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Theory of Computation: Functions, Closure, Decision Problems

Understand functions, closure properties, and decision problems in theory of computation. Explore concepts such as cardinality, sets, relations, and equivalence relations. Discuss questions from the syllabus, reading assignments, and more.

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Theory of Computation: Functions, Closure, Decision Problems

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  1. MA/CSSE 474 Theory of Computation Functions, Closure, Decision Problems

  2. Your Questions? • Syllabus • Yesterday's discussion • Reading Assignments • HW1 or HW2 • Anything else Must not be a FSM

  3. Responses to Reading Quiz 1 • From #4:℘(∅) = { ∅ } (not ℘(∅) = ∅)What is ℘(℘(∅))? • From #4: {a, b} X {1, 2, 3} X ∅ = ∅ • #10: (representing {1, 4, 9, 16, 25, 36, …} in the form: {xA : P(x)} {x ∊ ℕ : x>0 ∧ ∃y∊ℕ (y*y = x)}Why not {x ∊ ℕ : x>0 ∧ sqrt(x) ∊ ℕ} ? • From #15: x ℕ (y ℕ (y < x)).Why is this not satisfiable? (e, g, by x=3, y=2)

  4. Responses to Reading Quiz 1 #16: Let ℕ be the set of nonnegative integers. Let A be the set of nonnegative integers x such that x3 0. Show that |ℕ| = |A|.Define a function f : ℕ →A by f(n) = 3n. f is one-to-one: if f(n) = f(m), then 3n = 3m, so m=n.f is onto: Let k ∊ A. Then k = 3m for some m ∊ ℕ. So k = f(m).

  5. Responses to Reading Quiz 1 #18: Prove by induction:n>0 (n!  2n-1). Why is the following "proof" of the induction step shaky at best, perhaps wrong? (n+1)!  2n what we're trying to show (n+1)n!  2(2n-1)definitions of ! And exponents (n+1) 2 induction hypothesis (n!  2n-1) Since n is at least 1, this statement is true, therefore (n+1)!  2n is true.

  6. Relations on Strings: • aaa is a substring of aaabbbaaa • aaaaaa is not a substring of aaabbbaaa • aaa is a proper substringof aaabbbaaa • Every string is a substring of itself. • is a substring of every string. • s is a prefix of tiff: x * (t = sx). • s is a proper prefixof tiff: s is a prefix of t and s t. • Examples: • The prefixes of abba are: , a, ab, abb, abba. • The proper prefixes of abba are: , a, ab, abb. • Every string is a prefix of itself. •  is a prefix of every string. • s is a suffix of tiff: x * (t = xs) then continue as above: proper suffix, self, 

  7. Defining a Language A language is a (finite or infinite) set of strings over a finite alphabet . Examples: Let  = {a, b} Some languages over : , {}, {a, b}, {, a, aa, aaa, aaaa, aaaaa} If  is nonempty, the language * contains an infinite number of strings, including: , a, b, ab, ababaa.

  8. Example Language Definitions 1. L = {x {a, b}* : all a’s precede all b’s} , a, aa, aabbb, and bb are in L. aba, ba, and abc are not in L. 2. L = {x : y {a, b}* : x = ya} Simple English description: 3. L = {x#y: x, y {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}* and, when x and y are viewed as the decimal representations of natural numbers, square(x) = y}. Examples: 3#9, 12#144, 3#8, 12, 12#12#12, # • L = {an : n 0} uses replication, simpler description of L? • L = Ø = { } • L = {ε} Are the last two different languages?

  9. Natural Languages are Tricky L = {w: w is a sentence in English}. Examples: Kerry hit the ball. Colorless green ideas sleep furiously. The window needs fixed. Ball the Stacy hit blue.

  10. A Halting Problem Language • L = {w: w is a Java program that, when given any finite input string, is guaranteed to halt}. • Is this language well specified? • Can we decide which strings L contains?

  11. Languages and Prefixes What are the following languages: L = {w {a, b}*: no prefix of w contains b} L = {w {a, b}*: no prefix of w starts with a} L = {w {a, b}*: every prefix of w starts with a}

  12. Sets and Relations

  13. Cardinality of a set. The cardinality of every set we will consider is one of the following : • a natural number (if S is finite), • “countably infinite” (if S has the same number of elements as there are integers), or • “uncountably infinite” (if S has more elements than there are integers).

  14. Sets of Sets • The power set of S is the set of all subsets of S. Let S = {1, 2, 3}. Then: P(S) = {, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. •  P(S) is a partition of a set S iff: • Every element of  is nonempty, • Every pair of elements of  is disjoint , and • the union of all the elements of  equals S. Some partitions of S: {{1}, {2, 3}} or {{1, 3}, {2}} or {{1, 2, 3}}. How many different partitions of S?

  15. Closure • A set S is closed under binary operation opiff x,yS ( x op y  S) ,closed under unary function f iff xS (f(x)  S) • Examples • ℕ+ (the set of all positive integers) is closed under addition and multiplication but not negation, subtraction, or division. • What is the closure of N+ under subtraction? Under division? • The set of all finite sets is closed under union and intersection. Closed under infinite union? If S is not closed under unary function f, a closureof S is a set S' such that S is a subset of S' S' is closed under f No proper subset of S' contains S and is closed under f

  16. Equivalence Relations A relation on a set A is any set of ordered pairs of elements of A. A relation RAA is an equivalence relation iff it is: • reflexive, • symmetric, and • transitive. Examples of equivalence relations: • Equality • Lives-at-Same-Address-As • Same-Length-As • Contains the same number of a's as Show that ≡₃ is an equivalence relation

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