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Math Preliminaries: Alphabets, Strings, Languages, Automata, Numbers, Sets, Set Formers, Proofs

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- 1. Theory of Computation Mathematical Preliminaries Vladimir Kulyukin Department of Computer Science Utah State University www.vkedco.blogspot.com
- 2. Outline ● What is Theory of Computation? ● Alphabets, Strings, Languages ● Numbers, Sets, Set Formers ● Proofs www.vkedco.blogspot.com
- 3. What is Theory of Computation? www.vkedco.blogspot.com
- 4. What is Theory of Computation? ● Theoretical Computer Science can be broadly divided into algorithms and computability (aka theory of computation) ● The field of algorithms answers the question – how something can be computed? ● The field of computability answers the question – can something be computed? www.vkedco.blogspot.com
- 5. Basic Methodology ● Abstraction of hardware details ● Focus on what can be solved, not on how it can be solved ● Problem analysis in terms of devices (aka automata) and inputs (aka strings, languages) www.vkedco.blogspot.com
- 6. Basic Methodology ● Abstraction of hardware details ● Focus on what can be solved, not on how it can be solved ● Problem analysis in terms of devices (aka automata) and inputs (aka strings, languages) www.vkedco.blogspot.com
- 7. Abstraction of Hardware Details Junun Mark III Robot Smartphone Personal Computer While these devices have very different hardware, they have the same computational model www.vkedco.blogspot.com
- 8. Focus on What, Not How ● Problem: sorting a sequence of numbers from smallest to highest ● Algorithmic answer: merge sort, heap sort, quick sort, etc ● Computability answer: sorting is primitive recursive www.vkedco.blogspot.com
- 9. Automata & Languages ● Finite State Automata/Regular Expressions – Regular Languages ● Push Down Automata/Stack Machines – Context-Free Languages ● Linear Bounded Automata – Context Sensitive Languages ● Turing Machines/Universal Programs – Recursively Enumerable Languages www.vkedco.blogspot.com
- 10. Automata & Languages www.vkedco.blogspot.com
- 11. Alphabets, Languages, Strings www.vkedco.blogspot.com
- 12. Alphabets ● An alphabet is a finite set of symbols ● The Greek letter Σ is typically used to denote an alphabet ● Examples: Σ1 = {a, b}, Σ2 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. ● The symbols in the alphabet do not have any meaning in and of themselves www.vkedco.blogspot.com
- 13. Strings ● A string is a finite sequence of symbols ● ε is the empty string; ε is not a symbol in any alphabet; it denotes the string with zero symbols ● In the book (Davis et al.), the empty string is denoted as 0 ● In formal language theory, strings are typically written without quotation marks: aab and 010001 instead of “aab” and “010001” www.vkedco.blogspot.com
- 14. Strings ● The length of a string is the number of symbols/characters in it ● The length of a string is denoted with a pair of matching vertical lines around it ● Examples: – if x = aab, then |x| = 3; – if x = ε, then |x| = 0 www.vkedco.blogspot.com
- 15. String Concatenation ● The concatenation of two strings x and y is the strings containing the symbols of x followed by the symbols of y ● Examples: if x = ab and y = 100, then xy = ab100; if x = ε and y = abc, then xy = abc ● For any string, xε = εx = x www.vkedco.blogspot.com
- 16. Power Notation in String Concatenation ● When a natural number n is used as an exponent on a string, it denotes the concatenation of that string with itself n times ● Examples: – x0 = ε – x1 = x – x2 = xx – (ab)3 = ababab www.vkedco.blogspot.com
- 17. Languages ● A language is a set strings over a alphabet ● Note that we can define multiple languages over the same alphabet ● Example: Σ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. L1 = the set of all strings over Σ that end in 0 (e.g., 0, 110, 213450, etc.); L2 = the set of all strings over Σ that end in 1 (e.g., 1, 01, 0001, 91, etc.) www.vkedco.blogspot.com
- 18. 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 a's and b's, including the empty string www.vkedco.blogspot.com
- 19. Numbers, Sets, Set Formers www.vkedco.blogspot.com
- 20. Natural Numbers ● N is a set of natural numbers. ● N includes 0, 1, 2, 3, … ● Some texts exclude 0 from the set of natural numbers, but we will keep it in ● In the texts, the word number refers to natural number www.vkedco.blogspot.com
- 21. Sets { } { } { }ε≠ ∅ sets.emptyareor www.vkedco.blogspot.com
- 22. Set Former Notation { }{ } { } { } { } { } { }{ } .orbyfollowedisor wherestringsofsettheis,,,| s.'ofnumberthe toequaliss'ofnumbertheands'precedes'that such,overstringsempty-nonofsettheis1| 3.or2,1,0,islengthwhose ,overstringsallofsettheis3|, * ccaaba ccaaybaxxy b aba banba baxbax nn ∈∈ ≥ ≤∈ www.vkedco.blogspot.com
- 23. Subsets . :seteveryofsubsetaissetemptyThe .andif)onlyand(ififf R RSSRSR ⊆∅ ⊆⊆= www.vkedco.blogspot.com
- 24. Proper Subsets .andiff SRSRSR ≠⊆⊂ www.vkedco.blogspot.com
- 25. Set-Theoretic Equalities s.complementtheofuniontheis onintersectitheofcomplementthei.e., s.complementtheofonintersectithe isuniontheofcomplementthei.e., ).( .andofonintersecti- SRSR SRSR SRRSR SRSR ∪=∩ ∩=∪ ∩−=− ∩ www.vkedco.blogspot.com
- 26. Sets and N-Tuples ( ) ( ) ( ).,,,,,, :matterdoessequencesainelementsoforderThe }.,,{},,{},,{ :matternotdoessetainelementsoforderThe set.ais},...,,{ 21 bacacbcba cabbcacba aaa n ≠≠ == www.vkedco.blogspot.com
- 27. Sets and N-tuples }.,...,,|),...,,{( ... :followsasdefinedissettheseofproduct CartesianThen thesets.are,...,,Let 221121 21 21 nnn n n SaSaSaaaa SSS SSS ∈∈∈ =××× www.vkedco.blogspot.com
- 28. Predicates .0)(or1)(or)(or)(either eachforsuch thatonfunction valued-BooleantotalaispredicateA ==== ∈ aPaPFaPTaP SaS P www.vkedco.blogspot.com
- 29. Predicates R.offunctionsticcharacteriais)( }1)(|{ Then if0 if1 )( set.abeLet xP xPxR Rx Rx xP R == ∉ ∈ = www.vkedco.blogspot.com
- 30. Proofs www.vkedco.blogspot.com
- 31. 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 methods www.vkedco.blogspot.com
- 32. 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 them www.vkedco.blogspot.com
- 33. 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 applicable ● In CS, a prominent proof technique is induction www.vkedco.blogspot.com
- 34. 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 proofs www.vkedco.blogspot.com

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