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Chapter 1 Automata: the Methods & the Madness

Chapter 1 Automata: the Methods & the Madness. Angkor Wat, Cambodia. Outline. 1.1 Why Study Automata Theory? 1.2 Introduction to Formal Proof 1.3 Additional Forms of Proof 1.4 Inductive Proofs 1.5 Central Concepts of Automata Theory. 1.1 Why Study Automata Theory?.

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Chapter 1 Automata: the Methods & the Madness

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  1. Chapter 1 Automata: the Methods & the Madness Angkor Wat, Cambodia

  2. Outline • 1.1 Why Study Automata Theory? • 1.2 Introduction to Formal Proof • 1.3 Additional Forms of Proof • 1.4 Inductive Proofs • 1.5 Central Concepts of Automata Theory

  3. 1.1 Why Study Automata Theory? • 1.1.1 Introduction to Automata • Properties of finite-state systems • Having a finite number of states • Entire history cannot be remembered. • Good to implement with a fixed set of resources • Can be modeled by finite automata (FA)

  4. push start off push e t h n t th on start the then 1.1 Why Study Automata Theory? • Example 1.1 --- an FA modeling an on/off switch • Example 1.2 --- an FA modeling recognition of the keyword “then” in a lexical analyzer • The double circles specify the final or “accepting state.”

  5. 1.1 Why Study Automata Theory? • 1.1.2 Structural Representation • Two other models (not automaton-like) • Grammar --- processing data with a recursive structure • E.g., grammatical rule E  E + E for generating arithmetic expressions • Regular expression --- describing text strings • E.g., UNIX-style regular expression ’[A-Z][a-z]*[ ][A-Z][A-Z]’ describes Ithaca NY, Lafayette IN…

  6. 1.1 Why Study Automata Theory? • 1.1.3 Automata and Complexity • What can a computer do? --- Computability: • Studying problems which can be solved by computer, called decidable problems. • Decidability is the main topic in computability.

  7. 1.1 Why Study Automata Theory? • 1.1.3 Automata and Complexity • What can a computer do efficiently? --- Computational Complexity: • Studying tractable problems solvable with some slowly growing function (like polynomial) of input size, & intractableproblems solvable with fast growing function (like exponential). • Intractability is the main topic of computational complexity.

  8. 1.2 Introduction to Formal Proof • 1.2.0 Concepts --- • Formal proof techniques are indispensable for proving theorems in theory of computation. • Understanding workings of correct programs is equivalent to proving theorems by induction.

  9. 1.2 Introduction to Formal Proof • 1.2.1 Deductive Proofs (演繹法) Given facts (e.g. axioms), previous statements Hypothesis deduced statements  …  conclusion Logic principles • Logic principles: modus ponens (正前律、離斷規則、肯定前件), syllogism (三段論)… • 1.2.2~1.2.4 --- read by yourself

  10. 1.2 Introduction to Formal Proof • Modus ponens (正前律、離斷規則、肯定前件) • An example: • If it rains, the sidewalk will be wet. • It rains now. • So the sidewalk is wet. • Syllogism (三段論) (supplemental) • An example: • All humans will die. • Socrates is a human. • So, Socrates will die.

  11. 1.3 Additional Forms of Proof • 1.3.0 Some more proof techniques --- • Proofs about sets • Proofs by contradiction • Proofs by counterexamples • 1.3.1~1.3.4 --- read by yourself

  12. 1.4 Inductive Proofs (歸納法) • 1.4.1 Induction on Integers • To prove a statement S(n) about an integer n by induction, we do: • Basis --- show S(i) true for a particular basis integeri (0 or 1 usually) • Inductive step--- assume n i (basis integer), show that the statement “if S(n), then S(n + 1)” is true. • 1.4.2~1.4.4 --- read by yourself

  13. 1.5 Central Concepts of Automata Theory • 1.5.0 Three basic concepts • Alphabet --- a set of symbols • Strings --- a sequence of symbols from an alphabet • Language --- a set of strings from the same alphabet

  14. 1.5 Central Concepts of Automata Theory • 1.5.1 Alphabets • Definition --- An alphabet is a finite, nonempty set of symbols. • Conventional notation --- S

  15. 1.5 Central Concepts of Automata Theory • 1.5.1 Alphabets • The term “symbol” is usually undefined. • Examples --- • Binary alphabet S = {0, 1}. • S = {a, b, …, z} …

  16. 1.5 Central Concepts of Automata Theory • 1.5.2 Strings • Definition --- A string (or word) is a finite sequence of symbols from an alphabet. • Example --- • 1011 is a string from alphabet S = {0, 1}

  17. 1.5 Central Concepts of Automata Theory • 1.5.2 Strings • Empty string e--- a string with zero occurrences of symbols • Length |w| of string w --- the number of positions for symbols in w • Examples --- |0111|=4, |e|=0, …

  18. 1.5 Central Concepts of Automata Theory • 1.5.2 Strings • Power of an alphabet Sk --- a set of all strings of length k • Examples --- • givenS= {0, 1}, we have • S0 = {e}, S2 = {00, 01, 10, 11} • Supplemental --- 10 = e, (01)0 = e, …

  19. 1.5 Central Concepts of Automata Theory • 1.5.2 Strings • Set of all strings over S --- denoted as S* • It is not difficult to know that S* = S0 ∪S1 ∪S2 ∪…

  20. 1.5 Central Concepts of Automata Theory • 1.5.2 Strings • S+ = set of nonempty strings from S = S* {e} • Therefore, we have • S+ = S1 ∪S2 ∪S3 ∪… • S* = S+ ∪{e}

  21. 1.5 Central Concepts of Automata Theory • 1.5.2 Strings • Concatenation of two strings x and y --- xy • Example --- • if x = 01101, y = 110, then xy = 01101110, xx = x2= 0110101101, … • e is the identity for concatenation since ew = we = w.

  22. 1.5 Central Concepts of Automata Theory • 1.5.2 Strings (supplemental) • Powerof a string --- • Defined by concatenation --- • xi = xx…x (x concatenated i times) • Defined by recursion --- • x0 = e (by definition) • xi = xxi-1

  23. 1.5 Central Concepts of Automata Theory • 1.5.3 Languages • Definition --- a language is a set of strings all chosen from some S* • If S is an alphabet, and LS*, then L is a language over S.

  24. 1.5 Central Concepts of Automata Theory • 1.5.3 Languages • Examples --- • The set of all legal English words is a language. Why? What is the alphabet here? Answer: the set of all letters • A legal program of C is a language. Why? What is the alphabet here? Answer: a subset of the ASCII characters.

  25. 1.5 Central Concepts of Automata Theory • 1.5.3 Languages • More examples of languages --- • The set of all strings of n 0’s followed by n 1’s for n 0: {e, 01, 0011, 000111, …} • S* is aninfinite language for any alphabet S.

  26. 1.5 Central Concepts of Automata Theory • 1.5.3 Languages • More examples of languages (cont’d) --- •  = the empty language (not the empty string e) is a language over any alphabet. • {e} is a language over any alphabet (consisting of only one string, the empty string e).

  27. 1.5 Central Concepts of Automata Theory • 1.5.3 Languages • Ways to describe languages (1/3) --- • Description by exhaustive listing --- • L1 = {a, ab, abc} (finite language; listed one by one) • L2 = {a, ab, abb, abbb, ...} (infinite language; listed partially) • L3 = L(ab*) (infinite language; expressed by a regular expression)

  28. 1.5 Central Concepts of Automata Theory • 1.5.3 Languages • Ways to define languages (2/3) --- • Description by generic elements --- • L4 = {x | x is over V = {a, b}, begins with a, followed by any number of b, possible none} (note: L4 = L3 = L2)

  29. 1.5 Central Concepts of Automata Theory • 1.5.3 Languages • Ways to define languages (3/3) --- • Description by integer parameters --- • L5 = {abn | n  0} (note: L5 = L4 = L3 = L2) *** bn = power of a symbol

  30. 1.5 Central Concepts of Automata Theory • 1.5.3a Operations on Languages (supplemental)(1/2) • Languages are sets, and operations of sets may be applied to them: (1) union --- A∪B = {a | aA or aB} (2) intersection --- A∩B = {a | aA and aB} (3) difference --- AB = {a | aA and aB}

  31. 1.5 Central Concepts of Automata Theory • 1.5.3a Operations on Languages (supplemental)(2/2) • Languages are sets, and operations of sets may be applied to them: (4) product --- AB = {(a, b) | aA and bB} (5) complement --- Ā= {a | aU and aA} (6) power set --- 2A = {B | BA} Note:Uis the universal set, like S*which is the closure of an alphabet

  32. 1.5 Central Concepts of Automata Theory • 1.5.3b More Operations on Languages (supplemental)(1/2) • Concatenation of two languages L1 and L2 --- • L1L2 = {x1x2 | x1 L1 and x2 L2} • Power of a language L --- • Defined directly --- • Lk = {x1x2…xk | x1, x2, …, xkL} • Defined by recursion --- • L0 = {e} • Li= LLi-1

  33. 1.5 Central Concepts of Automata Theory • 1.5.3b More Operations on Languages (supplemental)(2/2) • Closure of language L --- • L* = = L0 ∪L1 ∪L2 ∪… • Positive closure of a language L --- • L+ = L1 ∪L2 ∪… • It can be shown that L* L0 = L* – {e} when e L

  34. 1.5 Central Concepts of Automata Theory • 1.5.4 Problems • A problem in automata theory --- a question of deciding whether a given string is a member of some particular language. • That is, if S is an alphabet, and L is a language over S, the problem L is: given a string w in S*, decide ifw  L or not.

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