J.H. Conway, \Unpredictable iterations." In Proc. 1972.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= 1 In dynamical systems terms, is all of N in the basin of attraction of the periodic orbit f1; 4; 2g We can generalize this as follows. Let f(x) a i x b i where x i mod p (1) for some base p and constants a i , b i for 0 i p. We will call any such f a Collatz function. J.H. Conway [3] showed that it is undecidable in general whether f (x) 1 for some t. Based on this fact, Burckel [2] has shown that it is undecidable whether certain functional equations have non trivial solutions; in addition, the record holding small Turing machines in the Busy Beaver competition ....

J.H. Conway, \Unpredictable iterations." In Proc. 1972.

....defines the cycle. For particular cycle unification classes see [45, 51] In this paper we will show that the two problems are undecidable for append like programs. The proof technic of [19, 20] is based on an original encoding of the unpredictable iterations of J. Conway within number theory [8] which are close to Minsky machines [39] An alternative proof of undecidability of the emptiness problem can be found in [28] It has been made independently and it is based on an encoding of the Post correspondence problem. We will present both proofs. We also study some particular subcases ....

....of the basic notions in the following sections. 4. MINSKY MACHINES AND CONWAY ITERATIONS In the following, the expression It is undecidable whether or not. stands for There exists no algorithm that always decides, whether or not. 4.1. Minsky Machines Presentation. The Minsky machines [39, 8] are deterministic machines with registers and instructions. Registers (finitely many of them) can hold arbitrary large non negative integers. A machine executes a program composed of instructions sequentially. Instructions are labeled by Q 1 , Q 2 , Delta Delta Delta, Qn (for a program of n ....

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Conway J.H. "Unpredictable Iterations." Proc. 1972 Number Theory Conference. University of Colorado, pp 49--52. 1972.

....of the function f . It can be reformulated by a closed formula of monadic second order logic. An abstract of this paper will be presented at MFCS 2000 1 We will consider here the transition graphs of labelled congruential systems, called the congruential graphs. It has been proved by Conway [Con72] that the termination problem of congruential functions is undecidable. On the other hand, Courcelle [Cou90] has shown that the regular graphs have a decidable monadic second order theory : any monadic second order sentence can be verified by an algorithm for those graphs. Originally, it has been ....

....and its degree is dG (s) d G (s) d Gamma G (s) The degree of a graph is the maximum degree of its vertices. 1 Congruential Systems 1. 1 Definition The congruential systems form a non deterministic generalization of the congruential (or partially linear) functions presented by [Con72] and [Bur94] Definition 1 A congruential system C is a finite set of rules of the form (p; r) a Gamma (q; s) where a 2 A and p; q; r; s 2 IN with r p and s q. The graph G(C) of any congruential system C is defined by G(C) ae pn r a Gamma qn s j (p; r) a Gamma C (q; ....

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J.H. Conway. Unpredictable iterations. In Number Theory, pages 49--52, 1972.

....and have proved it decidable for some particular cases (see also [16, 18] In [7] we have proved the halting problem to be undecidable in the general case. In this paper, using a similar proof technique based on the codification of the unpredictable iterations of J.H. Conway within number theory [3] which code Minsky machines [14] we will show that the emptiness problem is undecidable in the general case (another proof, established independently at the same time, of this result can be found in [10] Let us note, that although the basic technique (our (original) codification of Conway ....

....above one, the non linearity of the terms, the existence of some variables on one side of the clause, and the permutation of variables during inference generally make intuitive comprehension of behaviour impossible. 3 Theoretical Tools 3.1 Minsky Machines 3.1. 1 Presentation The Minsky machines [14, 3] are state register machines, the registers (in finite number) may hold non negative integers and two types of transitions are allowed : ffl in the state Q i , add 1 to register a and proceed to state Q j . ffl in the state Q i , if j a j 0 (where j a j denotes the content of the register ....

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Conway J.H. "Unpredictable Iterations." Proc. 1972 Number Theory Conference. University of Colorado, pp 49--52. 1972.

.... A4 : where A i are positive first order literals. This class is shown here to be as expressive as Turing machines and all simpler classes would be trivial. The proof is based on a remarkable and not enough known codification of any computable function by unpredictable iterations proposed by [5]. Then, we prove effectively by logical transformations that all conjunctive formulas of Horn clauses can be translated into an equivalent conjuctive 4 formula (as above) Some consequences are presented in several contexts (mathematical logic, unification modulo a set of axioms, compilation ....

....way and A 4 corresponds to the input datum. This very restrictive class is Turing complete. The proof is based on two different techniques. The first part (Section 2) is an encoding of periodically linear functions whose iterations have been proved to be equivalent to Minsky machines by [5]. Standard methods for decision problems and computational power use Minsky or Turing machines. It seems to be surprising that the remarkable result proposed by Conway was rarely used. Its formalism is of higher level and much easier to encode. The second part (Section 3) is based on logical ....

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Conway J.H. "Unpredictable Iterations." Proc. 1972 Number Theory Conference. University of Colorado, pp 49--52. 1972.

....conjectures: CD) T has no divergent trajectories. CC) The only cycle of T is f1 2 1g. Remark 1.2. Statements (CD) and (CC) simple and natural as they are, might well turn out to be algorithmically undecidable, as is their rather straightforward arithmetical generalization due to John H. Conway [1972]; hence the problem: PAD) Is the 3x 1 conjecture algorithmically decidable Past and present research on the 3x 1 problem has centered around the three themes (CD) CC) and (PAD) with their quite different and almost unrelated methods and techniques. This paper attempts to contribute to ....

.... 3 k . This gives the inequality n d k C Gamma 3 2 Delta k Gamma1 : Hence the result. 4. A NORMALIZED VERSION OF T We now consider a reformulation of the 3x 1 problem that has certain formal advantages. When we passed form T to S, we lost the periodically linear character of T [Conway 1972; Lagarias 1985] yet, the previously mentioned undecidability result of [Conway 1972] concerns just such functions. Another formal difficulty is that S is defined as a composition of the functions T 0 : n 7 n=2 and T 1 : n 7 (3n 1) 2 acting outside the domain D of definition of S. Both ....

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J. H. Conway, "Unpredictable iterations ", pp. 49--52 in Proceedings of the Number Theory Conference (Boulder, 1972), Univ. Colorado, Boulder, Colo., 1972.

....i and (a i =q i ) p for each i. For a Conway function g let C ( g) be a statement asserting that there exists a natural number i such that g i (2) j 1 (mod p) Theorem 2.1 (Conway) The problem: given a Conway function g. Does C(g) hold is undecidable. Proof of the theorem can be found in [C72] as well as in [L85] or [DLR93] Our main tool is the following refined version of Theorem 2.1 : Theorem 2.2 There exists a computable sequence fgng of Conway functions such that: i) For each g n , if a i and q i are coefficients from the definition of the function gn then (a i =q i ) 2. ii) ....

J.H. Conway, Unpredictable Iterations, Proceedings of 1972 Number Theory Conference, University of Colorado, pp. 49-52,1972.

....In Section 2, we introduce binary Horn clauses and their resolution, then a codification of the famous 3x 1 conjecture is given. In Section 3, a generalization of this conjecture is presented and based on the works of J.H. Conway. In Section 4, we show how the unpredictable iterations of [Con72] can be simulated by binary clauses and we use it to prove the undecidability of halting problem of binary clauses. 2 Binary Clauses Let F be a set of function symbols (which contains at least one constant and one symbol whose arity is greater than 1) and V ar be an infinite countable set of ....

....if the 3x 1 program is finite from the input n. In other words, the 3x 1 conjecture is equivalent to prove that, given any goal p(L; L; L) where L is a list of the form [a; Delta Delta Delta ; a j ] the resolution is finite. 3 A Generalization of the 3x 1 Problem J.H. Conway [Con72] considers the class of periodically piecewise linear functions g : IN IN having the structure : 8 0 k d Gamma 1; if n (mod d) k ; g(n) a k n : where a 0 ; Delta Delta Delta ; a d Gamma1 are rational numbers such that g(n) 2 IN. These are exactly the functions g : IN IN such that ....

Conway J.H. "Unpredictable Iterations." Proc. 1972 Number Theory Conference. University of Colorado, pp 49--52. 1972.

....p for each i. For a Conway function g and given natural number N let C(g; N) be a statement asserting that there exists a natural number i such that g i (N) 1. See Section 2.3 to find a nice example giving an idea of what a Conway function. is. Proof of the following theorem can be found in [4], in [14] or in [7] Theorem 2.1. Conway) The problem: given a Conway function g, and a natural number N . Does C(g; N) hold is undecidable. Our main tool is the following refined version of Theorem 2.1 : Theorem 2.2. 1. There exists a computable sequence fgn g of Conway functions such ....

J.H. Conway, Unpredictable Iterations, Proceedings of 1972 Number Theory Conference, University of Colorado, pp. 49-52,1972.

....= 1 In dynamical systems terms, is all of N in the basin of attraction of the periodic orbit f1; 4; 2g We can generalize this as follows. Let f(x) a i x b i where x = i mod p (1) for some base p and constants a i , b i for 0 i p. We will call any such f a Collatz function. J.H. Conway [3] showed that it is undecidable in general whether f t (x) 1 for some t. Based on this fact, Burckel [2] has shown that it is undecidable whether certain functional equations have non trivial solutions; in addition, the record holding small Turing machines in the Busy Beaver competition ....

J.H. Conway, "Unpredictable iterations." In Proc. 1972 Number Theory University of Colorado (1972) 49-52.

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