3, 4, 5 } Proper Subset: B A

26. A B = U Disjoint Sets A = { 1, 2, 3 } B = { 5, 6} A B

27. Set Cardinality • For finite sets A = { 2, 5, 7 } |A| = 3

28. Powersets A powerset is a set of sets S = { a, b, c } Powerset of S = the set of all the subsets of S 2S = { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} } Observation: | 2S | = 2|S| ( 8 = 23 )

29. Cartesian Product A = { 2, 4 } B = { 2, 3, 5 } A X B = { (2, 2), (2, 3), (2, 5), ( 4, 2), (4, 3), (4, 4) } |A X B| = |A| |B| Generalizes to more than two sets A X B X … X Z

30. FUNCTIONS domain range B A f(1) = a a 1 2 b c 3 f : A -> B If A = domain then f is a total function otherwise f is a partial function

31. RELATIONS R = {(x1, y1), (x2, y2), (x3, y3), …} xi R yi e. g. if R = ‘>’: 2 > 1, 3 > 2, 3 > 1 In relations xi can be repeated

32. Equivalence Relations • Reflexive: x R x • Symmetric: x R y y R x • Transitive: x R Y and y R z x R z • Example: R = ‘=‘ • x = x • x = y y = x • x = y and y = z x = z

33. Equivalence Classes For equivalence relation R equivalence class of x = {y : x R y} Example: R = { (1, 1), (2, 2), (1, 2), (2, 1), (3, 3), (4, 4), (3, 4), (4, 3) } Equivalence class of 1 = {1, 2} Equivalence class of 3 = {3, 4}

34. e b node d a edge c GRAPHS A directed graph • Nodes (Vertices) • V = { a, b, c, d, e } • Edges • E = { (a, b), (b, c), (c, a), (b, d), (d, c), (e, d) }

35. 6 e 2 b 1 3 d a 6 5 c Labeled Graph

36. e b d a c Walk Walk is a sequence of adjacent edges (e, d), (d, c), (c, a)

37. e b d a c Path Path is a walk where no edge is repeated Simple path: no node is repeated

38. Cycle e base b 3 1 d a 2 c Cycle: a walk from a node (base) to itself Simple cycle: only the base node is repeated

39. 8 base e 7 1 b 4 6 5 d a 2 3 c Euler Tour A cycle that contains each edge once

40. Hamiltonian Cycle 5 base e 1 b 4 d a 2 3 c A simple cycle that contains all nodes

41. e b d a c Finding All Simple Paths f

42. Step 1 e b f d a c (c, a) (c, e)

43. Step 2 e b f d a c (c, a) (c, a), (a, b) (c, e) (c, e), (e, b) (c, e), (e, d)

44. Step 3 e b f d a c (c, a) (c, a), (a, b) (c, e) (c, e), (e, b) (c, e), (e, d) (c, e), (e, d), (d, f) Repeat the same for each starting node

45. Trees root parent leaf child Trees have no cycles

46. root Level 0 Level 1 Height 3 leaf Level 2 Level 3

47. Binary Trees

48. PROOF TECHNIQUES • Proof by induction • Proof by contradiction

49. Induction We have statements P1, P2, P3, … • If we know • for some k that P1, P2, …, Pk are true • for any n >= k that • P1, P2, …, Pn imply Pn+1 • Then • Every Pi is true

50. Proof by Induction • Inductive basis • Find P1, P2, …, Pk which are true • Inductive hypothesis • Let’s assume P1, P2, …, Pn are true, • for any n >= k • Inductive step • Show that Pn+1 is true