1 Introduction
We initiated the ultimate intelligence research program in 2014 inspired by Seth Lloyd’s similarly titled article on the ultimate physical limits to computation [6], intended as a booklength treatment of the theory of generalpurpose AI. In similar spirit to Lloyd’s research, we investigate the ultimate physical limits and conditions of intelligence. A main motivation is to extend the theory of intelligence using physical units, emphasizing the physicalism inherent in computer science. This is the second installation of the paper series, the first part [13]
proposed that universal induction theory is physically complete arguing that the algorithmic entropy of a physical stochastic source is always finite, and argued that if we choose the laws of physics as the reference machine, the loophole in algorithmic information theory (AIT) of choosing a reference machine is closed. We also introduced several new physically meaningful complexity measures adequate for reasoning about intelligent machinery using the concepts of minimum volume, energy and action, which are applicable to both classical and quantum computers. Probably the most important of the new measures was the minimum energy required to physically transmit a message. The minimum energy complexity also naturally leads to an energy prior, complementing the speed prior
[15] which inspired our work on incorporating physical resource limits to inductive inference theory.In this part, we generalize logical depth and conceptual jump size to stochastic sources and consider the influence of volume, space and energy. We consider the energy efficiency of computing as an important parameter for an intelligent system, forgoing other details of a universal induction approximation. We thus relate the ultimate limits of intelligence to physical limits of computation.
2 Notation and Background
Let us recall Solomonoff’s universal distribution [17]. Let be a universal computer which runs programs with a prefixfree encoding like LISP; denotes that the output of program on is where and are bit strings. ^{1}^{1}1A prefixfree code is a set of codes in which no code is a prefix of another. A computer file uses a prefixfree code, ending with an EOF symbol, thus, most reasonable programming languages are prefixfree. Any unspecified variable or function is assumed to be represented as a bit string. denotes the length of a bitstring . refers to function rather than its application.
The algorithmic probability that a bit string is generated by a random program of is:
(1) 
which conforms to Kolmogorov’s axioms [5]. considers any continuation of , taking into account nonterminating programs.^{2}^{2}2We used the regular expression notation in language theory.
is also called the universal prior for it may be used as the prior in Bayesian inference, for any data can be encoded as a bit string.
We also give the basic definition of Algorithmic Information Theory (AIT), where the algorithmic entropy, or complexity of a bit string is
(2) 
We shall now briefly recall the wellknown Solomonoff induction method [17, 18]. Universal sequence induction method of Solomonoff works on bit strings drawn from a stochastic source . eq:alp is a semimeasure, but that is easily overcome as we can normalize it. We merely normalize sequence probabilities
(3) 
eliminating irrelevant programs and ensuring that the probabilities sum to , from which point on yields an accurate prediction. The error bound for this method is the best known for any such induction method. The total expected squared error between and is
(4) 
which is less than according to the convergence theorem proven in [19], and it is roughly [22]. Naturally, this method can only work if the algorithmic complexity of the stochastic source
is finite, i.e., the source has a computable probability distribution. The convergence theorem is quite significant, because it shows that Solomonoff induction has the best generalization performance among all prediction methods. In particular, the total error is expected to be a constant independent of the input, and the error rate will thus rapidly decrease with increasing input size.
Operator induction is a general form of supervised machine learning where we learn a stochastic map from question and answer pairs
sampled from a (computable) stochastic source . Operator induction can be solved by finding in available time a set of operator models such that the following goodness of fit is maximized(5) 
for a stochastic source where each term in the summation is
(6) 
and are question/answer pairs in the input dataset, and is a computable conditional pdf (cpdf) in eq:opindgofterm. We can use the found operators to predict unseen data [22]
(7) 
The goodness of fit in this case strikes a balance between high a priori probability and reproduction of data like in minimum message length (MML) method, yet uses a universal mixture like in sequence induction. The convergence theorem for operator induction was proven in [21] using Hutter’s extension to arbitrary alphabet.
Operator induction infers a generalized conditional probability density function (cpdf), and Solomonoff argues that it can be used to teach a computer anything. For instance, we can train the question/answer system with physics questions and answers, and the system would then be able to answer a new physics question, dependent upon how much has been taught in the examples; a future user could ask the system to describe a physics theory that unifies quantum mechanics and general relativity, given the solutions of every mathematics and physics problem ever solved in literature. Solomonoff’s original training sequence plan proposed to instruct the system first with an English subset and basic algebra, and then venture into more complex subjects. The generality of operator induction is partly due to the fact that it can be used to learn any kind of association, i.e., it models an ideal contentaddressable memory, but it also generalizes any kind of law therein implicitly, that is why it can learn an implicit principle (such as of syntax) from linguistic input, enabling the system to acquire language; it can also model complex translation problems, and all manners of problems that require additional reasoning (computation). In other words, it is a universal problem solver model. It is also the most general of the three kinds of induction, which are sequence, set, and operator induction, and the closest to machine learning literature. The popular applications of speech and image recognition are covered by operator induction model, as is the wealth of pattern recognition applications, such as describing a scene in English. We think that, therefore, operator induction is an AIcomplete problem – as hard as solving the humanlevel AI problem in general. It is with this in mind that we analyze the asymptotic behavior of an optimal solution to operator induction problem.
3 Physical Limits to Universal Induction
In this section, we elucidate the physical resource limits in the context of a hypothetical optimal solution to operator induction. We first extend Bennett’s logical depth and conceptual jump size to the case of operator induction, and show a new relation between expected simulation time of the universal mixture and conceptual jump size. We then introduce a new graphical model of computational complexity which we use to derive the relations among physical resource bounds. We introduce a new definition of physical computation which we call selfcontained computation, which is a physical counterpart to selfdelimiting program. The discovery of these basic bounds, and relations, exact, and asymptotic, give meaning to the complexity definitions of Part I.
Please note that Schmidhuber disagrees with the model of the stochastic source as a computable pdf [15], but Part I contained a strong argument that this was indeed the case. A stochastic source cannot have a pdf that is computable only in the limit, if that were the case, it could have a random pdf, which would have infinite algorithmic information content, and that is clearly contradicted by the main conclusion of Part I. A stochastic source cannot be semicomputable, because it would eventually run out of energy and hence the ability to generate further quantum entropy, especially the selfcontained computations of this section. That is the reason we had introduced selfcontained computation notion at any rate. Note also that Schmidhuber agrees that quantum entropy does not accumulate to make the world incompressible in general, therefore we consider his proposal that we should view a cpdf as computable in the limit as too weak an assumption. As with Part I, the analysis of this section is extensible to quantum computers, which is beyond the scope of the present article.
3.1 Logical depth and conceptual jump size
Conceptual Jump Size (CJS) is the time required by an incremental inductive inference system to learn a new concept, and it increases exponentially in proportion to the algorithmic information content of the concept to be learned relative to the concepts already known [20]. The physical limits to OOPS based on Conceptual Jump Size were examined in [14]. Here, we give a more detailed treatment. Let be the computable cpdf that exactly simulates with respect to , for operator induction.
(8) 
The conceptual jump size of inductive inference () can be defined with respect to the optimal solution program using Levin search [16]:
(9) 
where is the running time of a program on .
(10)  
(11) 
where is the time for solving an induction problem from source with sufficient input complexity (), we observe that the asymptotic complexity is
(12) 
for fixed . Note that corresponds to the stochastic extension of Bennett’s logical depth [1], which was defined as: “the running time of the minimal program that computes ”. Let us recall that the minimal program is essentially unique, a polytope in program space [3].
Definition 1
Stochastic logical depth is the running time of the minimal program that accurately simulates a stochastic source .
(13) 
which, with eq:time2, entails our first bound.
Lemma 1
(14) 
Lemma 2
is related to the expectation of the simulation time of the universal mixture.
(15) 
where is the input data to sequence induction, without loss of generality.
Proof
Rewrite as . Observe that lefthand side of the inequality is merely a term in the summation in the right.
3.2 A Graphical Analysis of Intelligent Computation
Let us introduce a graphical model of computational complexity that will help us visualize physical complexity relations that will be investigated. We do not model the computation itself, we just enumerate the physical resources required. Present treatment is merely classical computation over sequential circuits.
Definition 2
Let the computation be represented by a directed bipartite graph where vertices are partitioned into and which correspond to primitive operations and memory cells respectively, . Function assigns time to vertices and edges. ^{3}^{3}3Time as discrete timestamps, as opposed to duration. Edges correspond to causal dependencies. and correspond to input and output vertices interacting with the rest of the world. We denote acccess to vertex subsets with functions over , e.g., .
def:lattice is a lowlevel computational complexity model where the physical resources consumed by any operation, memory cell, and edge are the same for the sake of simplicity. Let be the unit spacetime volume, be the unit energy, and be the unit space.
Definition 3
Let the volume of computation be defined as which measures the spacetime volume of computation of on in physical units, i.e., .
For def:lattice, it is . Volume of computation measures the extent of the spacetime region occupied by the dynamical evolution of the computation of on
. We do not consider the theory of relativity. For instance, the space of a Turing Machine is the Instantaneous Description (ID) of it, and its time corresponds to
. A Turing Machine derivation that has an ID of length at time and takes steps to complete would have a volume of .^{4}^{4}4If the derivation is , it has volume.Definition 4
Let the energy of computation be defined as which measures the total energy required by computation of on in physical units, e.g, .
For def:lattice, it is .
Definition 5
Let the space of computation be defined as which measures the maximum volume of a synchronous slice of the spacetime of computation on in physical units, e.g., .
For def:lattice, it is
(16) 
Definition 6
In a selfcontained physical computation all the physical resources required by computation should be contained within the volume of computation.
Therefore, we do not allow a selfcontained physical computation to send queries over the internet, or use a power cord, for instance.
Using these new more general concepts, we measure the conceptual jump size in spacetime volume rather than time (spacetime extent might be a more accurate term). Algorithmic complexity remains the same, as the length of a program readily generalizes to spacetime volume of program at the input boundary of computation, which would be for def:lattice. If , bitstring and y correspond to , and respectively. A program corresponds to a vertex set usually, and its size is denoted as . We use bitstrings for data and programs below, but measure their sizes in physical units using this notation. It is possible to eliminate bit strings altogether using a volume prior, we mix notations only for ease of understanding.
Let us generalize logical depth to the logical volume of a bit string :
(17) 
Let us also generalize stochastic logical depth to stochastic logical volume:
(18) 
which entails that Conceptual Jump Volume (CJV), and logical volume of a stochastic source may be defined analogously to CJS
(19) 
where lefthand side corresponds to spacetime extent variant of . Likewise, we define logical energy for a bit string, and stochastic logical energy:
(20) 
Which brings us to an energy based statement of conceptual jump size, that we term conceptual jump energy, or conceptual gap energy:
Lemma 3
.
The inequality holds since we can use bounds in universal search instead of time. We now show an interesting relation which is the case for selfcontained computations.
Lemma 4
If all basic operations and basic communications spend constant energy for a fixed spacetime extent (volume), then:
One must spend energy to conserve a memory state, or to perform a basic operation (in a classical computer). We may assume the constant complexity of primitive operations, which holds in def:lattice. Let us also assume that the space complexity of a program is proportional to how much mass is required. Then, the energy from the resting mass of an optimal computation may be taken into account, which we call total energy complexity (in metric units):
Lemma 5
where is the speed of light, energy density , and mass density for the graphical model of complexity.
Lemma 6
Conceptual jump total energy (CJTE) of a stochastic source is:
(21) 
As a straightforward consequence of the above lemmas, we show a lower bound on the energy required, that is related to the volume, and space linearly, and algorithmic complexity of a stochastic source exponentially, for optimal induction.
Theorem 3.1
Proof
We assume that the energy density is constant; we can use for resource bounds in Levin search. The inequality is obtained by substituting lem:toteng into the definitional inequality.
The last inequality gives bounds for the total energy cost of inferring a source in relation to spacetime extent (volume of computation), space complexity, and an exponent of algorithmic complexity of . This inspires us to define priors using , , and which would extend Levin’s ideas about resource bounded Kolmogorov complexity, such as complexity. In the first installation of ultimate intelligence series, we had introduced complexity measures and priors based on energy and action. We now define the one that corresponds to CJE and leave the rest as future work due to lack of space.
Definition 7
Energybounded algorithmic entropy of a bit string is defined as:
(22) 
3.3 Physical limits, incremental learning, and digital physics
Landauer’s limit is a thermodynamic lower bound of J for erasing bit where is the Boltzmann constant and is the temperature [4]. The total number of bitwise operations that a quantum computer can evolve is operations where is average energy, and thus the physical limit to energy efficiency of computation is about operations/J [8]. Note that the MargolusLevitin limit may be considered a quantum analogue of our relation of the volume of computation with total energy, which is called “action volume” in their paper, as it depends on the quantum of action which has units. Bremermann discusses the minimum energy requirements of computation and communication in [2]. Lloyd [6] assumes that all the mass may be converted to energy and calculates the maximum computation capacity of a 1 kilogram “blackhole computer”, performing operations over bits. According to an earlier paper of his, the whole universe may not have performed more than operations over bits [7].
Corollary 1
for any where the logical volume is .
Proof
. Assume that . ^{5}^{5}5Although the assumption that it takes only 1 unit of spacetime volume to simulate the minimal program that reproduces the pdf is not realistic, we are only considering this for the sake of simplicity, and because 1 is close to the volume of a personal computer, or a brain. For many pdfs, it could be much larger in practice. .
Therefore, if has a greater algorithmic complexity than about bits, it would have been unguaranteed to discover it without any a priori information. Digital physics theories suggest that the physical law could be much simpler than that however, as there are very simple universal computers in the literature [9], a survey of which may be found in [10], which means interestingly that the universe may have had enough time to discover its basic law.
This limit shows the remarkable importance of incremental learning as both Solomonoff [23] and Schmidhuber [14] have emphasized, which is part of ongoing research. We proposed previously that incremental learning is an AI axiom [12]. Optimizing energy efficiency of computation would also be an obviously useful goal for a selfimproving AI. This measure was first formalized by Solomonoff in [21], which he imagined would be optimizing performance in units of bits/sec.J as applied to inductive inference, which we agree with, and will eventually implement in our Alpha Phase 2 machine; Alpha Phase 1 has already been partially implemented in our parallel incremental inductive inference system [11].
Acknowledgements
Thanks to anonymous reviewers whose comments substantially improved the presentation. Thanks to Gregory Chaitin and Juergen Schmidhuber for inspiring the mathematical philosophy / digital physics angle in the paper. I am forever indebted for the highquality research coming out of IDSIA which revitalized interest in humanlevel AI research.
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