ON ONTIC ORACLES AND EPISTEMIC ARTIFICIAL LIFE SYSTEMS

Transcription

1 ON ONTIC ORACLES AND EPISTEMIC ARTIFICIAL LIFE SYSTEMS PALEM GOPALAKRISHNA Research Scholar, Computer Science & Engineering Dept., IIT-Bombay, Mumbai , 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 counterparts 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 solution 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 coincide 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

2 2 The notion of ontic reality refers to the fundamental truth that is independent 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 realities 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 combination. The importance of this distinction can be realized by the array of limitation theories it leads to, such as, the uncompleteness theorem of Gödel 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 ontic/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). Exhaustive 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 transformation. 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.

3 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 U be the universal space, the ensemble of sequents from all possible Sequences, defined over the finite symbol set A = {a 1, a 2,..., a n }. U = (S S is a Sequence over A). We assume, without explicit statement, that U is effectively enumerable, and that every Sequence discussed is a subset of U.

4 4 Definition 2.2. A set z U defines a zone if there exists a function f : A {0, 1} such that for each s i U : { 1 if si z, f(s i ) = 0 if s i / z. Let Z = {z z U and z is a zone} be the set of all possible zones. Definition 2.3. Given two zones z i, z j Z, a subset of z i z j defines a transformation over Z. A transformation t from z i to z j is denoted by t zi z j. In infix notation it is written as z i t z j or simply as z i z j. If t z i z j and t z j z k are two transformations defined over Z, then their (binary) composition, indicated by t t, is a transformation t zi z k, defined as, t zi z k = {(x, y) (x z i, y z k, w z j )[(x, w) t z i z j (w, y) t z j z k ]}. 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) transformation system. Definition 2.5. By a transformation process in a transformation system Γ = (Z, T ) it is meant a finite series s 1, s 2,..., s m of sequents such that the following holds true i {1, m} z i Z [s i z i ] j {1, m 1} t zj z j+1 T [(s j, s j+1 ) t zj z j+1 ]. Each of the s i is called a step of the transformation process. Further, s 1 is called the initial step and s m is called the final step. Definition 2.6. A sequent s is said to be a result of Γ, indicated by Γ s, if s is 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 S Γ = {x Γ x}.

5 5 Definition 2.8. The decision problem for a transformation system Γ is recursively solvable or unsolvable, according as S Γ is or is not a recursive set. Theorem 2.1. Every effectively enumerable set is generated by a transformation system. Theorem 2.2. There exists a transformation system whose decision problem is recursively unsolvable. Definition 2.9. An Artificial Life system is defined by the tuple (f, c 1,..., c m ), where f : (A ) m (A ) m is a partial function referred to as the observer function, and each c i, 1 i m, called an element of the system, is defined by the pair (a i, Γ i ) where Γ i is a transformation system, and a i A is a result of Γ i, called as the observable state of c i. 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 transformation process over its observable state, followed by the (2) application of the observer function to the results of the transformation process obtained in step 1. At the end of each step, the tuple generated by f( Γ1 a 1,..., Γm a m ) defines the configuration of the system for the next step. We use the notation < Υ a i1,..., a im > < Υ a j1,..., a jm > to indicate that an Artificial Life system Υ has stepped from configuration < a i1,..., a im > to configuration < a j1,..., a jm >. By the decision problem for an ALsytem, we mean the problem of determining, of a given tuple < a i1, a i2,..., a im >, whether or not it is a configuration of the system. An ALsystem is said to be recursively unsolvable if its decision problem is recursively unsolvable. Theorem 2.3. There exists an ALsystem whose decision problem is recursively unsolvable.

6 6 3. Oracles and O-machines In this section we extend the transformation framework to include the formalization of oracles and o-machines. Definition 3.1. By an oracle for a transformation system Γ = (Z, T ), it is meant a transformation system Γ = (Z, T ) such that there exists a total-injective mapping ψ : (z i z i Z) (z j z j Z ) satisfying the condition that for every transformation process s l,..., s m in Γ the following holds true s l s m [(s l ψs l) (s m ψs m) t z l z m T [s l z l s m z m (s l, s m) t z l z m ]]. We use the notation Φ Γ to indicate that Φ is an oracle for Γ. The mapping ψ between Φ and Γ is referred to as an o-mapping, indicated by Φ ψ Γ. All possible o-mappings between Φ and Γ form an o-map class, [Φ Γ], given by [Φ Γ] = {ψ Φ ψ Γ}. An oracle class for Γ, written as [ Γ], is given by [ Γ] = {Φ ψ Φ ψ Γ}. Definition 3.2. Let Φ = (Z, T ) be an oracle for Γ = (Z, T ). Then, by an oracle set of Φ it is meant the set O Φ = {x y t z i z j T [y z i x z j (y, x) t z i z j ]}. Definition 3.3. An o-machine is an effective procedure f : A {0, 1} that, when associated with an oracle set O Φ of Φ ψ Γ, can decide if a sequent s U is a result of Γ or not as follows. For every sequent s U, Γ s f(s), where f(s) is defined as, { 1, if s O f(s) = Φ [s ψ s ]; 0, otherwise. We use the notation f X to indicate an o-machine that is an effective procedure f associated with oracle set X. An o-machine f X is 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.

7 7 Proof. Consider a recursively enumerable set S generated by a transformation system Γ. Let O Φ be an oracle set of Φ ψ Γ. By definition, for every transformation process s l,..., s m in Γ, Hence it follows that, s m O Φ [s m ψs m]. Γ s m s m O Φ [s m ψs m]. Now, consider an o-machine f OΦ. By Definition 3.3, s U [f(s) s O Φ [sψs ]]. We prove f OΦ can decide S as follows. Let x be an element of S. Then, x S Γ x x O Φ [xψx ] f(x). Thus, f when associated with O Φ can decide S. Hence proved. An o-machine is said to be universal if it can decide all recursively enumerable sets. Theorem 3.2. There exists no universal o-machine that can decide all recursively enumerable sets. Proof. For contradiction, assume f X to be an universal o-machine. Let S and S = U\S be two recursively enumerable sets generated by transformation systems Γ and Γ respectively. Since f X has been assumed to be universal, it follows that for all x U, However, which leads to (x S) f(x) and (x S) f(x). x U [(x S) (x S)], x U [f(x) f(x)], a contradiction. Thus our assumption that f X is 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].

8 8 Theorem 3.3. The decision problem of every closed ALsystem is solvable. Proof. Consider a closed ALsystem Υ = (f, c 1,..., c m ), where f( x) = x for all x, and c i = (a i, Γ i ), a i A, 1 i m. Considering a tuple < s 1,..., s m >, < Υ s 1,..., s m > Γ1 s 1 Γm s m. Let f X1 1,..., f Xm m be o-machines such that for all s U, f i (s) Γi s, where X i is an oracle set of Φ i Γ i. Let g : (A ) m {0, 1} be a function defined as, for every x 1,..., x m U, g(x 1,..., x m ) = m i=1f i (x i ). To prove that the decision problem of Υ is solvable, < Υ s 1,..., s m > Γ1 s 1 Γm s m f 1 (s 1 ) f m (s m ) g(s 1,..., s m ). Thus g can decide if a tuple < s 1,..., s m > 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, c 1,..., c m ), where c i = (a i, Γ i ), a i A, Γ i = (Z i, T i ), 1 i m. Let S = S 1 S 2 S m = {s 1, s 2,..., s l }, l N, where S i = (z z Z i ), 1 i m. Let g : (A ) m S be a function that converts an m-tuple into the corresponding element of S. Now, construct a transformation system Γ = (Z, T ) as follows. Define Z = {z 1, z 2,..., z l }, as z i = {s i}, 1 i l. For every step < Υ a 1,..., a m > < Υ a 1,..., a m > of ALsystem Υ, define a new transformation {(g(a 1,..., a m ), g(a 1,..., a m))} in T such that for every x 1,..., x m U, < Υ x 1,..., x m > Γ g(x 1,..., x m ).

9 9 Thus the decision problem of Υ is solvable if and only if the decision problem of Γ is solvable. Next, define a closed ALsystem Υ = (f, c 1) with single element c 1 = (s 1, Γ ) and identity observer function f. At the end of each step of Υ it holds that for all x U, < Υ x > Γ x. Thus the decision problem of Γ is solvable if and only if the decision problem of Υ is solvable. However, from Theorem 3.3, the decision problem of Υ is solvable. Consequently, the decision problem of Υ is solvable as well. While o-machines can solve the decision problems of ALsystems, the solutions 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 otherwise. 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 being 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,

10 10 without requiring any intelligence or insight. For such a procedure, assuming 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 intelligence 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 epistemic 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 exist 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 7 terms, 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,

11 11 what class of abilities can be regarded as being attributable to individual mathematicians that can cross the barrier of effective computability while still holding the view of being a mechanical procedure? (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 hypermachine 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 < 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

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