Smooth Maps of the Interval and the Real Line Capable of Universal Computation

Transcription

1 Smooth Maps of the Interval and the Real Line Capable of Universal Computation Cristopher Moore SFI WORKING PAPER: SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. SANTA FE INSTITUTE

2 Smooth maps of the interval and the real line capable of universal computation Cristopher Moore Santa Fe Institute January 2, 1993 Abstract We construct two classes oc maps: once-differentiable maps oc the unit interval which can simulate a. wide variety of operations on sequences, including cellular automata, generaliz~d shifts, and Tnring machines;.and.analytic maps'of R which simulate Turing machines. This brings down the dimensionality in which a smooth dynamical system can be computa. tionally universal from 2 [1] to 1. 1 Introduction Many-dimensional models ofuniversal computation have long been known, including cellular automata [2], neural networks [3], partial differential equations [4], and billiard models [5] to name a few. In [1] we construct"d maps and flows in 2 and 3 dimensi.ons conjugate to universal Thring maclijnes, and showed that they could be made infinitely differentiable. 'Thus, in principle, universal computation can be embedded in smooth, low-dilllensional dynamical systems. This paper goes one step further, constructing two types maps in ~ dimension which can simulate a wide variety of computational systems, including Turing machines, cellular automata, and the generalized shifts introduced.in [1].. The first type is once-differentiable, the second is analytic (and can be.written in closed form). Since these maps inherit the same typ~ of unpredictability and undecidability that Turing machines possess, they show that simple questions about one"dimensional maps can be undecidable. For instance, the following question is equivalent to the Ualting Problem: given one of these maps, an initial point x, and an open set A, will x ever fall into A as the map is iterated? This undecidability persists even if the initial conditions are known exactly. Given that these systems are highly contrived and almost certainly not "physical". we hope that they at least serve to illustrate something about the

3 nature of computation and how it can be embedded in contiilljous dynamical systems. 2 C l maps of the interval Suppose (a;) is a one-sided sequence, where a; E {O,...,n - I) and I ranges from 0 to 00;' Then let <Xl " x =-- a-ii: aiq'- '.. n-l. (1) i=o If a ~ n, the values x E X describe a Cantor set in the unit interval, whose largest gaps are of width I - n/a. If a = n, X is the whole interval, and ai is the base-n expansion of x. Suppose we have some dynamical system ci> defined on the sequences a. This could include a simple shift map ai = ahi. a one-dimeilsional cellular automatonai =f(ai_ro.., ai,.. " ai+r), ora generalized shift as defined in [1). This will then induce '8. map on'x, which we will also call <1>. We now try to extend this map to the whole interval by'interj>olating across the gaps ofx. We.use some sigmoid function, for instance a pair of parabolic arcs; we want the derivative at each end of the gap to be zero. If Xl and X2 are the left and right ends of the gap, the maximum derivative will then be i.e., the rise over the run. The constant of proportionality C (2 if we use parabolic arcs) will stay the same as we scale the sigmoid down to fit each gap. Thus the derivative is zero on X, and bounded,!jverywhere else. This is illustrated in' figure h." Now suppose that ci>'s action on sequences has a finite radius; i.e., suppose that ci>(a); depends on elements of a no more than r spaces to the right. Then if u, = bj for all i < k, ci>(a); ::: <I>(b), for all i < k - r. This is true for cellular automata, where r.is the same radius usually referred to, and for generalized shifts, where r == maxf(a)... We can then put an upper bound on the derivative. If the size of the gap is n-k(l - n/a), Xl and X2 agree on the first k digits; if n = 2, for instance, Xl =.aolll... and X2 =.al000..., where a is a common initial sequence k digits lolig. The rise, or the width of the gap's image, is then bo\lnded by a- k + r, and so the derivative is bounded by (2) a r dci>/dx :5 C / (3) 1 - n a for some constallt C. (This is'minimized by setting a = n(i-i/,'):) Second and higher derivatives, however, diverge exponentially as k goes to infinity, since the nth derivative is inversely proportional to the nth power of the gap size. 2

4 If we wish to work with two-siued sequences (a;) where i ranges over the integers, we can fold them up into aoa_iala_2a2"': I(a); = i even a_(;+i)/2 i odd Turing machines can then be simulated as shown in [I); the finite state can be re-written directly onto the tape at the origin, and dependence on it then becomes membership in a finite number of blocks of X. In figure 2, we show two examples: a folding of the two-sided shift u(a); = a;+i, and a simple additive cellular automaton f(a); = a; + ai+1 mod 2. These maps scale nicely, and their values on X are the same as the global mappings defined in [6).. 3 Analytic maps on R We now construct computationally universal. analytic maps in one dimension, using a construction of Minsky's [3). Suppose we give a finite-state machine access to a finite number of integer registers, and allow it to increment, decrement, and branch on zero; i.e., it can increment or decrement any of the registers, but the only dependence it is allowed to have on their values is to check whether they are zero or not. How many such registers does it need to simulate an arbitrary Turing machine? In fact, two are sufficient; and if we allow the machine to multiply or divide by a constant, and to.check whether a register is zero mod p for constant p, even just one will do. We briefly review this result. First, let's break the Turing machine's tape into left and right halves, and associate them with integers Land R where the least significant digits are those closest to the head. We can assume the tape alphabet has two symbols. Then Land R are simply the two halves of the tape sequence when written in binary, except that R is in reverse order. (L and R are finite since we can assume the tape is blank beyond a finite distance from the origin.) Now to shift the machine's head to the right, say, we effectively shift the tape to the left. We need t~ halve R and double L; to check on what the value of the tape at the head is we need to calculate R mod 2; and finally we need to add 0 or I to [, to write the new value.. If our finite-state machine is allowed only to increment, decrement, anp branch on zero, we can do all this with an auditional register, which we'll call W. For instance, to double L, we first clear out W by decrementing it until it is zero; then we do a loop (dec L, inc W, inc W) until L = 0, incrementing W twice every time we decrement L; then we feed W back into L with (dec W,inc L) until W = O. We can halve R in a similar way.. To find R mod 2, set III' a two-state loop, with both states doing (dec R, inc Wl If R hits zero on the first state, R mod 2 = I; if it hits zero on the second state, ~" 3

5 R mod 2 = O. Then feed W back into R to rest.ore il Finally, a simple (inc f,) will suffice t.o write the new symbol on the t.ape if necessary. So we've shown t.hat 3 increment.-decrement. registers are sufficient for the job. But if we combine these three int.o a single int.eger, by writ.ing t.hen we can simulate them all, as long as our machine is allowed t.o mult.iply, divide, and check for zero mod p; to increment. or decrement. the p-register, i.e. the power of p in I's prime decomposition, we just multiply or divide I by p. To check the p-register for zero, we t.ake I mod p; this will be zero if and only if the p-register is nonzero. (We said two, not. t.hree, increment.decrement registers were sufficient; this is because we can now simulat.e the mult.iply-di~ide machine on I wit.h an increment-decrement machine with an additional regist.er W' using the above t.echniques.) Now to get. rid of the finite-state machine, we combine the finit.e st.at.e with I to give one integer. If t.here are n internal st.at.es, label t.hem s = 0, I,..., n - 1 and write x = ni + s. (5) Our machine now corresponds t.o a map on x, which we will now embed in an analytic map of the reals. First. we need t.o check whether or not we're in stat.e s. Define h,(x) = ( Sill 71'X )2.. x«-,) (6) nsln Then h, = I' on integer values if x mod n = sand 0 ot.herwise, Similarly, for each s, we want. to check wh'ether some regisber p, is nonzero, where p, =;2,' 3, or 5. Define n (4) Then g, = 1 if I mod p, = 0, i,e. the p,-regist.er is nonzero, and g, == 0 if I mod p, i: 0, i.e. t.he p.-regist.er is zero. Now if t.he p,-register is nonzero, we want. t.o millt.iply I by some rat.ional number l'"n, (say), incrementing and decrementing t.he desired regist.ers (for inst.ance, 3/5 if we want. to increment. L and decrement. W),and cliange t.he state to s: n" If it 'is zero, we perform a different set. of increments and decrement.s by m'ult.iplying by r." instead, and challge t.he st.at.e to s:". To accomplish this, writ.e x - s ( ') X - S )( ) ') ( ) f,(x)=g.(--)(x-s)r,n,+s,n, +(I-g,(--) (x-sr",+s". 8, n. In'

6 Finally, write n-l f(x) = L h,(x)f,(x),,=0 and we're done. As written, the derivative of f on the integer values will be the Illultiplicative fact.ors r. It is easy to modify the function so that l' is zero on these values, by replacing the terms (x - S),-, in f, above with (9) (x - S - (1/27f)si7127fx)r,. (10) Then the computation will actually be stably attractive, and so would take place on a set of positive measure, at least for any measure on the reals with some integers in its support. This is interesting, since the maps in [IJ and even some infinite-dimensional models such as [5] only succeed in computing on a set of measure zero. By encoding the Turing tape in integers with a bounded distance between them, rather than in a Cantor set, we can make the computation "robust" (although such an oscillating function over all the reals still greatly stretches the definition of a "physical" dynamical system). Of course, we can squeeze this map into the interval by conjugating it with some z that sends [0,1) to [0, +00), such as z(x) = tan(7fx/2). This map will be analytic except at x = 1, and its derivatives will diverge in that vicinity. To illustrate this construction, in figure 2 we show an analytic function, on R and condensed into the interval, that implements the famous dynamical system on the integers: I' = 1/2 I even I odd. It is not known whether or not all integers converge on the cycle (4,2,1). Several authors have observed that the general question for maps of this type is undecidable, since we can use Minsky's construction to simulate a Turing machine. To encode a universal Turing machine (such as Minsky's with 7 states and 4 tape symbols) wonld take about 100 terms, although I'm sure that cleverer and more economical coding methods exist. 4 Conclusion We have exhibited two ways ofembedding universal computation in one-dimensional maps; one which is once-differentiable, and one which is analytic (and can be v V written in closed form). As stated above, the dynamics of these maps then inherit the undecidability of the Halting Problem; even if initial conditions are known exactly, there is no algorithm to determine their eventual fate. Other 5

7 propert.ies of t.heir long-t.erm dynamics (e.g., ergodicit.y on X for t.he first. class) are probably also undecidable, as they are for the maps in [1]. There is no denying that these maps are cont.rived; we have deliberately const.ructed t.hem so that their behavior is provably undecidable. However, the existence of Sl1ch maps, even as isolated examples, sayssomet.hing abou t. t.he complexity of t.he-space of one-dimensional maps in general. Acknowledgements: I am grateful t.o Mats Nordahl and Seth Lloyd for helpful discussions. References [1] C. Moore, Unpredictability and Undecidability in Dynamical Systems. Phys. Rev. LelC64, 2354 (1990), and Nonlinear'ily 4, 199 (1991). [2] see for instan~e M. G. Nordahl and K. Lindgren Complex Systems , or J. Albert and R. Culik II, ComplexSyslems [3] M.Minsky, Computation: Finite and Infinile Machines (Prentice-lIall, Englewood Cliffs, NJ, 1967). [4] S. Omohundro, Physica lod, (1984). [5] E. Fredkin and T. ToffoJi, Int. J. Theo,:. Phys. 21, (1982). [6J R. Feldberg, C. Knudsen, and S. Rasmussen, Re~ursive Definition of Global Cellular Automata Mappings (preprint).."

8 Two-sided shifl, a=2.s Two-sided shift, a=2.\ Additive CA, a=2.s Additive CA, a=2.l Figure I. Two examples of embedding operations on sequences in a C' map of the interval.

9 Figure 2. dynamical An analytic function embedding the "x/2 or 3x+ \" integer system. and its compression into the interval. ".. " " o. 10