\section{Introduction}
Intelligence, according to Chollet, is a measure of how little information one requires to attain a skill \cite{l6,chollet_2019}. Accordingly a more intelligent agent would need less information to make correct predictions, would predict at least as accurately as less intelligence agents given the same information and so adapt more effectively to changing circumstances. More succinctly, intelligence is the ability to make accurate generalisations \cite{l6,chollet_2019}. For the purposes of this paper an AGI is the most intelligent agent by this definition. 
There are compelling arguments to be made for the development of this sort of AGI. Such an agent would not only more effectively learn, adapt and even ascribe purpose to what it observes \cite{bennett2021,bennettmaruyama2021b,l5}, but may yield social benefits in comparison to methods popular today. For example, only large organisations have the resources to train models that require a lot of data \cite{wangwiebe2016}. AGI would be more accessible. 
Yet until now there existed only one mathematical formalism which, arguably and under specific conditions, satisfied this definition of artificial general intelligence \cite{b27,l0}. It was named AIXI, and while it was formulated to address a different definition of intelligence, the universal prior \cite{solomonoff_1964a,solomonoff_1964b} AIXI employed also allowed it to make correct predictions from minimal data. 
Unfortunately that universal prior was incomputable, ensuring AIXI could only ever be approximated. To make matters worse, AIXI's performance was later shown to be subjective, because it could be affected by the Universal Turing Machine (UTM) on which it ran \cite{leike2015}.
Nevertheless AIXI remains an important result. Given Deepmind co-founder Shane Legg's PhD thesis was on AIXI \cite{l0} it is arguable that it shaped the entire AI research sector. However, construction of an AGI requires an alternative theory which addresses these problems. This paper puts forward such a theory, proves it is an artificial general intelligence as defined above, and provides experimental evidence in support of these claims. Background material likely to be unfamiliar is introduced below.
\label{review}

\subsection{Semantic Theories of Meaning}
\paragraph{\it{Extension.}} Gottlob Frege, a 19$^{th}$ century philosopher and mathematician, tried to formalise language so that linguistic expressions could be treated as mathematical expressions \cite{frege1906}. Theories of reference were the result.   
These posit each sentence has a truth value, and each subsentential expression within a sentence contributes to that truth value. Assume it is true that ``magpies can fly", but not that ``pigs can fly". The predicate ``can fly" behaves as a boolean function that accepts an object and returns true if and only if that object is in the set of things which ``can fly". The set of things to which an expression refers is called its \textit{extension}. ``Pigs can fly" yields an empty set, because there are no pigs of which ``can fly" is true.

\paragraph{\it{Intension.}} Reference alone failed to capture subtle distinctions between logically equivalent sentences \cite{speaks2021}.
Quine gave an example using animals with hearts (chordates) and animals with kidneys (renates) \cite{quine1986}. Consider the following expressions:
\begin{enumerate}
    \item ``all chordates are chordates"
    \item ``all chordates are renates"
\end{enumerate}
In our world chordates and renates share the same extension, and so both expressions have the same truth value. Yet the first expression is an obvious tautology, while the second reveals the potentially useful fact that all animals with hearts also have kidneys. Their content, or meaning as a human might interpret it, is different.
Another way to say two logically equivalent expressions have different content is to say they have different intensions.
Logically equivalent sentences share an extension, but may differ in their intension. This notion underpins a formulation of tasks, which argues the meaningful difference between logically equivalent sentences lies in the extensions of their subsentential parts \cite{bennett2021}.

\subsection{A Formulation of Tasks}
 
Given a tool, a human will tend to look for problems that the tool might address \cite{l2}. Yet it is arguably better to first identify an urgent problem and only then consider what tools might be needed \cite{spradlin2012}.
If intelligence is a tool, then the problem it addresses is a task \cite{bennett2021}. The greater the intelligence \cite{l6,chollet_2019}, the more complex the tasks with which it can cope.
This begs the question, if it is better to begin with the problem then what, exactly, is a task?
To answer this question a model of an arbitrary task \cite{bennett2021} was formulated. It suggested intelligence is characterised by the ability to learn the purpose of a task, rather than the means by which one is completed \cite{bennettmaruyama2021b}. Both purpose and means were sets of rules \cite{bennett2021}, but means to an end were very specific rules. It was argued that what differentiated extension from intension was how ``weak" (nonspecific) the rules involved were \cite{bennettmaruyama2021a}. The purpose of a task together with the circumstances in which it was undertaken then formed an intension, and all means by which that purpose could be fulfilled the extension \cite{bennettmaruyama2021b}. By altering the weakness of those rules involved, the purpose of a task could be constructed from the means by which it was completed.
Further details are given below of how the formulation of tasks related to intelligent agents. However, the original formulation lacked rigour, and details needed for implementation. This paper fleshes out those details. 

\paragraph{\it{Agency.}}
An agent must have agency. Typically, intelligent agents are understood using the agent-environment paradigm \cite{russellandnorvig2021}, in which an agent maps percepts to actions. Nowhere in this paradigm was a task mentioned, so it was modified (see figure \ref{fig:1}) to account for the existence of tasks.
\label{agi}

\begin{figure}[h] 
  \centering
  \includegraphics[width=\textwidth]{images/fulfig1.png}
  \caption{Agent environment paradigm, modified to account for tasks.}
  \label{fig:1}
\end{figure}

In this modified paradigm (which amounts to enactive cognition \cite{thompson2007}), a task would be completed as follows: an embodied mind would be presented with a situation, about which it would make a decision, which could set in motion actions, affecting the environment.

\paragraph{\it{Correctness.}} That decision would be correct if sufficiently likely to cause the task's completion. Primitives of cognition such as ``pleasure" and ``hunger" were assumed to be built into the agent's body. These specified a partial ordering of states and a threshold beyond which the utility is ``good enough" to indicate completion \cite{bennettmaruyama2021b}. Such criteria would need to be imposed upon the agent through natural selection or design. Because a decision could specify a sequence of actions, the formulation of tasks accounted for agency \cite{bennettmaruyama2021b}.

\paragraph{\it{Rulesets.}}
To distinguish correct from incorrect implied a set of rules. Any ruleset that specified the desired end of a task would be sufficient. A ruleset would be constructed from a set of only positive examples \cite{bennettmaruyama2021b}, known as an ostensive definition \cite{gupta2021}. As Russell put it ``all nominal definitions, if pushed back far enough, must lead ultimately to terms having only ostensive definitions" \cite{russell1948}. This paper assumes an ostensive definition is given, but one might also be constructed through repeated interaction in the manner of a reinforcement learning agent \cite{bennettmaruyama2021b} given a means of assessing whether expected reward exceeds the threshold for task completion described above. Rulesets constructed from both positive and negative examples are discussed in section \ref{twoclass}, and formally defined in appendix
\ref{supplemental}.

\paragraph{\it{Language.}}
Rules correspond unambiguously to the state of a machine (the embodied agent). The state of a machine is just a set of facts, each pertaining to self evident phenomena one could detect mechanically, such as the position of a mechanical arm or the electrical current on a wire.
Not to be confused with physical symbol systems \cite{nilsson_2007}, preceding work \cite{bennett2021} defined a physically implementable language as one in which statements
\begin{enumerate}
    \item unambiguously correspond to the state of hardware (dyadic), 
    \item can be mechanically verified (implementable), which means
    \item there is one and only one true interpretation of such a statement.
\end{enumerate}

\section{Definitions and Proofs}
To predict the world, an agent must have a model of the world. AIXI modelled the world as a program able to perfectly reproduce what AIXI had experienced. There would always be many such programs, each of which served to completely explain the past, yet they would differ in how well they predicted the future. What made AIXI intelligent was that it had a means of discerning which of those programs would predict the future accurately. The lower a program's Kolmogorov Complexity $k$ \cite{kolmogorov_1963}, the more likely it would  correctly predict the future. This gave AIXI what is called a universal prior. 
Yet Kolmogorov Complexity also introduced problems. It was incomputable, and it equated plausibility with program length (true only sometimes), making AIXI's performance subjective \cite{leike2015}. 
Problems with Kolmogorov Complexity aside, the underlying ideas remained compelling. All that was needed was a different means of discerning which model (of a set of models equally capable of explaining the past) might best predict the future. This was possible using the aforementioned formulation of tasks.
Under this formulation \cite{bennett2021}, models took the form of rules rather than programs. What was modelled was a task, not the environment. A universal prior would still be required, but could not use Kolmogorov Complexity.
Chollet formulated a measure of intelligence, but it too relied upon Kolmogorov Complexity \cite{chollet_2019} and so could not serve (I would suggest this makes it as subjective as AIXI's performance). However, his argument that intelligence is a measure of the ability to generalise accurately remained compelling. The idea would just need to be reformulated as weakness \cite{bennett2021}, with which rulesets could be evaluated, to give a universal prior.

\subsection{Definitions}
\label{definitions}

To aid understanding, informal explanations accompany some of the mathematical definitions below.

\paragraph{\bf{Definition 0 - Fact:}} A measurable property of the world which is, at present, true (independent of any observer).


\paragraph{\it{Informal Explanation.}} For example, a fact might be a specific value held at specific memory address in a computer, or the occurrence of a pattern of values at positions relative to any memory address in a computer, or even a pattern over time such as a bit changing from $1$ to $0$.

\paragraph{\bf{Definition 1 - State:}} The present state of a system is \textit{the set of all facts} about that system. 

\paragraph{\it{Informal Explanation.}} 
Naturally, the set of all facts includes what must logically follow or have preceded the present. 
A set of facts is also a fact, and so there exists a fact for each and every possible representation, combination or subset all facts about a system. Further, the present state of a system is itself a fact.
Importantly, a state cannot contradict itself. For example, a bit in a conventional computer cannot be $0$ if it is $1$ (the latter prevents the former from occurring). 

\paragraph{\bf{Definition 2 - Hardware:}}
A set $H$ of possible states. There exists a probability distribution over these states, and it is assumed to be a uniform distribution.

\paragraph{\it{Informal Explanation.}}  Hardware is the body of the embodied agent. Hardware exists in one state at a time (only the present state is a fact). $H$ is the set of all states a piece of hardware may occupy.

\paragraph{\bf{Definition 3 - Physically Implementable Language:}} A triple $\mathcal{L} = \langle H, L, \lambda \rangle$, where:
\begin{itemize}
    \item $H$ is hardware. 
    \item $\lambda \subset \bigcup\limits_{h \in H} h$ is a finite set, named the \textbf{vocabulary}
    \item $L = \{ l \in 2^\lambda : \exists h \in H \ (l \subseteq h) \}$, the elements of which are \textbf{statements}.
\end{itemize}

    
\paragraph{\it{Informal Explanation.}}  There may be infinitely many states, and a state may be composed of infinitely many facts, so to avoid undecidability $\lambda$ must be finite. If $\lambda$ is finite, then it follows that $L$ and every member $l \in L$ is also finite. 
A statement is true at a particular time if it is a subset of the state at that time. The reason each statement must be a subset of at least one possible state, is so that statements do not describe impossible combinations of facts. In other words only statements that {\it{can}} be true are permitted. Not all subsets of $\lambda$ are statements, because $\lambda$ may include things which cannot be true at the same time ($\lambda$ is not a fact, but a set of things which can be facts if they are in the present state).

\paragraph{\bf{Definition 4 - Extension of a Statement:}} Given statement $a \in L$, the extension $Z_a$ of $a$ is defined as $Z_a = \{b \in L : a \subseteq b\}$. See figure \ref{fig:2}.

\paragraph{\bf{Definition 5 - Extension of a Set of Statements:}} Given a set $A \subseteq L$, the extension $Z_A$ of $A$ is defined as $Z_A = \bigcup\limits_{a \in A} Z_a$. See figure \ref{fig:2}.

\begin{figure}[h]
    \centering
    \includegraphics[width=\textwidth]{images/fulfig2.png}
   
    \caption{A diagramatic representation of definitions 4 and 5.}
  \label{fig:2}
\end{figure}

\paragraph{\bf{Conventions of Notation:}} To simplify presentation I'll use the following conventions for notation, except where stated otherwise:
\begin{itemize}
    \item $\textbf{Statements}$ are denoted by Latin lower case letters, for example $x$. 
    \item $\textbf{Sets of statements}$ are denoted by upper case, for example $X$.
    \item The $\bf{extension}$ of an object is denoted by the capital letter Z with the object subscripted. For example the extension of $a$ would be denoted $Z_a$, and the extension of $A$ would be denoted $Z_A$. 
\end{itemize}

\paragraph{\bf{Definition 6 - Task:}} A task (see figure \ref{fig:3}) is a triple $\mathcal{T} = \langle S, G, C \rangle$ where:

\begin{itemize}
    \item $S \subset L$ is a set of statements called \textbf{situations}, where $Z_S$ is the set of all possible \textbf{decisions} which can be made in those situations.
    \item $G \subset Z_S$ is the set of \textbf{goal satisficing decisions} for this task. 
    \item $C \subset L$ is the set of all \textbf{rulesets} for the task, distinguishing between decisions in $\mathcal{G}$ and those in $\overline{\mathcal{G}}$, where $$
    C = \{c \in L : Z_S \cap Z_c \equiv G, \forall z \in Z_c \ (z \subseteq \bigcup\limits_{g \in G} g)\}$$
\end{itemize}

\begin{figure}[h]
    \centering
    \includegraphics[width=\textwidth]{images/fulfig3.png}
   
    \caption{A diagramatic representation of definition 6.}
  \label{fig:3}
\end{figure}

\paragraph{\bf{Definition 7 - Process by Which a Decision is Made:}} An agent trying to complete a task is first 
\begin{enumerate}
    \item presented with a situation $s \in S$, then
    \item selects $z \in Z_s$, called a decision.
    \item If $z \in G$, then the agent has made a correct decision.
\end{enumerate} 

Note that $\forall c \in C : G \equiv Z_S \cap Z_c$. To abduct a correct decision given a situation, an agent would require a ruleset $c \in C$. 

\paragraph{\bf{Definition 8 - Generalise:}} Given two tasks $\mathcal{T}_1 = \langle S_1, G_1, C_1\rangle$ and $\mathcal{T}_2 = \langle S_2, G_2, C_2 \rangle$, a ruleset $c \in C_1$ generalises to task $\mathcal{T}_2$ if $c \in C_2$.

An equivalent alternative definition is that $c \in C_1$ generalises to $\mathcal{T}_2$ if $Z_{S_2} \cap Z_{c} \equiv G_2$. See figure \ref{fig:4}.

\begin{figure}[h]
    \centering
    \includegraphics[width=\textwidth]{images/fulfig4.png}
   
    \caption{Diagramatic representation of definition 8.}
    \label{fig:4}
\end{figure}

\paragraph{\bf{Definition 9 - Sub-task and Parent-task:}} A task $\mathcal{T}_2 = \langle S_2, G_2, C_2 \rangle$ is a subtask of $\mathcal{T}_1 = \langle S_1, G_1, C_1\rangle$ if:
\begin{enumerate}
    \item $S_2 \subset S_1$
    \item $G_2 \subset G_1$ 
\end{enumerate}
$\mathcal{T}_1$ is then a parent task of $\mathcal{T}_2$.

\paragraph{\bf{Definition 10 - Weakness:}} $w : C \rightarrow \mathbb{N}$ is a function which accepts a ruleset and returns the cardinality of its extension:
$$w(c) = | Z_{c} |$$

This expressed the weakness of a given ruleset (the greater the cardinality, the weaker the ruleset).
 
 


\subsection{Theoretical Results}
\label{proofs}

\paragraph{\bf{Proposition 1:}} The probability that a ruleset generalises to a parent task increases monotonically with its weakness.

\paragraph{\it{Proof.}}  
Let $\mathcal{T}_{m} = \langle S_{m}, G_{m}, C_{m}\rangle$ be a task of which the complete definition is known.
Let $\mathcal{T}_n = \langle S_n, G_n, C_n\rangle$ be a task to which I wish to generalise. All I know of $\mathcal{T}_n$ is that it is a parent task of $\mathcal{T}_m$, meaning $G_m \subset G_n$ and $S_m \subset S_n$.
\begin{enumerate}
    \item The set of all decisions which may potentially be required to address the situations in $S_n$, and which are not required for $S_m$, is $\overline{Z_{S_m}} = A$.
    \item For any given $c_m \in C_m$, the set of decisions $c_m$ implies which fall outside the scope of what is required for the known task $\mathcal{T}_m$ is $\overline{Z_{S_m}} \cap Z_{c_m} = B$.
    \item $2^{|A|}$ is the number of tasks which fall outside of what it is necessary for a ruleset of $\mathcal{T}_m$ to generalise to, and $2^{|B|}$ is the number of those tasks to which a given $c_m$ does generalise.
    \item Therefore the probability that a given ruleset $c_m \in C_m$ generalises to the unknown parent task $\mathcal{T}_n$ is $$p(c_m \in C_n) = \frac{2^{|B|}}{2^{|A|}}$$
\end{enumerate}
$p(c_m \in C_n)$ increases monotonically with the weakness of $c_m$. $\square$

\paragraph{\bf{Proposition 2:}} Weakness is necessary for generalisation.

\paragraph{\it{Proof.}}  
Once again let $\mathcal{T}_{m}$ be a task of which the complete definition is known, and $\mathcal{T}_n$ be a parent task of $\mathcal{T}_{m}$ to which I wish to generalise, for which the complete definition is not known.
If $Z_{S_n} \cap Z_{c_m} \equiv G_n$ then it must be he case that $G_n \subseteq Z_{c_m}$. The weaker a ruleset is, the more likely it is that $G_n \subseteq Z_{c_m}$. Therefore a sufficiently weak ruleset is necessary for generalisation. $\square$

A universal prior is one that assigns belief to every possible hypothesis, or in this case ruleset. Weakness is a means of accurately predicting $p(c)$, how likely one ruleset $c$ is to generalise relative to others.

\paragraph{\bf{Corollary - A Computable Universal Prior:}} $p(c) = \frac{2^{w(c)}}{2^{|L|}}$

\paragraph{\bf{Proposition 3:}} Agent which chooses the weakest rulesets is an artificial general intelligence.

\paragraph{\it{Proof.}}  
Intelligence is a measure of the ability to generalise, and an artificial general intelligence is the most intelligent agent by this measure. Propositions 1 and 2 show that a preference for weaker rulesets is both necessary and sufficient to maximise the probability of generalisation. Therefore the most intelligent agent is one that constructs the weakest rulesets.  $\square$






\section{Experiments}
Included with this paper is a simple program \cite{bennett_2022b}. It computes rulesets to solve 8-bit string prediction problems. With this, experiments were performed (see
appendix \ref{appendix_experiments}).

\subsection{Methods}
\label{methods}
The purpose of these experiments was to try and falsify proposition 3. Minimum description length (MDL) rulesets were compared with the weakest rulesets. MDL was chosen because it is arguably the only plausible alternative to weakness \cite{baker_2016,budhathokivreeken2018,evans_2020a,evans_2020b,evans_2021b}. As this was a comparison between rulesets, a MDL ruleset was defined as $c_{mdl} \in \argmin_{c \in C} |c|$. The hardware $H$ contained 256 states, one for every possible 8-bit string. Propositional logic was employed for the physically implementable language (to clarify, a written example of this is also provided in the appendix), meaning $L$ was a set of 256 different statements in propositional logic. A task was specified by choosing $G \subset L$ such that all $g \in G$ conformed to the rules of either binary addition or multiplication with 4-bits of input, followed by 4-bits of output (the included code may be re-run with any alternative operation the reader wishes). The experiments were made up of trials. 
The parameters of each trial were ``operation" (a function), and an even integer ``number\_of\_trials" between 4 and 14 which determined the cardinality of the set $G_k$ (defined below).
Each trial was divided into training and testing phases. The training phase proceeded as follows:
\begin{enumerate}
    \item A task $\mathcal{T}_n$ was generated: 
    \begin{enumerate}
        \item First, every possible 4-bit input for the chosen binary operation was used to generate an 8-bit string. These 16 strings then formed $G_n$.
        \item A bit between 0 and 7 was then chosen. $S_n$ was then created by cloning $G_n$ and deleting the chosen bit from every string (meaning $S_n$ was composed of 16 different 7-bit strings, each of which could be found in an 8-bit string in $G_n$).
    \end{enumerate} 
    \item A subtask $\mathcal{T}_k =  \langle S_k, G_k, C_k  \rangle$ was sampled from the parent task $\mathcal{T}_n$. Recall, $|G_k|$ was determined as a parameter of the trial.
    \item From $\mathcal{T}_k$ two rulesets were then generated; a weakest $c_w$, and a MDL $c_{mdl}$. 
\end{enumerate}
For each ruleset $c$, the testing phase was as follows:
\begin{enumerate}
    \item The extension $Z_c$ of $c$ was then generated.
    \item A prediction $G_{recon}$ was then constructed s.t. $G_{recon} = \{z \in Z_c : \exists s \in S_n \ (s \subset z ) \}$.
    \item $G_{recon}$ was then compared to the ground truth $G_n$, and results recorded. 
\end{enumerate}
Between $75$ and $256$ trials were run for each value of the parameter $|G_k|$. Fewer trials were run for larger values of $|G_k|$ as these took longer to process. The largest and smallest values of $|G_k|$ were discarded due to time and resource constraints. The results of these trails were then averaged for each value of $|G_k|$.

\begin{figure}[h]
    \centering
    \includegraphics[width=\textwidth]{images/tables_med.png}
    \label{fig:tables}
\end{figure}

\paragraph{\it{Rate at which rulesets generalised completely.}} Generalisation was deemed to have occurred where $G_{recon} = G_n$. The number of trials in which generalisation occurred was measured, and divided by $n$ to obtain the rate of generalisation for each ruleset $c_w$ and $c_{mdl}$. Error was computed as a Wald 95\% confidence interval. Intuitively, generalisation meant the task was correctly understood (rate being the portion of trials in which the ruleset in question perfectly matched the ground truth).

\paragraph{\it{Average Extent to which rulesets generalised.}} Where $G_{recon} \neq G_n$, the extent to which rulesets generalised could still be ascertained. $\frac{|G_{recon} \cap G_n|}{|G_n|}$ was measured for each ruleset, averaged for each value of $|G_k|$, and the standard error computed.

\subsection{Results}
\label{experimentalresults}
The results (displayed in tables 1, 2 and figure \ref{fig:5}) were consistent with propositions 1, 2 and 3. That there was not a single trial in which $c_{mdl}$ outperformed $c_w$. 

\begin{comment}
\begin{table}
  \caption{Results for Binary Addition}
  \label{table1}
  \centering
  \begin{tabular}{lllllllll}
    \toprule
    \multicolumn{1}{l}{$ \quad \quad \quad c_w$} 
    \multicolumn{6}{r}{$c_{mdl}$}  \\
    \cmidrule(r){2-3}
    \cmidrule(r){4-5}
    \cmidrule(r){6-7}
    \cmidrule(r){8-9}
    $|G_k|$     & Rate & $\pm 95 \%$    & AvgExt & StdErr & Rate & $\pm 95 \%$       & AvgExt & StdErr \\
    \midrule
    6 & .11 & .039 & .75 & .008 & .10 & .037 & .48 & .012    \\
    10 & .27 & .064 & .91 & .006  & .13 & .048 & .69 & .009     \\
    14 & .68 & .106 & .98 & .005  & .24 & .097 & .91 & .006  \\
    \bottomrule
  \end{tabular}
\end{table}
\begin{table}
  \caption{Results for  Binary Multiplication}
  \label{table2}
  \centering
  \begin{tabular}{lllllllll}
    \toprule
    \multicolumn{1}{l}{$ \quad \quad \quad c_w$} 
    \multicolumn{6}{r}{$c_{mdl}$}  \\
    \cmidrule(r){2-3}
    \cmidrule(r){4-5}
    \cmidrule(r){6-7}
    \cmidrule(r){8-9}
    $|G_k|$     & Rate & $\pm 95 \%$    & AvgExt & StdErr & Rate & $\pm 95 \%$       & AvgExt & StdErr \\
    \midrule
    6 & .05  & .026 & .74 & .009 & .01  & .011 & .58 & .011    \\
    10 & .16 & .045 & .86 & .006  & .08 & .034 & .78 & .008     \\
    14 & .46 & .061 & .96 & .003  & .21 & .050 & .93 & .003  \\
    \bottomrule
  \end{tabular}
\end{table}
\end{comment}


The generalisation rate for $c_w$ exceeded that of $c_{mdl}$. Reviewing the raw output data, there was one instance of $c_{mdl}$ generalising where $c_w$ did not. 
The extent to which rulesets generalised was also greater for weaker rulesets. $c_w$ always generated as many or more true positives than $c_{mdl}$. The performance difference between $c_{mdl}$ and $c_w$ was smaller than it was for generalisation rate.

\begin{figure}[h] 
    \centering
    \includegraphics[width=\textwidth]{images/fig5.png}
   
    \caption{Plot of the experimental results. The horizontal axis is $|G_k|$.}
    \label{fig:5}
\end{figure}

\section{Discussion}
\begin{comment}
Just as AIXI assumes a programming language, the above assumes a physically implementable language is provided. 
However the language used is an important aspect of an agent's design. From a conceptual point of view, it reflects the machinery of which an agent is composed. An elephant trying to open a jar faces a very different task to a human attempting the same. Consider how AIXI might cope if a programming language $\delta$ contained an inbuilt function for every possible program written in language $\gamma$, each of which calls the appropriate program written in $\gamma$. The Kolmogorov complexity of a program written in $\delta$ may then be the length of the function call, which is completely unrelated to the length of the program it calls. Any agent employing that language, if equating compression with plausibility, may perform no better than sequence of uniformly distributed random choices. Indeed, the choice of UTM is a choice of language and it was, in 2015, found to have a significant impact on AIXI's performance \cite{leike2015}. In other words, AIXI's performance is subjective.
\end{comment}

\subsection{On Subjectivity and The Measure of Intelligence}
AIXI's performance is subjective because the universal prior it employs equates plausibility with Kolmogorov Complexity, the truth of which depends upon the Universal Turing Machine (UTM) on which AIXI runs \cite{leike2015}.
The computable universal prior presented here involves neither Kolmogorov Complexity nor UTMs, and so does not suffer this flaw. Chollet's formalism to measure intelligence also depends upon Kolmogorov Complexity, and so I suggest it is as subjective as AIXI's performance. Weakness, however, may facilitate an objective measure of intelligence. 

\subsection{Limitations}
\label{limitations}
There are two important limitations to take note of. First, the problem of determining correctness is non-trivial, but is discussed at length in related work \cite{bennettmaruyama2021b}. Second, the vocabulary $\lambda$ affects how weak rulesets can be.
For example, one might write that ``address $109$ contains $1$ and address $110$ contains $0$", or that ``there exists an address $i$ containing $1$ such that address $i+1$ contains $0$, and $i=109$". These two sentences are equivalent, but the extensions of their subsentential expressions are different. Remove ``$i=109$" from the 2nd sentence, and what remains could be interpreted as a convolution over all $i$. 
In other words, these two sentences have different intensions but the same extension. This suggests that what separates such intensions from one another is the extensions of their subsentential expressions. Weaker subsentential expressions may be recombined to form weaker sentences than stronger subsentential expressions.

Given any $\beta \in \lambda$, let $H_\beta = \{h \in H : \beta \in h \}$.
This is like the extension of $\beta$, but in terms of hardware states rather than statements in $L$. 
The members of $\lambda$ are like subsentential expressions. If $H_\alpha \subset H_\beta$, and if $\{\beta\}$ and $\{\alpha \}$ are both rulesets, then it must be the case that $w(\{\beta\}) \ge w(\{\alpha\})$.
Therefore $\lambda$ which contains $\beta$ but not $\alpha$ will permit construction of weaker rulesets than $\lambda$ containing $\alpha$ but not $\beta$.
All else being equal, it follows that construction of the weakest rulesets possible would always be possible if the following rule, \textbf{strict objectivity}, were applied to $\lambda$: $$\forall \beta \in \lambda \ \lnot( \exists \gamma \in \overline{\lambda} \ ( H_\beta \subset H_\gamma)).$$

The ruleset most likely to generalise is the weakest, but the ability to construct weaker rulesets still depends upon $\lambda$. An agent which always chooses the weakest rulesets meets the definition of AGI to the extent possible given $\lambda$. However in the context of every possible $\lambda$, strict objectivity would ensure no more intelligent agent could exist.

\subsection{Connectionist Interpretations}
\label{twoclass}
The theory also accounts for rulesets constructed from both positive and negative examples. Appendix
\ref{supplemental} formalises such rulesets and how their weakness is measured. However the ability to learn from both positive and negative feedback was deemed less important (it is far easier to achieve good results in two-class classification than one), and so the results associated with the rulesets described in appendix
\ref{supplemental} were relegated to a paper following this one, discussed in 
\ref{connectionist}.

\subsection{Connectionist Interpretations}
\label{connectionist}
This is not necessarily an endorsement of either symbolic or connectionist methods. Both work with this theory. I used symbolic methods here because they're easier to interpret, but the theory is as easy to apply in the context of neural networks (and can be explained in kind with category theory). For example, the rules determining correctness could be modelled by training a classifier $o(l) \in [0, 1]$ such that $l \in Z_S$ and $o(l) = 1$ if $l \in G$ and $o(l) = 0$ otherwise. $o$ must behave in accordance with the definition of ruleset, the weakness of which could then be maximised. $o$ could be used as a loss function to train another network $n : S \rightarrow G$ which outputs correct decisions. $o$ learns the desired end of the task, while $n$ learns means by which it can be completed. An upcoming paper of mine includes an implementation of this, along with results for rulesets constructed from positive and negative examples as described in appendix 
\ref{supplemental}. 

\subsection{Potential Negative Consequences}
\label{negatives}
The potential negative consequences of AGI are the subject of both popular fiction and extensive research \cite{bennettmaruyama2021b}. A scaleable implementation of this theory is not given. Nevertheless this research may precipitate the automation of jobs, the development of autonomous weapons and greater instability in financial markets. 

\subsection{Concluding Remarks}
\label{conclusion}
The proofs in \ref{proofs} show that the optimal choice of ruleset for generalisation is the weakest. The experimental results in \ref{experimentalresults} support this claim, as there was not a single trial in which $c_{mdl}$ outperformed $c_w$. Weakness is a computable, objective measure of how likely a ruleset is to generalise. Subsequently an agent that prefers the weakest rulesets is an artificial general intelligence (by the definition discussed in this paper's introduction \cite{l6,chollet_2019}).
According to preceding work, the weakest ruleset is the purpose of a task \cite{bennett2021}. Because it is the optimal choice for generalisation, it follows that general intelligence is characterised by the ability to learn the ends, not just the means. 
However, general intelligence alone is insufficient to emulate human intelligence, as human intelligence employs various inductive biases \cite{l2,evans_2020a}. The work of Evans et. al.  \cite{evans_2020a,evans_2020b,evans_2021b} of Deepmind illustrates how a curated vocabulary (in which to construct rules) can provide such an inductive bias (which would make weaker rulesets easier to find in some contexts). 
Finally, the formulation of tasks employed here has been used to explain how human language functions \cite{bennett2021} and lends itself to an artificial theory of mind \cite{bennettmaruyama2021b,l5}. The more rigorous mathematical treatment given here has also been applied to those derivatives in an upcoming publication.

\section*{Acknowledgements}
\label{acknowledgements}
The following people took the trouble to either unofficially peer review, or at least give useful feedback on this paper: Vincent Abbott, Matthew Aitchison, David Quarel and Yoshihiro Maruyama all of the ANU, Badri Vellambi of UC and Sean Welsh. Their contributions significantly improved the clarity and rigour of this work, for which I am sincerely grateful. In particular, Vincent's suggestions on simplifying the formulation, and Matthew's suggestions regarding the communication of experimental evidence, were very much appreciated.

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\begin{comment}
\section*{Checklist}

\begin{enumerate}


\item For all authors...
\begin{enumerate}
  \item Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
    \answerYes{A definition of AGI is given, and a theory is described which fulfils that definition. That claim is then both formally proven and supported with experimental evidence.}
  \item Did you describe the limitations of your work?
    \answerYes{See Section~\ref{limitations} and ~\ref{conclusion}.}
  \item Did you discuss any potential negative societal impacts of your work?
    \answerYes{See \ref{negatives}. That said, this work is about a very specific definition of AGI which addresses one of the ethical quandaries of current generation machine learning; that the technology is inaccessible to those with limited resources (as discussed in the introduction).}
  \item Have you read the ethics review guidelines and ensured that your paper conforms to them?
    \answerYes{}
\end{enumerate}


\item If you are including theoretical results...
\begin{enumerate}
  \item Did you state the full set of assumptions of all theoretical results?
    \answerYes{See section \ref{definitions}}
        \item Did you include complete proofs of all theoretical results?
    \answerYes{See section \ref{proofs}}
\end{enumerate}


\item If you ran experiments...
\begin{enumerate}
  \item Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?
    \answerYes{The instructions to rerun the experiments are in appendix
\ref{appendix_experiments}, but really all that's needed is to run experiments.py, which is included in the supplementary materials. This will generate a dataset and run experiments. The parameters can be changed at the end of the file. The results are also included in genrates.xlsx.}
  \item Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
    \answerYes{Yes, this is stipulated in \ref{methods}, and the parameter variables included in experiments.py are commented to indicate default values.}
        \item Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?
    \answerYes{Wald interval was computed for the portion of trials in which generalisation occurred. Standard error was also computed for the average extent of generalisation (These numbers are reported in Table 1, Table 2).}
        \item Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?
    \answerYes{See appendix \ref{appendix_experiments}}
\end{enumerate}


\item If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
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  \item If your work uses existing assets, did you cite the creators?
    \answerNA{}
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    \answerNA{}
  \item Did you include any new assets either in the supplemental material or as a URL?
    \answerNA{}
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    \answerNA{}
  \item Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?
    \answerNA{}
\end{enumerate}


\item If you used crowdsourcing or conducted research with human subjects...
\begin{enumerate}
  \item Did you include the full text of instructions given to participants and screenshots, if applicable?
    \answerNA{}
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    \answerNA{}
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    \answerNA{}
\end{enumerate}


\end{enumerate}



\end{comment}

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