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Discrete dynamical systems
and intrinsic computability
Marco Giunti
University of Cagliari, Italy
[email protected]
http://edu.supereva.it/giuntihome.dadacasa
Outline
General thesis – Computation theory is a special branch of
dynamical systems theory, and its objects (the
computational systems) are a special kind of discrete
dynamical systems. The specific difference of these
objects, i.e. the property of being computational, can be
thought as an intrinsic property of their dynamics.
1. Dynamical systems: definition and examples.
2. Computational systems: definition1 and why, by this
definition, being computational is not intrinsic.
3. Dynamically effective representations of discrete systems
and a refined, intrinsic, definition2 of a computational
system.
4. Possible consequences for computability theory: (some)
non-recursive functions may turn out to be computable by
a particular class of intrinsic computational systems.
A Dynamical System (DS ) is a mathematical
model that expresses the idea of a
deterministic system (discrete/continuous,
revers./irrevers.)

A Dynamical System (DS) is a set theoretical
structure (M, (gt)tT) such that:
1. the set M is not empty; M is called the statespace of the system;
2. the set T, is either Z, Z+ (integers) or R, R+
(reals); T is called the time set;
3. (gt)tT is a family of functions from M to M;
each function gt is called a state transition or
a t-advance of the system;
4. for any t and w T, for any x M,
a. g0(x) = x;
b. gt+w(x) = gw(gt(x)).
Intuitive meaning of the definition of
dynamical system
gt
x
gt(x)
t
t0
t0+t
g0
gt+w
x
x
gt
gw
Example of a continuous DS (Galilean
model of free fall)

Explicit specification
Let F = (M, (gt)tT) such that
 M = SV and S = V = T = real numbers
 gt(s, v) = (s + vt + at2/2, v + at)

Implicit specification
Let F = (M, (gt)tT) such that
 M = SV and S = V = T = real numbers
 ds(t)/dt = v(t), dv(t)/dt = a
A standard functional scheme of a
Turing machine

A physical realization of a Turing machine is any
concrete system which satisfies (implements, works
according to) the abstract functional scheme below.
aj
External
memory
qi
Internal
memory
ak
L
Read/write
head
qi
qm
Read/write/move
head
aj
..:. ..
qiaj:akLqm
.. . : . . .
.. . : . . .
Control unit
Example of a discrete DS (Functional
scheme of a Turing machine)

The abstract functional scheme of a Turing
machine can be identified with the discrete
dynamical system T = (M, (gt)tT) such that:
 M = PCS, where P = Z (integers) is the set
of the possible relative positions of the
read/write/move head, C is the set of the
possible contents of the whole external
memory, and Q is the set of the possible
contents of the internal memory;
 T = Z+ (non-negative integers);
 let g be the function from M to M determined
by the machine table of the functional scheme;
then, g0 is the identity function on M and, for
any t >0, gt is the t-th iteration of g.
Computational systems: intuitive
concept

Extensional characterization: by the term
computational system I refer to any device of the
kind studied by standard computation theory;
e.g. Turing machines, register machines, cellular
automata, finite state automata, etc.



Discreteness and determinism are two properties shared
by all such devices;
thus, so called analog computers are not computational
systems is this sense.
Intensional characterization: the computational
systems can be identified with those discrete,
deterministic dynamical systems that can be
represented effectively.
The crucial question: What is an effective
representation of a discrete dynamical system?


A natural definition (perhaps the most
natural definition?) of effective
representation is as follows:
an effective representation of a discrete
dynamical system DS = (M, (gt)tT) is a
pair (u, DS#) such that:
1. DS# = (N, (ht)tT) is a discrete
dynamical system, where N is either Z+
or a finite initial segment of Z+;
2. u: N  M is an isomorphism of DS# in
DS;
3. for any t  T, ht is a recursive function.
The first definition of a computational
system. Is it intrinsic?



If we buy the previous definition of an effective
representation of a discrete dynamical system,
we can then define:
DS is a computational1 system iff DS is a discrete
dynamical system, and there is an effective
representation of DS.
Question: is the property of being computational1
intrinsic to the dynamics of the discrete system
DS? In fact, DS might admit two isomorphic
numeric representations, such that one is
recursive and the other is not. In this case, the
property of being computational1 could not be
said to be intrinsic to the dynamics of DS, for it
would depend on the numeric representation of
the dynamics we choose.
Being computational1 is not intrinsic

There is a discrete DS such that:



it is obviously computational1 (i.e., it has an
effective representation = it has a recursive
numeric representation);
but, it also has a numeric representation that
is not recursive (i.e., the first two conditions of
the definition of effective representation are
satisfied, but not the third).
Surprisingly enough, this system is DS1 =
(Z+, (s n)nZ+), i.e., the discrete dynamical
system generated by iterating the
successor function s.
DS1 = (Z+, (s n)nZ+) is computational1,
but not intrinsic. Sketch of proof (1/2)


Obviously, a recursive numeric representation of
DS1 = (Z+, (s n)nZ+), is (i, DS1), where i: Z+  Z+
is the identity function.
Consider an arbitrary bijection p: Z+  Z+ and
the “new successor function” sp on Z+
corresponding to the order induced by p:
0
1
2
3
4
5
6
7
8
9
75
123
48
3 1,003 98 87,561 23
0
s
p
sp
35,101
FIGURE 1 A hypothetical initial segment of p
DS1 = (Z+, (s n)nZ+) is computational1,
but not intrinsic. Sketch of proof (2/2)

Thus, (p-1, DSp), where DSp is the discrete
dynamical system generated by sp, is a numeric
representation of DS1.

How many representations (p-1, DSp) are there?

As many as the number of bijections p of the
non-negative integers.

But the number of such bijections is uncountable.

Therefore, there is p* such that (p*-1, DSp*) is a
non-recursive numeric representation of DS1.
Q.E.D.
The previous proof is surprising
It is odd to realize that a dynamical system
like DSp*, which has exactly the same
structure as the sequence of the natural
numbers, is generated by a non-recursive
pseudo-successor function sp*, and that
(p*-1, DSp*) thus constitutes a bona fide
non-recursive numeric representation of
DS1, which, in contrast, is generated by
the authentic successor function that is
obviously recursive.
Might ( p*-1, DSp*) be not a bona fide numeric
representation of the dynamics of DS1?

Compare the “good” representation (i, DS1) with
the “odd” one (p*-1, DSp*):

if we are given the whole structure of DS1 (i.e.,
the successor function s: Z+  Z+), we can
mechanically produce the identity function i by
simply starting from state 0 and counting 0, then
moving to state s(0) = 1 and counting 1, and so
forth;

but it seems that, for any starting state, moving
back and forth along the structure of DS1 and
counting whenever we reach a new state won’t
allow us to produce such a complex p*-1.
The odd representation ( p*-1, DSp*) is not
dynamically effective
Thus, it seems that the “good”
representation (i, DS1) can be constructed
effectively by means of a mechanical
procedure that takes as given the whole
structure of the state space M of DS1;
 while the “odd” one (p*-1, DSp*) cannot be
constructed effectively in this way.
 To distinguish the two kinds of
representations, let us then introduce the
concept of a dynamically effective
representation.

Dynamically effective representation
(condition 3 is not formal)

A dynamically effective representation of a
discrete dynamical system DS = (M, (gt)tT) is a
pair (u, DS#) such that:
1.
DS# = (N, (ht)tT) is a discrete dynamical
system, where N is either Z+ or a finite initial
segment of Z+;
2.
u: N  M is an isomorphism of DS# in DS;
3.
the enumeration u: N  M can be constructed
effectively by means of a mechanical
procedure that takes as given the whole
structure of the state space M of DS (and
nothing more).
Lines for a formal analysis of
condition 3

Condition 3 of the previous definition can be
analyzed once we make clear what we mean by:




whole structure of the state space;
mechanical procedure that takes such a structure as
given.
In extreme synthesis: the state-space structure
can be identified with a special kind of connected
(infinite) graph, which can assume nine types of
general forms;
the mechanical procedure is the one executed by
a special kind of ideal machine, which can move
back and forth along the edges of such graphs
and “count” 0, 1, 2, ... , n, ... whenever it
reaches a new node.
The second definition of a
computational system. Is it intrinsic?




Thus, we now have two possible formal
explications of the intuitive idea of an effective
representation of a discrete DS;
the first definition is the basis for the concept of a
computational1 system. But this concept is not
intrinsic to the dynamics of DS, for it depends on
the way we numerically represent such dynamics;
on the basis of the second definition, we can now
define: DS is a computational2 system iff DS is a
discrete dynamical system, and there is a
dynamically effective representation of DS.
Question: is the property of being computational2
intrinsic to the dynamics of the discrete system
DS?
Being computational2 is intrinsic

First, being computational2 is intrinsic to the
dynamics of a discrete DS in an obvious, but not
trivial, sense: for DS has a numeric
representation (u, DS#) whose enumeration u: N
 M is constructed effectively by means of a
mechanical procedure that takes as given the
whole structure of the state space M of DS, i.e.,
the dynamics of DS.

Second, there is a strong informal argument in
favor of the conjecture that any two dynamically
effective representations of the same DS are
either both recursive or both non-recursive.
Two scenarios for computability theory

If (i) we buy the second definition of a
computational system and (ii) the previous
conjecture is true, there are two possible
scenarios:
1.
any computational2 system DS is intrinsically
recursive, i.e., for any dynamically effective
representation (u, DS# = (N, (ht)tT)) of DS, the
dynamics (ht)tT turns out to be recursive;
2.
some computational2 system DS is intrinsically
non-recursive, i.e., for any dynamically effective
representation (u, DS# = (N, (ht)tT)) of DS, the
dynamics (ht)tT turns out to be non-recursive.
Consequences for Turing-Church’s
thesis as a mathematical thesis
Turing-Church’s thesis (TC-thesis) can be
interpreted in many different ways. The
Mathematical TC-thesis (MTC-thesis) can
be expressed as follows:
 any numeric function that can be
computed by a computational system (in
the intuitive sense) is recursive.
 But then, provided that computational2 is
a good explication for the intuitive concept
of a computational system, it is clear that
the truth of either scenario (1) or scenario
(2) entails, respectively, the truth or
falsity of MTC-thesis.

That’s all
Thank you