Presented By : Abhinav Aggarwal CSI-IDD, V th yr Indian Institute of Technology Roorkee. Joint work with: Prof. Padam Kumar

Transcription

1 Presented By : Abhinav Aggarwal CSI-IDD, V th yr Indian Institute of Technology Roorkee Joint work with: Prof. Padam Kumar A Seminar Presentation on Recursiveness, Computability and The Halting Problem

2 2 A quick glance ü Nature of computation ü Classical Recursion Theory Theory of functions and sets of natural numbers ü Goes back to Dedekind defining functions using recurrence ü Recursiveness Church, Gödel, Kleene, Turing, Post ü Effectively computable functions = Recursive functions ü Development of programming languages ü Termination of Programs

3 3 Recursiveness Primary references : [1] Odifreddi, P., Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers, Elsevier Science, 1980 [2] Post, Absolutely unsolvable problems and relatively undecidable propositions, The Undecidable, Raven Press, 1965, pp [3] Gödel, On undecidable propositions of formal mathematical systems, The Undecidable, Raven Press, 1965, pp [4] Kleene, General recursive functions of natural numbers, The Undecidable, Raven Press, 1965, pp [5] Turing, A. M., On computable numbers with an application to the Entscheidungsproblem, Proc. London Math. Soc. 42, 1936, pp

4 4 A new word ü Recursion, recurrence perhaps they were already taken ü Closer look at functions from ω to ω ω = { 0,1,2,3,... } ü Characterization of natural numbers Dedekind s formalization 0, S(0), S(S(0)), S(S(S(0))), ü Formal arithmetic models loads of axioms Initial Functions

5 5 Where does it take us? ü Dedekind s Induction Principle ü Least Number Principle ü Primitive Recursive functions ü µ-recursive functions ü Class of recursive functions

6 6 Examples ü Commonly encountered functions are recursive bounded sum, bounded product, factorial, prime number generation f(x,y) = x + y ü Coding (numbering) of the entities of a set Countability is recursive Cantor s coding of the plane of ordered pairs

7 7 Curious Bob ü Is every function recursive? No ü What about mathematical functions? - Yes ü But where is the re-occurrence? - In the initial functions ü OK. But why can t I compute them directly, instead of using recursion? Formal arithmetic ü How is it useful at all? Doesn t it complicate things? It gives them a structure Facilitates generalization ü Fine. But how do I know if this will work? Do you have a magic wand? - Computability

8 8 Computability Primary references : [1] Odifreddi, P., Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers, Elsevier Science, 1980 [2] Soare, R. I., Computability and Recursion, 10 th Int. Cong. for Logic, Methodology and Philosophy of Science, Sec. 3 : Recursion Theory and Constructivism, 1995

9 9 A function is computable (also called effectively calculable or simply calculable ) if it can be calculated by a finite mechanical procedure. f ( x) = f 1 x = 0 ( ) +1 x > 0 x 1 g( x) = g x +1 1 x = 0 ( ) +1 x > 0 Can we devise an algorithm for the given function (task) and check if it can be implemented using the resources (theoretical or practical) available at that time? Code C/C++, Java, PASCAL, Haskell, Scheme, LISP Algorithm Flowcharts, Pseudocode, Textual Description Mathematical Models Turing Machine, Lambda expression

10 10 The first thought ü Written Calculations on a sheet of paper ü Perceive what is written by reading it ü Make a decision for what to write next ü Read, Think, Write Turing Machine A function is Turing Machine Computable if there is a Turing Machine which reaches a final configuration with f(x) represented in unary notation on its tape, when started with only x in unary notation on the tape. ü Machine-dependent languages

11 11 I 1 1 ( x) = x

12 12 The next step ü Machine independent abstract core ü Emphasis on sequence of steps for a general input ü Making a flowchart for the process involved Flowchart Program A function is Flowchart Computable if there exists a flowchart which modifies the input variables x and exits (halts) when the output variable takes the value f(x). ü General Purpose Simulation Systems ü For-programs and While-programs ü Algol, Pascal, Basic, Fortran

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14 14 A higher vision ü Can functions act as arguments? First class citizens ü No distinction between functions and data λ - abstractions ü Any other evidence? Gödel Numbering Enter Church n λ f.λx.f ( n) ( x) Church Numerals A function f is λ-definable if there is a λ-term F such that f(a) = b holds if and only if F(a) = b is true, upto β-reductions. Functional Programming Languages LISP, Scheme, Haskell, SML, Scala

15 15 Putting it all together ü Church s Thesis Every computable function is recursive ü Post - General recusiveness = Effective computability ü Absolute unsolvability of functions ü Restricted versions of computability Polynomial Time Computability ü Complexity Classes ü Equivalence and Inclusion relations between complexity classes ü Effective optimized computability

16 16 Extending a little ü Defining functions for only some inputs Partial Recursive Functions ü Kleene Recursive functions are exactly the partial recursive functions which happen to be total ü Index e to every p.r.f Enumeration Theorem ü Data can be effectively incorporated into a program and can effectively code programs as well Parametrization (Smn) Theorem ü Kleene Universal Partial Recursive Function, f(e,x) = g(x) ü Universal Turing Machine computes f ü Recursively enumerable relations domain of a p.r.f ü f(e) = e - Quines (Kleene s Fixpoint Theorem such an e exists) eval s= print eval s= ;p s

17 17 Curious Bob II ü Wait a minute! So many definitions? They are all equivalent ü How can if there exists be verified? No general way ü Is there any way? Write programs! Hire coders! Google! ü What if my code never stops running? :P ü Can I check if this would happen? Sometimes ü Come on! Sometimes won t suffice. In general? No ü Who says that? Alan Turing ü Is he the authority here? Yes, he is known as the Father of Computer Science

18 18 The Halting Problem Primary references : [1] Floyd, R. W., Assigning meaning to programs, American Math. Soc. 1967, pp [2] Aho, A. V., Ullman J. D., The theory of parsing, translation and compiling, Vol. 2, Prentice Hall, 1973 [3] Katz, S., Manna, Z., A closer look at termination, Acta Informatica 5, 1975, pp [4] Manna Z., Dershowitz, N., Proving termination with multiset orderings, Memo AIM-310, Stanford Intelligence Laboratory, 1978 [5] Blass, A., Gurevich, Y., Proving termination and well-partial orderings, ACM Transactions on computational logic, 2006, pp. 1 26

19 19 The actual problem ü Optimized resource management and determinism of events ü Program must halt for a given set of inputs ü Existence of infinite computations in a program ü Generalized Halting Problem is algorithmically unsolvable Undecidable ü Solving restricted instances

20 20 Known approaches ü Floyd s approach No infinitely-descending-chain ü Loop Approach ü Exit Approach ü Burstall s Approach ü Well-partial orderings ü Multiset orderings

21 21 Floyd s Approach

22 22 Loop Approach

23 23 Exit Approach

24 24 Structural Induction

25 25 Multiset Ordering Using well-founded set ω Using multiset theory

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