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Register a new userPrinciples of Solomonoff Induction and AIXI
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Added by
Marcus Hutter
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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We identify principles characterizing Solomonoff Induction by demands on an agents external behaviour. Key concepts are rationality, computability, indifference and time consistency. Furthermore, we discuss extensions to the full AI case to derive AIXI.
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We consider two combinatorial principles, ${sf{ERT}}$ and ${sf{ECT}}$. Both are easily proved in ${sf{RCA}}_0$ plus ${Sigma^0_2}$ induction. We give two proofs of ${sf{ERT}}$ in ${sf{RCA}}_0$, using different methods to eliminate the use of ${Sigma^0_2}$ induction. Working in the weakened base system ${sf{RCA}}_0^*$, we prove that ${sf{ERT}}$ is equivalent to ${Sigma^0_1}$ induction and ${sf{ECT}}$ is equivalent to ${Sigma^0_2}$ induction. We conclude with a Weihrauch analysis of the principles, showing ${sf{ERT}} {equiv_{rm W}} {sf{LPO}}^* {<_{rm W}}{{sf{TC}}_{mathbb N}}^* {equiv_{rm W}} {sf{ECT}}$.
User defined recursive types are a fundamental feature of modern functional programming languages like Haskell, Clean, and the ML family of languages. Properties of programs defined by recursion on the structure of recursive types are generally proved by structural induction on the type. It is well known in the theorem proving community how to generate structural induction principles from data type declarations. These methods deserve to be better know in the functional programming community. Existing functional programming textbooks gloss over this material. And yet, if functional programmers do not know how to write down the structural induction principle for a new type - how are they supposed to reason about it? In this paper we describe an algorithm to generate structural induction principles from data type declarations. We also discuss how these methods are taught in the functional programming course at the University of Wyoming. A Haskell implementation of the algorithm is included in an appendix.
We develop an approach to choice principles and their contrapositive bar-induction principles as extensionality schemes connecting an intensional or effective view of respectively ill-and well-foundedness properties to an extensional or ideal view of these properties. After classifying and analysing the relations between different intensional definitions of ill-foundedness and well-foundedness, we introduce, for a domain $A$, a codomain $B$ and a filter $T$ on finite approximations of functions from $A$ to $B$, a generalised form GDC$_{A,B,T}$ of the axiom of dependent choice and dually a generalised bar induction principle GBI$_{A,B,T}$ such that: GDC$_{A,B,T}$ intuitionistically captures the strength of $bullet$ the general axiom of choice expressed as $forall aexists b R(a, b) Rightarrowexistsalphaforall alpha R(alpha,alpha(a))$ when $T$ is a filter that derives point-wise from a relation $R$ on $A times B$ without introducing further constraints, $bullet$ the Boolean Prime Filter Theorem / Ultrafilter Theorem if $B$ is the two-element set $mathbb{B}$ (for a constructive definition of prime filter), $bullet$ the axiom of dependent choice if $A = mathbb{N}$, $bullet$ Weak K{o}nigs Lemma if $A = mathbb{N}$ and $B = mathbb{B}$ (up to weak classical reasoning) GBI$_{A,B,T}$ intuitionistically captures the strength of $bullet$ G{o}dels completeness theorem in the form validity implies provability for entailment relations if $B = mathbb{B}$, $bullet$ bar induction when $A = mathbb{N}$, $bullet$ the Weak Fan Theorem when $A = mathbb{N}$ and $B = mathbb{B}$. Contrastingly, even though GDC$_{A,B,T}$ and GBI$_{A,B,T}$ smoothly capture several variants of choice and bar induction, some instances are inconsistent, e.g. when $A$ is $mathbb{B}^mathbb{N}$ and $B$ is $mathbb{N}$.
Higher inductive-inductive types (HIITs) generalize inductive types of dependent type theories in two ways. On the one hand they allow the simultaneous definition of multiple sorts that can be indexed over each other. On the other hand they support equality constructors, thus generalizing higher inductive types of homotopy type theory. Examples that make use of both features are the Cauchy real numbers and the well-typed syntax of type theory where conversion rules are given as equality constructors. In this paper we propose a general definition of HIITs using a small type theory, named the theory of signatures. A context in this theory encodes a HIIT by listing the constructors. We also compute notions of induction and recursion for HIITs, by using variants of syntactic logical relation translations. Building full categorical semantics and constructing initial algebras is left for future work. The theory of HIIT signatures was formalised in Agda together with the syntactic translations. We also provide a Haskell implementation, which takes signatures as input and outputs translation results as valid Agda code.
This article describes an action rule induction algorithm based on a sequential covering approach. Two variants of the algorithm are presented. The algorithm allows the action rule induction from a source and a target decision class point of view. The application of rule quality measures enables the induction of action rules that meet various quality criteria. The article also presents a method for recommendation induction. The recommendations indicate the actions to be taken to move a given test example, representing the source class, to the target one. The recommendation method is based on a set of induced action rules. The experimental part of the article presents the results of the algorithm operation on sixteen data sets. As a result of the conducted research the Ac-Rules package was made available.
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