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Background for Decidability
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Decidability Truth
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Decidability Truth
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Contents
Background for Decidability
Turing Machines briefly defined
Procedures and Functions
Procedures equivalent to TM's
More functions than procedures
Problems
A problem is a yes-no question
Reducibility
A reducible to B means A at least as hard as B
Encoding and Algorithms
Problem instances as strings, problem as language
Recursive and Recursively Enumerable Languages
Recursively Enumerable Languages
r.e. languages enumerable by TMs
Enumeration means listing only valid strings
Some r.e. languages have no always-halting TM
TMs always halt for
w
in
L
(
M
)
Recursive Languages
Recursive languages = TMs that always halt
Are All Languages Recursively Enumerable?
Some languages not r.e.
Decidability
problems with recursive languages decidable
Trivial Decidability
one string = recursive language
Decidable Doesn't Mean Solved
e.g.
Fermat's conjecture
decidable doesn't mean we know how to solve it
Decidability Theorems
Theorem 1
The complement of a recursive language is recursive
Theorem 2
The union of two recursive languages is recursive
Theorem 3
If a language and it's complement are r.e, they are both recursive
Implications of Theorem 3
Decidability analysis
Strategy for decidability
first principles or reduction
A non-r.e. language:
L
_{d}
definition of
w
_{i}
and
M
_{j}
acceptance table
The language
L
_{d}
is not r.e.
A r.e, non-recursive language:
L
_{u}
The Universal Language
Rice's Theorem
Rice's theorem applies to languages, not to TMs
Post's Correspondence Problem
root
6/10/1998