The Power and the Limits of Computation

1
The Power and the Limits of Computation

• Elaine Rich

2
The Earliest Digital Computers
1945 ENIAC
Stored 20 10-digit decimal numbers
3
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
4
(No Transcript)
5
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
6
The iPod Nano
2005 Can store 4 gigabytes (1000 songs). Cost
about 250.
7
Compute Power Increases Over Time
From Hans Moravec, Robot Mere Machine to
Transcendent Mind 1998.
8
Moores Law
9
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?
10
The Traveling Salesman Problem
11
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.

12
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

13
The Growth Rate of n!
n! n(n-1)(n-2)(1)
14
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
15
The Post Correspondence Problem
16
The Post Correspondence Problem
2 X b a b b b Y b a
17
The Post Correspondence Problem
2 1 X b a b b b b Y b a b b b
18
The Post Correspondence Problem
2 1 1 X b a b b b b b Y b a b b b b
b b
19
The Post Correspondence Problem
2 1 1 3 X b a b b b b b b a Y b
a b b b b b b a
20
The Post Correspondence Problem
21
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 .
22
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?
23
A Tiling Problem
24
A Tiling Problem
25
A Tiling Problem
26
A Tiling Problem
27
A Tiling Problem
28
A Tiling Problem
29
A Tiling Problem
30
A Tiling Problem
31
A Tiling Problem
32
A Tiling Problem
33
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
34
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
35
Other Kinds of Loops
result 0 count 0 until result gt 100 do
read n count count 1
result resultn print count
36
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.
37
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.
38
The Halting Problem Is Not Solvable

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

39
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.

40
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.

41
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.