15 453 flac lecture 19 turing machines and real life reductions l.
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  1. 15-453 FLACLecture 19Turing Machines and Real Life*Reductions Mihai Budiu March 3, 2000

  2. A Turing Machine solves one problem • We must distinguish between • The set of all strings • A problem (e.g. the language to be accepted) • A problem instance (e.g. a word from the language) • A TM solves each instance presented as input S* L w FLAC: Reductions

  3. First computers: custom computing machines Input tape (read only) Control Work tape (memory) 1950 -- Eniac: the control is hardwired manually for each problem. Output tape (write only) FLAC: Reductions

  4. Input tape (read only) Control Work tape (memory) Output tape (write only) TMs can be described using text • Program • n states • s letters for alphabet • transition function: • d(q0,a) = (q1, b, L) • d(q0,b) = (q1, a, R) • …. Consequence: There is a countable number of Turing Machines FLAC: Reductions

  5. There is a TM which can simulate any other TM(Alan Turing, 1930) Input tape Program Universal TM Work tape (memory) Output tape Interpreter FLAC: Reductions

  6. The Digital Computer: a UTM Machine code Input Universal TM Memory + disk Output Machine-code Interpreter FLAC: Reductions

  7. Random Access Memories x x Row access Sequential access tape Column access Address FLAC: Reductions

  8. Turing machines as functions S* w Input word TM f S* Output word f(w) A function f is a computable function if a TM with word w as input halts with f(w) as output. FLAC: Reductions

  9. Computable functions • Many functions encountered in everyday life are computable: • Addition : binary strings * binary strings -> binary strings • Sort : sequence of strings -> sequence of strings • Roots : polynomial -> sequence of integer roots [pg 144] • A TM which decides a language computes a function from the set of all words to {y,n} • There are many uncomputable functions: • |funs : N -> N| = uncountable • |TMs| = countable FLAC: Reductions

  10. Compilers: computable functions Translator TM Compiler FLAC: Reductions

  11. Multiple levels of virtualization Ex: running a BASIC interpreter written in Java Basic program Basic interpreter Java interpreter CPU FLAC: Reductions

  12. The operating system: another (multi-tape) universal turing machine OS Process 1 Process 2 Operating system CPU FLAC: Reductions

  13. The operating system = dovetailing We used this technology in the proof of the last theorem during the last lecture: Theorem: L is decidable if and only if L and ~L are recognizable. Proof: execute in parallel recognizers for L and ~L and stop when the first stops. Process 1 Process 2 Operating system CPU FLAC: Reductions

  14. Reconfigurable Hardware Interconnection network Universal gates and/or storage elements Switches 0 0 0 1 a0 data a0 0 1 a1 & a2 a1 a1 1 1 Universal gate = RAM A switch is controlled by a 1-bit RAM cell FLAC: Reductions

  15. Reconfigurable hardware If (a) c = 2; Program Compiler Compiler Custom hardware (configuration) Generate a TM for a specific problem! Back to ENIAC! FLAC: Reductions

  16. < m Reductions between languages f : S* ->S*, computable f reduces L1 to L2 iff w in L1 <=> f(w) in L2 S* S* f L1 L2 f is called reduction. We write L1L2. FLAC: Reductions

  17. < m Reductions are useful • Notice that neither L1 or L2 must be decidable. • Theorem: if L1L2 and L2 is decidable • then L1 is decidable. • Proof: If M decides L2 and N computes a reductionf from L1 to L2, build a TM P which on input w: • Computes f(w) using N • Runs M on f(w) • Outputs the result of M. FLAC: Reductions

  18. < m Using reductions to prove undecidability • Use the converse of the previous theorem:if AB and A is undecidable, then B is undecidable. • We have seen two techniques to proveundecidability • The “diagonalization” proof • Reductions using this theorem. FLAC: Reductions

  19. Undecidability • We know that most problems are undecidable • Turing exhibited one natural undecidable problem: the Halting Problem (does a TM halt when given an input w?) • More than that: many important problems are undecidable • When you face a problem, you should be aware that it may be unsolvable: • You can search a solution • You can try to prove it has no solution (usu. by reduction) FLAC: Reductions

  20. Some undecidable problems • Does the TM M accept the word w? • Is a mathematical statement true? • Does a Diofantine equation have any roots? • Does a TM accept a regular language? • Does a CFG generate all words in S*? • Will a parallel system of processes ever deadlock? • Is there a smaller TM implementing the same function? • Are two programs computing the same thing? FLAC: Reductions

  21. Compilers and undecidability • The HP is about a language describing TMs • Many other problems concerning TMs are undecidable • Compilers cannot prove some properties of programs: • Does a C program access an array out of bounds? • Will ever a print instruction be executed? • Can these two pointers point to the same location? • Will a LISP program crash with a type error? • Can this memory location be garbage collected at some point? • Compilers behave conservatively FLAC: Reductions

  22. The full-employment theorem for compiler-writers Theorem: There is no best optimizing compiler for a general-purpose language. Proof: by reduction to the HP. Theorem: For any compiler, there’s a better one. Proof: we can always hardcode a program which doesn’t terminate. FLAC: Reductions

  23. Research in programming languages • Contrast the decision problems for regular and context-free languages with Turing machines. • Research in programming languages tries to get the best of both worlds: • Languages powerful enough to express useful computations • But restricted enough to guarantee program properties FLAC: Reductions

  24. < < m m Reduction examples Definition:FIN = { M | L(M) is finite } Theorem: FIN is not Turing-recognizable. ~FIN is not Turing-recognizable. Proof: We will show ~HP FIN ~HP ~FIN Notice that HP is Turing-recognizable HP is not Turing-decidable So ~HP is not Turing-recognizable. FLAC: Reductions

  25. A reduction from ~HP to FIN S* S* M#x f(M#x) f ~HP FIN • f(M#x) is the description of a TM M’ which on input y: • Erases the input y • Writes x on the input tape • Runs M on x • Accepts if M halts on x FLAC: Reductions

  26. Reducing ~HP to FIN (end) • f(M#x) is the description of a TM M’ which on input y: • Erases the input y • Writes x on the input tape • Runs M on x • accepts if M halts on x Notice that M halts on x L(M’) = S* M does not halt on x L(M’) = F (finite) Notice that f is computable: f does not run M, f just writes down a machine M’ which calls M on x FLAC: Reductions

  27. Part 2: Reducing ~HP to ~FIN S* S* M#x g g(M#x) ~HP FIN ~FIN • g(M#x) is the description of a TM M’’ which on input y: • Saves y • Writes x on the input tape • Runs M on x for |y| steps • Accepts if M does not halt before |y| steps FLAC: Reductions

  28. Reducing ~HP to ~FIN (end) • g(M#x) is the description of a TM M’’ which on input y: • Saves y • Writes x on the input tape • Runs M on x for |y| steps • Accepts if M does not halt before |y| steps M halts on x M does not halt on x L(M’’) = { y | |y| < run-time of M on input x } finite L(M’’) = S* Again: g is a computable function it just manipulates machine descriptions. FLAC: Reductions

  29. To remember • The theoretical notion of TM has profoundly influenced computer engineering • There are many important undecidable problems • Reductions are a tool to prove problems being undecidable (Note: we will see reductions again in complexity theory; we will use them to prove a problem is the hardest in a class of problems) FLAC: Reductions