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On Ontic Oracles and Epistemic Artificial Life Systems



Oracles are hypothetical machines that can compute any function in unit time. They are representatives of the ontic truth that is independent of any theory or model. Artificial life systems are synthetic models that try to capture the ontic truth through epistemic inferences. Their descriptions are epistemic perceptions of the observer. Understanding the distinctions between these ontic-epistemic coun-terparts is vital for building successful models of nature. Towards this extent this paper formalizes the notions of oracles and o-machines and establishes their role in solving the decision problem for artificial life systems. It further examines the converse of Church-Turing thesis that holds the key to the possibility of practical realization of those hypothetical machines.
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Research Scholar, Computer Science & Engineering Dept.,
IIT-Bombay, Mumbai 400076, India
Oracles are hypothetical machines that can compute any function in unit time.
They are representatives of the ontic truth that is independent of any theory or
model. Artificial life systems are synthetic models that try to capture the ontic
truth through epistemic inferences. Their descriptions are epistemic perceptions of
the observer. Understanding the distinctions between these ontic-epistemic coun-
terparts is vital for building successful models of nature. Towards this extent this
paper formalizes the notions of oracles and o-machines and establishes their role
in solving the decision problem for artificial life systems. It further examines the
converse of Church-Turing thesis that holds the key to the possibility of practical
realization of those hypothetical machines.
1. Introduction
By an Artificial Life system, ALsystem for short, we essentially mean a
program that is aimed at mimicking or modeling the natural world. Any
system, natural or artificial, can be expressed as a state machine, with each
state representing an unique instance of the system, and the transitions
across the states representing the processes occurring within the system.
When a system is viewed as a state machine, a problem occurring within
the system becomes, but just one of the possible configurations which the
observer would like to analyze, understand and/or avoid. Likewise, a so-
lution becomes another configuration, not necessarily different from the
problem configuration, which the observer would like the system to enter
to. However, no system can be understood completely when the observer
himself is a part of the system. Thus, we would like to move the observer
out of the system and study it as it performs ”when no one looks at it”.
That is, we would like to identify the models whose epistemic inferences co-
incide with the underlying ontic reality. Langton expresses this as ”locating
life-as-we-know-it within the larger picture of life-as-it-could-be16.
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The notion of ontic reality refers to the fundamental truth that is inde-
pendent of the model or theory that proposes it. Contrariwise, the epistemic
reality refers to the notion of fact that appears to be the truth given some
suitable model or theory. This distinction between ontic and epistemic re-
alities is crucial for the success of any model of a system, for it poses a
restriction on what could be achieved through that system-model combi-
nation. The importance of this distinction can be realized by the array of
”limitation” theories it leads to, such as, the uncompleteness theorem of
odel from the theory of logical systems, or the problem of Halting from
the theory of computation, or the principle of uncertainty by Heisenberg
from Physics etc. . . 20,2,22 . Hence understanding and analyzing these on-
tic/epistemic distinctions is vital for all fields of study, and Artificial Life
is no exception.
In the following we analyze the concept of solving the decision problem
of ALsystems using oracles. An oracle is a hypothetical machine that can
solve any problem in unit time. It represents an ontic procedure that can
describe all properties of a system exhaustively (or completely). ”Exhaus-
tive” in this context means without referring to any epistemic knowledge or
ignorance. In our recent work 10, we have proved that the decision problem
for ALsystems is unsolvable within the framework of effectively computable
functions. Toward that extent, we created a transformation framework that
captured the notion of ”change” through a formal concept of transforma-
tion. In this paper we extend that framework to formally define oracles and
o-machines and prove that the decision problem of ALsystems is solvable
using o-machines.
While an oracle is a powerful machine that can solve any problem in
unit time, there exists, however, two limitations that need to be addressed
before one could actually benefit from such machine.
(1) Only limited amount of information can be communicated to/from
oracle at any time. That is, though one has access to an oracle
that could answer all infinite possible questions, one could ask only
few questions at a time. It is merely a restriction posed by the
communication channel and the information gathering capacity of
observer and hence possibly can not be overcome by any practical
(2) The ontic descriptions provided by the oracle may not be compatible
with the epistemic descriptions that the observer expects and hence
need to be mapped and interpreted properly before actually used.
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This second limitation can be overcome by having a machine that when
provided with necessary details about the oracle can interpret the ontic
description given by the oracle and provide the corresponding epistemic
details. We call such a machine as the o-machine and the information it
requires about the oracle as the oracle set.
As could be easily understood, the o-machine acts as a bridge between
ontic and epistemic levels of descriptions. However, due to the limitation
of finite communication mentioned above, the o-machine has access to only
limited part of the oracle knowledge and hence can be only as powerful.
We formalize these concepts in the following sections. In Sec. 2, we
present the required preliminaries of our transformation framework from
10, followed by the formalization of oracles and o-machines in Sec. 3. In
Sec. 4 we examine the conceptions offered by the converse of Church-Turing
thesis that could make the theory of o-machines a possible practicality. We
complete this discussion by presenting the conclusions in Sec. 5 followed
by relevant references.
2. Preliminaries
In this section we briefly review our framework for solving artificial life
systems using the notion of effective enumerable sets and transformation
systems. The observable functionality of natural life systems is so tightly
woven around the concept of ”change” that its role cannot be overlooked
in designing ALsystems either. In this framework we capture this intuitive
notion of change through the formal concept of transformation and use it in
formalizing the definition for ALsystems. For this discussion we ignore any
subtle differences between the notions of recursive and effective procedures
and use both terms interchangeably.
Definition 2.1. Given a set of symbols A, any effectively enumerable set
SAdefines a Sequence.
Elements of a Sequence are called sequents. The number of sequents in
a Sequence gives the length of the Sequence. For a Sequence S, its length
is indicated by ]S.
Let Ube the universal space, the ensemble of sequents from all possible
Sequences, defined over the finite symbol set A={a1, a2, . . . , an}.
U=[(S|Sis a Sequence over A).
We assume, without explicit statement, that Uis effectively enumerable,
and that every Sequence discussed is a subset of U.
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Definition 2.2. A set zUdefines a zone if there exists a function
f:A→ {0,1}such that for each siU:
f(si) = 1 if siz,
0 if si/z.
Let Z={z|zUand zis a zone}be the set of all possible zones.
Definition 2.3. Given two zones zi, zjZ,a subset of zi×zjdefines a
transformation over Z.
A transformation tfrom zito zjis denoted by tzizj.In infix notation it
is written as zi
tzjor simply as zizj.
If t0
zizjand t00
zjzkare two transformations defined over Z,then their
(binary) composition, indicated by t0·t00,is a transformation tzizk,defined
tzizk={(x, y)|∀(xzi, y zk, w zj)[(x, w)t0
zizj(w, y)t00
Definition 2.4. A pair (Z, T ) consisting of a set of zones ZZ,and a set
of transformations T defined over Z, is called an (abstract) transforma-
tion system.
Definition 2.5. By a transformation process in a transformation sys-
tem Γ = (Z, T ) it is meant a finite series s1, s2, . . . , smof sequents such
that the following holds true
i∈ {1, m} ∃ziZ[sizi]
j∈ {1, m 1} ∃tzjzj+1 T[(sj, sj+1 )tzjzj+1 ].
Each of the siis called a step of the transformation process. Further, s1is
called the initial step and smis called the final step.
Definition 2.6. A sequent sis said to be a result of Γ,indicated by `Γs ,
if sis the final step of a transformation process in Γ.
By the decision problem for a transformation system, we mean the problem
of determining, of a given sequent, whether or not it is a result of the
Definition 2.7. Let Γ be a transformation system. Then, by the set of
sequents generated by Γ we shall mean the set
SΓ={x| `Γx}.
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Definition 2.8. The decision problem for a transformation system Γ is re-
cursively solvable or unsolvable, according as SΓis or is not a recursive
Theorem 2.1. Every effectively enumerable set is generated by a transfor-
mation system.
Theorem 2.2. There exists a transformation system whose decision prob-
lem is recursively unsolvable.
Definition 2.9. An Artificial Life system is defined by the tuple
(f, c1, . . . , cm),where
f: (A)m(A)mis a partial function referred to as the observer
function, and
each ci,1im, called an element of the system, is defined by
the pair (ai,Γi) where
Γiis a transformation system, and
aiAis a result of Γi,called as the observable state of ci.
By a step of Artificial Life system we mean a two fold process that involves,
(1) each element of the system carrying out the respective transforma-
tion process over its observable state, followed by the
(2) application of the observer function to the results of the transfor-
mation process obtained in step 1.
At the end of each step, the tuple generated by f(`Γ1a1, . . . , `Γmam) de-
fines the configuration of the system for the next step. We use the notation
<Υ|ai1, . . . , aim><Υ|aj1, . . . , ajm>
to indicate that an Artificial Life system Υ has stepped from configuration
< ai1, . . . , aim>to configuration < aj1, . . . , ajm> .
By the decision problem for an ALsytem, we mean the problem of de-
termining, of a given tuple < ai1, ai2, . . . , aim>, whether or not it is a
configuration of the system. An ALsystem is said to be recursively unsolv-
able if its decision problem is recursively unsolvable.
Theorem 2.3. There exists an ALsystem whose decision problem is recur-
sively unsolvable.
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3. Oracles and O-machines
In this section we extend the transformation framework to include the for-
malization of oracles and o-machines.
Definition 3.1. By an oracle for a transformation system Γ = (Z, T),
it is meant a transformation system Γ0= (Z0, T 0) such that there exists a
total-injective mapping ψ:(zi|ziZ)→ ∪(z0
jZ0) satisfying the
condition that for every transformation process sl, . . . , smin Γ the following
holds true
m)⇒ ∃t0
l, s0
We use the notation ΦΓ to indicate that Φ is an oracle for Γ.The mapping
ψbetween Φ and Γ is referred to as an o-mapping, indicated by Φ ψΓ. All
possible o-mappings between Φ and Γ form an o-map class, Γ],given
Γ] = {ψ|ΦψΓ}.
An oracle class for Γ,written as [Γ],is given by
[Γ] = {Φ| ∃ψΦψΓ}.
Definition 3.2. Let Φ = (Z0, T 0) be an oracle for Γ = (Z, T ).Then, by an
oracle set of Φ it is meant the set
OΦ={x| ∃yt0
j(y, x)t0
Definition 3.3. An o-machine is an effective procedure f:A→ {0,1}
that, when associated with an oracle set OΦof Φ ψΓ,can decide if a
sequent sUis a result of Γ or not as follows. For every sequent sU,
where f(s) is defined as,
f(s) = 1,if s0OΦ[s ψ s0];
We use the notation fXto indicate an o-machine that is an effective pro-
cedure fassociated with oracle set X. An o-machine fXis said to decide
a set SU, if and only if sU[sSf(s)].
Theorem 3.1. Every recursively enumerable set is decidable by an o-
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Proof. Consider a recursively enumerable set Sgenerated by a transfor-
mation system Γ.Let OΦbe an oracle set of Φ ψΓ.By definition, for every
transformation process sl, . . . , smin Γ,
Hence it follows that,
∀ `Γsms0
Now, consider an o-machine fOΦ.By Definition 3.3,
sU[f(s)⇔ ∃s0OΦ[sψs0]] .
We prove fOΦcan decide Sas follows. Let xbe an element of S. Then,
xS⇔ `Γx
⇔ ∃x0OΦ[xψx0]
Thus, fwhen associated with OΦcan decide S. Hence proved.
An o-machine is said to be universal if it can decide all recursively enumer-
able sets.
Theorem 3.2. There exists no universal o-machine that can decide all
recursively enumerable sets.
Proof. For contradiction, assume fXto be an universal o-machine.
Let Sand ¯
S=U\Sbe two recursively enumerable sets generated by trans-
formation systems Γ and Γ0respectively.
Since fXhas been assumed to be universal, it follows that for all xU,
(xS)f(x) and (x¯
xU[(xS)⇔ ¬(x¯
S)] ,
which leads to
xU[f(x)⇔ ¬f(x)] ,
a contradiction. Thus our assumption that fXis universal is invalid. Hence
An ALsystem is said to be closed if its observer function is an identity
function of the form ~x [f(~x) = ~x].
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Theorem 3.3. The decision problem of every closed ALsystem is solvable.
Proof. Consider a closed ALsystem Υ = (f, c1, . . . , cm),where f(~x) = ~x
for all ~x, and ci= (ai,Γi), aiA,1im. Considering a tuple
< s1, . . . , sm>,
<Υ|s1, . . . , sm>⇔ `Γ1s1∧ · · · ∧ `Γmsm.
Let fX1
1, . . . , f Xm
mbe o-machines such that for all sU,
fi(s)⇔ `Γis ,
where Xiis an oracle set of ΦiΓi.
Let g: (A)m→ {0,1}be a function defined as, for every x1, . . . , xmU,
g(x1, . . . , xm) = m
To prove that the decision problem of Υ is solvable,
<Υ|s1, . . . , sm>⇔ `Γ1s1∧ · · · ∧ `Γmsm
f1(s1)∧ · ·· ∧ fm(sm)
g(s1, . . . , sm).
Thus gcan decide if a tuple < s1, . . . , sm>is configuration of Υ or not.
Hence proved.
An ALSystem with a non-identity observer function is referred to as an
open ALsystem.
Theorem 3.4. The decision problem of every open ALsystem is solvable.
Proof. Consider an open ALsystem Υ = (f, c1, . . . , cm),where ci=
(ai,Γi), aiA,Γi= (Zi, Ti),1im. Let S=S1×S2× · · · × Sm=
{s1, s2, . . . , sl}, l N,where Si=(z|zZi),1im.
Let g: (A)mSbe a function that converts an m-tuple into the
corresponding element of S.
Now, construct a transformation system Γ0= (Z0, T 0) as follows. Define
1, z0
2, . . . , z0
l},as z0
For every step <Υ|a1, . . . , am><Υ|a0
1, . . . , a0
m>of ALsystem
Υ, define a new transformation {(g(a1, . . . , am), g(a0
1, . . . , a0
m))}in T0such
that for every x1, . . . , xmU,
<Υ|x1, . . . , xm>⇔ `Γ0g(x1, . . . , xm).
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Thus the decision problem of Υ is solvable if and only if the decision problem
of Γ0is solvable.
Next, define a closed ALsystem Υ0= (f0, c0
1) with single element c0
(s1,Γ0) and identity observer function f0.At the end of each step of Υ0it
holds that for all xU,
<Υ0|x > ⇔ `Γ0x .
Thus the decision problem of Γ0is solvable if and only if the decision prob-
lem of Υ0is solvable.
However, from Theorem 3.3, the decision problem of Υ0is solvable.
Consequently, the decision problem of Υ is solvable as well.
While o-machines can solve the decision problems of ALsystems, the so-
lutions offered by them may not be recursive in the traditional sense of
effective computability. This could become a practical limitation on the
usefulness of oracles. However, the possibility of non-effective mechanical
procedures offered by the converse of Church-Turing thesis promises other-
wise. The following section discusses those details.
4. The Converse of Church-Turing Thesis
In general terms the Church-Turing thesis asserts that every effectively
calculable function is computable by Turing machine. A function is said to
be effectively calculable if there exists an effective or mechanical method
for calculating the values of the function. In this regard, a method, or
procedure, M, for achieving some desired result is termed ’effective’ or
’mechanical’ if
(1) M is set out in terms of a finite number of exact instructions, where
each instruction is expressed by means of a finite number of symbols;
(2) M will, if carried out without error, produce the desired result in a
finite number of steps;
(3) M can, in practice or in principle, be carried out by a human being
unaided by any machinery save paper and pencil;
(4) M demands no insight or ingenuity on the part of the human be-
ing carrying it out except that which is needed to understand and
execute the instructions.
In essence an effective procedure is a procedure that can be carried out
in finite means by a human mathematician, any human mathematician,
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without requiring any intelligence or insight. For such a procedure, assum-
ing an appropriate book keeping facility, if one mathematician pauses the
computation at any point, then any other mathematician should be able
to resume the computation from that point and complete it without any
difficulty, no matter how much these two mathematicians differ in their in-
telligence and experience. That is, an effective procedure never relies upon
a particular ability of one particular mathematician. Instead, it relies upon
something that all mathematicians are expected to have in common - the
ability to compute. In this respect, a procedure that can be carried out only
by a particular mathematician or by some special group of mathematicians
cannot be termed effective.
The notion of effective procedure essentially aims at minimizing epis-
temic dependencies in the procedural descriptions. Does this mean the
procedure would get closer to ontic reality? We do not believe it to be so
for at least two reasons.
(1) The ontic truth may not always be the same as what all epistemic
facts claim it to be. There exists no known way of comparing ontic
truth with epistemic facts other than through observation. This is
apparent from the results provided by Turing that there can ex-
ist a universal machine that can mimic the behavior of all Turing
machines but there can not exist any universal machine that can
predict the behavior of all Turing machines correctly 24,25 .
(2) Accounting for being compliant with all epistemic views may render
the procedure move further away from ontic reality. It is seldom
true that two radically different epistemic views can agree upon the
same fact, and when they do it is always possible that they both
miss some fundamental point hitherto unknown. Views of classical
and quantum mechanics are good example for this.
Of particular interest for us in this regard would be the converse of Church-
Turing thesis that raises the question, can Turing machines compute only
effective procedures? Stated in Copeland 7terms, we have the question,
Are there mechanical procedures that are not mechanical?
While this might seem a paradox at first, given our above explanation
of what constitutes a mechanical procedure and effective calculability, it
nonetheless is a valid question in that its answer could provide means for
the success of artificial intelligence. The issues that need to be explored in
this regard are,
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what class of abilities can be regarded as being attributable to indi-
vidual mathematicians that can cross the barrier of effective com-
putability while still holding the view of being a mechanical proce-
dure? (Needless to say that intelligence and experience will be the
obvious first candidates to go into that class.)
How does a mechanical procedure use such an ability? What would
be the requirements and restrictions for such specification?
Given one such non-effective mechanical procedure, would it always
be possible to come up with an equivalent effective procedure and
vice versa?
It is worth noting that the notion of effective procedure does not speak
about the efficiency of the human mathematician. Thus we are free to
choose between a lazy mathematician who would take one minute rest after
each step, and a hard working mathematician who could increase his speed
with every step. The effectiveness of the procedure guarantees that both
would eventually solve the problem in the same way, i.e. either both would
halt with same results or both would not halt.
In fact, this concept of increasing the speed with every step forms the
basis for one of the hypermachines known as accelerated Turing machine
that can arguably solve even the Halting problem 19,8. Typically a hy-
permachine is a machine that could compute functions that are beyond
the power of Turing machines. An accelerated Turing machine works by
doubling its speed with every step, performing its first step in one unit
time and each subsequent step in half the time of the step before. Since
1 + 1
8+. . . < 2,such a process could complete an infinity of steps in
two time units. These machines can compute any function within constant
time and are well suitable for being o-machines. Exploring these notions
further can improve our perceptions about oracles and may provide insight
into the methods that can make them practical.
5. Conclusions
Artificial life systems are synthetic counter parts of natural life systems.
Solving them entails the notion of being able to predict the effects of past
upon the future. It can be achieved by abstracting the logic of life through
the mechanism of computation. However, it turns out that such a notion
of abstracting a common logical framework for all hierarchical levels of
life systems is unsolvable within the framework of effective computable
functions (in the sense of Turing computability) and requires machines that
June 18, 2005 16:58 Proceedings Trim Size: 9in x 6in ALOracles
could compute well beyond the power of ordinary Turing machines.
In this paper we have extended our transformation framework to al-
low the formalizations for oracles and o-machines. Oracles stand for the
underlying truth that is independent of the observer while ALsystems rep-
resent the epistemic perceptions with their descriptions depending on the
state (knowledge) of the observer. O-machines act as communication bridge
between these ontic and epistemic counterparts. The converse of Church-
Turing thesis offers conceptions that could make the theory of o-machines a
possible practicality by providing insights into the non-effective alternatives
of mechanical procedures. A measured exploration of these alternatives
could open new possibilities for solving artificial life systems.
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Full-text available
Mathematically a system is said to be solved if its future states can be predicted from the information provided by the present and past state history. In this paper, we examine the problem of solving Artificial Life systems using the principles of state-machines. The observable functionality of natural life systems is so tightly woven around the concept of "change" that its role cannot be overlooked in designing Arti-ficial Life systems either. Keeping this in view, we define a framework that captures this intuitive notion of change through a more formal concept of transformation, and use it in formalizing the definition for Artificial Life systems. We examine the feasibil-ity of solving Artificial Life systems within the framework of effectively computable functions and provide a proof for their unsolvability. Finally, we relate the complex-ity of Artificial Life systems with the underlying algorithmic complexity and establish that open-ended artificial evolution requires programs with ever increasing algorith-mic complexity.
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Foundational controversies in artificial life and artificial intelligence arise from lack of decidable criteria for defining the epistemic cuts that separate knowledge of reality from reality itself, e.g., description from construction, simulation from realization, mind from brain. Selective evolution began with a description-construction cut, i.e., the genetically coded synthesis of proteins. The highly evolved cognitive epistemology of physics requires an epistemic cut between reversible dynamic laws and the irreversible process of measuring initial conditions. This is also known as the measurement problem. Good physics can be done without addressing this epistemic problem, but not good biology and artificial life, because open-ended evolution requires the physical implementation of genetic descriptions. The course of evolution depends on the speed and reliability of this implementation, or how efficiently the real or artificial physical dynamics can be harnessed by nondynamic genetic symbols.
Algorithmic information content is equal to the size — in the number of bits — of the shortest program for a universal Turing machine which can reproduce (i.e., plot with the requisite accuracy) a state of a physical system. In contrast to the statistical Boltzmann-Gibbs-Shannon entropy, which measures ignorance, the algorithmic information content is a measure of the available information. It is defined without a recourse to probabilities and can be regarded as a measure of randomness of a definite microstate. I suggest that the physical entropy S — that is, the quantity which determines the amount of the work ΔW which can be extracted in the cyclic isothermal expansion process through the equation ΔW = kBTΔS — is a sum of two contributions: (i) The missing information measured by the usual statistical entropy and (ii) the known randomness measured by the algorithmic information content. The sum of these two contributions is a “constant of motion” in the process of a dissipationless measurement on an equilibrium ensemble. This conservation under a measurement, which can be traced back to the noiseless coding theorem of Shannon, is necessary to rule out existence of a successful Maxwell's demon.
It is argued, by use of specific examples, that mathematical understanding is something which cannot be modelled in terms of entirely computational procedures. Our conception of a natural number (a non‐negative integer: 0, 1, 2, 3,…) is something which goes beyond any formulation in terms of computational rules. Our ability to perceive the properties of natural numbers depends upon our awareness, and represents just one of the many ways in which awareness provides an essential ingredient to our ability to understand. There is no bar to the quality of understanding being the result of natural selection, but only so long as the physical laws contain a non‐computational ingredient.
Reduction systems are widely used in theoretical computer science. Symbolic manipulations on syntactic objects are based upon their providing models for computation in an operational setting.
Basic notions and results are presented in this chapter. We start with elementary concepts of set theory and algebra in Section 1.1. Closure systems are the key tool in Section 1.2. The fundamental concepts of generation, induction, recursion and deduction are based upon them. Section 1.3 deals with relations. Union, product and star of relations are called regular operations. It turns out that they are essential for our development. Furthermore, equivalence relations and partial orders are studied. Terminating relations will be needed, in particular, for proving termination of term rewriting systems later. Because termination is generally an undecidable property, special proof techniques are required. Well-founded and well-behaved relations provide such methods. In Section 1.4, trees and labelled trees are considered within the framework of well-founded partially ordered sets. Finally, ω-complete partially ordered sets and fix-point methods are studied in Section 1.5 as a prerequisite of semantics for recursive program schemes.
There is a need for a unified, mathematical theory of complexity that is capable of expressing the structures and processes common to different phenomena. This account proposes three elements for such a theory. (1) Universal measurement generalises the concept of measurement to include many formal systems of observation. (2) Graphs, which are inherent in the structure and behaviour of all complex systems, provide possible units of measurement that are the equivalent of numbers for organisation. (3) Syntactic homomorphism, which identifies the underlying "programs" common to classes of patterns and processes, provides at least one analytical tool. As well as suggesting new techniques, the above elements imply a need to re-interpret many current ideas and methods. They also suggest a series of open questions for future research.
This book provides general and systematic introduction to the semantics of concurrent systems. The author presents his own theory of the behavioral semantics of processes ( testing equivalence ) and original results in example languages for distributed processes. The problems addressed are motivated from the standpoint of computer science, and all the required algebraic concepts are covered.
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Can we physically implement a hypercomputer? So far no one has.