intro automata
TRANSCRIPT
7/29/2019 Intro Automata
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CS142
Introduction to Theory of
Computing
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Components of Computer Theory
• Theory of Mathematical Logic– Georg Cantor (1845-1918) – Theory of Sets
– David Hilbert (1862-1943) – Algorithm and Rules ofInference
– Kurt Godel (1906-1978) – Incompleteness Theorem• True statements without any possible proof
– Alonzo Church, Stephen Cole Kleene, Emil Post• That there are problems that no algorithm could solve
– Alan Mathison Turing (1912-1954)
• Developed the concept of a theoretical “universal-algorithm”machine
• Theory of Computer Languages– Noam Chomsky
• Mathematical models for the description of languages
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• Theoretically explore the capabilities
and limitations of computers– Complexity theory
• What makes some problems computationallyhard and others easy?
– Computability theory• What problems can be solved by a computer?
– Automata theory
• How can we mathematically modelcomputation?
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Sets, multisets and sequences
• Set– Order and repetition don’t matter
• {7,4,7,3} = {3,4,7}
• Multiset– Order doesn’t matter, repetition does
• {7,4,7,3} = {3,4,7,7} {3,4,7}
• Sequence
– Order and repetition matter• (7,4,7,3) (3,4,7,7)• Finite sequence of k elements may be called
a k -tuple
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Sets, multisets and sequences
• Set– Subset A B– Proper subset A B– Infinite set { 1, 2, 3, . . . }– Empty set { } or – Venn Diagram
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Set notation
• Union: AB• Intersection: AB
• Complement: A
• Cartesian Product: AB– Also called cross product
• Power set: P
(A)
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Example
• A = {1,2}, B={2,3}, U = {x N|x < 6}– AB =
– AB =
– A =– AB =
– P (A) =
• A = {1,2}, B={2,3}, U = {x N|x < 6}– AB = {1,2,3}
– AB = {2}
– A = {3,4,5}– AB = {(1,2), (1,3), (2,2), (2,3)}
– P (A) = {Ø, {1}, {2}, {1,2}}
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Function
• Mechanism associating each input value with exactly one output value– Domain: set of all possible input values
– Range: set containing all possible outputvaluesf : D R
n f (n )
1
2
3
4
2
4
2
4
f : {1, 2, 3, 4} {2, 4}
f : {1, 2, 3, 4} {1, 2, 3, 4}
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Relation
• Predicate: function whose outputvalue is always either true or false
• Relation: predicate whose domain is
the set A×A×…×A– If domain is all k -tuples of A, the
relation is a k -ary relation on A
– Properties of Relations (Reflexive,Irreflexive, Symmetric, Asymmetric,Transitive)
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Graphs
Nodes
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Graphs
Edges
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Graphs
Degree = 2Degree = 1Degree = 3
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Graphs
Binary tree Subgraph
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Directed graphs
1
5
43
2
{(2,1),(3,1),(4,3),(5,2)}
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Alphabets and strings
• Alphabet: any finite set1 = {1,2,3}
2 = {,,}
• String: finite sequence of symbolsfrom the given alphabet1212123
• Empty string, ε, contains no symbols of the
alphabet
• Language: a set of strings
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Boolean logic
• Conjunction (and) • Disjunction (or )
• Negation (not)
• Exclusive or (xor ) • Equality
• Implication
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Proof techniques
• Construction– Prove a “there exists” statement by
finding the object that exists
• Contradiction– Assume the opposite and find a
contradiction
• Induction– Show true for a base case and show thatif the property holds for the value k ,then it must also hold for the value k + 1
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Find the error in the following proof that 2 = 1.
Consider the equation
a = b
Multiply both sides by a to obtain
a2 = ab
Subtract b2 from both sides to get
a2 –
b2 = ab –
b2 Now factor each side,
(a+b)(a-b) = b(a-b)
and divide each side by (a-b), to get
a+b = b
Finally, let a and b equal 1, which shows that
2 = 1