Chapter 1, Part 3
Regular Expression
The regular expressions are equivalent to the nite automata.
CSC527, Chapter 1, Part 3 c 2012 Mitsunori Ogihara
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Regular Expression
An expression R is a regular expression if R is
1.
2.
3.
4.
5.
6.
a for some a in som
Chapter 0: Fundamental Concepts
Fundamental Concepts
CSC527, Chapter 0 c 2012 Mitsunori Ogihara
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Sets
a S: a is an element of S; a is a member of S.
Example 1 cfw_1, 2, 3.
S T : S is a subset of T ; S is contained in T . This means
that every member of
Chapter 4, Part 2
The Halting Problem
CSC527 Chapter 4, Part 2 c 2012 Mitsunori Ogihara
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The Halting Problem
Dene ATM = cfw_ M, w | M is a Turing machine and accepts w.
Theorem. ATM is not decidable.
CSC527 Chapter 4, Part 2 c 2012 Mitsunori Ogihara
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Th
Chapter 4, Part 1
Decidability
CSC527 Chapter 4, Part 1 c 2012 Mitsunori Ogihara
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Decidable Problems About Regular Languages
The Acceptance Problem for DFA
Dene ADFA to be:
cfw_ B, w | B is a DFA that accepts input string w.
Theorem. ADFA is decidable.
P
Chapter 2, Part 3
Pushdown Automata and CFLs Are
Equivalent
CSC527, Chapter 2, Part 3 c 2012 Mitsunori Ogihara
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Properties of Context-Free Languages
Theorem. The context-free languages are closed under
union, concatenation, and star.
Proof Let S1 and S2
Chapter 2, Part 1
Context-free Languages
CSC527, Chapter 2, Part 1 c 2012 Mitsunori Ogihara
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Context-free Languages
A context-free grammar is a 4-tuple G = (V, , R, S). Here
1. V is the set of variables (or nonterminals),
2. is the set of terminals,
3. R
Chapter 2, Part 2
Pushdown Automata
The machine model for the context-free language.
CSC527, Chapter 2, Part 2 c 2012 Mitsunori Ogihara
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Pushdown Automata
A pushdown automaton is an NFA with a last-in, rst-out storage
device called stack.
CSC527, Chapter
Chapter 1, Part4
Nonregular Languages
How can we show that a language is not regular?
CSC527, Chapter 1, Part 4 c 2012 Mitsunori Ogihara
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The Pumping Lemma
Theorem. (Pumping Lemma) Let L be an arbitrary regular
language. Then there exists a positive inte
Chapter 3, Part 1
Computability Theory
CSC527, Chapter 3, Part 1 c 2012 Mitsunori Ogihara
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Turing Machines
A Turing machine is a step-wise computing device consisting of:
1. an innitely long tape divided into tape squares (or tape
cells),
2. a head for s
Chapter 2, Part 4
Non-context-free Languages
How do we show something is not context free?
CSC527, Chapter 2, Part 4 c 2012 Mitsunori Ogihara
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The Pumping Lemma
Theorem. (Pumping Lemma) Let L be context free. There
exists a positive integer p with the fo
Chapter 6
Advanced Topics in Computability Theory
CSC527, Chapter 6 c 2010 Mitsunori Ogihara
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The Recursion Theorem
A self-reproducing machine SELF is a machine that disregards its
input and produces its description on the input.
We will contruct such a
Chapter 5, Part 2
PCP and Mapping Reducibility
CSC527, Chapter 5, Part 2 c 2012 Mitsunori Ogihara
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Post Correspondence Problem (PCP)
We have a collection of domino pieces, each of which has a string
in the top half and a string in the bottom half. Suppos
Chapter 5, Part 1
Reducibility
CS527, Chapter 5, Part 1 c 2012 Mitsunori Ogihara
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The Halting Problem
Based on undecidability of one language, A, undecidability of
another language, B, can be shown
CS527, Chapter 5, Part 1 c 2012 Mitsunori Ogihara
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The