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Theory of Computation: Lecture 02
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Theory of Computation: Lecture 02

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1. Sets ...

1. Sets
2. Tuples
3. Formal Languages
4. Predicates
5. Characteristic Functions
6. Bounded and Unbounded Quantifiers
7. Why Bother with Proofs?
8. Class home page is at http://vkedco.blogspot.com/2011/08/theory-of-computation-home.html

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Theory of Computation: Lecture 02 Presentation Transcript

  • 1. CS 5000: Theory of Computation Lecture 02 Vladimir Kulyukin Department of Computer Science Utah State Universitywww.youtube.com/vkedco www.vkedco.blogspot.com
  • 2. Outline ● Mathematical Preliminaries – Kleene Closure, Sets, Tuples – Predicates and Characteristic Functions – Quantifiers – Mathematical Proofswww.youtube.com/vkedco www.vkedco.blogspot.com
  • 3. Kleene Closures, Sets, Tupleswww.youtube.com/vkedco www.vkedco.blogspot.com
  • 4. Review: Kleene Closures of Alphabets ● The Kleene closure of an alphabet is the set of all strings over it ● If Σ is an alphabet, its Kleene closure is * written as Σ * ● Example: Σ = {a, b}. Σ = the set of all strings consisting of as and bs, including the empty stringwww.youtube.com/vkedco www.vkedco.blogspot.com
  • 5. Review: Sets ∅ or { } are empty sets. { } ≠ {ε }www.youtube.com/vkedco www.vkedco.blogspot.com
  • 6. Review: Set Former Notation {x ∈ {a, b} * } | x ≤ 3 is the set of all strings over { a, b} whose length is 0, 1, 2, or 3. {a b n n } | n ≥ 1 is the set of non - empty strings over { a, b} such that a s precede b s and the number of a s is equal to the number of b s. { xy | x ∈ {a, b}, y ∈ {aa, cc}} is the set of strings where a or b is followed by aa or cc.www.youtube.com/vkedco www.vkedco.blogspot.com
  • 7. Subsets R = S iff (if and only if) R ⊆ S and S ⊆ R. The empty set is a subset of every set : ∅ ⊆ R.www.youtube.com/vkedco www.vkedco.blogspot.com
  • 8. Proper Subsets R ⊂ S iff R ⊆ S and R ≠ S .www.youtube.com/vkedco www.vkedco.blogspot.com
  • 9. Set-Theoretic Equalities R ∩ S - intersection of R and S . R − S = R − ( R ∩ S ). R ∪ S = R ∩ S , i.e. the complement of the union is the intersection of the complements. R ∩ S = R ∪ S , i.e. the complement of the intersection is the union of the complements.www.youtube.com/vkedco www.vkedco.blogspot.com
  • 10. Sets and N-Tuples {a1 , a2 ,..., an } is a set. The order of elements in a set does not matter : {a, b, c} = {a, c, b} = {b, a, c}. The order of elements in a sequences does matter : ( a , b, c ) ≠ ( b, c , a ) ≠ ( c , a , b ) .www.youtube.com/vkedco www.vkedco.blogspot.com
  • 11. Sets and N-tuples Let S1 , S 2 ,..., S n are sets. Then the Cartesian product of these set is defined as follows : S1 × S 2 × ... × S n = {( a1 , a2 ,..., an ) | a1 ∈ S1 , a2 ∈ S 2 ,..., an ∈ S n }.www.youtube.com/vkedco www.vkedco.blogspot.com
  • 12. Predicates and Characteristic Functionswww.youtube.com/vkedco www.vkedco.blogspot.com
  • 13. Predicates A predicate P is a total Boolean - valued function on S such that for each a ∈ S either P (a ) = T or P (a ) = F or P (a ) = 1 or P (a ) = 0.www.youtube.com/vkedco www.vkedco.blogspot.com
  • 14. Predicates Let R be a set.  1 if x ∈ R P( x) =   0 if x ∉ R Then R = {x | P ( x ) = 1} P ( x) is a characteristic function of R.www.youtube.com/vkedco www.vkedco.blogspot.com
  • 15. Quantifierswww.youtube.com/vkedco www.vkedco.blogspot.com
  • 16. Bounded Existential Quantifier: Definition ( ∃ t ) ≤ y P( t , x1 ,..., xn ) is 1 (TRUE) if and only if P( t , x1 ,..., xn ) = 1, for at least one t ∈ [ 0, y ].www.youtube.com/vkedco www.vkedco.blogspot.com
  • 17. Unbounded Existential Quantifier: Definition ( ∃ t ) P( t , x1 ,..., xn ) is 1 (TRUE) if and only if P ( t , x1 ,..., xn ) = 1, for at least one t ∈ N .www.youtube.com/vkedco www.vkedco.blogspot.com
  • 18. Bounded Universal Quantifier: Definition ( ∀ t ) ≤ y P( t , x1 ,..., xn ) is 1 (TRUE) if and only if P( t , x1 ,..., xn ) = 1,0 ≤ t ≤ y.www.youtube.com/vkedco www.vkedco.blogspot.com
  • 19. Unbounded Universal Quantifier: Definition ( ∀ t ) P( t , x1 ,..., xn ) is 1 (TRUE) if and only if P ( t , x1 ,..., xn ) = 1, t ∈ N .www.youtube.com/vkedco www.vkedco.blogspot.com
  • 20. Proofswww.youtube.com/vkedco www.vkedco.blogspot.com
  • 21. Proof Methods ● In CS, there are, broadly speaking, two methods of proving things: formal and empirical ● Formal methods are used in theory of computation, algorithms, operations research, etc. ● Empirical methods are used in many applied branches of CS ● Many R&D projects combine formal and empirical methodswww.youtube.com/vkedco www.vkedco.blogspot.com
  • 22. Mathematical Proofs ● The corner stone of the formal method is the mathematical proof ● Many online and printed CS materials contain proofs ● It is of vital importance for a CS practitioner to read at least some proofs ● The good news is that reading proofs is significantly easier than doing themwww.youtube.com/vkedco www.vkedco.blogspot.com
  • 23. Proof Techniques ● Proof techniques are independent of their subject matter: valid proofs in calculus use the same proof techniques as valid proofs in algorithms or theory of computation ● Common proof techniques can be identified ● Learning to identify common proof techniques will enable you to study many areas of CS independently ● The ability to identify proof techniques is based on your ability to understand how the technique works and when it is likely to be applicablewww.youtube.com/vkedco www.vkedco.blogspot.com
  • 24. Learning to Love the P-Word ● General advice: Do not be afraid of proofs; one can be a mediocre theorem prover but a very good proof reader ● The first step in mastering the art of mathematical proof is to read and do proofs of known facts; do not think of it as a waste of time ● When you read some CS material, do not shy away from it, if it contains proofswww.youtube.com/vkedco www.vkedco.blogspot.com
  • 25. Induction ● The induction method should be seriously considered every time you have a statement of the form: for every n∈ N , n≥k , k ∈ N , k ≥0, some statement S istrue.www.youtube.com/vkedco www.vkedco.blogspot.com
  • 26. Reading ● Ch. 1, Davis, Sigal, Weyuker. Computability, Complexity, and Languages, 2nd Ed.www.youtube.com/vkedco www.vkedco.blogspot.com
  • 27. Feedback Comments, typos, bugs to vladimir dot kulyukin at gmail dot comwww.youtube.com/vkedco www.vkedco.blogspot.com