The Power and the Limits of Computation

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The Power and the Limits of Computation

• Elaine Rich

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The Earliest Digital Computers
1945 ENIAC
Stored 20 10-digit decimal numbers
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The IBM 7090
A dual 7090 system at NASA in about 1962. Could
store 32,768 36-bit words. Thats about .00015
gigabytes. Cost about 3,000,000. or
19,794,000 2005 dollars
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(No Transcript)
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The Earth Simulator
The Earth Simulator (ES) is a project of Japanese
agencies to develop a 40 TFLOPS system for
climate modeling. The ES site is a new location
in an industrial area of Yokohama, an hour drive
west of Tokyo. The facility became operational in
late 2001. Hardware The ES is based on 5,120
(640 8-way nodes) 500 MHz NEC CPUs 8 GFLOPS per
CPU (41 TFLOPS total) 2 GB (4 512 MB FPLRAM
modules) per CPU (10 TB total) shared memory
inside the node 640 640 crossbar switch
between the nodes 16 GB/s inter-node bandwidth
20 kVA power consumption per node
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The iPod Nano
2005 Can store 4 gigabytes (1000 songs). Cost
about 250.
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Compute Power Increases Over Time
From Hans Moravec, Robot Mere Machine to
Transcendent Mind 1998.
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Moores Law
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What Can We Do With All that Power?
Overheard in ACES last week Genomics has
turned biology into an information science.
Is there anything we cant do?
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The Traveling Salesman Problem
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Finding a Solution to the TSP

• Given n cities
• n choices for a starting point.
• n-1 for the next city
• n-2 for the next city
• For a total of n! paths to be considered.

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Finding a Solution to the TSP

• Given n cities
• n choices for a starting point.
• n-1 for the next city
• n-2 for the next city
• For a total of n! paths to be considered.
• We notice that it doesnt matter where we start
  (since we need to make a loop).
• And the cost is the same forward or backward. So
  we can cut the number of paths down to
• (n 1)!/2

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The Growth Rate of n!
n! n(n-1)(n-2)(1)
2 2 11 479001600
3 6 12 6227020800
4 24 13 87178291200
5 120 14 1307674368000
6 720 15 20922789888000
7 5040 16 355687428096000
8 40320 17 6402373705728000
9 362880 18 121645100408832000
10 3628800 19 2432902008176640000
11 39916800 36 3.6 ?1041
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Putting that Rate into Perspective
Speed of light 3 ?108 m/sec Width of a
proton 10-15 m If we perform one operation
in the time light crosses a proton, we
can perform 3 ? 1023 ops/sec Seconds since
the big bang 3 ? 1017 Operations since the big
bang 9 ? 1040 Compared to 36! 3.6 ?1041
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The Post Correspondence Problem
i X Y
1 b bbb
2 babbb ba
3 ba a
4 bbbaa babbb
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The Post Correspondence Problem
i X Y
1 b bbb
2 babbb ba
3 ba a
4 bbbaa babbb
2 X b a b b b Y b a
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The Post Correspondence Problem
i X Y
1 b bbb
2 babbb ba
3 ba a
4 bbbaa babbb
2 1 X b a b b b b Y b a b b b
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The Post Correspondence Problem
i X Y
1 b bbb
2 babbb ba
3 ba a
4 bbbaa babbb
2 1 1 X b a b b b b b Y b a b b b b
b b
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The Post Correspondence Problem
i X Y
1 b bbb
2 babbb ba
3 ba a
4 bbbaa babbb
2 1 1 3 X b a b b b b b b a Y b
a b b b b b b a
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The Post Correspondence Problem
i X Y
1 10 101
2 011 11
3 101 011
i X Y
1 1101 1
2 0110 11
3 1 110
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The Post Correspondence Problem
A program to solve this problem Until a
solution is found do Generate the next candidate
solution. Test it. If it is a solution, halt and
report yes. So, if there are say 4 rows in the
table, well try 1 2 3
4 1,1 1,2 1,3 1,4
1,5 2,1 1,1,1 .
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The Post Correspondence Problem
A program to solve this problem Until a
solution is found do Generate the next candidate
solution. Test it. If it is a solution, halt and
report yes. So, if there are say 4 rows in the
table, well try 1 2 3
4 1,1 1,2 1,3 1,4
1,5 2,1 1,1,1 . But what if
there is no solution?
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A Tiling Problem
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A Tiling Problem
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A Tiling Problem
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A Tiling Problem
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A Tiling Problem
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A Tiling Problem
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A Tiling Problem
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A Tiling Problem
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A Tiling Problem
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A Tiling Problem
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Programs Debug Programs
Given an arbitrary program, can it be guaranteed
to halt?
read n if 2int(n/2) n then print
even else print odd read n result
1 for i 2 to n do result result
i print result
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A Problem That Cannot be Solved in Any Amount of
Time
Given an arbitrary program, can it be guaranteed
to halt?
read n if 2int(n/2) n then print
even else print odd read n result
1 for i 2 to n do result result
i print result
result 1 i 2 2 i 3 6 i 4
24 i 5 120
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Other Kinds of Loops
result 0 count 0 until result gt 100 do
read n count count 1
result resultn print count
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Other Kinds of Loops
result 0 count 0 until result gt 100 do
read n count count 1
result resultn print count Suppose all
the inputs are positive integers.
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Other Kinds of Loops
result 0 count 0 until result gt 100 do
read n count count 1
result resultn print count Suppose some
of the integers are negative.
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The Halting Problem Is Not Solvable

• Suppose that the following program existed
• Halts(program, string) returns
• True if program halts on string
• False otherwise

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The Halting Problem is Not Solvable

• Consider the following program
• Trouble(string)
• If Halts(string, string) then loop forever
• else halt.
• Now we invoke Trouble(ltTroublegt).
• What should Halts(ltTroublegt, ltTroublegt) say? If
  it
• Returns True (Trouble will halt) Trouble
  loops
• Returns False (Trouble will not halt) Trouble
  halts
• So there is no answer that Halts can give that
  does not lead to a contradiction.

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Other Unsolvable Problems

• PCP
• We can encode a ltprogramgt,ltinputgt pair as an
  instance of PCP so that the PCP problem has a
  solution iff ltprogramgt halts on ltinputgt.
• So if PCP were solvable then Halting would be.
• But Halting isnt. So neither is PCP.

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Other Unsolvable Problems

• Tiling
• We can encode a ltprogramgt,ltinputgt pair as an
  instance of a tiling problem so that there is an
  infinite tiling iff ltprogramgt halts on ltinputgt.
• 00010000111000000111110000000000000
  00010000111010000111110000000000000
• 00010000111011000111110000000000000
• So if the tiling problem were solvable then
  Halting would be.
• But Halting isnt. So neither is the tiling
  problem.