1. 1 CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 14 Mälardalen University 2006

2. 2 Rices sats Om  är en mängd av Turing-accepterbara språk som innehåller något men inte alla sådana språk, så kan ingen TM avgöra för ett godtyckligt Turing-accepterbart språk L om L tillhör  eller ej. (Varje icke-trivial egenskap av Turing- accepterbara språk är oavgörbar.)

3. 3 Avgörbart? Motivera! a) ”Är L(M) oändlig?” Givet att M är en godtycklig DFA. b) ”Är L(M) oändlig?” Givet att M är en godtycklig TM. Svar a) AVGÖRBART! Man behöver bara kontrollera om M innehåller någon slinga på väg till acceptans, vilket kan göras i ändligt många steg. Se Sallings bok uppgift 7.2. b) OAVGÖRBART! Följer av Rices sats, om man väljer  som mängden av alla oändliga Turingaccepterbara språk, eftersom denna mängd är icketrivial. Exempel

4. 4 Recursion In computer programming, recursion is related to performing computations in a loop.

5. 5 Recursion in Problem Modelling Reducing the complexity by breaking up computational sequences into its simplest forms. synthesizing components into more complex objects by replicating simple component sequences over and over again.

6. 6 "A reduction is a way of converting one problem into another problem in such a way that a solution to the second problem can be used to solve the first problem." Michael Sipser, Introduction to the Theory of Computation

7. 7 Recursion can be seen as concept of well- defined self-reference. We use recursion frequently. Consider, for example, the following hypothetical “definition of a Jew”. I found it on web, as a joke. “Somebody is a Jew if she is Abraham's wife Sarah, or if his or her mother is a Jew.” (My digression: I wonder what about Abraham?)

8. 8 So if I want to know if I am a Jew, I look at this definition. I'm not Sarah, so I need to know whether my mother is a Jew. How do I know about my mother? I look at the definition again. She isn't Sarah either, so I ask about her mother. I keep going back through the generations - recursively.

9. 9 Self-referential definitions can be dangerous if we're not careful to avoid circularity. The definition ''A rose is a rose'‘* just doesn't cut it. This is why our definition of recursion includes the word well-defined. *Know Gertrude Stein? '' A rose is a rose is a rose''

10. 10 We can write pseudocode to determine whether somebody is an immigrant: FUNCTION isAnImmigrant(person): IF person immigrated herself, THEN: return true ELSE: return isAnImmigrant(person's parent) END IF This is a recursive function, since it uses itself to compute its own value. [According to some authors ( Rudbeckius ) Adam and Eve were Swedish.] Yet another recursive definition: an immigrant…

11. 11 Functions From math classes, we have seen many ways of defining and combining numerical functions. –Inversef -1 –Compositionf ◦ g –Derivativesf´(x), f´´(x), … –Iterationf 1 (x), f 2 (x), f 3 (x), f 4 (x), … –…

12. 12 Functions Look at what happens when we use only some of these. –How can we define standard interesting functions? –How do these relate to e.g. TM computations? We have seen TMs as functions. They are cumbersome! As alternative, look at a more intuitive definition of functions.

13. 13 Notation For brevity, limit to functions on natural numbers N = {0,1,2,…} Notation will also use n-tuples of numbers (m 1, …, m n )

14. 14 Natural Numbers Start with standard recursive definition of natural numbers (remember Peano?): A natural number is either 0, or successor(n), where n is a natural number.

15. 15 What is a recurrence? A recurrence is a well-defined mathematical function written in terms of itself. It is a mathematical function defined recursively.

16. 16 Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,... The first two numbers of the sequence are both 1, while each succeeding number is the sum of the two numbers before it. (We arrived at 55 as the tenth number, since it is the sum of 21 and 34, the eighth and ninth numbers.)

17. 17 F is called a recurrence, since it is defined in terms of itself evaluated at other values. F(0) = 1F(1) = 1 (base cases) F(n) = F(n - 1) + F(n - 2)

18. 18 A recursive process is one in which objects are defined in terms of other objects of the same type. Using some sort of recurrence relation*, the entire class of objects can then be built up from a few initial values and a small number of rules. Recursion & Recurrence (*Recurrence is a mathematical function defined recursively.)

19. 19 Computable Function Any computable function can be programmed using while-loops (i.e., "while something is true, do something else"). For-loops (which have a fixed iteration limit) are a special case of while-loops. Computable functions could also be coded using a combination of for- and while-loops.

20. 20 Total Function A function defined for all possible input values. Primitive Recursive Function A function which can be implemented using only for-loops.

21. 21 An example function DomainRange

22. 22 We need a way to define functions. We need a set of basic functions.

23. 23 Zero function: Successor function: Projection functions: Basic Primitive Recursive Functions

24. 24 Building functions Composition

25. 25 Composition, Generally Given g 1 : N k  N. g m : N k  N f : N m  N h(n 1,…,n k ) = f(g 1 (n 1,…,n k ), …, g m (n 1,…,n k )) h = f ◦ (g 1,…,g m )Alternate notation. Create h : N k  N

26. 26 Primitive Recursion “Template” N.B. For primitive recursive functions recursion in only one argument.

27. 27 Any function built from the basic primitive recursive functions is called Primitive Recursive Function.

28. 28 Basic Primitive Zero function (a constant function) Example

29. 29 Basic Primitive Identity function Recursive definition

30. 30 Basic Primitive Successor function

31. 31 Using Basic Primitive Zero function and a Successor function we can construct Constant functions etc..

32. 32 Example

33. 33 A Primitive Recursive Function (projection) (successor function)

34. 34 Example

35. 35 Example

36. 36 Basic Primitive Predecessor function

37. 37 Predecessor Predecessor is a primitive recursive function with no direct self-reference.

38. 38 Subtraction

39. 39 Example

40. 40 A Primitive Recursive Function

41. 41 Example

42. 42 A Primitive Recursive Function

43. 43 Example

44. 44 Primitive Recursion: Logic A predicate (Boolean function) with output in the set {0,1} which is interpreted as {yes, no}, can be used to define standard functions. –Logical connectives , , , , … –Numeric comparisons=, <, , … –Bounded existential quantification  i  n, f(i) –Bounded universal quantification  i  n, f(i) –Bounded minimizationmin i i  n, f(i) where result = 0 if f(i) never true within bounds.

45. 45 Recursive Predicates and

46. 46 More Recursive Predicates

47. 47 More Recursive Predicates...

48. 48 Example Recursive predicates can combine into powerful functions. What does this compute? Tests primality. ???(n) =  i  n,  j  n, ((i=1  j=n)  (j=1  i=n)  i  j  n)

49. 49 prime(n) = n  2   i 0) mod(m,n) = if n>0 then (min i i  m, div(m,n)  n+i=m) else 0 div(m,n) = if n>0 then (min i i  m, (i+1)  n>m) else 0 Example Another version of prime(n)

50. 50 Function

51. 51 our construction primitive recursive template

52. 52 Division example: x/4

53. 53 Division as Primitive Recursion

54. 54 Division example: x/4

55. 55 Division as Primitive Recursion

56. 56 Recursive Predicate

57. 57 Recursive Predicate

58. 58 Theorem The set of primitive recursive functions is countable. Proof Each primitive recursive function can be encoded as a string. Enumerate all strings in proper order. Check if a string is a function.

59. 59 There is a function that is not primitive recursive. Proof Enumerate the primitive recursive functions Theorem

60. 60 Define function differs from every is not primitive recursive END OF PROOF

61. 61 A specific function that is not primitive recursive: Ackermann’s function: Grows very fast, faster than any primitive recursive function

62. 62 The Ackermann function is the simplest example of a well defined total function which is computable but not primitive recursive, providing a counterexample to the belief in the early 1900s that every computable function was also primitive recursive.

63. 63 Recursive Functions Ackermann’s function is a Recursive Function

64. 64 Primitive recursive functions Recursive Functions

65. 65 Primitive Recursion: Extended Example Needs following building blocks: –constants –addition –multiplication –exponentiation –subtraction A polynomial function: f(x,y) = 3x 7 + xy – 7y 2.

66. 66 Addition add(m,n) = m+n add(0,n)= add(m+1,n)= n succ(add(m,n)) Multiplication: mult(m,n) = m  n mult(0,n)= mult(m+1,n)= 0 add(mult(m,n),n)

67. 67 Exponentiation: exp(m,n) = n m exp(0,n)= exp(m+1,n)= 1 mult(exp(m,n),n) = one(n) Subtraction sub(m,n) = m-n sub(0,n)= sub(m+1,n)= 0=zero(n) succ(sub(m,n))

68. 68 Primitive Recursion: Extended Example f(x,y) = (3x 7 + xy) - 7y 2 f = sub◦ (add ◦ (f 1,f 2 ), f 3 ) f 1 (x,y) = mult(3,exp(7,x))f 1 = mult ◦ (three, exp ◦ (seven)) f 2 (x,y) = mult(x,y)f 2 = mult f 3 (x,y) = mult(7,exp(2,y))f 3 = mult ◦ (seven, exp ◦ (two)) f(x,y) = sub(add(f 1 (x,y),f 2 (x,y)),f 3 (x,y))

69. 69 Primitive Recursion All primitive recursive functions are total. I.e., they are defined for all values. Primitive recursion lack some interesting functions. “True” subtraction– when using natural numbers. “True” division– undefined when divisor is 0. Trigonometric functions– undefined for some values. …

70. 70 Partial Recursive A function is partial recursive  it can be defined by the previous constructions. A function is recursive  it is partial recursive and total.

71. 71 Division: div(m,n) =  m  n  div(m,n) = min i, sub(succ(m),add(mult(i,n),n)) = 0 div(m,n) = minimum i such that i  mni  mn i  n  m-(n-1) i  n+n  m+1 (m+1) – (i  n+n)  0 (m+1)  (i  n+n) = 0 Example

72. 72 Relations Among Function Classes Functions  TMs –Define TMs in terms of the function formers. –Straightforward, but long. TMs  Functions –Define functions where subcomputations encode TM behavior. –Simple encoding scheme. –Straightforward, but very messy. partial recursive = recognizable recursive = decidable primitive recursive

73. 73 More Examples of Primitive Recursion A recursive function is a function that calls itself (by using its own name within its function body). Even

74. 74 Factorials

75. 75 Is a number a square? Forward recursion (  -recursion)

76. 76

77. 77

78. 78 Midterm Exam 3 Restriction-Free Languages Place: the LAMBDA examination hall Time: on Tuesday 2004-05-30, 10:15-12:00 It is OPEN BOOK. (This means you are allowed to bring in one book of your choice.) It will cover Turing Machines/Restriction-free Languages). You will have the two hours to do the test.