1' Overview of Automata Theory and Proof Methods

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• 1. Overview of Automata Theory and Proof Methods
• Automata theory studies abstract computing
  devices, or machines
• Turing machines were proposed by A. Turing in
  the 1930s as an abstraction for general-purpose
  computing machines
• Finite automata, which are simpler (less
  powerful) machines, were proposed in the 1940s
  and 1950s to model brain functions
• Formal languages, which are closely related the
  pushdown automata machines, were proposed by
  linguist N. Chomsky in the 1950s
• The study of time and space efficiency of
  programs led to models for tractable and
  intractable problems, e.g. NP-complete and
  NP-hard problems

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• Applications of automata theory include
• compiler construction (lexical analyzer and
  parser)
• string searches (web search engines, string
  search tools used in hard disk analysis by
  computer crime investigators)
• program design (the control structure of program
  modules)
• verification of communication protocols used in
  networking
• Example. A finite automata for the wc (word
  count) program

nChar, nLine, nWord initially 0 whiteCharacter
space, tab, or eol (end-of-line) character eof
end-of-file indicator, not a character
!eof and !whiteCharacter (nChar nWord)
start
eof
eof
outOfWord
inWord
end
whiteCharacter (nChar if eol then nLine)
!eof and !whiteCharacter (nChar)
whiteCharacter (nChar if eol then nLine)
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• Formal Proof Methods
• Statements in the if-then form
• if x ? 4 and x is an integer then 2x ? x2.
• if A ? B and B ? C then A ? C.
• Use definitions and theorems
• Let S ? U, where S is finite and U is infinite.
  Prove U S is infinite. (Definition A set A is
  finite if A n, a natural number (more
  precisely, if there is a bijection between A and
  the set 1, 2, , n). A set is infinite is it
  is not finite. Theorem Let A ? B U and A ? B
  ?. If both A and B are finite, then A B
  U.)
• Statements in the iff (if-and-only-if) form
• Let x be a real number. Then x is an integer
  iff ?x? ?x?.

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• Formal Proof Methods (contd)
• Statements without assumptions
• sin2? cos2? 1.
• R ? (S ? T) (R ? S) ? (R ? T). (There is a
  typographical error on the right-hand side of
  this identity on p. 14 of the text, in two
  places.)
• Proof by contradiction
• (The pigeonhole principle) If n1 pigeons nest
  in n holes, then there exists one hole that has
  at least two pigeons.
• Counterexamples
• (Conjecture) The expression n2 n 41 is a
  prime for all natural number n. (A
  counterexample. Try n 41. The value is 412
  41 41 412, not a prime.)

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• Formal Proof Methods (contd)
• Use of universal (for all, ?) and existential
  (there exists, ?) quantifiers
• (Definition) A set S is infinite if ? n ? N ? A
  ? S such that A n that is, for all natural
  number n the set S contains a subset A whose size
  equals n.
• Given two positive functions f(n) and g(n) that
  are defined on natural numbers n. We say f(n)
  O(g(n)), or f O(g) in short, if ? constants c
  and k such that ? n ? k, f(n) ? c g(n) that is,
  f(n) ? c g(n) for all n that is larger than some
  threshold value k, where c and k are some
  constants (independent of n).
• (DeMorgans Laws) ?(?n P(n)) ? ?n (?P(n)) ?(?n
  P(n)) ? ?n (?P(n)). For example, the negation of
  all apples in the bag are red is there exists a
  non-red apple in the bag the negation of there
  exists a rainy day last month is every day of
  last month is non-rainy.

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• Induction and Recursive Construction (or
  Definition)
• A recursive or inductive definition
  (construction) of non-empty rooted trees
• (Basis, or the base step) A single node is a
  tree, called its root

(Induction, or the recursive step) Suppose T1,
T2, , and Tk are trees, where k ? 1, N is a
single node, we can construct a tree by adding
edges from N to the roots of each of the trees T1
through Tk . Node N is the root of the resulting
tree. (Closure) All trees must be constructed by
using the base step followed by zero or more
recursive steps.
N
T1
T1
Tk
A tree with root N and k subtrees T1 through
Tk
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• Induction and Recursive Construction (contd)
• Every (rooted) tree has one more node than it
  has edges.
• (Proof) We use induction on the number of
  recursive steps used in constructing a tree. We
  use the notations n and e for the numbers of
  nodes and edges, respectively.
• (Basis) Suppose a tree is constructed by the
  base step only. In this case, the tree has one
  single node and no edges thus, n 1 and e 0,
  so n e 1.
• (Induction) We assume every tree satisfies the
  node-edge relation if it is constructed using ? m
  steps. We now consider a tree T which uses m1
  steps. The last step must be connecting k
  subtrees T1 through Tk to a root N. Since each
  subtree uses ? m steps (because the whole tree
  uses m1 steps, one of which is the last step),
  each subtree Ti satisfies the relations ni ei
  1, where ni and ei denote tree Tis numbers of
  nodes and edges, resp. Thus,
• ?1 ? i ?n ni ?1? i ?n (ei 1). Since tree T
  has (1 ?1 ?i ?n ni) nodes and has ?1 ? i ?n (ei
  1) edges, so the node-edge relation is
  satisfied by tree T.

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• Strings and Languages
• (Definitions and notations) An alphabet is a
  finite, non-empty set of symbols. A string (or a
  word) is a finite sequence of symbols from some
  alphabet. The number of symbols in a string w is
  the length of the string, denoted w. Thus, if
  the alphabet ? 0, 1, a string w 011 has a
  length 011 3. Two strings u and v can be
  concatenated, denoted uv, by writing out the
  symbols of u followed by those of v. Thus, uv
  u v . Note that uv ? vu in general. The
  empty string (a string of no symbols) is denoted
  ? (sometimes ?). Note that w 0 iff w ?.
• (Definition related to languages) Let ? be an
  alphabet. Then ?1 ?, ?2 the set of strings of
  length 2 over ?, similarly for ?3, ?4, etc. By
  convention, ?0 ?. The set of all (finite)
  strings over ? is denoted ? ?0 ? ?1 ? ?2 ? ,
  which is an infinite union. Any subset L ? ? is
  called a language over ?. In particular, the
  empty set ? is a language. Note that ? ? ?.

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• Some Sample languages
• Define L 0n1n n ? 0 ?, 01, 0011,
  000111, the set of strings that start with
  any number of 0s followed by the same number of
  1s, including the empty string ?.
• (A recursive or inductive definition of a
  language S)
• (Base step) The empty string ? belongs to S.
• (Recursive step) If a string w belongs to S,
  then the string 0w1 also belongs to S.
• (Closure) No other strings belong to S unless
  they are constructed by applying zero or more
  recursive steps after the base step.
• To prove L S, we prove two parts (1) S ? L and
  (2) L ? S. Proving (1) means proving x ? S
  implies x ? L, which can be done by induction on
  the steps used in constructing string x. To
  prove (2), use induction on n to prove 0n1n ? S
  for all n ? 0.

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• Some Laws Related to Strings and Languages
• Let A, B, and C denote sets of strings over an
  alphabet ?.
• (Definition) The concatenation set AB uv u
  ? A and v ? B that is, AB contains all possible
  concatenations of any string of A followed by any
  string of B. The Kleene star A A0 ? A1 ? A2 ?
  , where A0 ? thus, A contains all strings
  formed by concatenating an arbitrary number of
  strings of A including multiple occurrences of
  the same string in the concatenation.
• The following are commonly used theorems
• if A ? B then AC ? BC and A ? B
• A(B ? C) AB ? AC (B ? C)A BA ? CA
• A(B ? C) ? AB ? AC however, the two sides are
  in general not equal, e.g., when A a, ab, B
  b, C bb, A(B ? C) A? ?, but AB ? AC
  ab, abb?abb, abbb abb.