Module 10

1
Module 10

• Universal Algorithms
• moving beyond one problem at a time
• operating system/general purpose computer

2
Observation

• So far, each program solves one specific problem
• Divisor
• Sorting
• Multiplication
• Language L

3
Universal Problem/Program

• Universal Problem (nonstandard term)
• Input
• Program P
• Input x to program P
• Task
• Compute P(x)
• Univeral Program
• Program which solves universal problem
• Universal Turing machine

4
Example Input

• int main(A6)
  Input
• int i,temp
• for (i1ilt3i) A1 6
• if (Ai gt Ai3) A2 4
• temp Ai3 A3 2
• Ai3 Ai A4 3
• Ai temp A5 5
• A6 1
• for (i1 ilt5 i)
• for (ji1jlt6j)
• if (Aj-1 gt Aj)
• temp Aj
• Aj Aj-1
• Aj-1 temp

5
Organization
Universal Programs Memory

• Program P
• int main(A6)
• int i,temp
• for (i1ilt3i)
• if (Ai gt Ai3)
• temp Ai3
• Ai3 Ai
• Ai temp
• for (i1 ilt5 i)
• for (ji1jlt6j)
• if (Aj-1 gt Aj)
• temp Aj
• Aj Aj-1
• Aj-1 temp

6 4 2 5 3 1
int A6,i,temp
Line 1
6
Description of Universal Program

• Basic Loop
• Find current line of program P
• Execute current line of program P
• Update program Ps memory
• Update program counter
• Return to Top of Loop

7
Past, Present, Future

• Turing came up with the concept of a universal
  program (Universal Turing machine) in the 1930s
• This is well before the invention of the general
  purpose computer
• People were still thinking of computing devices
  as special-purpose devices (calculators, etc.)
• Turing helped move people beyond this narrow
  perspective
• Turing/Von Neumann perspective
• Computers are general purpose/universal
  algorithms
• Focused on computation
• Stand-alone
• Today, we are moving beyond this view
• Computation, communication, cyberspace
• However, results in Turing perspective still
  relevant

8
Halting Problem Revisited

• Halting Problem is half-solvable
• Modified Universal Program (MUP) half-solves H
• Run P on x
• Output yes
• This step only executed if first step halts
• Behavior
• What does MUP do on all yes instances of H?
• What does MUP do on all no inputs of H?

9
Debuggers

• How are debuggers like gdb or ddd related to
  universal programs?
• How do debuggers simplify the debugging process?

10
RE and REC

• We now have a problem that is half-solvable but
  not solvable
• What do we now know about the complement of the
  Halting Problem?
• What additional fact about RE and set complement
  does this prove?

11
RE and REC
All Languages
RE
12
Summary

• Universal Programs
• 1930s, Turing
• Introduces general purpose computing concept
• Not a super intelligent program, merely a precise
  follower of instructions
• Halting Problem half-solvable but not solvable
• RE not closed under set complement