Can machine learning deliver AI? Theoretical results, inspiration from the brain and cognition, as well as machine learning experiments suggest that in order to learn the kind of complicated functions that can represent high-level abstractions (e.g. in vision, language, and other AI-level tasks), one would need deep architectures. Deep architectures are composed of multiple levels of non-linear operations, such as in neural nets with many hidden layers, graphical models with many levels of latent variables, or in complicated propositional formulae re-using many sub-formulae. Each level of the architecture represents features at a different level of abstraction, defined as a composition of lower-level features. Searching the parameter space of deep architectures is a difficult task, but new algorithms have been discovered and a new sub-area has emerged in the machine learning community since 2006, following these discoveries. Learning algorithms such as those for Deep Belief Networks and other related unsupervised learning algorithms have recently been proposed to train deep architectures, yielding exciting results and beating the state-of-the-art in certain areas. Learning Deep Architectures for AI discusses the motivations for and principles of learning algorithms for deep architectures. By analyzing and comparing recent results with different learning algorithms for deep architectures, explanations for their success are proposed and discussed, highlighting challenges and suggesting avenues for future explorations in this area.

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Learning Deep Architectures for AI

Future users of large data banks must be protected from having to know how the data is organized in the machine (the internal representation). A prompting service which supplies such information is not a satisfactory solution. Activities of users at terminals and most application programs should remain unaffected when the internal representation of data is changed and even when some aspects of the external representation are changed. Changes in data representation will often be needed as a result of changes in query, update, and report traffic and natural growth in the types of stored information.Existing noninferential, formatted data systems provide users with tree-structured files or slightly more general network models of the data. In Section 1, inadequacies of these models are discussed. A model based on n-ary relations, a normal form for data base relations, and the concept of a universal data sublanguage are introduced. In Section 2, certain operations on relations (other than logical inference) are discussed and applied to the problems of redundancy and consistency in the user's model.

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A relational model of data for large shared data banks

A relational model of data for large shared data banks. 1970

https://www.control.isy.liu.se/student/graduate/hybrid_course/oh_BemMor.pdf

Control of systems integrating logic, dynamics, and constraints

https://www.fil.ion.ucl.ac.uk/spm/doc/papers/NicholsBrett.pdf

Valid conjunction inference with the minimum statistic.

Retaining all the key features of the previous editions, Introduction to Mathematical Logic, Fifth Edition explores the principal topics of mathematical logic. It covers propositional logic, first-order logic, first-order number theory, axiomatic set theory, and the theory of computability. The text also discusses the major results of Gdel, Church, Kleene, Rosser, and Turing.New to the Fifth Edition A new section covering basic ideas and results about nonstandard models of number theoryA second appendix that introduces modal propositional logicAn expanded bibliography Additional exercises and selected answers This long-established text continues to expose students to natural proofs and set-theoretic methods. Only requiring some experience in abstract mathematical thinking, it offers enough material for either a one- or two-semester course on mathematical logic.

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Introduction to Mathematical Logic

Annual Meeting of the Association for Symbolic Logic

Introduction COMBINATORIAL GAMES Definition of Combinatorial Game The Fundamental Theorem for Combinatorial Games Nim Hex and Other Games Tree Games Grundy Functions Bogus Nim-Sums TWO-PERSON ZERO-SUM GAMES Games in Normal Form Saddle Points and Equilibrium Pairs Maximin and Minimax Mixed Strategies 2 x 2 Matrix Games 2 x n, m x 2, and 3 x 3 Matrix Games Linear Programming. Von Neumann's Theorem THE SIMPLEX METHOD. THE FUNDAMENTAL THEOREM OF DUALITY. SOLUTION OF TWO-PERSON ZERO-SUM GAMES. Slack Variables. Perfect Canonical Linear Programming Problems The Simplex Method Pivoting The Perfect Phase of the Simplex Method The Big M Method Bland's Rules to Prevent Cycling Duality and the Simplex Method Solution of Game Matrices Proofs of Facts 1-4 NON-ZERO-SUM GAMES AND k-PERSON GAMES The General Setting Nash Equilibria Graphical Method for Finding Nash Equilibria for 2 ' 2 Matrices Inadequacies of Nash Equilibria in Non-Zero-Sum Games. Cooperative Games The Nash Arbitration Procedure Games with Two or More Players Coalitions Games in Coalition Form The Shapley Value Strategic Equivalence Stable Sets APPENDICES Finite Probability Theory Utility Theory Nash's Theorem Answers to Selected Exercises Bibliography INDEX

Introducing Game Theory and Its Applications

1. The purpose of this paper is to show that, if the axioms of a system G of set theory are consistent, then it is impossible to prove from them the following weak form of the axiom of choice: (H1) For every denumerable set x of disjoint two-element sets, there is a set y, called a choice set for x, which contains exactly one element in common with each element of x. Among the axioms of the system G. we take, with minor modifications, Axioms A, B, C of G6del [6]. Clearly, the independence of H, implies that of all stronger propositions, including the general axiom of choice and the generalized continuum hypothesis. The proof depends upon ideas of Fraenkel and Mostowski,2 and proceeds in the following manner. Let a be a denumerable set of objects AO, Al, A2. -the exact nature of which will be specified later. Let it, {?42,, L24+1} for each j, c {o, ,1, Y2, . . . }, and b = [the sum set of a]. By transfinite induction, construct the class V. which is the closure of b under the power-set operation. For each j, it is possible to define a 1-1 mapping Q, of V, onto itself with the following properties. The mapping preserves the e-relation, or, more precisely, Q,(x) = {&(y) I y E x}. Also, Qj(A2L) A2,+1 and M,(A2+1,) = A2i, while, for i different from 2j and 2j+ 1, Q(A4) 4 Ai. Thus, Q, permutes the members of ys1, while leaving the other ,ui's unaffected. The Q, are constructed somewhat as follows: Because of the special way in which the Ai's will be constructed, Q, can be defined easily on the A4's and on all the ancestors of the A's with respect to the e-relation. Then, since V. is defined by transfinite induction starting with the A's, one can extend Q, throughout V, by a similar induction based upon the property: &(y) {j,(x) Ix E y}. Now, let E be the subclass of V, consisting of those x for which there is a finite set d of pi's such that x is fixed under each Q, which leaves unaffected all the pi's of d. Roughly speaking, we can characterize E as the class of those sets in V, which are "symmetric" with respect to all but a finite number of the pi's. In a simple manner, we can define a class gwhich is included in E and which can serve as the universe of sets of a model for Axioms A, B, C. The set c will be a set of !:, but c cannot have a choice set in 0, since any choice set for c will not

Elliott Mendelson

Papers

Introduction to Mathematical Logic

Annual Meeting of the Association for Symbolic Logic

Introducing Game Theory and Its Applications

The Independence of a Weak Axiom of Choice

Theory and problems of differential and integral calculus