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What is a Physical Realization of a Computational System?

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Abstract

The concept of a physical realization of a computational system is one of the key notions of both functionalism and the symbolic approach to cognitive science or AI. Notwithstanding a widespread consensus on the theoretical importance of this concept, it comes somewhat as a surprise that a precise analysis or shared definition does not exist yet, either in the philosophical camp, or in the cognitive science and AI community. This paper develops an original approach to such an analysis.
Marco Giunti, “What is a Physical Realization of a Computational System?”
in Flavia Marcacci and Maria Grazia Rossi, Reasoning, metaphor and science, pp. 177-192.
© 2017 Isonomia, On-line Journal of Philosophy—Epistemologica—ISSN 2037-4348
University of Urbino Carlo Bo
http://isonomia.uniurb.it/epistemologica
What is a Physical Realization
of a Computational System?
Marco Giunti
ALOPHIS -- Applied LOgic, Philosophy and HIstory of Science*
Università di Cagliari
giunti@unica.it
1. Introduction
The concept of a physical realization of a computational system is one of the
key notions of both functionalism and the symbolic approach to cognitive
science or AI (Pylyshyn 1980: 113, 1984; Chalmers 1996: 309).
Notwithstanding a widespread consensus on the theoretical importance of
this concept, it comes somewhat as a surprise that a precise analysis or
shared definition does not exist yet, either in the philosophical camp, or in
the cognitive science and AI community1. In this paper, I will present my
attempt at such an analysis. Before doing this, however, I will say a few
words on why I believe that a seemingly promising alternative strategy is
misguided.
*Postal address: Dipartimento di Pedagogia Psicologia Filosofia, via Is Mirrionis 1, 09123
Cagliari, Italy.
1 Endicott (2005) distinguishes at least three different technical meanings of the term
‘realization’ among philosophers. Putnam claims that “every ordinary open system is a
realization of every abstract finite automaton” (1988: 121) and, in the same vein, Searle
(1990) argues that any physical system can be seen to realize any computation. Both
Putnam’s and Searle’s arguments are clearly flawed (Chalmers 1996), but they do show
that a precise explication of the notion of realization of a computational system is badly
needed.
178 Reasoning, Metaphor and Science
According to this strategy, both a computational system and its multiple
physical realizations are to be thought as dynamical systems2, and the
realization relation is then analyzed as a special kind of structure preserving
mapping from the state space of the computational system into the state
space of the physical one (Giunti 1997: ch. 1; for a similar approach, see
Fano et al. 2014). As the same kind of structure preserving mapping (i.e., an
emulation function)3 may very well exist between a single computational
system and many physical systems, it might seem that we have a clearcut
explanation of the basic feature of the realization relation, namely, multiple
(low level) realizability of the same (high level) system.
However, there are two main reasons why the realization relation
cannot be analyzed in terms of some form of emulation of a computational
system by a physical one. The first is that emulation is a structure preserving
mapping between two mathematical systems (i.e., dynamical systems), and
thus this kind of approach merely shifts the problem, for the realizing
system (the physical one) is not a real or concrete system, but another
mathematical or abstract system. So the question now is: what is a
realization of a physical system? This question is not unsolvable in
principle, but the only solution I can think of is subscribing to a form of
Platonism, according to which at least some mathematical systems (those at
the lowest level) are real.
Second, the emulation based strategy provides us with an in principle
solution, which makes us think of realization as a relation between two
dynamical systems — a high level computational system on the one hand,
and a low level physical one on the other. However, no one would maintain
that we could in fact know the details of the low level physical system, for
2 Giunti and Mazzola (2012: Definition 1) define a deterministic dynamical system on any
time model L = (T, +) that satisfies the minimal requirement of being a monoid, as follows.
DS is a dynamical system on L := DS 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; (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 identity element of L; (b) gv+t(x) = gv(gt(x)).
3 Let DS1 = (M, (gt)t
T
) be a dynamical system on L = (T, +), and DS2 = (N, (hv)v
V
) be a
dynamical system on P = (V, ). An emulation function u of DS2 in DS1 is defined as
follows (Giunti 2010a: sec. 2, 2014: sec. 2, Definition 8); u is an emulation of DS2 in DS1 :=
u is an injective function from N to M and, for any v V, for any c N, there is t T such
that u(hv(c)) = gt(u(c)). Furthermore, we say that DS1 emulates DS2 just in case there is an
emulation u of DS2 in DS1.
GIUNTI, What is a Physical Realization of a Computational System? 179
those details would admittedly be too complex. I believe that this kind of in
principle solution is not in agreement with the question we have in mind
when we ask what it means for a computational system to be realized by a
physical one. This question does not ask for abstract in principle answers,
but for detailed and concrete ones. For, in many cases, we can very well tell
that a real system is a physical realization of a computational one, even
though we do not have any low level description of the real system, nor do
we even mention such description, think of, or make any reference to it.
2. Computational systems as setup interpreted dynamical systems
My present strategy envisages that an adequate solution to the realization
problem will emerge once the quite complex nature of computational
systems is fully recognized and brought to light. According to this view,
computational systems are more similar to empirically correct dynamical
models than to dynamical systems tout court. Thus, the solution of the
realization problem is to be sought among the modeling relations between
dynamical systems and phenomena, and not among the emulation relations
between purely mathematical dynamical systems.4
In my view a computational system CS = (DS, H, IDS,H) is a complex
object which consists of three parts. First, a mathematical part DS =
(M, (gt)tT), which is a discrete5 n-component6 dynamical system.7 Second, a
4 While emulation is the wrong kind of relation for an analysis of realization, emulation is
indeed an adequate basis for a representational analysis of reduction of empirically
interpreted dynamical systems (Giunti 2010a, 2014). The two issues are closely related but,
in order to avoid conceptual confusion, they cannot be lumped together. The first one has to
do with a relation between a computational system and its realizers, where each realizer is a
concrete or real system. The second one deals with a relation between two mathematical
dynamical systems, both of which are models of real phenomena.
5 A dynamical system DS = (M, (gt)t
T
) is discrete := (i) the time model of DS is the
additive monoid (T, +), where the time set T is either the set of the non-negative integers
Z≥0 or the integers Z, and + is the usual operation of sum of two integers; (ii) the state space
M is at most countably infinite (Giunti 2010a: sec. 2, 2014: sec. 2).
6 For any i (1 in), let Xi be a non-empty set, and let DS = (M, (gt)t
T
) be a dynamical
system whose state space M X1× . . . ×Xn; for any i, the set Ci = {xi : for some n-tuple x
M, xi is the i-th element of x} is called the i-th component of M, and DS is called an
n-component dynamical system (Giunti 2014: sec. 4.1, 2016: Definition 2).
7 The two requirements of (i) being a discrete dynamical system with (ii) a finite number of
state space components make up a necessary, but not sufficient condition for the
mathematical part DS of a computational system. The debate on which other requirements
should be considered for an adequate definition of the purely mathematical part of a
computational system is still open (Gandy 1980; Giunti 1997, 2006; Giunti & Giuntini
180 Reasoning, Metaphor and Science
computational setup8 H = (F, BF), which is made up of a theoretical part F,
and a real part BF. And, third, an interpretation IDS,H, which links the
dynamical system DS with the setup H.
2.1. The computational setup H
Let us now see in more detail what the computational setup H = (F, BF)
looks like. First, its theoretical part F is a functional description which
provides a sufficiently detailed specification of:
a) the internal constitution and organization, or functioning, of any real
system of a certain type ASF;
b) a causal scheme CSF of the external interactions of any real system of
type ASF. In particular, the description of the causal scheme CSF must
include the specification of:
(b.1) the initial conditions that an arbitrary temporal evolution of any
real system of type ASF must satisfy;
(b.2) the boundary conditions during the whole subsequent evolution;
(b.3) and, possibly, the final conditions under which the evolution
terminates.
Second, the real part BF of the computational setup H is the set of all
real or concrete systems which satisfy the functional description F or, in
other words, BF is the set of all real systems of type ASF whose temporal
evolutions are all constrained by the causal scheme CSF. BF is called the
realization domain9 (or application domain) of H. Any real system bF BF is
called an F-realizer.
As H = (F, BF) is a computational setup, any temporal evolution of an
arbitrary F-realizer bF BF proceeds in discrete time steps, which
correspond to basic operation cycles of bF. A basic operation cycle can
always be interpreted as the (serial or parallel) execution, by bF, of a finite
2007; Sieg 1999, 2002a, 2002b; Shagrir 2002; Dershowitz & Gurevich 2008; Fano et al.
2014).
8 A computational setup is in fact a special kind of dynamical phenomenon, see Giunti
(2010a: sec. 4, 2010b: Appendice, 2014: sec. 4.1, 2016: sec. 3).
9 Since the functional description F typically contains several idealizations, no real system
exactly satisfies F, but it rather fits F up to a certain degree. Thus, from a formal point of
view, the realization domain BF of a computational setup H = (F, BF) would be more
faithfully described as a fuzzy set.
GIUNTI, What is a Physical Realization of a Computational System? 181
number of instructions for symbol manipulation. Therefore, the temporal
evolutions of an F-realizer are in fact its computations.
A clear example of a computational setup is provided by a typical
informal presentation of a Turing machine. An arbitrary Turing machine
setup, HTM = (FTM, BFTM), is detailed below.
FTM: Functional description of a Turing machine (theoretical part of a
Turing machine setup HTM)
(A) SPECIFICATION OF ANY REAL SYSTEM OF TURING MACHINE TYPE ASFTM
A real system of Turing machine type ASFTM is made up of two main parts:
an external part and an internal one. The external part consists of two
devices. First, the external memory, which may be thought as a linear
arrangement of a finite number of cells (e.g., a tape divided into squares).
Each cell always contains exactly one symbol taken from a finite alphabet
A = {b, a1, ... , am} of m+1 (m ≥ 1) different symbols. The number of cells is
always finite, but new cells can be added as needed (see below), either to
the left of the leftmost cell, or to the right of the rightmost one. However,
when a cell is added, it always contains the special symbol b, called the
blank.
Second, a read/write/move head, which is a device that, at each instant,
is located on exactly one cell of the external memory, and is capable of
doing three operations: (1) read the symbol contained in the cell where it is
located (scanned symbol), and send it to the control unit (in the internal
part) to which it is constantly connected; (2) replace the scanned symbol
with a symbol received from the control unit; (3) move one cell to the right,
one cell to the left, or stay put, according to the moving command +1, −1, or
0 that it receives from the control unit. If the head receives the command +1
(−1) when it is located on the rightmost (leftmost) cell, it first adds one
blank cell to the right (left), and then moves to the newly added cell.
The internal part of the system also consists of two devices. First, an
internal memory, which may be thought as a single cell that always contains
exactly one symbol (internal state) taken from a finite alphabet Q =
{q1, ... , qn} of n (n ≥ 1) different symbols. The internal memory is
constantly connected to the control unit and, besides being just a container
for an internal state, it is also capable of two operations: (1) read its internal
state, and send it to the control unit; (2) replace its internal state with an
internal state received from the control unit.
The second internal device is a control unit, which is a deterministic
input/output mechanism. Its inputs are state-symbol pairs qiaj, while its
outputs are symbol-movement-state triples akXql. (X stands for one of the
182 Reasoning, Metaphor and Science
three commands +1, −1, 0.) For any possible input pair qiaj, the control unit
always responds with a fixed output triple akXql. The input-output behavior
of the control unit is thus completely described by three finite tables, T1,
T2, T3, in which each input pair qiaj is listed together with, respectively, the
corresponding output symbol ak, the movement command X, or the new
internal state ql. Both the symbol ak and the command X of an output triple
are always sent to the head, while the new internal state ql is sent to the
internal memory.
An operation cycle of the whole system is as follows. First, the current
internal state q and the symbol a in the cell where the head is located are
simultaneously read and sent to the control unit, which responds with an
output triple a’Xq’. Second, the new symbol a’ and the moving command X
are sent to the head, which first replaces a with a’ and then moves according
to X; at the same time, the new internal state q’ is sent to the internal
memory, where it replaces q. The operation cycle is thus complete, and the
system is ready to start a new cycle.
(B) SPECIFICATION OF THE CAUSAL SCHEME CSFTM OF THE EXTERNAL
INTERACTIONS OF ANY REAL SYSTEM OF TURING MACHINE TYPE ASFTM
(1) Initial conditions. A sequence of operation cycles (i.e., a temporal
evolution, or a computation) of a system of type ASFTM starts as soon
as the following settings are performed: (i) an initial internal state is
fixed, (ii) a symbol in each cell of the external memory is fixed, and
(iii) the cell where the head is initially located is fixed.
(2) Boundary conditions. During the whole subsequent computation
there is no further interaction with the external environment.
(3) Final conditions. The computation terminates immediately after an
operation cycle that satisfies all the following conditions: (i) q’ = q,
(ii) a’ = a, (iii) X = 0.
BFTM : Realization domain (real part of a Turing machine setup HTM)
The realization domain BFTM of a Turing machine setup HTM is the (fuzzy)10
set of all real or concrete systems that satisfy the given specification of the
Turing machine type ASFTM, and whose temporal evolutions all satisfy the
given specification of the Turing machine interaction scheme CSFTM. Any
system bFTM BFTM is called a Turing machine realizer.
10 See footnote 9.
GIUNTI, What is a Physical Realization of a Computational System? 183
2.2. The discrete n-component dynamical system DS
So far, we have mainly focused on the setup part H of a computational
system. But, as mentioned above, a computational system is a complex
object that also includes a purely mathematical part DS, and an
interpretation IDS,H of the mathematical part DS on the setup H. In particular,
we have seen that the mathematical part DS is a discrete n-component
dynamical system. Let us now see what this dynamical system looks like in
the special case of a Turing machine. An arbitrary Turing machine
dynamical system, DSTM, is described below.
An arbitrary Turing machine dynamical system DSTM
From the purely mathematical point of view, an arbitrary Turing machine
can be identified with the discrete, 3-component dynamical system DSTM =
(Q×P×C, (gt)tZ≥0 ), which is the one completely specified by the difference
equation stated below (Table 2.3). Note first that this dynamical system is
discrete, for its time model is the additive monoid (Z≥0, +) of the non-
negative integers, and its state space Q×P×C is countably infinite (because
of the finitary restriction on the functions in C, see next paragraph).
Second, DSTM is a 3-component dynamical system, for its state space
has the three components Q, P, and C. Q = {q1, ... , qn}, where qi is an
arbitrary internal state of the Turing machine; P = Z is the set of all integers,
which is intended to represent all the possible positions of the head, and C is
the set of all functions c: P → {b, a1, ... , am} that satisfy the finitary
restriction: c(x) ≠ b for at most finitely many x P. Any such function is
intended to represent a possible content of the Turing machine external
memory. Let us now consider the five variables and the six functions
defined in Tables 2.1 and 2.2 below.
Table 2.1 The variables needed for describing an arbitrary Turing machine
dynamical system DSTM = (Q×C×P, (gt)tZ 0).
Variable Domain State variable
a A := {b, a1, ... , am} NO
q Q := {q1, ... , qn} YES
p P := Z, the integers YES
c
C := the set of all functions c: P A
such that c(p) ≠ b for at most finitely
many p
P
YES
m M := {−1, 0, +1} NO
184 Reasoning, Metaphor and Science
Then, an arbitrary Turing machine dynamical system is the
3-component discrete dynamical system DSTM := (Q×P×C, (gt)tZ 0), whose
time set is Z≥0, whose state space is Q×P×C, and whose family of state
transition functions (gt : Q×P×C Q×P×C)tZ≥0 is defined by the
3-component difference equation shown in Table 2.3.11
Function Codomain Definition
READ(c, p) A
READ
(c, p) := c(p)
WRITE(a, c, p) C
WRITE(a, c, p) := c
C such that, for
any x
P, if x = p, c’(x) = a;
else, c’(x) = c(x)
MOVE(p, m) P
MOVE
(p, m) := p + m
A(q, a) A
A
(q, a) := for any q and a,
A
(q, a) is
listed in a given finite table T1
M(q, a) M
M
(q, a) := for any q and a,
M
(q, a) is
listed in a given finite table T2
Q(q, a) Q
Q
(q, a) := for any q and a,
Q
(q, a) is
listed in a given finite table T3
Table 2.2 The functions needed for describing an arbitrary Turing machine
dynamical system DSTM = (Q×C×P, (gt)tZ 0).
State variable Difference equation
q q’ =
Q
(q,
READ
(c, p))
p p’ =
MOVE
(p,
M
(q,
READ
(c, p)))
c c’ =
WRITE
(
A
(q,
READ
(c, p)), c, p)
Table 2.3 The 3-component difference equation that univocally individuates
an arbitrary Turing machine dynamical system DSTM = (Q×C×P, (gt)tZ 0).
2.3. The interpretation IDS, H
The final step of my analysis focuses on the nature and role of the third part
of a computational system, that is, the interpretation IDS,H of the
mathematical part DS on the computational setup H. Let DS = (M, (gt)tT)
11 More precisely, the family of state transition functions (gt: Q×P×C Q×P×C)tZ≥0 is
defined as follows: (i) g0 is the identity function; (ii) g1 is the function defined by the 3-
component difference equation in Table 2.3; for any t Z≥1, gt is the t-th iteration of g1.
GIUNTI, What is a Physical Realization of a Computational System? 185
be a discrete n-component dynamical system, and H = (F, BF) be a
computational setup. An interpretation IDS,H of DS on H consists in stating
that (i) each component Ci of the state space M is included in, or is equal to,
the set V(Mi) of all the possible values of a magnitude Mi of setup H, and
(ii) the time set T of DS is equal to the set V(T) of all the possible values of
the time magnitude T of setup H.12 In other words, an interpretation IDS,H
can always be identified with a particular set of n+1 statements. We thus
define:
IDS,H is an interpretation of DS on H := IDS,H = {C1 V(M1), ... , Cn
V(Mn), T = V(T)}, where Ci is the i-th component of M, Mi is a magnitude
of the setup H, V(Mi) is the set of all possible values of Mi, T is the time
magnitude of H and, for any i and j, if i j, then Mi Mj.
For example, let us consider a Turing machine dynamical system
DSTM := (Q×P×C, (gt)tZ 0) and a Turing machine setup HTM = (FTM, BFTM).
Then, the intended interpretation of DSTM on HTM is described below.
The intended interpretation of DSTM on HTM
Let Q be the content of the internal memory of an arbitrary Turing machine
realizer bFTM BFTM; then, Q = V(Q).
Let P be the position of the head of bFTM. The head position is always
individuated by the cell where it is located; as the cells are linearly arranged
and their number is finite, if we choose one of them as the origin, we obtain
a one-one correspondence between the cells and an initial segment of Z (in
either the positive or the negative direction). Once such an integer
coordinate system is fixed, an arbitrary possible value of P can be identified
with the coordinate of the cell where the head is located; thus, V(P) = Z = P.
Let C be the whole content of the external memory of bFTM, i.e., the
distribution of symbols over each of its cells. Then, with respect to the fixed
integer coordinate system for the cells (see previous paragraph), an arbitrary
12 In general, we take a magnitude of a computational setup H = (F, BF) to be a property Mj
of every F-realizer bF
BF such that, at different instants, it can assume different values.
The set of all possible values of magnitude Mj is indicated by V(Mj). We further assume
that, among the magnitudes of any computational setup H, there always is its time
magnitude, which we denote by T. The set of all possible values (instants or durations) of
the time magnitude of H is indicated by V(T), Since the time of a computational setup
evolves in discrete steps that correspond to its basic operation cycles, we take V(T) = Z≥0
(or V(T) = Z, only if the setup is reversible).
186 Reasoning, Metaphor and Science
possible value of magnitude C can be thought as any of the functions
c C;13 therefore, C = V(C).
Finally, let T be the temporal ordering according to which an arbitrary
temporal evolution of bFTM occurs. Since any such evolution is in fact a
sequence of successive operation cycles, each operation cycle corresponds
to a non-negative integer; thus, Z≥0 = V(T).
In short, the intended interpretation of DSTM on HTM consists of the
following set of four identities.
IDSTM,HTM
:= {Q = V(Q), P = V(P), C = V(C), Z≥0 = V(T)}.
Turing machines as setup interpreted dynamical systems
Finally, as a typical example of a computational system, we can define a
Turing machine as follows.
TM is a Turing machine := TM is a triple (DSTM, HTM , IDSTM,HTM), where
DSTM = (Q×C×Z, (gt)tZ≥0) is a Turing machine dynamical system, HTM =
(FTM, BFTM) is a Turing machine setup, IDSTM,HTM is the intended interpretation
of DSTM on HTM , and the three tables T1, T2, T3 which, respectively, define
the three functions A, M, Q of DSTM are identical to the three tables T1, T2,
T3 that completely describe the input-output behavior of the control unit of
an arbitrary real system of Turing machine type ASFTM, as specified by the
functional description FTM of setup HTM .
3. Physical realizations of a computational system
Once an interpretation IDS,H = {C1 V(M1), ... , Cn ⊆V(Mn), T = V(T)} is
given, we define the possible states of the setup H = (F, BF) as follows.
x is a possible state of H relative to IDS,H := x V(M1)× ... ×V(Mn).
V(M1)× ... ×V(Mn) is called the state space of H relative to IDS,H, and is
indicated by M.
The interpretation IDS,H also allows us to define the instantaneous state
of an arbitrary F-realizer of H. Let bF BF be an arbitrary F-realizer of setup
H, and j T an arbitrary instant. Then:
x is the state of bF at instant j relative to IDS,H := x = (x1, ... , xn), where
xi is the value at instant j T of magnitude Mi of bF (if, at instant j, such a
value exists).
13 Recall that, for any c C, c: ZA = {b, a1, ..., am}.
GIUNTI, What is a Physical Realization of a Computational System? 187
Obviously, if x is the state of bF at instant j, then x M. Note, however,
that, depending on the instant j, the value of magnitude Mi of bF may not
exist.14 If this happens, the state of bF at instant j relative to IDS,H is not
defined.
Now, relative to the interpretation IDS,H, we may define the set CF of all
those possible states of H (if any) that actually are initial states of H.
CF := {x : for some bF BF, for some temporal evolution e of bF, for
some j T, j is the initial instant of e and x is the state of bF at j relative to
IDS,H}. CF is called the set of all initial states of H, relative to interpretation
IDS,H.
Intuitively, the set CF may be thought as the set of all those states in M
that are consistent with the initial conditions specified by the causal scheme
CSF and are in fact initial states of some realizer bF BF. Also note that,
depending on the interpretation IDS,H, CF may be empty, or CF may not be a
subset of the state space M of DS.15 The definition of an admissible
interpretation (see below) will exclude these somewhat pathological
interpretations.
Let CF. Let us now define, with respect to interpretation IDS,H, the
set of all initial instants of the evolutions of a given F-realizer bF BF,
whose initial state x CF be fixed. We call this set JbF,x .
JbF,x := {jbF,x : jbF,x is the initial instant of some evolution of bF, and x is
the state of bF at instant jbF,x}. JbF,x is called the set of the initial instants of
bF whose initial state is x, relative to IDS,H.
Note that, for some bF BF and x CF, JbF,x may be empty.16 However,
by the definition of CF, for any x CF, there is bF BF such that JbF,x.
As the setup H is usually taken to be deterministic, the existence and
identity of the instantaneous state, at any fixed stage of an evolution of any
realizer bF, is not intended to depend on either the initial instant, or the
14 If, for some reason, bF no longer exists at instant j T, then a fortiori the value at j of
magnitude Mi of bF does not exist either. Furthermore, we are not making any assumption
about the continuous existence of the values of a magnitude during any interval of time.
Thus, it is in principle possible that the value of magnitude Mi of bF exists at some instant j
of bF’s existence, but does not exist at some other instant k of its existence.
15 In fact, by the definition of instantaneous state, CF is empty if, for any bF BF and any
state evolution e of bF, some magnitude Mi does not have a value at the initial instant of e.
Also recall that, according to interpretation IDS,H, each component Ci of the state space M is
in general a subset of V(Mi). Thus, if for some x CF, its i-th component xi V(Mi) is not
a member of Ci, then CF M.
16 In fact, JbF,x is empty if x is not the state of bF at the initial instant of any of its evolutions.
188 Reasoning, Metaphor and Science
identity of bF, but only on the initial state. Thus, any admissible
interpretation IDS,H should at least ensure that the condition below holds.
Condition D (Determinism). For any bF, dF BF, for any x CF, for
any jbF,x JbF,x, for any kdF,x JdF,x, for any t T, if t + jbF,x is an instant of the
evolution of bF that starts at jbF,x and the state of bF at instant t + jbF,x exists,
then t + kdF,x is an instant of the evolution of cF that starts at kdF,x, the state of
cF at instant t + kdF,x exists as well, and the state of bF at instant t + jbF,x = the
state of dF at instant t + kdF,x.
Let CF . For any initial state x CF, let us consider the set of all
F-realizers whose initial state is x. This set, denoted by BFx, is in other words
the collection of all F-realizers bF whose set JbF, x is not empty. Note that
also this definition, as the previous ones, depends on the interpretation IDS,H.
BFx := {bF BF such that JbF, x}. BFx is called the set of all
F-realizers whose initial state is x, relative to interpretation IDS,H.
We noticed above that, for any x CF, there is bF BF such that
JbF, x. Therefore, by the definition just stated, for any x CF, BFx.
Suppose CF . Then, for any x CF, for any bF BFx, for any jbF,x
JbF,x, we define the following set of durations:
qbF,jbF,x(x) := {t: t T, t + jbF,x is an instant of the evolution of bF that
starts at jbF,x, and there is y M such that y is the state of bF at t + jbF,x,
relative to IDS,H}.
Note that this definition, like the previous ones, is relative to the
interpretation IDS,H. Furthermore, qbF,jbF,x(x) , for 0 qbF,jbF,x(x).
Also note that, whenever Condition D above holds, qbF,jbF,x(x) depends
on x, but does not depend on either bF or jbF,x; therefore, if Condition D
holds, we simply write qF(x)” instead of “qbF,jbF,x(x)”.
By Condition D and the previous definition, for any x CF, qF(x) is the
set of all durations t that transform the initial state x of an arbitrary
F-realizer bF BFx into some other state of bF. More briefly, we call qF(x)
the set of all durations that transform the initial state x of H into some other
state.
As we are not interested in any interpretation IDS,H such that (a) CF = ,
or (b) CF M, or (c) Condition D does not hold17, we define:
17 If either (a), (b), or (c), the interpretation IDS,H, is obviously incorrect, for: if (a) holds, no
evolution of any F-realizer bF can be represented by means of the state transition family
(gt)t
T
of DS = (M, (gt)t
T
); if (b) holds, some evolution of some F-realizer bF cannot be
represented by (gt)t
T
; if (c) holds, some evolution of some F-realizer bF cannot be
correctly represented by (gt)t
T
.
GIUNTI, What is a Physical Realization of a Computational System? 189
IDS,H is an admissible interpretation of DS on H := (i) CF and
(ii) CF M and (iii) Condition D holds.
We can now precisely state the conditions for an interpretation IDS,H to
be correct. The intuitive idea is this. As soon as an interpretation IDS,H is
fixed, the dynamical system DS = (M, (gt)tT) provides us with a
representation of the real systems (F-realizers) in the realization domain of
computational setup H. Such a representation is in fact provided by the state
transition family (gt)tT of dynamical system DS. The interpretation IDS,H
will thus turn out to be correct if the representation, provided by (gt)tT, of
all temporal evolutions of all F-realizers of H is correct. This intuitive idea
is formally expressed by the definition below.
IDS,H is a correct interpretation of DS on H := (i) IDS,H is an admissible
interpretation of DS on H and (ii) for any x CF, for any t qF(x), for any
bF BFx, for any jbF,x JbF,x, gt(x) = the state of bF at instant t + jbF,x relative
to IDS,H.
Let CS = (DS, H, IDS,H) be a computational system, and BF be the
realization domain of H. We can thus finally define:
b is a physical realization of CS := b BF and IDS,H is a correct
interpretation of DS on H.
It is now easy to prove that, in the special but important case of an
arbitrary Turing machine TM, the intended interpretation IDSTM,HTM of the
Turing machine dynamical system DSTM on the Turing machine setup HTM
is indeed a correct interpretation of DSTM on HTM , so that any Turing
machine realizer bFTM BFTM turns out to be a physical realization of TM, and
conversely.
Theorem
If TM = (DSTM , HTM , IDSTM,HTM) is a Turing machine, then IDSTM,HTM
is a correct interpretation of DSTM on HTM .
Proof
The thesis is a straightforward consequence of the definitions of
Turing machine and correct interpretation of DS on H, in
conjunction with the theoretical part FTM of the Turing machine
setup HTM, the specification of DSTM , and the intended
interpretation IDSTM,HTM. Q.E.D.
Corollary
For any Turing machine TM, b is a physical realization of TM
iff b BFTM.
190 Reasoning, Metaphor and Science
Proof
By the previous Theorem and the definition of physical
realization of a computational system. Q.E.D.
4. Concluding remarks
It is my contention that the previous Theorem is not peculiar to Turing
machines, but that an analogous theorem holds for each specific type of
computational system (for instance, finite automata, register machines,
cellular automata, and so on). If this is true, all computational systems are
then characterized by a form of a-priori (or purely theoretical)
interpretation of the mathematical part on the setup part, for the correctness
of the interpretation can be proved. The a-priori character of the
interpretation of a computational system distinguishes this kind of system
from ordinary dynamical models of phenomena (Giunti 2010a: sec. 4,
2010b: Appendice, 2014: sec. 4.1, 2016: secs. 3 and 4), as found in
empirical science. In fact, for this second kind of dynamical system, the
correctness of the interpretation of the mathematical part on the intended
phenomenon (which is the analog of the setup of a computational system)
cannot be established a-priori—cannot be proved—but only a-posteriori or
empirically.
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... However, a fundamental approach for designing by definition and even identifying instances of morphological computation is not yet presented or agreed upon. However, a good starting point might be employing the methods discussed in the theoretical physics and computer science research on the definition and modeling of a computing physical phenomenon [26,27], despite their differences [25]. For the first time in this study, we try to extend a recently proposed definition for physical computational systems by Giunti [27], (i) to define morphological computational systems, and (ii) to realize the computational capability of a natural structure, i.e. spider webs. ...
... However, a good starting point might be employing the methods discussed in the theoretical physics and computer science research on the definition and modeling of a computing physical phenomenon [26,27], despite their differences [25]. For the first time in this study, we try to extend a recently proposed definition for physical computational systems by Giunti [27], (i) to define morphological computational systems, and (ii) to realize the computational capability of a natural structure, i.e. spider webs. ...
... Giunti [27] has proposed a precise analysis for physical realization of a computational system, by looking at the modeling relation between a dynamic system and phenomena, as a complex object. A computational system contains three parts (Fig 1.a); 1) a mathematical part in the form of a discrete n-component dynamic system (DS = (M, (g t ) t∈T )) with a state space (M ), a transition function (g t ) and a time set (T ); 2) a computational setup (H = (F, B F )) with a theoretical part (F ) and a real (physically feasible) part (B F ); and 3) an interpretation (I DS,H ) linking the aforementioned parts. ...
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Currently there is growing interest in the application of dynamical methods to the study of cognition. Computation, Dynamics, and Cognition investigates this convergence from a theoretical and philosophical perspective, generating a provocative new view of the aims and methods of cognitive science. Advancing the dynamical approach as the methodological frame best equipped to guide inquiry in the field's two main research programs--the symbolic and connectionist approaches--Marco Giunti engages a host of questions crucial not only to the science of cognition, but also to computation theory, dynamical systems theory, philosophy of mind, and philosophy of science. In chapter one Giunti employs a dynamical viewpoint to explore foundational issues in computation theory. Using the concept of Turing computability, he precisely and originally defines the nature of a computational system, sharpening our understanding of computation theory and its applications. In chapter two he generalizes his definition of a computational system, arguing that the concept of Turing computability itself is relative to the kind of support on which Turing machine operate. Chapter three completes the book's conceptual foundation, discussing a form of scientific explanation for real dynamical systems that Giunti calls "Galilean explanation." The book's fourth and final chapter develops the methodological thesis that all cognitive systems are dynamical systems. On Giunti's view, a dynamical approach is likely to benefit even those scientific explanations of cognition which are based on symbolic models. Giunti concludes by proposing a new modeling practice for cognitive science, one based on "Galilean models" of cognitive systems. Innovative, lucidly-written, and broad-ranging in its analysis, Computation, Dynamics, and Cognition will interest philosophers of science and mind, as well as cognitive scientists, computer scientists, and theorists of dynamical systems. This book elaborates a comprehensive picture of the application of dynamical methods to the study of cognition. Giunti argues that both computational systems and connectionist networks are special types of dynamical systems. He shows how this dynamical approach can be applied to problems of cognition, information processing, consciousness, meaning, and the relation between body and mind.