ComputabilityUncomputability - PowerPoint PPT Presentation
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ComputabilityUncomputability
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Computability/Uncomputability. Is every number-theoretic function Turing computable? ... Isomorphic Machines Will Be Identified. TMn Is Finite ... – PowerPoint PPT presentation
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Title: ComputabilityUncomputability
1
Computability/Uncomputability
- Is every number-theoretic function Turing
computable? - No (see below).
- Does this mean that there are uncomputable
functions? (That depends.)
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TMn (with n ³ 0)
- TMn class of deterministic, single-tape,
- (n 1)-state Turing machines with input
alphabet ? and tape alphabet 1 - Machines all start scanning a blank on a
completely blank tape.
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Isomorphic Machines Will Be Identified
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TMn Is Finite
- Â Â Within any state diagram with n 1 nodes
there are only 2 ? 4 8 possible arc labels and,
from any given state, only n 1 states to which
such an arc may be directed.
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The Big Question
- If started scanning a square on a completely
blank tape, does M halt and, if so, how many 1s
are on the tape when M halts? - We call this number the productivity of M
- If M never halts, then productivity 0.
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?(n)
- ?(n) the maximum productivity of any of member
of TMn - Busy Beaver function
- Unary number-theoretic function
- Total (Why?)
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?(2) gt 2 and, in general, ?(n) gt n
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Monotone Increasing
- if n ? m, then ?(n) ? ?(m)
- Suppose that M is a most productive (n 1)-state
machine - M halts and hence has some state q with outdegree
lt 2 - Add a length-(m n) path of no-ops to obtain
new (m 1)-state machine with same productivity
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?(n) Is Not Turing-Computable.
- Rado 1962
- Example 2.7.1 Chris Nielsens Busy Beaver shows
that ?(5) ? 21
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