Table of Contents
T-720-ATAI-2022 Main
Lecture Notes 2022
WORLDS
"The World"
| What is it? | A set of constraints on what is and isn't possible. We perceive the physical world through sensors. |
| Sensors are physical | Our sensors are made from the same stuff that they perceive. |
| Descartes | René Descartes, the French philosopher, famously claimed that “I think, therefore I am.” He recognized that the only certainty we have of anything is that we perceive in the here-and-now. |
| Artificial Worlds | We may conceive of any “world” which follows different rules of our own. These worlds are potential worlds for AI systems, just as the physical world is. |
| However … | Any implemented world, whether abstract or otherwise, but bow to the nature of the physical universe, because implementation means physical incarnation. |
| Hence | The nature of our physical universe is important in AI. |
| A Question of (Un)certainty | The physical world, and in fact many artificial ones also, are uncertain, meaning that there is a lot about them that we don't know. |
| A Requirement of Certainty | To do anything reliably takes certainty. |
| How To Address That | Figuring out what can reliably be achieved in uncertain worlds. |
| AI Boils Down To | Building machines that can figure out what can be reliably achieved in uncertain worlds. |
| Abstract Worlds | We may of course define any kind of “world” of our choosing. However, if it is to be implemented it must obey physical laws. |
Worlds: How it Hangs Together
| <m>W</m>: A World | <m>W = {lbrace V,F,S_0,R rbrace}</m> | |||||
| <m>V</m>: Variables | <m>V = {lbrace v_1, ~v_2, ~. . . , ~v_{ | V | } rbrace}</m> | |||
| <m>F</m>: Transition Functions | <m>F</m> is a set of transition functions / rules describing how the variables can change. The dynamics can intuitively be thought of as the world’s “laws of nature”, continually transforming the world’s current state into the next: <m>S_{t+δ} = F(S_t)</m>. | |||||
| <m>S_0</m>: Initial State | <m>S_0</m> is the State that <m>W</m> started out in. In any complex world this is unlikely to be known; for artificial worlds this may be defined. | |||||
| <m>R</m>: Relations | <m>R</m> are the relations between variables in the world. These may be unknown or partially known to an Agent in the world. | |||||
| Static World | Changes State only through Agent Action. | |||||
| Dynamic World | Changes State through Agent Action and through other means. | |||||
| State | <m>s_t~subset~V_t</m>. A set of variables <m>x</m> with a set of values, specified to some particular precision (with constraints, e.g. error bounds), for relevant to a World. For all practical purposes, in any complex World “State” refers by default to a sub-state, since it is a practical impossibility to know its full state (values of the complete set of variables) of a world; there will always be a vastly higher number of “don't care” variables than the variables listed for e.g. a Goal State (a State associated with a Goal). | |||||
| State definition | <m>s_t~subset~V_t</m> where <m>lbrace_x_l_x_u_rbrace ~{ | }~x_l_x_x_u</m> define lower and upper bounds on acceptable range for each <m>x</m> to count towards the State, respectively. | ||||
| Exposable Variables | Variables in <m>V</m> that are measurable and/or manipulatable in principle. | |||||
| Observable Variables | Variables in <m>V</m> that can be measured for a particular interval in time are observable during that interval. | |||||
| Manipulatable Variables | Variables in <m>V</m> whose value can be affected, either directly or indirectly (by an Agent or something else). | |||||
The Physical World
Laplace's Daemon
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| Concept | If a world is deterministic, and everything in it is caused from the ground up, from the smallest parts, then everything in that world is pre-determined based on its starting state. |
| Laplace | “In the history of science, Laplace's demon was the first published articulation of causal or scientific determinism, by Pierre-Simon Laplace in 1814. According to determinism, if someone (the demon) knows the precise location and momentum of every atom in the universe, their past and future values for any given time are entailed; they can be calculated from the laws of classical mechanics.” source: Wikipedia |
Determinism vs. Non-Determinism
| Is our universe deterministic? | This is a major question for physics, but ultimately is not of much consequence for those building GMI (general machine intelligence). This is because any agent situated in the physical world will never know the precise position, direction and momentum of all its smallest particles, and thus must always deal with uncertainty. |
| Regularity | A world with no regularity is pure noise. In such a world intelligence is useless. |
| Pure Determinism | A world that is completely deterministic is pre-determined at all levels for all eternity; in such a world there is no concept of choice, and hence there can be no relevance for intelligence. |
| “Axiomatic AI” | Some mathematicians believe the universe to be fundamentally mathematical, and see the role of science (and mathematics) to find its “ultimate formula”. Many AI folks seem to subscribe to such a view. We'll come back to that in a bit. |
Formalization
| The Physical World | <m>V = {lbrace}{x_1, ~x_2, ~… ~x_n}{rbrace}.</m> <m>F={lbrace}{f_1, ~f_2 ~… ~f_n}{rbrace}</m>. <m>x</m> are real-valued variables, <m>f</m> are transition functions, and <m>V_{t+delta} = F(V_{t})</m> and <m>{lbrace}{x}over{.}_1, ~{x}over{.}_2, ~… ~{x}over{.}_n {rbrace}</m> represent the first derivative of the variables during continuous change. Note that <m> | V | ~=~infty</m> and <m> | F | ~=~infty</m>. | ||||
Time Scales of the Physical World Relevant to AI
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| From Card, Moran & Newell et al. The Psychology of Human-Computer Interaction (1983) & K. R. Thórisson PhD Thesis, MIT, 1996. |
Maxwell's Demon
 Source: Wikipedia By User:Htkym - Own work, CC BY 2.5, REF |
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| A Thought Experiment | Imagine a container divided into two parts, A and B. Both parts are filled with the same gas at equal temperatures and placed next to each other. Observing the molecules on both sides, an imaginary demon guards a trapdoor between the two parts. When a faster-than-average molecule from A flies towards the trapdoor, the demon opens it, and the molecule will fly from A to B. Likewise, when a slower-than-average molecule from B flies towards the trapdoor, the demon will let it pass from B to A. The average speed of the molecules in B will have increased while in A they will have slowed down. Since average molecular speed corresponds to temperature, the temperature decreases in A and increases in B, contrary to the second law of thermodynamics. A heat extractor operating between the thermal reservoirs A and B could extract energy from this temperature difference, creating a perpetual motion machine. [ Adapted from Wikipedia ] |
| The Error | The thought experiment is flawed because the demon must be part of the same system that the container is part of; thinking (or computation, if the demon is a robot) requires time and energy, and so whatever heat is saved in the container will be spent to run the demon's thinking processes. (This was first proposed in 1929 by Leo Szilard.) |
The Physical World & AI Research
| Physical World | The field of physics has been systematically studying the physical world since the ancient Greeks. Over 2000 years later physics is now the most advanced scientific field of all of science. Any proper scientific theory of anything must ultimately rest on its shoulders. | |
| What is it? | Physics seeks to uncover the “ultimate” rules which determine how the universe behaves, including life, intelligence and everything else. No (general or limited) intelligence in a complex environment such as the physical world can be granted access to a full set of axioms of the system it’s controlling, let alone the ⟨agent,environment⟩ tuple, and thus the behavior of a practical generally intelligent artificial agent as a whole simply cannot be captured formally. (see Steunebrink et al 2016) | |
| Useful AI | To be useful, an AI must do something. Ultimately, to be of any use, it must do something in the physical world, be it building skyscrapers, auto-generating whole movies from scratch, doing experimental science or inventing the next AI. | |
| What this means | A key target environment of the present work is the physical world. | |
| Uncertainty | Since any agent in the physical universe will never know everything: Some things will always be uncertain. | |
| Novelty | In such a world novelty abounds—most things that a learner encounters will contain some form of novelty. | |
| Learning | Some unobservable exposable variables can be made observable (exposed) through manipulation. For any Agent with a set of Goals and limited knowledge but an ability to learn, which variables may be made observable and how is the subject of the Agent's learning. | |
Task-Environments
Task
| Task | A Problem that can be assigned in an Environment. Typically comes with Instructions (Some form of encoding of the task must exist for it to be assignable to a performing agent). An assigned Task has a designated Agent to do it, start-time, end-time, and maximum duration. |
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| Problem | A Goal with (all) relevant constraints imposed by a particular Task-Environment (where particular ranges of variable values, and particular groupings of elements, can be assumed). | |
| Goal | <m>G subset S</m>. A (future) (sub-) State to be attained during period t, plus other optional constraints on the Goal. | |
| LTE | All tasks have a limited time & energy: No Task exists that can be performed with infinite energy, or for for which infinite time is available for achieving. | |
Environment
| Environment | <m>e ~=~ {lbrace {V ~subset ~V_W}, ~{F ~subset ~F_W} rbrace} ~+ ~C</m> where <m>C</m> are additional constraints on <m>V,~ F</m> and some values for <m>V</m> are (typically) fixed within ranges. |
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| Task-Environment | <m>T_e ~subset ~W</m> An Environment in which one or more Tasks may be assigned and performed. |
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Task-Environments: Key Concepts
| Task Family | A set of tasks that are similar in some (important) ways. Since there exists no proper “Task Theory” as of yet, comparisons of similarity or other dimensions are still an art form. |
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| Problem | A Goal with (all) relevant constraints (≈ requirements). | |
| Problem Family | A set of problems that are similar in some (important) ways; a Problem plus variations of that Problem. | |
| Goal | A (future) State to be attained, plus optional constraints on the Goal. | |
| Goal State | A set of values (with error bounds) for a set of variables relevant to a Goal. | |
| Goal Family | A set of goals that are similar in some (important) ways. | |
| Environment | A set of constraints relevant to a Task but not counted as part of a Task proper. Also conceived as negative Goals (i.e. States to be avoided). |
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| Constraint | A set of factors that limit the flexibility (state space) of that which it constrains. | |
| Solution | The set of (atomic) actions that can achieve a Goal. | |
| Action | The changes an Agent can make to variables relevant to a Task-Environment. | |
| Plan | A partial way (incomplete Instructions) to accomplish a Task. | |
| Instructions | Partial Plan for accomplishing a Task, typically given to an Agent along with a Task by a Teacher. | |
| Teacher | The Agent (or interactive process) assigning a Task to another Agent (student), optionally in charge of Instructions. | |
Other High-Level Concepts
| Constraint | A set of factors that limit the flexibility of that which it constrains. Best conceived of as a negative Goal (State that is to be avoided). |
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| Family | A set whose elements share one or more common trait (within some “sensible” (pre-defined or definable) allowed variability on one or more of: the types of variables, number of variables, the ranges of these variables). | |
| Domains | A Family of Environments, <m>D ⊂ W</m>. The concept of ‘domains’ as subsets of the world, where a particular bias of distributions of variables, values and ranges exists, may be useful in the context of tasks that can be systematically impacted by such a bias (e.g. gravity vs. zero-gravity). Each variable <m>v ~∈~ D</m> may take on any value from the associated domain <m>D</m>; for physical domains we can take the domain of variables to be a subset of real numbers, <m>bbR</m>. |
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| Solution | The set of (atomic) Actions that can achieve one or more Goals in one or more Task-Environments. | |
| Action | The changes an Agent can make to variables relevant to a Task-Environment. | |
| Plan | Description of a partial Solution to accomplishing a Task. | |
| Teacher | The Agent assigning a Task to another Agent (student). | |
| Instructions | Plan (partial) for accomplishing a Task typically given to an Agent by a Teacher, along with a Task. Guide to Solutions (partial or full) - at some level of maximum available detail. |
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'Problem'
| Definition | In everyday use the term is a flexible one that generally be applied to any part of a task that, more or less, prevents goals (or subgoals) from being achieved effortlessly or directly. It assumes a non-zero set of goals of agents for which it is relevant (otherwise anything and everything would be a 'problem'). | |
| Closed Problem | A Problem that may be assigned as a Task to a particular Agent with known time & energy for producing a Solution (achieving a Goal). | |
| Alternative formulation | A known Goal for which there exists a known {Agent, Task} tuple that can achieve it, where the Task is fully specified in the {Agent Knowledge}+{Instructions} tuple (including all required parameters for assigning its completion). | |
| Example | Doing the dishes. | |
| Plans for Closed problems | Can be reliably produced, e.g. in the form of Instructions. | |
| Open Problem | A Problem whose solution is unknown and cannot be obviously assumed from analogy with similar problems whose solution is known. Cannot be guaranteed a solution with LTE. | |
| Example | Any Problem that is the subject of basic research and for which no known solution yet exists. | |
| Plans for Open problems | Cannot be reliably produced. | |
Task-Environment Constraints on Tasks
| Solution Constraint | Reduces the flexibility for producing a Solution. |
| Task Constraint | Limits the allowed Solution Space for a Problem. Can help or hinder a Task to be achieved. |
| Solution Space | The amount of variation allowed on a State while still counting as a Solution to a Problem. |
| Task Space | The size of variations on a Task that would have be explored with no up-front knowledge or information about the Solution Space of a Problem. |
| Limited time | No Task, no matter how small, takes zero time. ≈ “Any task worth doing takes time.” ≈ All implemented intelligences will be subject to the laws of physics, which means that for every Task, time is limited. |
| Limited energy | “Any task worth doing takes energy.” ≈ All implemented intelligences will be subject to the laws of physics, which means that for every Task, energy is limited. |
| No task takes zero time or zero energy | If <m>te</m> is a function that returns time and energy, an act of perception (reliable measurement) <m>te(p in P) > 0</m> a commitment to a measurement <m>m</m>, <m>com(m in M)</m>, that is, the measurement is deemed reliable, in <m>{te({com(m in M)}) > 0}</m>, and an action <m>te(a in A) > 0</m> to record it (to memory) or announce it (to the external world, e.g. by writing). It follows deductively that any Task requires at least one of these, possibly all. For a task <m>T</m> whose Goals have already been achieved at the time <m>T</m> is assigned to an agent <m>A</m>, <m>A</m> still needs to do the measurement of the Goal State <m>s</m>, <m>s in S</m>, or at least* commit to the measurement of this fact, and then record it. Even in this case we have <m>te(T) > 0</m>. Anything that takes zero time or zero energy is by definition not a Task. |
| Limited Time & Energy | <m>forall ~{lbrace T, ~T_e rbrace}~: ~te(T)~>~0</m> where <m>T_e subset W</m> is a task-environment in a world, <m>T</m> is a task, and <m>te</m> is a function that returns time and energy. |
*This could be the case if, for instance, task <m>T_1</m> assigned to agent <m>A</m> at time <m>t_1</m> comes with Instructions telling <m>A</m> that <m>T_1</m> has already been done at the time of its assignment <m>t_1</m> . Sort of like you were to get a shopping list for use on your upcoming shopping trip where at least one item was crossed out.
What Kinds of Task-Environments are Relevant for AI?
| Intelligence = Learning | Completely static: No (or little) need for learning. Completely random: Learning is of no use. |
| Worlds | Worlds where intelligence is relevant: - Sit somewhere between deterministic and completely random (pure noise). - Strike a balance between completely dynamic and completely static. - Some regularity must exist at a level that is observable initially by a learning agent. |
| Hierarchy | Any world that is hierarchically decomposable into smaller detail along time and space. (See “Time Scales of the Physical World”, above.) |
| Environment | Large number of potentially relevant variables. |
| Task | Ditto (large number of potentially relevant variables). |
| Medium or few number of Solutions. | |
| Instructions of varying detail are possible. | |
| Novelty | Novelty is unavoidable. In other words, unforeseen circumstances will be encountered by any AI operating in such circumstances. Accordingly, it should be able to handle it, since that is essentially why intelligence exists in the first place. |
| What that means | 1. Since no agent will ever know everything, no agent (artificial or natural) can assume axiomatic knowledge. 2. It cannot be assumed that all knowledge that the AI system needs to know is known by its designer up front. This means it must acquire its own knowledge. All advanced AI systems must be cumulative learners. 3. Since it must acquire its own knowledge, incrementally, knowledge acquisition will introduce knowledge gaps and inconsistencies. 4. A cumulative learning agent will continuously live in a state of insufficient knowledge and resources (with respect to perfect knowledge), due to the physical world's limited time and energy. |
2022©K.R.Thórisson