Content uploaded by Gopalakrishna Palem

Author content

All content in this area was uploaded by Gopalakrishna Palem

Content may be subject to copyright.

June 18, 2005 16:58 Proceedings Trim Size: 9in x 6in ALOracles

ON ONTIC ORACLES AND EPISTEMIC ARTIFICIAL LIFE

SYSTEMS

PALEM GOPALAKRISHNA

Research Scholar, Computer Science & Engineering Dept.,

IIT-Bombay, Mumbai 400076, India

E-mail: krishna@cse.iitb.ac.in

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-be”16.

1

June 18, 2005 16:58 Proceedings Trim Size: 9in x 6in ALOracles

2

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

G¨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

means.

(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.

June 18, 2005 16:58 Proceedings Trim Size: 9in x 6in ALOracles

3

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

S⊆A∗deﬁnes 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.

June 18, 2005 16:58 Proceedings Trim Size: 9in x 6in ALOracles

4

Deﬁnition 2.2. A set z⊆Udeﬁnes a zone if there exists a function

f:A∗→ {0,1}such that for each si∈U:

f(si) = 1 if si∈z,

0 if si/∈z.

Let Z={z|z⊆Uand zis a zone}be the set of all possible zones.

Deﬁnition 2.3. Given two zones zi, zj∈Z,a subset of zi×zjdeﬁnes a

transformation over Z.

A transformation tfrom zito zjis denoted by tzi→zj.In inﬁx notation it

is written as zi→

tzjor simply as zi→zj.

If t0

zi→zjand t00

zj→zkare two transformations deﬁned over Z,then their

(binary) composition, indicated by t0·t00,is a transformation tzi→zk,deﬁned

as,

tzi→zk={(x, y)|∀(x∈zi, y ∈zk, w ∈zj)[(x, w)∈t0

zi→zj∧(w, y)∈t00

zj→zk]}.

Deﬁnition 2.4. A pair (Z, T ) consisting of a set of zones Z⊆Z,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} ∃zi∈Z[si∈zi]∧

∀j∈ {1, m −1} ∃tzj→zj+1 ∈T[(sj, sj+1 )∈tzj→zj+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

system.

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}.

June 18, 2005 16:58 Proceedings Trim Size: 9in x 6in ALOracles

5

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

set.

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,1≤i≤m, called an element of the system, is deﬁned by

the pair (ai,Γi) where

–Γiis a transformation system, and

–ai∈A∗is 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.

June 18, 2005 16:58 Proceedings Trim Size: 9in x 6in ALOracles

6

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|zi∈Z)→ ∪(z0

j|z0

j∈Z0) satisfying the

condition that for every transformation process sl, . . . , smin Γ the following

holds true

∃s0

l∃s0

m[(slψs0

l)∧(smψs0

m)⇒ ∃t0

z0

l→z0

m∈T0

[s0

l∈z0

l∧s0

m∈z0

m∧(s0

l, s0

m)∈tz0

l→z0

m]].

We use the notation Φ◦Γ 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, [Φ ◦Γ],given

by

[Φ ◦Γ] = {ψ|Φ◦ψΓ}.

An oracle class for Γ,written as [◦Γ],is given by

[◦Γ] = {Φ| ∃ψΦ◦ψΓ}.

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| ∃y∃t0

z0

i→z0

j∈T0[y∈z0

i∧x∈z0

j∧(y, x)∈t0

z0

i→z0

j]}.

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 s∈Uis a result of Γ or not as follows. For every sequent s∈U,

`Γs⇔f(s),

where f(s) is deﬁned as,

f(s) = 1,if ∃s0∈OΦ[s ψ s0];

0,otherwise.

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 S⊆U, if and only if ∀s∈U[s∈S⇔f(s)].

Theorem 3.1. Every recursively enumerable set is decidable by an o-

machine.

June 18, 2005 16:58 Proceedings Trim Size: 9in x 6in ALOracles

7

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,

∃s0

m∈OΦ[smψs0

m].

Hence it follows that,

∀ `Γsm∃s0

m∈OΦ[smψs0

m].

Now, consider an o-machine fOΦ.By Deﬁnition 3.3,

∀s∈U[f(s)⇔ ∃s0∈OΦ[sψs0]] .

We prove fOΦcan decide Sas follows. Let xbe an element of S. Then,

x∈S⇔ `Γx

⇔ ∃x0∈OΦ[xψx0]

⇔f(x).

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 x∈U,

(x∈S)⇔f(x) and (x∈¯

S)⇔f(x).

However,

∀x∈U[(x∈S)⇔ ¬(x∈¯

S)] ,

which leads to

∀x∈U[f(x)⇔ ¬f(x)] ,

a contradiction. Thus our assumption that fXis universal is invalid. Hence

proved.

An ALsystem is said to be closed if its observer function is an identity

function of the form ∀~x [f(~x) = ~x].

June 18, 2005 16:58 Proceedings Trim Size: 9in x 6in ALOracles

8

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), ai∈A∗,1≤i≤m. Considering a tuple

< s1, . . . , sm>,

<Υ|s1, . . . , sm>⇔ `Γ1s1∧ · · · ∧ `Γmsm.

Let fX1

1, . . . , f Xm

mbe o-machines such that for all s∈U,

fi(s)⇔ `Γis ,

where Xiis an oracle set of Φi◦Γi.

Let g: (A∗)m→ {0,1}be a function deﬁned as, for every x1, . . . , xm∈U,

g(x1, . . . , xm) = ∧m

i=1fi(xi).

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), ai∈A∗,Γi= (Zi, Ti),1≤i≤m. Let S=S1×S2× · · · × Sm=

{s1, s2, . . . , sl}, l ∈N,where Si=∪(z|z∈Zi),1≤i≤m.

Let g: (A∗)m→Sbe 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

Z0={z0

1, z0

2, . . . , z0

l},as z0

i={si},1≤i≤l.

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, . . . , xm∈U,

<Υ|x1, . . . , xm>⇔ `Γ0g(x1, . . . , xm).

June 18, 2005 16:58 Proceedings Trim Size: 9in x 6in ALOracles

9

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

1=

(s1,Γ0) and identity observer function f0.At the end of each step of Υ0it

holds that for all x∈U,

<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,

June 18, 2005 16:58 Proceedings Trim Size: 9in x 6in ALOracles

10

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,

June 18, 2005 16:58 Proceedings Trim Size: 9in x 6in ALOracles

11

•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

2+1

4+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

12

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.

References

1. Chris Adami. On modelling life. In Artiﬁcial Life IV, pages 269–276. MIT

Press, 1994.

2. Harald Atmanspacher. Between Chance and Choice, chapter Determinism is

Ontic, Determinability is Epistemic, pages 49–74. Imprint Academic, 2002.

3. Harald Atmanspacher and Hans Primas. Epistemic and ontic quantum real-

ities. In Andrei Khrennikov, editor, Foundations of Probability and Physics

- 3, volume 750 of AIP Conference Proceedings, pages 49–62. American In-

stitute of Physics, 2005.

4. Mark A. Bedau. Blackwell Guide to the Philosophy of Computing and Infor-

mation, chapter Artiﬁcial Life, pages 197–211. Blackwell Philosophy Guides.

Blackwell Publishers, 2003.

5. Mark A. Bedau, John S. McCaskill, Norman H. Packard, Steen Rasmussen,

Chris Adami, David G. Green, Takashi Ikegami, Kunihiko Kaneko, and

Thomas S. Ray. Open problems in artiﬁcial life. Artiﬁcial Life, 6:363–376,

2000.

6. E. Bernstein and U. Vazirani. Quantum complexity theory. SIAM Journal

on Computing, 26(5):1411–1473, 1997.

7. Jack Copeland. The broad conception of computation. American Behavioral

Scientist, 40:690–716, 1997.

8. Jack Copeland. Unconventional Models of Computation, chapter Even Turing

Machines Can Compute Uncomputable Functions, pages 150–164. Springer-

Verlag, 1998.

9. Martin Davis. Computability &Unsolvability. Mc Graw Hill Series in Infor-

mation Processing and Computers. Mc Graw Hill, New York, 1958.

10. Palem GopalaKrishna. A transformation framework for solving artiﬁcial life

systems. Submitted to Evolutionary Computation, 2005.

11. David G. Green. Towards a mathematics of complexity. Complex Systems,

3:97–105, 1996.

12. Joel Hamkins and Andy Lewis. Inﬁnite time turing machines. Journal of

Symbolic Logic, 65:567–604, 1998.

June 18, 2005 16:58 Proceedings Trim Size: 9in x 6in ALOracles

13

13. Matthew Hennessy. Algebraic Theory of Processes. MIT Press Series in the

Foundations of Computing. MIT Press, London, 1988.

14. Chris Hunter and Paul Strooper. Systematically deriving partial oracles for

testing concurrent programs. In ACSC ’01: Proceedings of the 24th Aus-

tralasian Conference on Computer science, pages 83–91, Washington, DC,

USA, 2001. IEEE Computer Society.

15. Neil D. Jones. Computability Theory An Introduction. ACM Monograph Se-

ries. Academic Press, New York, 1973.

16. C. G. Langton. Studying artiﬁcial life with cellular automata. Physica D, 22,

1986.

17. Elliott Mendelson. Number Systems and the Foundations of Analysis. Acad-

emic Press, New York, 1973.

18. John Von Neumann. Theory of Self-Reproducing Automata. University of

Illinois Press, 1966.

19. Toby Ord. Hypercomputation: Computing more than the turing machine.

Technical report, University of Melbourne, Melbourne, Australia, September

2002.

20. Howard H. Pattee. Advances in Artiﬁcial Life, chapter Artiﬁcial life needs a

real epistemology, pages 23–38. Springer-Verlag, 1995.

21. R. Penrose. On understanding understanding. International Studies in the

Philosophy of Science, 11(1):7–20, 1997.

22. Hans Primas. Between Chance and Choice, chapter Hidden Determinism,

Probability, and Times Arrow, pages 89–113. Imprint Academic, 2002.

23. Christof Teuscher and Moshe Sipper. Hypercomputation: Hype or computa-

tion? Communications of the ACM, 45(8):23–24, 2002.

24. Alan M Turing. On computable numbers, with an application to the entschei-

dungs problem. Proc. Lond. Math. Soc., 43(2), 1936.

25. Alan M Turing. Systems of logic based on the ordinals. Proceedings of the

London Mathematical Society, 45:161–228, 1939.

26. Wolfgang Wechler. Universal Algebra for Computer Scientists. EATCS Mono-

graphs on Theoritical Computer Science. Springer-Verlag, New York, 1992.

27. J. Wiedermann and J. van Leeuwen. Artiﬁcial Life 2001, 6th European Con-

ference, chapter Emergence of Super-Turing Power in Artiﬁcial Living Sys-

tems, pages 55–65. LNAI 2159. Springer-Verlag, 2001.

28. J. Wiedermann and J. van Leeuwen. The emergent computational potential of

evolving artiﬁcial living systems. Tech. Report UU/CS/2002-002, University

of Utrecht, 2002.

29. Lucian Wischik. Non-ﬁnite computation in malament-hogarth spacetimes.

M.phil. dissertation, University of Cambridge, London, 1997.

30. David H. Wolpert. Computational capabilities of physical systems. Physical

Review E, 65(016128), 2001.

31. W. H. Zurek. Algorithmic randomness and physical entropy. Physical Review

A, 40(8):4731–4751, 1989.