A Connection between Clone Theory and FCA Provided by Duality Theory

The syntax and semantics of entailment in duality theory

Journal of Symbolic Logic ◽

10.2307/2275875 ◽

1995 ◽

Vol 60 (4) ◽

pp. 1087-1114 ◽

Cited By ~ 15

Author(s):

B. A. Davey ◽

M. Haviar ◽

H. A. Priestley

Keyword(s):

Duality Theory ◽

Finite Algebra ◽

Partial Functions ◽

Distributive Variety ◽

Positive Formula ◽

Primitive Positive Formula ◽

Congruence Distributive Variety ◽

Finite Set ◽

Clone Theory ◽

Congruence Distributive

AbstractBoth syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formula and a new derivation of the corresponding result in clone theory, viz. the syntactic description of Inv(Pol(R)) for a given set R of unitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given in the form of an algorithm. It shows, in the special case of a purely relational type, that duality-theoretic entailment is describable in terms of five constructs, namely trivial relations, intersection, repetition removal, product, and retractive projection. All except the last are concrete, in the sense that they are described by a quantifier-free formula. It is proved that if the finite algebra M generates a congruence-distributive variety and all subalgebras of M are subdirectly irreducible, then concrete constructs suffice to describe entailment. The concept of entailment appropriate to strong dualities is also introduced, and described in terms of coordinate projections, restriction of domains, and composition of partial functions.

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Multiplicative Duality Theory in Production Economics

SSRN Electronic Journal ◽

10.2139/ssrn.2644581 ◽

2015 ◽

Cited By ~ 1

Author(s):

Walter Briec ◽

Paola Ravelojaona

Keyword(s):

Duality Theory ◽

Production Economics

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A Mathematical Introduction to Property Theory: The Fundamental Theorem and Duality Theory for Penalties

SSRN Electronic Journal ◽

10.2139/ssrn.633721 ◽

2004 ◽

Author(s):

David Ellerman

Keyword(s):

Fundamental Theorem ◽

Duality Theory ◽

Property Theory

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Fully distributed actor-critic architecture for multitask deep reinforcement learning

The Knowledge Engineering Review ◽

10.1017/s0269888921000023 ◽

2021 ◽

Vol 36 ◽

Author(s):

Sergio Valcarcel Macua ◽

Ian Davies ◽

Aleksi Tukiainen ◽

Enrique Munoz de Cote

Keyword(s):

Neural Network ◽

Reinforcement Learning ◽

Duality Theory ◽

Deep Neural Network ◽

Original Problem ◽

Almost Sure Convergence ◽

Continuous Control ◽

Access Data ◽

Central Station ◽

Common Policy

Abstract We propose a fully distributed actor-critic architecture, named diffusion-distributed-actor-critic Diff-DAC, with application to multitask reinforcement learning (MRL). During the learning process, agents communicate their value and policy parameters to their neighbours, diffusing the information across a network of agents with no need for a central station. Each agent can only access data from its local task, but aims to learn a common policy that performs well for the whole set of tasks. The architecture is scalable, since the computational and communication cost per agent depends on the number of neighbours rather than the overall number of agents. We derive Diff-DAC from duality theory and provide novel insights into the actor-critic framework, showing that it is actually an instance of the dual-ascent method. We prove almost sure convergence of Diff-DAC to a common policy under general assumptions that hold even for deep neural network approximations. For more restrictive assumptions, we also prove that this common policy is a stationary point of an approximation of the original problem. Numerical results on multitask extensions of common continuous control benchmarks demonstrate that Diff-DAC stabilises learning and has a regularising effect that induces higher performance and better generalisation properties than previous architectures.

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