An improved connectionist activation function for energy minimization

Symmetric networks that are based on energy minimization, such as Boltzmann machines or Hopfield nets, are used extensively for optimization, constraint satisfaction, and approximation of NP-hard problems. Nevertheless, finding a global minimum for the energy function is not guaranteed, and even a local minimum may take an exponential number of steps. We propose an improvement to the standard activation function used for such networks. The improved algorithm guarantees that a global minimum is found in linear time for tree-like subnetworks. The algorithm is uniform and does not assume that the network is a tree. It performs no worse than the standard algorithms for any network topology. In the case where there are trees growing from a cyclic subnetwork, the new algorithm performs better than the standard algorithms by avoiding local minima along the trees and by optimizing the free energy of these trees in linear time. The algorithm is self-stabilizing for trees (cycle-free undirected graphs) and remains correct under various scheduling demons. However, no uniform protocol exists to optimize trees under a pure distributed demon and no such protocol exists for cyclic networks under central demon

Improving Connectionist Energy Minimization

Symmetric networks designed for energy minimization such as Boltzman machines and Hopfield nets are frequently investigated for use in optimization, constraint satisfaction and approximation of NP-hard problems. Nevertheless, finding a global solution (i.e., a global minimum for the energy function) is not guaranteed and even a local solution may take an exponential number of steps. We propose an improvement to the standard local activation function used for such networks. The improved algorithm guarantees that a global minimum is found in linear time for tree-like subnetworks. The algorithm, called activate, is uniform and does not assume that the network is tree-like. It can identify tree-like subnetworks even in cyclic topologies (arbitrary networks) and avoid local minima along these trees. For acyclic networks, the algorithm is guaranteed to converge to a global minimum from any initial state of the system (self-stabilization) and remains correct under various types of schedulers. On the negative side, we show that in the presence of cycles, no uniform algorithm exists that guarantees optimality even under a sequential asynchronous scheduler. An asynchronous scheduler can activate only one unit at a time while a synchronous scheduler can activate any number of units in a single time step. In addition, no uniform algorithm exists to optimize even acyclic networks when the scheduler is synchronous. Finally, we show how the algorithm can be improved using the cycle-cutset scheme. The general algorithm, called activate-with-cutset, improves over activate and has some performance guarantees that are related to the size of the network's cycle-cutset.Comment: See http://www.jair.org/ for any accompanying file

Cruise Report 69-S-3: Pelagic fish and trawling survey of Santa Barbara oil spill

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First-Order Logic Proofs using Connectionist Constraints Relaxation

This paper considers the problem of expressing predicate calculus in connectionist networks that are based on energy minimization. Given a first-order-logic knowledge base and a bound k, a symmetric network is constructed (like a Boltzman machine or a Hopfield network) that searches for a proof fora given query. If a resolution-based proof is length no longer k exists, then the global minima of the energy function that is associated with the network represent such proofs. If no proof exist then the global minima indicate the lack of a proof. The network that is generated is of size polynomial in the bound k and the knowledge size. There are no restrictions on the type of logic formulas that can be represented. An extension enables the mechanism to copy with inconsistency in the knowledge base; i.e. a query is entailed if there exists a proof supporting the query and no better (or equally good ) proof exists supporting its negation. Fault tolerance is obtained since symbolic roles are dynamically assigned to units and many units are competing for those roles

Tempting of Speech in Music Education: Reflections on Thomas Mann's Doktor Faustus

The article deals with the temptation of speech in the teaching of music, as highlighted in Thomas Mann’s Doktor Faustus. Within the framework of the teacher-pupil, pupil-narrator and narrator-reader dialogue in Thomas Mann’s Doktor Faustus, a new perspective on the perception of music is formed without a single note being played. Words are meant to take us to imagery. Correspondingly, is it possible to talk about the music that exists, to construct ideas logically, but to be distant from the identity of what is heard? Such questions are the basis of the hermeneutic spiral of analysis-interpretation and the subject of teacher-pupil dialogue, where true insight can be born

Representing and Learning Propositional Logic in Symmetric Connectionist Networks

The chapter presents methods for efficiently representing logic formulas in connectionist networks that perform energy minimization. Algorithms are given for transforming any formula into a network in linear time and space and for learning representations of unknown formulas by observing examples of satisfying truth assignments. The relaxation process that underlies networks of energy minimization reveals an efficient hill climbing algorithm for satisfiability problems. Experimental results indicate that the parallel implementation of the algorithm with give extremely good average-case performance, even for large-scale, hard satisfiability problems (randomly generated)

Converting Binary Thresholds Networks into Equivalent Symmetric Networks

We give algorithms to convert any network of binary threshold units (that does not oscillate) into an equivalent network with symmetric weight matrix (like Hopfield networks [Hopfield 82] or Boltzmann machines [Hinton, Sejnowski 88]). The motivation for the transformation is dual: a) to demonstrate the expressive power of symmetric networks; i.e. binary threshold networks (that do not oscillate) are subsumed in the energy minimization paradigm; 2) to use network modules (developed for the spreading activation paradigm for example), within the energy minimization paradigm. Thus optimization [Tank, Hopfield 88] and approximation of hard problems can be combined with efficient modules, that solve tractable sub-problems; 3)to unify a large class of networks under one paradigm. For acyclic networks we give an algorithm that generates an equivalent symmetric network that is of the same size and performs as efficiently as the original network. For the conversion of recurrent networks, we introduce several techniques; however, the generated networks may be larger (in size) then the original

A Fault Tolerant Connectionist Architecture for Construction of Logic Proofs

This chapter considers the problems of expressing logic and constructing proofs in fault tolerant connectionist networks that are based on energy minimalism. Given a first-order-logic knowledge base and a bound k, a symmetric network is constructed (like a Boltzman machine or a Hopfield network) that searches for a proof for a given query. If a resolution-based proof of length no longer than k exists, then the global minima of the energy function that is associated with the network represent such proofs. If no proof exist then the global minima indicate the lack of a proof. The network that is generated is of size polynomial in the bound k and the knowledge size. There are no restrictions on the type of logic formulas that can be represented. Most of the chapter discusses the representation of propositional formulas and proofs; however, an extension is presented that allows the representation of unrestricted first-order logic formulas (predicate calculus). Fault tolerance is obtained using a binding technique that dynamically assigns symbolic roles to winner takes all units