A Complete Theory of Everything (will be subjective)

1 A Complete Theory of Everything (will be subjective )Marcus HutterSoCS, RSISE, IAS, CECSA ustralian National UniversityCanberra, ACT, 0200, AustraliaAugust 2010 AbstractIncreasingly encompassing models have been suggested for our world. The-ories range from generally accepted to increasingly speculative to apparentlybogus. The progression of theories from ego- to geo- to helio-centric models touniverse and multiverse theories and beyond was accompanied by a dramaticincrease in the sizes of the postulated worlds, with humans being expelled fromtheir center to ever more remote and random locations.

2 Rather than leadingto a true Theory of Everything , this trend faces a turning point after which thepredictive power of such theories decreases (actually to zero). Incorporatingthe location and other capacities of the observer into such theories avoids thisproblem and allows to distinguish meaningful from predictively meaninglesstheories. This also leads to a truly Complete Theory of Everything consisting ofa (conventional objective) Theory of Everything plus a (novel subjective ) ob-server process. The observer localization is neither based on the controversialanthropic principle, nor has it anything to do with the quantum-mechanicalobservation process.

3 The suggested principle is extended to more practical(partial, approximate, probabilistic, parametric) world models (rather thantheories of Everything ). Finally, I provide a justification of Ockham s razor,and criticize the anthropic principle, the doomsday argument, the no freelunch theorem, and the falsifiability Introduction22 Theories of Something, Everything & Nothing43 Predictive Power & Observer Localization74 Complete ToEs (CToEs)95 Complete ToE - Formalization126 Universal ToE - Formalization147 Extensions158 Justification of Ockham s Razor199 Discussion23 References24A List of Notation26 Keywordsworld models; observer localization; predictive power; Ockham s razor.

4 Uni-versal theories; inductive reasoning; simplicity and complexity; universal self-sampling; no-free-lunch; .. in spite of it s incomputability, Algorithmic Probability can serve asa kind of Gold Standard for induction systems Ray Solomonoff (1997) There is a Theory which states that if ever anyone discovers exactlywhat the Universe is for and why it is here, it will instantly disappearand be replaced by something even more bizarre and inexplicable. Thereis another Theory which states that this has already happened. Douglas Adams, Hitchhikers guide to the Galaxy (1979)1 IntroductionThis paper uses aninformation-theoreticandcomputationala pproach for addressingthephilosophicalproblem of judging theories (of Everything ) inphysics.

5 In order tokeep the paper generally accessible, I ve tried to minimize field-specific jargon andmathematics, and focus on the core problem and its mean anymodelwhich can explain describe predict compress[Hut06a] our observations, whatever the form of the model. Scientists often saythat their modelexplainssome phenomenon. What is usually meant is that themodeldescribes(the relevant aspects of) the observations more compactly than theraw data. The model is then regarded as capturing a law (of nature), which isbelieved to hold true also for unseen/future process of inferring general conclusions from example instances is callinduc-tive reasoning.

6 For instance, observing 1000 black ravens but no white one supportsbut cannot prove the hypothesis that all ravens are black. In general, induction isused to find properties or rules or models of past observations. The ultimate pur-pose of the induced models is to use them for making predictions, that the nextobserved raven will also be black. Arguably inductive reasoning is even more impor-tant than deductive reasoning in science and everyday life: for scientific discovery,in machine learning, for forecasting in economics, as a philosophical discipline, incommon-sense decision making, and last but not least to find theories of , some famous, but apparently misguided philosophers [Sto82, Gar01],including Popper and Miller, even disputed the existence, necessity or validity ofinductive reasoning.

7 Meanwhile it is well-known how minimum encoding lengthprinciples [Wal05, Gr u07], rooted in (algorithmic) information Theory [LV08], quan-tify Ockham s razor principle, and lead to a solid pragmatic foundation of inductivereasoning [Hut07]. Essentially, one can show that the more one cancompress, thebetter one canpredict, and vice deterministic Theory /model allows from initial conditions to determine an ob-servation sequence, which could be coded as a bit string. For instance, Newtonmechanics maps initial planet positions+velocities into a time-series of planet posi-tions.

8 So a deterministic model with initial conditions is just a compact represen-2tation of an infinite observation string. A stochastic model is just a probabilitydistribution over observation models in physics are essentially differential equations describing thetime-evolution of some aspects of the world. A Theory of Everything (ToE) modelsthe whole universe or multiverse, which should include initial conditions. As Iwill argue, it can be crucial to also localize the observer, to augment the ToEwith a model of the properties of the observer, even for non-quantum-mechanicalphenomena.

9 I call a ToE with observer localization, aComplete ToE(CToE).That the observer itself is important in describing our world is well-known. Mostprominently in quantum mechanics, the observer plays an active role in collapsingthe wave function . This is a specific and relatively well-defined role of the observerfor a particular Theory , which isnotmy concern. I will show that (even the local-ization of) the observer is indispensable forfindingor developingany(useful) , the anthropic principle is invoked for this purpose (our universe is as it isbecause otherwise we would not exist).

10 Unfortunately its current use is rather vagueand limited, if not outright unscientific [Smo04]. In Section 6 I extend Schmidhuber sformal work [Sch00] on computable ToEs toformallyinclude observers. Schmidhu-ber [Sch00] already discusses observers and mentions sampling universes consistentwith our own existence, but this part stays informal. I give a precise and formal ac-count of observers by explicitly separating the observer s subjective experience fromthe objectively existing universe or multiverse, which besides other things shows thatwe also need to localize the observer within our universe (not only which universethe observer is in).