Lecture 10

1. Lecture 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(A[6]) { Input int i,temp; for (i=1;i<=3;i++) A[1] = 6 if (A[i] > A[i+3]) { A[2] =4 temp = A[i+3]; A[3] = 2 A[i+3] = A[i]; A[4] = 5 A[i] = temp; A[5] = 3 } A[6] = 1 for (i=1; i<=5; i++) for (j=i+1;j<=6;j++) if (A[j-1] > A[j]) { temp = A[j]; A[j] = A[j-1]; A[j-1] = temp; } }

5. Questions • Which of the following two people is most like a universal program? • Brilliant, creative scientist • Efficient, obedient soldier • How does a universal program work? • What is a universal program in real life?

6. Program P Program P’s Memory Program Counter Organization Universal Program’s Memory Program P int main(A[6]){ int i,temp; for (i=1;i<=3;i++) if (A[i] > A[i+3]) { temp = A[i+3]; A[i+3] = A[i]; A[i] = temp; } for (i=1; i<=5; i++) for (j=i+1;j<=6;j++) if (A[j-1] > A[j]) { temp = A[j]; A[j] = A[j-1]; A[j-1] = temp; } } 6 4 2 5 3 1 int A[6],i,temp; Line 1

7. Description of Universal Program • Basic Loop • Find current line of program P • Execute current line of program P • Update program P’s memory • Update program counter • Return to Top of Loop

8. Past, Present, Future • Turing came up with the concept of a universal program (Universal Turing machine) in the 1930’s • 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

9. 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 • MUP halts and accepts all yes instances of H • MUP loops on all no instances of H • Stuck in “Run P on x” step

10. Debuggers • How do debugger’s like gdb or ddd work? • They basically operate like a universal program • Key properties of debugger’s to aid in debugging • Breakpoints • Display memory commands

11. 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? • It is not even half-solvable • This proves RE is not closed under the set complement operation • H is counterexample

12. H REC Hc RE and REC All Languages RE

13. Summary • Universal Programs • 1930’s, 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