Dynamical Systems on Monoids: Toward a General Theory of Deterministic Systems and Motion

Abstract

Dynamical systems are mathematical structures whose aim is to describe the evolution of an arbitrary deterministic system through time, which is typically modeled as (a subset of) the integers or the real numbers. We show that it is possible to generalize the standard notion of a dynamical system, so that its time dimension is only required to possess the algebraic structure of a monoid: first, we endow any dynamical system with an associated graph and, second, we prove that such a graph is a category if and only if the time model of the dynamical system is a monoid. In addition, we show that the general notion of a dynamical system allows us not only to define a family of meaningful dynamical concepts, but also to distinguish among a cluster of otherwise tangled notions of reversibility, whose logical relationships are finally analyzed.

Full text

173
DYNAMICAL SYSTEMS ON MONOIDS:
TOWARD A GENERAL THEORY OF DETERMINISTIC
SYSTEMS AND MOTION
MARCO GIUNTI, CLAUDIO MAZZOLA
Dipartimento di Filosofia e Teoria delle Scienze Umane, Università di Cagliari
Via Is Mirrionis 1, 09123 Cagliari, Italy
E-mail: giunti@unica.it mazzola.c@gmail.com
Dynamical systems are mathematical structures whose aim is to describe the evolution of an
arbitrary deterministic system through time, which is typically modeled as (a subset of) the integers
or the real numbers. We show that it is possible to generalize the standard notion of a dynamical
system, so that its time dimension is only required to possess the algebraic structure of a monoid:
first, we endow any dynamical system with an associated graph and, second, we prove that su ch a
graph is a category if and only if the time model of the dynamical system is a monoid. In addition,
we show that the general notion of a dynamical system allows us not only to define a family of
meaningful dynamical concepts, but also to distinguish among a cluster of otherwise tangled notions
of reversibility, whose logical relationships are finally analyzed.
Keywords: dynamical system, reversibility, irreversibility, category theory.
1 Introduction
A dynamical system is a kind of mathematical model that purports to formally capture the
intuitive notion of an arbitrary deterministic system, either reversible or irreversible, with
discrete or continuous time or state space (Arnold 1977 [1]; Szlenk 1984 [6]; Giunti 1997
[3]; Hirsch, Smale and Devaney, 2004 [2]). Let Z be the integers, Z+ the non-negative
integers, R the reals and R+ the non-negative reals; below is a standard definition of a
dynamical system.
Definition 0: DS is a dynamical system iff DS is a pair (M, (gt)t T) such that
(i) T is either Z, Z+, R, or R+. Any t T is called a duration of the system, and T is
called its time set;
(ii) M is a non-empty set. Any x M is called a state of the system, and M is called its
state space;
(iii) (gt)t T is a family indexed by T of functions from M to M. For any t T, the
function gt is called the state transition of duration t (briefly, t-transition, or
t-advance) of the system;
(iv) for any v, t T, for any x M,
(a) g0(x) = x;
(b) gv+t(x) = gv(gt(x)).
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174 M. Giunti and C. Mazzola
Examples of dynamical systems with discrete time and discrete state space are Turing
machines and cellular automata; with discrete time and continuous state space: systems
specified by difference equations (e.g. iterated mappings on R); with continuous time and
continuous state space: systems specified by ordinary differential equations.
Definition 0 captures the intuitive notion of a deterministic system in the following
sense. In the first place, condition (iii) should be interpreted as telling us the state of the
system after an evolution of an arbitrary duration t T, provided that the state of the
system at the present time t0 T is known; in other words, if at instant t0 the system is in
state x M, then at instant t+t0 the system is in state gt(x). In addition, condition (iv.a)
tells us that, whatever state the system is in, the evolution of duration 0 does not modify
that state; and, finally, condition (iv.b) tells us that any evolution of duration v+t can
always be decomposed in two successive evolutions, the first one of duration t, and the
second one of duration v.
However, Definition 0 is not fully explicit, for it does not make clear exactly which
structure on the time set T is needed, in order to support appropriate dynamics for the
system. By condition (i), T is either Z, Z+, R, or R+. With respect to the addition
operation, these four models share the structure of a linearly ordered commutative
monoid; but it is by no means obvious that all this structure on T is needed for a general
definition of a dynamical system.
It is our contention that the minimal structure on the time set that underpins a
materially adequate definition of a dynamical system is just that of a monoid (sec. 2,
Definition 1). To substantiate our tenet we will focus first on the directed graph that any
dynamical system induces on its state space, and on a revealing link between this graph
and category theory. We will then prove that such a graph can be made into a category if,
and only if, the algebraic structure of the time set is that of a monoid (sec. 3, Theorem 1).
Finally, we will show how an algebraic time model as simple as a monoid is nevertheless
sufficient to support a variety of significant dynamical concepts, as well as a rich web of
relations among them (sec. 4).
2 Dynamical systems on monoids
To start with, we make the algebraic structure of the set T of durations explicit, by
requiring that a binary operation + be given on T and the pair L = (T, +) be a monoid. We
then define the notion of a dynamical system on L by modifying Definition 0 as follows.
Definition 1: DSL is a dynamical system on L iff DSL is a pair (M, (gt)t T) and L is a
pair (T, +) such that
(i) L = (T, +) is a monoid. Any t T is called a duration of the system, T is called its
time set, and L its time model;
(ii) M is a non-empty set. Any x M is called a state of the system, and M is called its
state space;
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Dynamical Systems on Monoids: Toward a General Theory … 175
(iii) (gt)t T is a family indexed by T of functions from M to M. For any t T, the
function gt is called the state transition of duration t (briefly, t-transition, or
t-advance) of the system;
(iv) for any v, t T, for any x M,
(a) g0(x) = x, where 0 is the unity of L;
(b) gv+t(x) = gv(gt(x)).
Definition 2: DSL is a possible dynamical system on L iff DSL is a pair (M, (gt)t T) and
L is a pair (T, +) such that
T is a non-empty set and + is a binary operation on T;
M is a non-empty set and (gt)t T is a family indexed by T of functions from M to M.
Definition 3: f is a ρ-isomorphism of DSL2 in DSL1 iff DSL2 = (N, (hv)v V) is a possible
dynamical system on L2 = (V, ), DSL1 = (M, (gt)t T) is a possible dynamical system on
L1 = (T, +), ρ is an isomorphism of L2 in L1, and f: N → M is a bijection such that, for any
x N and v V, f(hv(x)) = gρ(v)(f(x)).
Definition 4: DSL2 is isomorphic to DSL1 iff for some f and ρ, f is a ρ-isomorphism of
DSL2 in DSL1.
By Definition 4, it is easy to verify:
Proposition 1: Being isomorphic to is an equivalence relation on any given set of
possible dynamical systems.
It is also not difficult to show that the relation of isomorphism is compatible with the
property of being a dynamical system on a monoid, that is to say,
Proposition 2: If DSL2 is isomorphic to DSL1 and DSL2 is a dynamical system on L2, then
DSL1 is a dynamical system on L1.
This allows us to speak of abstract dynamical systems on monoids in exactly the
same sense we talk of abstract groups, fields, lattices, order structures, etc. We thus
define:
Definition 5: AS is an abstract dynamical system iff AS is an equivalence class of
isomorphic dynamical systems.
Definition 6: P is a dynamical property iff for any two possible dynamical systems DSL2
and DSL1,
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176 M. Giunti and C. Mazzola
(i) if DSL2 has P, then DSL2 is a dynamical system on L2;
(ii) if DSL2 has P, and DSL2 is isomorphic to DSL1, then DSL1 has P.
By Definition 6, a dynamical property is proper to dynamical systems and preserved
by isomorphism. Dynamical properties can thus be regarded as the specific structural
properties of dynamical systems. It is then easily shown:
Proposition 3: any two dynamical systems on monoids have exactly the same dynamical
properties iff they are isomorphic.
Proof: If two dynamical systems on monoids are isomorphic, then by condition (ii) of
Definition 6, they have exactly the same dynamical properties. Conversely, for any two
non-isomorphic dynamical systems on monoids, DSL1 and DSL2, there is a dynamical
property they do not share; namely, the property of being isomorphic to DSL1. Q.E.D.
By general dynamical systems theory we mean the mathematical theory whose
Suppes' style axiomatization (1957, ch. 12) is given by Definition 1. Since general
dynamical systems theory is programmatically concerned with the study of dynamical
properties, it regards any two isomorphic dynamical systems on monoids as identical.
3 Transition graphs and their relation to category theory
A graph in Lambek's sense (Marquis, 2010) is a quadruple (X, A, σ, τ), where X and A are
non-empty sets, while both σ and τ are functions from A to X. An element of X is called
an object, node, point or vertex of the graph, while a member of A is called an arrow or a
directed edge of the graph. For any arrow a A, σ(a) is called its source, and τ(a) its
target. With the notation x a→ y we mean that a is an arrow with source x and target y
(briefly, a is an arrow from x to y).
A deductive system is a graph together with a family (idx)x X of arrows and a partial
binary operation on A that satisfy (i) for any x X, σ(idx) = x and τ(idx) = x; (ii) for
any x, y, z X, for any a, b A, if x a→ y and y b→ z, then x b a→ z. For any x X, idx is
called the x-identity arrow, and the partial operation is called arrow composition. (For
ease of understanding, we suggest reading “b a” as “b following a”.)
Finally, a category is a deductive system that also satisfies (iii) for any w, x, y, z X,
for any a, b, c A, if w a→ x, x b→ y and y c→ z, then c(b a) = (c b)a; (iv) for any
a A, idτ(a)a = a and aidσ(a) = a.
There is an interesting link between general dynamical systems theory and category
theory. Let us first of all notice that the family of t-transitions of a possible dynamical
system DSL naturally gives rise to a particular graph in Lambek's sense. We call this
graph the transition graph of a possible dynamical system; the definition is below.
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Dynamical Systems on Monoids: Toward a General Theory … 177
Definition 7: G(DSL) is the transition graph of DSL iff DSL = (M, (gt)t T) is a possible
dynamical system on L = (T, +) and G(DSL) is the quadruple (M, A, σ, τ) such that
(i) A = {a: for some t T, x M, a = (x, t, gt(x))};
(ii) σ: A → M such that, for any triple a A, σ(a) is its first element;
(iii) τ: A → M such that, for any triple a A, τ(a) is its last element.
From now on we indicate an arbitrary arrow a = (x, t, gt(x)) of a transition graph with
the notation x t
→ gt(x). We remark that this notation exclusively applies to arrows of
transition graphs. It should not be confused with the more general notation y b→ z, which
instead applies to an arbitrary arrow b of any graph. Also note that the first notation is
just a different way of writing the triple (x, t, gt(x)) = a, while the general notation does
not stand for an arrow, but it is rather an abbreviation for the statement “b is an arrow
with source y and target z”.
A quasi-dynamical system DSL on L = (T, +) differs from a dynamical system just for
the fact that the binary operation + not necessarily is associative; in other words, the time
model L is not assumed to be a monoid, but a magma with unity. Below is the precise
definition.
Definition 8: DSL is a quasi-dynamical system on L iff DSL is a pair (M, (gt)t T) and L is
a pair (T, +) such that
(i) L = (T, +) is a magma with unity;
(ii) M is a non-empty set;
(iii) (gt)t T is a family indexed by T of functions from M to M;
(iv) for any v, t T, for any x M,
(a) g0(x) = x, where 0 is the unity of L;
(b) gv+t(x) = gv(gt(x)).
Let DSL be a quasi-dynamical system on L and G(DSL) = (M, A, σ, τ) be its transition
graph. We now equip G(DSL) with a family of x-identity arrows (idx)x M , where, for any
x M, idx = x 0
→ g0(x).
As for the operation of arrow composition, we define it as follows. For any two
arrows a = x t
→ gt(x) and b = gt(x) u
→ gu(gt(x)), b a = x u+t
→ gu+t(x).
We show below that the transition graph G(DSL) of a quasi-dynamical system DSL,
together with the family of x-identity arrows (idx)x M and the operation of arrow
composition , is a category if and only if L is a monoid.
Theorem 1: Let DSL = (M, (gt)t T) be a quasi-dynamical system on L = (T, +), and
G(DSL) = (M, A, σ, τ) be the transition graph of DSL; let (idx)x M be the family of
x-identity arrows, and be the operation of arrow composition, as defined above.
G*(DSL) = (M, A, σ, τ, (idx)x M , ) is a category iff L is a monoid.
Proof: We show first that G*(DSL) is a deductive system. Condition (i) of the definition
of a deductive system follows from the definition of the family (idx)x X and from
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178 M. Giunti and C. Mazzola
condition (iv.a) of Definition 8; as for condition (ii), it follows from the definition of the
operation and from condition (iv.b) of Definition 8.
Next, we show that, if L is a monoid, then G*(DSL) is a category as well. For any three
arrows a = x t
→ gt(x), b = gt(x) u
→ gu(gt(x)), c = gu(gt(x)) v
→ gv(gu(gt(x))), we have:
c(b a) = c(x u+t
→ gu+t(x)) = (1)
c(x u+t
→ gu(gt(x)) = x v+(u+t)→ gv(gu(gt(x)))
(c b)a = (gt(x) v+u
→ gv+u(gt(x))))a = (2)
(gt(x) v+u
→ gv(gu(gt(x))))a = x (v+u)+ t→ gv(gu(gt(x)))
As L is a monoid, from (1) and (2), and by associativity of +, c(b a) = (c b)a; thus,
condition (iii) of the definition of a category holds.
Finally, condition (iv) of the definition of a category holds as well, since for any three
arrows idσ(a) = x 0
→ g0(x), a = x t
→ gt(x), idτ(a) = gt(x) 0
→ g0(gt(x)), we have:
idτ(a)a = (gt(x) 0
→ g0(gt(x)))a = (3)
(gt(x) 0
→ gt(x)) a = x 0+t
→ gt(x) = x t
→ gt(x) = a
aidσ(a) = a(x 0
→ g0(x)) = a (x 0
→ x)) = (4)
x t+0
→ gt(x) = x t
→ gt(x) = a
Therefore, if L is a monoid, then G*(DSL) is a category.
Conversely, we show that, if L is not a monoid, then G*(DSL) is not a category, for
condition (iii) of the definition of a category fails. Suppose L is not a monoid. Then, as L
is a magma with unity, + is not associative. Hence, for some v, u, t T, v+(u+t) ≠
(v+u)+t, and thus x v+(u+t)→ gv(gu(gt(x))) ≠ x (v+u)+t→ gv(gu(gt(x))). On the other hand, for
any three arrows a = x t
→ gt(x), b = gt(x) u
→ gu(gt(x)), c = gu(gt(x)) v
→ gv(gu(gt(x))),
equations (1) and (2) above hold. Therefore, c(b a) ≠ (c b)a, so that condition (iii) of
the definition of a category fails. Q.E.D.
Theorem 1 provides us with a justification for our claim that the minimal structure on
the time set that supports a materially adequate definition of a dynamical system is at
least that of a monoid. For, if it is not, the transition graph of the system cannot even be
made into a category.
4 Motion, orbits, and types of reversibility/irreversibility in dynamical
systems on monoids
In this section we show that general dynamical systems theory, though based on a time
model as simple as a monoid, is nevertheless sufficient to define a variety of genuine
dynamical concepts, as well as to prove about them significant and sometimes even
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Dynamical Systems on Monoids: Toward a General Theory … 179
surprising results. Due to space limits, we will often skip details of the proofs, or we will
even omit some proofs.
We start with the observation that the minimal framework of general dynamical
systems theory provides us with a quite general concept of motion. Let Y and Z be any
two sets; let eval: ZYY Z such that, for any function f ZY, for any y Y, eval(f, y) =
f(y); we then define:
Definition 9: gx is the motion with initial state x of DSL iff DSL = (M, (gt)t T) is a
dynamical system on L = (T, +), for any x M, for any t T, gx: T M and
gx(t) = eval(gt, x). ( gx is also called the x-motion of DSL or the x-evolution of DSL.)
Intuitively, for any x M, the x-evolution gx represents the motion of the system
when the state at the initial time t0 is x; more precisely, if the state at t0 is x, then, for any
t T, gx(t) is the state at time t + t0.
Let DSL = (M, (gt)t T) be a dynamical system on L = (T, +), and x M; we then
define:
Definition 10:
(i) The orbit of x = orb(x) = {z: z = gt(x), for some t T};
(ii) r is an orbit iff for some x M, r = orb(x);
(iii) the phase portrait = {r: r is an orbit}.
Note that, by Definition 10.i and Definition 9, for any x M, orb(x) = the image
of gx.
From an intuitive point of view, if a system is deterministic and two orbits have a
state in common, then, from that state on, they must coincide; that is to say, deterministic
systems do not admit crossing orbits. The following proposition expresses exactly this
fact.
Proposition 4: Let DSL = (M, (gt)t T) be a dynamical system on L = (T, +). For any x, y,
z M, if z orb(x) and z orb(y), then orb(z) orb(x) and orb(z) orb(y).
Proof: By Definition 10.i and condition (iv.b) of Definition 1. Q.E.D.
As the time model L = (T, +) of a dynamical system DSL is a monoid, for any t T,
there is at most one inverse; when the inverse of t exists, we indicate it by −t. The next
proposition highlights three interesting consequences of the existence of the inverse −t of
a duration t T.
Proposition 5: Let DSL = (M, (gt)t T) be a dynamical system on L = (T, +). For any t T,
if the inverse −t of t exists, then
(i) gt is a bijection;
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180 M. Giunti and C. Mazzola
(ii) g−t is the inverse of gt with respect to the operation of function composition ;
(iii) g−t = (gt)−1, where (gt)−1 is the inverse function of gt.
Proof: both (i) and (ii) follow from conditions (iv.a) and (iv.b) of Definition 1; (iii)
follows from (ii) and the fact that (gt)−1 is the inverse of gt with respect to . Q.E.D.
Let DSL = (M, (gt)t T) be a dynamical system on L = (T, +), x M, and t T; we
then define:
Definition 11:
(i) x is a fixed point iff orb(x) = {x};
(ii) x is a periodic point iff for some t T, t ≠ 0 and gt(x) = x;
(iii) t is a period of x iff t ≠ 0 and gt(x) = x;
(iv) t is the period of x iff t is a period of x and, for any t* T, if t* is a period of x,
t* ≠ t and t* ≠ −t, then there is k 2 such that t* = t+t…k times or t* = −t + −t …k times .
Note that, by Definitions 11.iii and 11.ii, if t is a period of x, x is a periodic point;
conversely, if x is a periodic point, there is t such that t is a period of x, but not necessarily
is such a t unique. Also, by Definition 11.iii and Proposition 5, if t is a period of x and the
inverse −t of t exists, then −t is a period of x as well.
By Definition 11.iv, if t is the period of x, then (i) t is unique iff the inverse of t does
not exist, or −t = t; (ii) if t is not unique, then −t is the period of x as well, but nothing
else is the period of x.
Proposition 6: If x is a fixed point, then x is a periodic point and, for any t T, if t ≠ 0, t
is a period of x.
Proof: By Definition 11.i, 11.ii and 11.iii. Q.E.D.
For an arbitrary dynamical system DSL , we are now able to distinguish three
mutually exclusive and jointly exhaustive types of orbit: periodic, eventually periodic,
and aperiodic. Let DSL = (M, (gt)t T) be a dynamical system on L = (T, +); we define:
Definition 12:
(i) r is a periodic orbit iff for some x M, r = orb(x) and x is a periodic point;
(ii) r is an eventually periodic orbit iff r is an orbit, r is not a periodic orbit and, for
some x r, orb(x) is a periodic orbit;
(iii) r is an aperiodic orbit iff r is an orbit, r is not a periodic orbit, and r is not an
eventually periodic orbit.
Intuitively an orbit is merging when it shares a state with a different orbit and, from
that state on, the two orbits coincide; this idea is expressed by the next definition.
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Dynamical Systems on Monoids: Toward a General Theory … 181
Definition 13: r is a merging orbit iff for some x, y M, r = orb(x), orb(x)orb(y) ≠ ,
¬(orb(x) orb(y)) and ¬(orb(y) orb(x)).
By combining Definitions 12 and 13 we obtain a partition of the phase portrait of an
arbitrary dynamical system DSL into six orbit types. It is thus interesting to study how the
instantiation pattern of the six orbit types varies with specific characters of the systems
considered.
For example, if we only consider dynamical systems on Z or Z+, and we also take into
account whether they have finite or infinite state space M, we get the instantiation patterns
shown in Table 1.
Table 1. Instantiation of orbit types in four different types of
dynamical systems.
Dynamical System Types
Orbit
Types
DSZ and
M finite
DSZ+ and
M finite
DSZ and
M infinite
DSZ+ and
M infinite
(i.a)
0
0
0
0
(i.b)
1
1
1
1
(ii.a)
0
1
0
1
(ii.b)
0
1
0
1
(iii.a)
0
0
0
1
(iii.b)
0
0
1
1
Note: M = state space; (i.a) = periodic and merging, (i.b) = periodic and not merging, (ii.a) = eventually
periodic and merging, (ii.b) = eventually periodic and not merging, (iii.a) = aperiodic and merging, (iii.b) =
aperiodic and not merging; 1 = orbit type is instantiated, 0 = orbit type is not instantiated.
Let DSL = (M, (gt)t T) be a dynamical system on L = (T, +); the family of t-transitions
(gt)t T allows us to introduce purely dynamical concepts of past and future, as follows.
Let 0 be the unity of L, and x M:
Definition 14:
(i) Pt(x) is the t-past of x iff t T{0} and Pt(x) = {y: gt(y) = x};
(ii) Ft(x) is the t-future of x iff t T{0} and Ft(x) = {y: gt(x) = y};
(iii) P(x) is the past of x iff P(x) = t T {0} Pt(x);
(iv) F(x) is the future of x iff F(x) = t T {0} Ft(x).
Note that, by Definition 14.ii, for any x M and t T{0}, Ft(x) is a singleton.
Analogous definitions can be given for a set of states X M.
It is quite unexpected that general dynamical systems theory might provide us with
fine distinctions among many different concepts of reversibility/irreversibility. However,
within this theory we can in fact distinguish at least eight notions of reversibility. Let
DSL = (M, (gt)t T) be a dynamical system on L = (T, +); we define:
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182 M. Giunti and C. Mazzola
Definition 15:
(i) DSL is reversible iff for any x M, for any t T, for some w T, gw(gt(x)) = x;
(ii) DSL is strictly reversible iff for any t T, for some w T, for any x M,
gw(gt(x)) = x;
(iii) DSL is logically reversible iff for any t T, gt is injective;
(iv) DSL has complete past iff for any t T, gt is surjective;
(v) DSL is completely logically reversible iff for any t T, gt is bijective;
(vi) DSL is time symmetric iff DSL is completely logically reversible and there is
: M → M such that, for any x M, for any t T, (gt( (x))) = (gt)−1(x);
(vii) DSL is space invertible iff there is : M → M such that, for any x M, for any t T,
gt( (gt( (x)))) = x;
(viii) DSL is time invertible iff L = (T, +) is a group.
Obviously, by negating each of the previous concepts of reversibility, we obtain eight
corresponding concepts of irreversibility.
The order of the eight reversibility concepts does not correspond to their logical
strength; in particular, (i) and (viii) are not, respectively, the weakest and the strongest
concept; rather, (i) and (viii) are the weakest and the strongest proper concept of
reversibility there is only one more proper concept of reversibility, i.e. (ii). The
following proposition provides details about the exact logical relations among the eight
reversibility concepts of Definition 15.
Proposition 7:
(i) (15.viii) → (15.ii);
(ii) (15.ii) → (15.i);
(iii) (15.v) → (15.iv);
(iv) (15.v) → (15.iii);
(v) (15.vii) → (15.v);
(vi) (15.vii) ↔ (15.vi)
(vii) (15.ii) → (15.v).
Proof:
(i) Follows from Definitions 15.viii and 15.ii, and Proposition 5;
(ii) follows from Definitions 15.ii and 15.i;
(iii) follows from Definitions 15.v and 15.iv;
(iv) follows from Definitions 15.v and 15.iii;
(v) suppose DSL is space invertible; hence, by Definition 15.vii, there is : M → M such
that, for any x M, for any t T,
gt( (gt( (x)))) = x (5)
thus, by applying to both members of (5),
(gt( (gt( (x))))) = x (6)
consequently, by substituting (x) for x in (6),
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Dynamical Systems on Monoids: Toward a General Theory … 183
(gt( (gt( ( (x)))))) = (x)) (7)
on the other hand, by setting t = 0 in (5),
( (x)) = x (8)
so that, by (8), is an involution on M and, by (8) and (7), for any t T, the
composed function gt is also an involution on M. Therefore, both gt and are
bijections. Injectivity of gt and the fact that is an involution entail injectivity of
gt; also, surjectivity of gt, the fact that is an involution, and surjectivity of
entail surjectivity of gt. Therefore, by Definition 15.v, DSL is completely logically
reversible;
(vi) the implication from right to left follows from Definitions 15.vi and 15.vii; the
implication fron left to right follows from Definition 15.vii, thesis (v), and
Definition 15.vi;
(vii) logical reversibility follows easily from Definitions 15.ii and 15.iii by reduction; in
turn, also by reductio, complete past follows from Definitions 15.ii, 15.iv, and 15.iii.
Q.E.D.
Proposition 7 describes all the implications among the eight reversibility concepts of
Definition 15; in other words, no other implication holds, except for those licensed by
transitivity. To prove this claim, however, we need several counterexamples, which go
beyond the scope of this paper. We only show below some of the most significant
counterexamples.
Example 1: ¬((15.i) → (15.iii)) and ¬((15.i) → (15.iv)). This is shown by the dynamical
system DSL defined below. DSL is in fact reversible, logically irreversible, and with
incomplete past. Also note that L is a non-commutative monoid.
Let L = (T, +), where T = {0, 1, 2, 3} and the sum operation + is defined by table 2.
Table 2. The sum operation +.
+
0
1
2
3
0
0
1
2
3
1
1
1
2
3
2
2
1
2
3
3
3
1
2
3
Note: Read column+row, as reading order matters.
Let DSL = (M, (gt)t T), where M = {x1, x2, x3} and, for any t T, the t-transition gt is
defined by table 3.
Table 3. The t-transitions family (gt)t T .
g0
g1
g2
g3
x1
x1
x2
x1
x3
x2
x2
x2
x1
x3
x3
x3
x2
x1
x3
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184 M. Giunti and C. Mazzola
Example 2: ¬((15.v) → (15.i)). This is shown by the dynamical system DSL =
(Z, (sk)k Z+) on L = (Z+, +), where s0 is the identity function on Z and, for any k > 0, sk is
the k-th iteration of the successor function on Z. Then, obviously, DSL is both completely
logically reversible and irreversible.
Example 3: ¬((15.v) → (15.vi)). This is shown by the dynamical system DSL =
(Z, (gt)t) on L = (T, ), where T is the set of all bijective functions on Z, is the
operation of function composition and, for any t and any x gt(x)=t(x). On the
one hand, DSL is completely logically reversible, as all its state transitions are bijective
by definition.
On the other hand, suppose for reductio DSL is time-symmetric; then, by Definition 15.vi,
there is : Z → Z such that, for any x Z, for any t T,
(gt( (x))) = (gt)−1(x) (9)
hence, by setting in (9) t = 0 = the identity function on Z, turns out to be an involution
on Z, and thus a bijection on Z, so that T. Also, by applying to both sides of (9),
(gt( (x)))) = ((gt)−1(x)) (10)
by (10), as is an involution,
gt( (x)) = ((gt)−1(x)) (11)
since is an involution, = ( )−1, so that by (11),
gt( (x)) = ( )−1((gt)−1(x)) (12)
for any u, v T, (u)−1 (v)−1 = (v u)−1, thus by (12),
gt( (x)) = (gt−1(x) (13)
by (13), (9), and being an involution,
gt( (x)) = (gt( (x))) = (gt(((x)))) = (gt(x)) (14)
by applying to both sides of (14), and being an involution,
(gt( (x))) = ( (gt(x))) = gt(x) (15)
by (15) and (9),
(gt)−1(x) = gt(x) (16)
that is to say, all bijective functions on Z would be involutions, which is plainly false.
Hence, DSL is not time-symmetric.
Example 4: ¬((15.vii) → (15.i)). Let DSL = (Z, (gk)k) be the dynamical system on L =
(Z+, ) such that, for any z Z and any t Z+, gt(x) = t + x. DSL is space-invertible, for
consider the function :Z Z, (x) −x; then, for any x Z and any k Z+,
gk( (gk( (x)))) = k + (−(k + (−x))) = k −k + x = x (17)
on the other hand, DSL is not reversible, for let x and k > 0; then, gk(x) = k + x > x, and
thus, for any n Z+, gn(gk(x)) = n + k + x ≥ k + x > x.
A concept related to the notions of reversibility/irreversibility is that of a garden of
Eden, that is, a state without past. Let DSL = (M, (gt)t T) be a dynamical system on
L = (T, +), and x M; we define:
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Dynamical Systems on Monoids: Toward a General Theory … 185
Definition 16: x is a garden of Eden iff P(x) = .
It is then easy to show:
Proposition 8: Let DSL = (M, (gt)t T) be a dynamical system on L = (T, +). If DSL is
reversible, then for any x M, x is not a garden of Eden; the converse implication is
false.
Proof: If DSL is reversible, then for any x M, and any t T−{0}, there is w T such
that gw(gt(x)) = x; but then, if w ≠ 0, x is not a garden of Eden and, if w 0, gt(x) = x, so
that x is not a garden of Eden either. Hence, DSL has no garden of Eden. Falsity of the
converse is shown by Example 2 above. Q.E.D.
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