Followup to: Agents with Cartesian childhood and Physicalist adulthood

In previous posts I have defined a formalism for quantifying the general intelligence of an abstract agent (program). This formalism relies on counting proofs in a given formal system F (like in regular UDT), which makes it susceptible to the Loebian obstacle. That is, if we imagine the agent itself making decisions by looking for proofs in the same formal system F then it would be impossible to present a general proof of its trustworthiness, since no formal system can assert is own soundness. Thus the agent might fail to qualify for high intelligence ranking according to the formalism. We can assume the agent uses a weaker formal system the soundness of which is provable in F but then we still run into difficulties if we want the agent to be self-modifying (as we expect it to be). Such an agent would have to trust its descendants which means that subsequent agents use weaker and weaker formal systems until self-modification becomes impossible.

One known solution to this is Benja's parametric polymorphism. In this post I adapt parametric polymorphism to the updateless intelligence metric framework. The formal form of this union looks harmonious but it raises questions which I currently don't fully understand.

"Ineffable mystery" using oracles instead of timeouts

In the original parametric polymorphism, a constant κ is introduced (informally known as "the number of ineffable mystery") s.t. the agent has to prove its actions are "safe" for time period κ (i.e. have no "bad" consequences during this period). Since it's impossible to the agent to prove any upper bound on κ, effectively its actions have to be safe indefinitely. I found that for our purposes it's better to cast the formalism otherwise. Instead of κ playing the role of a "timeout", the agent is provided with an oracle that answers questions of the form "κ > n?" for any n. Contrary to what the agent thinks, the oracle provides an affirmative answer for all n.

Let's do some math. The formalism I introduce at this point is very close to Yudkowsky and Herreshoff, with some adjustments.

Consider an infinite chain of agents A0, A1, A2... where A{i+1} is created by Ai. Formally, each Ai is a program for a universal oracle machine which computes a_i = (a_i, A{i+1}). Here a_i is a "motor action". I'm ignoring sensory input for the sake of simplicity (of course sensory input will reappear when we return to intelligence metrics).

Denote T_κ the usual theory of parametric polymorphism. That is T_κ results from starting with a theory T (e.g. Peano Arithmetic), extending the language by the symbol κ (which behaves like a natural number) and adding the (self-referential) axiom schema 

(1) For all formulae φ: κ > 0 -> For all n: (@_{T_κ} "φ(n)" -> φ(n)[κ\κ-1])

Here '@_S' is a formula with the semantics of "provable in theory S" and φ(n)[κ\κ-1] is φ(n) with all occurrences of κ replaced by κ-1. Thus (1) is a self-referential soundness schema saved from paradoxicality by the "counter" κ. See Yudkowsky and Herreshoff for the proof that T_κ is sound (it has 1 model for each possible value of κ).

Assume the licensing condition for A0 to perform action b_0 is

(2) Exists n: T_κ | κ > n -> (A0() = b_0 -> G)

Here S | φ means that φ is provable in S, G is a formula denoting the "goal" of the agent (as in Yudkowsky and Herreshoff, the agent is for now assumed to be a satisficer), A0() is a formula meaning "the result of the computation A0", where the oracle is interpreted as answering the question "κ > x?" (in particular the formula A0() involves κ). Such a licensing condition can be implemented in practice by making A0 a proof searcher which is allowed to use the oracle to introduce assumptions of the form κ > n. In particular, our agent employs ambient control.

We then have the following proposition, showing that A0 is able to tile indefinitely (create an infinite chain of meaningful descendants). The proposition is formulated for A1 but the rest of the chain follows by induction since the form of the licensing condition is preserved (up to a certain twist) by the tiling.

Proposition

Assume b_0 = (b_0, B1) and B1 is s.t.

(3) T_κ | For all b_1: (B1() = b_1 -> (b_1 = 0 or Exists n: (κ > n and @_{T_κ} "(κ+1 > n and B1()[κ\κ+1] = b_1) -> G[κ\κ+1]")))

Here 0 is some default "harmless" action, performed if no appropriate theorem is proved.

Further assume that if B1 is created and does 0, the goal is achieved (in this framework "achieving the goal" is to be thought of as "not doing anything unsafe").

(4) T_κ | (A0() = b_0 and B1() = 0) -> G

Then A0 is allowed to create B1 i.e. licensing condition (2) holds.

B1's licensing condition contains a "twist", namely it believes that when the oracle receives input n it answers the question "κ+1 > n?" rather than "κ > n" and its goal is shifted in the same fashion. Note that the goal typically depends on κ since the agent itself and thus its κ-oracle is part of the universe in which this goal is to be attained. For example the goal can be a function of the numbers (a0, a1, a2...) defined as the output of corresponding computations with κ-oracles. Thus the shift in G can also be interpreted as a shifted interpretation of the oracle by B1.

Proof

(3) trivially implies that

T_κ | κ > 0 -> (For all b_1: (B1() = b_1 -> (b_1 = 0 or Exists n: (κ > n and @_{T_κ} "(κ+1 > n and B1()[κ\κ+1] = b_1) -> G[κ\κ+1]"))))

Using the premise κ > 0, we can apply the soundness schema (1). We obtain:

T_κ | κ > 0 -> (For all b_1: (B1() = b_1 -> (b_1 = 0 or Exists n: (κ > n and ((κ > n and B1() = b_1) -> G)))))

T_κ | κ > 0 -> (For all b_1: (B1() = b_1 -> (b_1 = 0 or Exists n: (B1() = b_1 -> G))))

T_κ | κ > 0 -> (For all b_1: (B1() = b_1 -> (b_1 = 0 or (B1() = b_1 -> G))))

T_κ | κ > 0 -> (For all b_1: (B1() = b_1 -> (b_1 = 0 or G)))

This trivially implies that

T_κ | κ > 0 -> (For all b_1: ((A0() = b_0 and B1() = b_1) -> (b_1 = 0 or G)))

Applying (4) we finally get (2) for n=0

Back to intelligence metrics

To apply parametric polymorphism to the updateless intelligence formalism, we need to do the following:

• Assume the machine M on which the agent is running is an oracle machine.
• Assume the Solomonoff measure of the ideal (Cartesian) universe X is defined using a universal oracle machine. The oracle in M has to correspond to the oracle in the hypothesis T describing X: this correspondence becomes part of the rules N.
• Assume the universal program u defining the Solomonoff measure for the physical universe is a universal oracle program, i.e. the hypotheses D describing the physical universe are also allowed to invoke the oracle.
• Assume the logical expectation value E_L is computed using T_κ extended by N applied to the given T (this is provable in T_κ anyway but we want the proof to be short) and the axiom schema {κ > n} for every natural number n. The latter extension is consistent since adding any finite number of such axioms admits models. The proofs counted in E_L interpret the oracle as answering the the question "κ > n?". That is, they are proofs of theorems of the form "if this oracle-program T computes q when the oracle is taken to be κ > n, then the k-th digit of the expected utility is 0/1 where the expected utility is defined by a Solomonoff sum over oracle programs with the oracle again taken to be κ > n".

Discussion

• Such an agent, when considering hypotheses consistent with given observations, will always face a large number of different compatible hypothesis with similar complexity. These hypotheses result from arbitrary insertions of the oracle (which increase complexity of course, but not drastically). It is not entirely clear to me how such an epistemology will look like.
• The formalism admits naturalistic trust to the extent the agent believes that the other agent's oracle is "genuine" and carries a sufficient "twist". This will often be ambiguous so trust will probably be limited to some finite probability. If the other agent is equivalent to the given one on the level of physical implementation then the trust probability is likely to be high.
• The agent is able to quickly confirm κ > n for any n small enough to fit into memory. For the sake of efficiency we might want to enhance this ability by allowing the agent to confirm that (Exist n: φ(n)) -> Exist n: (φ(n) and κ > n) for any given formula φ.
• For the sake of simplicity I neglected multi-phase AI development, but the corresponding construction seems to be straightforward.
• Overall I retain the feeling that a good theory of logical uncertainty should allow the agent to assign a high probability the soundness of its own reasoning system (a la Christiano et al). Whether this will make parametric polymorphism redundant remains to be seen.

Followup to: Updateless intelligence metrics in the multiverse

In the previous post I explained how to define a quantity that I called "the intelligence metric" which allows comparing intelligence of programs written for a given hardware. It is a development of the ideas by Legg and Hutter which accounts for the "physicality" of the agent i.e. that the agent should be aware it is part of the physical universe it is trying to model (this desideratum is known as naturalized induction). My construction of the intelligence metric exploits ideas from UDT, translating them from the realm of decision algorithms to the realm of programs which run on an actual piece of hardware with input and output channels, with all the ensuing limitations (in particular computing resource limitations).

In this post I present a variant of the formalism which overcomes a certain problem implicit in the construction. This problem has to do with overly strong sensitivity to the choice of a universal computing model used in constructing Solomonoff measure. The solution sheds some interesting light on how the development of the seed AI should occur.

Structure of this post:

• A 1-paragraph recap of how the updateless intelligence formalism works. The reader interested in technical details is referred to the previous post.
• Explanation of the deficiencies in the formalism I set out to overcome.
• Explanation of the solution.
• Concluding remarks concerning AI safety and future development.

TLDR of the previous formalism

The metric is a utility expectation value over a Solomonoff measure in the space of hypotheses describing a "Platonic ideal" version of the target hardware. In other words it is an expectation value over all universes containing this hardware in which the hardware cannot "break" i.e. violate the hardware's intrinsic rules. For example, if the hardware in question is a Turing machine, the rules are the time evolution rules of the Turing machine, if the hardware in question is a cellular automaton, the rules are the rules of the cellular automaton. This is consistent with the agent being Physicalist since the utility function is evaluated on a different universe (also distributed according to a Solomonoff measure) which isn't constrained to contain the hardware or follow its rules. The coupling between these two different universes is achieved via the usual mechanism of interaction between the decision algorithm and the universe in UDT i.e. by evaluating expectation values conditioned on logical counterfactuals.

Problem

The Solomonoff measure depends on choosing a universal computing model (e.g. a universal Turing machine). Solomonoff induction only depends on this choice weakly in the sense that any Solomonoff predictor converges to the right hypothesis given enough time. This has to do with the fact that Kolmogorov complexity only depends on the choice of universal computing model through an O(1) additive correction. It is thus a natural desideratum for the intelligence metric to depend on the universal computing model weakly in some sense. Intuitively, the agent in question should always converge to the right model of the universe it inhabits regardless of the Solomonoff prior with which it started. 

The problem with realizing this expectation has to do with exploration-exploitation tradeoffs. Namely, if the prior strongly expects a given universe, the agent would be optimized for maximal utility generation (exploitation) in this universe. This optimization can be so strong that the agent would lack the faculty to model the universe in any other way. This is markedly different from what happens with AIXI since our agent has limited computing resources to spare and it is physicalist therefore its source code might have side effects important to utility generation that have nothing to do with the computation implemented by the source code. For example, imagine that our Solomonoff prior assigns very high probability to a universe inhabited by Snarks. Snarks have the property that once they see a robot programmed with the machine code "000000..." they immediately produce a huge pile of utilons. On the other hand, when they see a robot programmed with any other code they immediately eat it and produce a huge pile of negative utilons. Such a prior would result in the code "000000..." being assigned the maximal intelligence value even though it is everything but intelligent. Observe that there is nothing preventing us from producing a Solomonoff prior with such bias since it is possible to set the probabilities of any finite collection of computable universes to any non-zero values with sum < 1.

More precisely, the intelligence metric involves two Solomonoff measures: the measure of the "Platonic" universe and the measure of the physical universe. The latter is not really a problem since it can be regarded to be a part of the utility function. The utility-agnostic version of the formalism assumes a program for computing the utility function is read by the agent from a special storage. There is nothing to stop us from postulating that the agent reads another program from that storage which is the universal computer used for defining the Solomonoff measure over the physical universe. However, this doesn't solve our problem since even if the physical universe is distributed with a "reasonable" Solomonoff measure (assuming there is such a thing), the Platonic measure determines in which portions of the physical universe (more precisely multiverse) our agent manifests.

There is another way to think about this problem. If the seed AI knows nothing about the universe except the working of its own hardware and software, the Solomonoff prior might be insufficient "information" to prevent it from making irreversible mistakes early on. What we would like to do is to endow it from the first moment with the sum of our own knowledge, but this might prove to be very difficult.

Solution

Imagine the hardware architecture of our AI to be composed of two machines. One I call the "child machine", the other the "adult machine". The child machine receives data from the same input channels (and "utility storage") as the adult machine and is able to read the internal state of the adult machine itself or at least the content of its output channels. However, the child machine has no output channels of its own. The child machine has special memory called "template memory" into which it has unlimited write access. There a single moment in time ("end of childhood"), determined by factors external to both machines (i.e. the human operator) in which the content of the template memory is copied into the instruction space of the adult machine. Thus, the child machine's entire role is making observations and using them to prepare a program for the adult machine which will be eventually loaded into the latter.

The new intelligence metric assigns intelligence values to programs for the child machine. For each hypothesis describing the Platonic universe (which now contains both machines, the end of childhood time value and the entire ruleset of the system) we compute the utility expectation value under the following logical counterfactual condition: "The program loaded into template memory at the end of childhood is the same as would result from the given program for the child machine if this program for the child machine would be run with the inputs actually produced by the given hypothesis regarding the Platonic universe". The intelligence value is then the expectation value of that quantity with respect to a Solomonoff measure over hypotheses describing the Platonic universe.

The important property of the logical counterfactual is that it doesn't state the given program is actually loaded into the child machine. It only says the resulting content of the template memory is the same as which would be obtained from the given program assuming all the laws of the Platonic universe hold. This formulation prevents exploitation of side effects of the child source code since the condition doesn't fix the source code, only its output. Effectively, the child agents considers itself to be Cartesian, i.e. can consider neither the side effects of its computations nor the possibility the physical universe will violate the laws of its machinery. On the other hand the child's output (the mature program) is a physicalist agent since it affects the physical universe by manifesting in it.

If such an AI is implemented in practice, it makes sense to prime the adult machine with a "demo" program which will utilize the output channels in various ways and do some "exploring" using its input channels. This would serve to provide the child with as much as possible information.

To sum up, the new expression for the intelligence metric is:

I(q) = E_HX[E_HY(Ec(X))[E_L[U(Y, Eu(X)) | Q(X, t(X)) = Q*(X; q)]] | N]

Here:

• q is the program priming the child machine
• HX is the hypothesis producing the Platonic universe X (a sequence of bits encoding the state of the hardware as a function of time and the end-of-childhood time t(X)). It is a program for a fixed universal computing model C.
• HY is the hypothesis producing the Physical universe (an abstract sequence of bits). It is a program for the universal computer program ("virtual machine") Ec(X) written into storage E in X.
• E_L is logical expectation value defined e.g. using evidence logic.
• Eu(X) is a program for computing the utility function which is written into storage E in X.
• U is the utility function which consists of applying Eu(X) to Y.
• Q(X, t(X)) is the content of template memory at time t(X).
• Q*(X; q) is the content that would be in the template memory if it was generated by program q receiving the inputs going into the child machine under hypothesis HX.
• N is the full ruleset of the hardware including the reprogramming of the adult machine that occurs at t(X).

Concluding Remarks

• It would be very valuable to formulate and prove a mathematical theorem which expresses the sense in which the new formalism depends on the choice of universal computing model weakly (in particular it would validate the notion).
• This formalism might have an interesting implication on AI safety. Since the child agent is Cartesian and has no output channels (it cannot create output channels because it is Cartesian) it doesn't present as much risk as an adult AI. Imagine template memory is write-only (which is not a problem for the formalism) and is implemented by a channel that doesn't store the result anywhere (in particular the mature program is never run). There can still be risk due to side effects of the mature program that manifest through presence of its partial or full versions in (non-template) memory of the child machine. For example, imagine the mature program is s.t. any person who reads it experiences compulsion to run it. This risk can be mitigated by allowing both machines to interact only with a virtual world which receives no inputs from the external reality. Of course the AI might still be able to deduce external reality. However, this can be prevented by exploiting prior bias: we can equip the AI with a Solomonoff prior that favors the virtual world to such extent that it would have no reason to deduce the real world. This way the AI is safe unless it invents a "generic" box-escaping protocol which would work in a huge variety of different universes that might contain the virtual world.
• If we factor finite logical uncertainty into evaluation of the logical expectation value E_L, the plot thickens. Namely, a new problem arises related to bias in the "logic prior". To solve this new problem we need to introduce yet another stage into AI development which might be dubbed "fetus". The fetus has no access to external inputs and is responsible for building a sufficient understanding of mathematics in the same sense the child is responsible to build a sufficient understanding of physics. Details will follow in subsequent posts, so stay tuned!

Followup to: Intelligence Metrics with Naturalized Induction using UDT

In the previous post I have defined an intelligence metric solving the duality (aka naturalized induction) and ontology problems in AIXI. This model used a formalization of UDT using Benja's model of logical uncertainty. In the current post I am going to:

• Explain some problems with my previous model (that section can be skipped if you don't care about the previous model and only want to understand the new one).
• Formulate a new model solving these problems. Incidentally, the new model is much closer to the usual way UDT is represented. It is also based on a different model of logical uncertainty.
• Show how to define intelligence without specifying the utility function a priori.
• Since the new model requires utility functions formulated with abstract ontology i.e. well-defined on the entire Tegmark level IV multiverse. These are generally difficult to construct (i.e. the ontology problem resurfaces in a different form). I outline a method for constructing such utility functions.

Problems with UIM 1.0

The previous model postulated that naturalized induction uses a version of Solomonoff induction updated in the direction of an innate model N with a temporal confidence parameter t. This entails several problems:

• The dependence on the parameter t whose relevant value is not easy to determine.
• Conceptual divergence from the UDT philosophy that we should not update at all.
• Difficulties with counterfactual mugging and acausal trade scenarios in which G doesn't exist in the "other universe".
• Once G discovers even a small violation of N at a very early time, it loses all ground for trusting its own mind. Effectively, G would find itself in the position of a Boltzmann brain. This is especially dangerous when N over-specifies the hardware running G's mind. For example assume N specifies G to be a human brain modeled on the level of quantum field theory (particle physics). If G discovers that in truth it is a computer simulation on the merely molecular level, it loses its epistemic footing completely.

UIM 2.0

I now propose the following intelligence metric (the formula goes first and then I explain the notation):

I_U(q) := E_T[E_D[E_L[U(Y(D)) | Q(X(T)) = q]] | N]

• N is the "ideal" model of the mind of the agent G. For example, it can be a universal Turing machine M with special "sensory" registers e whose values can change arbitrarily after each step of M. N is specified as a system of constraints on an infinite sequence of natural numbers X, which should be thought of as the "Platonic ideal" realization of G, i.e. an imagery realization which cannot be tempered with by external forces such as anvils. As we shall see, this "ideal" serves as a template for "physical" realizations of G which are prone to violations of N.
• Q is a function that decodes G's code from X e.g. the program loaded in M at time 0. q is a particular value of this code whose (utility specific) intelligence I_U(q) we are evaluating.
• T is a random (as in random variable) computable hypothesis about the "physics" of X, i.e a program computing X implemented on some fixed universal computing model (e.g. universal Turing machine) C. T is distributed according to the Solomonoff measure however the expectation value in the definition of I_U(q) is conditional on N, i.e. we restrict to programs which are compatible with N. From the UDT standpoint, T is the decision algorithm itself and the uncertainty in T is "introspective" uncertainty i.e. the uncertainty of the putative precursor agent PG (the agent creating G e.g. an AI programmer) regarding her own decision algorithm. Note that we don't actually need to postulate a PG which is "agenty" (i.e. use for N a model of AI hardware together with a model of the AI programmer programming this hardware), we can be content to remain in a more abstract framework.
• D is a random computable hypothesis about the physics of Y, where Y is an infinite sequence of natural numbers representing the physical (as opposed to "ideal") universe. D is distributed according to the Solomonoff measure and the respective expectation value is unconditional (i.e. we use the raw Solomonoff prior for Y which makes the model truly updateless). In UDT terms, D is indexical uncertainty.
• U is a computable function from infinite sequences of natural numbers to [0, 1] representing G's utility function. 
• L represents logical uncertainty. It can be defined by the model explained by cousin_it here, together with my previous construction for computing logical expectation values of random variables in [0, 1]. That is, we define E_L(d_k) to be the probability that a random string of bits p encodes a proof of the sentence "Q(X(T)) = q implies that the k-th digit of U(Y(D)) is 1" in some prefix-free encoding of proofs conditional on p encoding the proof of either that sentence or the sentence "Q(X(T)) = q implies that the k-th digit of U(Y(D)) is 0". We then define 
  E_L[U(Y(D)) | Q(X(T)) = q] := Σ_k 2^-k E_L(d_k). Here, the sentences and the proofs belong to some fixed formal logic F, e.g. Peano arthimetics or ZFC.  

Discussion

• G's mental architecture N is defined in the "ideal" universe X where it is inviolable. However, G's utility function U inhabits the physical universe Y. This means that a highly intelligent q is designed so that imperfect realizations of G inside Y generate as many utilons as possible. A typical T is a low Kolmogorov complexity universe which contains a perfect realization of G. Q(X(T)) is L-correlated to the programming of imperfect realizations of G inside Y because T serves as an effective (approximate) model of the formation of these realizations. For abstract N, this means q is highly intelligent when a Solomonoff-random "M-programming process" producing q entails a high expected value of U.
• Solving the Loebian obstacle requires a more sophisticated model of logical uncertainty. I think I can formulate such a model. I will explain it in another post after more contemplation.
• It is desirable that the encoding of proofs p satisfies a universality property so that the length of the encoding can only change by an additive constant, analogically to the weak dependence of Kolmogorov complexity on C. It is in fact not difficult to formulate this property and show the existence of appropriate encodings. I will discuss this point in more detail in another post.

Generic Intelligence

It seems conceptually desirable to have a notion of intelligence independent of the specifics of the utility function. Such an intelligence metric is possible to construct in a way analogical to what I've done in UIM 1.0, however it is no longer a special case of the utility-specific metric.

Assume N to consist of a machine M connected to a special storage device E. Assume further that at X-time 0, E contains a valid C-program u realizing a utility function U, but that this is the only constraint on the initial content of E imposed by N. Define

I(q) := E_T[E_D[E_L[u(Y(D); X(T)) | Q(X(T)) = q]] | N]

Here, u(Y(D); X(T)) means that we decode u from X(T) and evaluate it on Y(D). Thus utility depends both on the physical universe Y and the ideal universe X. This means G is not precisely a UDT agent but rather a "proto-agent": only when a realization of G reads u from E it knows which other realizations of G in the multiverse (the Solomonoff ensemble from which Y is selected) should be considered as the "same" agent UDT-wise.

Incidentally, this can be used as a formalism for reasoning about agents that don't know their utility functions. I believe this has important applications in metaethics I will discuss in another post.

Utility Functions in the Multiverse

UIM 2.0 is a formalism that solves the diseases of UIM 1.0 at the price of losing N in the capacity of the ontology for utility functions. We need the utility function to be defined on the entire multiverse i.e. on any sequence of natural numbers. I will outline a way to extend "ontology-specific" utility functions to the multiverse through a simple example.

Suppose G is an agent that cares about universes realizing the Game of Life, its utility function U corresponding to e.g. some sort of glider maximization with exponential temporal discount. Fix a specific way DC to decode any Y into a history of a 2D cellular automaton with two cell states ("dead" and "alive"). Our multiversal utility function U* assigns Ys for which DC(Y) is a legal Game of Life the value U(DC(Y)). All other Ys are treated by dividing the cells into cells O obeying the rules of Life and cells V violating the rules of Life. We can then evaluate U on O only (assuming it has some sort of locality) and assign V utility by some other rule, e.g.:

• zero utility
• constant utility per V cell with temporal discount
• constant utility per unit of surface area of the boundary between O and V with temporal discount 

Discussion

TLDR

The intelligence of a physicalist agent is defined to be the UDT-value of the "decision" to create the agent by the process creating the agent. The process is selected randomly from a Solomonoff measure conditional on obeying the laws of the hardware on which the agent is implemented. The "decision" is made in an "ideal" universe in which the agent is Cartesian, but the utility function is evaluated on the real universe (raw Solomonoff measure). The interaction between the two "universes" is purely via logical conditional probabilities (acausal).

If we want to discuss intelligence without specifying a utility function up front, we allow the "ideal" agent to read a program describing the utility function from a special storage immediately after "booting up".

Utility functions in the Tegmark level IV multiverse are defined by specifying a "reference universe", specifying an encoding of the reference universe and extending a utility function defined on the reference universe to encodings which violate the reference laws by summing the utility of the portion of the universe which obeys the reference laws with some function of the space-time shape of the violation.

Followup to: Intelligence Metrics and Decision Theory
Related to: Bridge Collapse: Reductionism as Engineering Problem

A central problem in AGI is giving a formal definition of intelligence. Marcus Hutter has proposed AIXI as a model of perfectly intelligent agent. Legg and Hutter have defined a quantitative measure of intelligence applicable to any suitable formalized agent such that AIXI is the agent with maximal intelligence according to this measure.

Legg-Hutter intelligence suffers from a number of problems I have previously discussed, the most important being:

• The formalism is inherently Cartesian. Solving this problem is known as naturalized induction and it is discussed in detail here.
• The utility function Legg & Hutter use is a formalization of reinforcement learning, while we would like to consider agents with arbitrary preferences. Moreover, a real AGI designed with reinforcement learning would tend to wrestle control of the reinforcement signal from the operators (there must be a classic reference on this but I can't find it. Help?). It is straightword to tweak to formalism to allow for any utility function which depends on the agent's sensations and actions, however we would like to be able to use any ontology for defining it.