4,601 research outputs found
Unifiability and Structural Completeness in Relation Algebras and in Products of Modal Logic S5
Unifiability of terms (and formulas) and structural completeness in the variety of relation algebras RA and in the products of modal logic S5 is investigated. Nonunifiable terms (formulas) which are satisfiable in varieties (in logics) are exhibited. Consequently, RA and products of S5 as well as representable diagonal-free n-dimensional cylindric algebras, RDfn, are almost structurally complete but not structurally complete. In case of S5ⁿ a basis for admissible rules and the form of all passive rules are provided
InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets
This paper describes InfoGAN, an information-theoretic extension to the
Generative Adversarial Network that is able to learn disentangled
representations in a completely unsupervised manner. InfoGAN is a generative
adversarial network that also maximizes the mutual information between a small
subset of the latent variables and the observation. We derive a lower bound to
the mutual information objective that can be optimized efficiently, and show
that our training procedure can be interpreted as a variation of the Wake-Sleep
algorithm. Specifically, InfoGAN successfully disentangles writing styles from
digit shapes on the MNIST dataset, pose from lighting of 3D rendered images,
and background digits from the central digit on the SVHN dataset. It also
discovers visual concepts that include hair styles, presence/absence of
eyeglasses, and emotions on the CelebA face dataset. Experiments show that
InfoGAN learns interpretable representations that are competitive with
representations learned by existing fully supervised methods
Almost structural completeness; an algebraic approach
A deductive system is structurally complete if its admissible inference rules
are derivable. For several important systems, like modal logic S5, failure of
structural completeness is caused only by the underivability of passive rules,
i.e. rules that can not be applied to theorems of the system. Neglecting
passive rules leads to the notion of almost structural completeness, that
means, derivablity of admissible non-passive rules. Almost structural
completeness for quasivarieties and varieties of general algebras is
investigated here by purely algebraic means. The results apply to all
algebraizable deductive systems.
Firstly, various characterizations of almost structurally complete
quasivarieties are presented. Two of them are general: expressed with finitely
presented algebras, and with subdirectly irreducible algebras. One is
restricted to quasivarieties with finite model property and equationally
definable principal relative congruences, where the condition is verifiable on
finite subdirectly irreducible algebras.
Secondly, examples of almost structurally complete varieties are provided
Particular emphasis is put on varieties of closure algebras, that are known to
constitute adequate semantics for normal extensions of S4 modal logic. A
certain infinite family of such almost structurally complete, but not
structurally complete, varieties is constructed. Every variety from this family
has a finitely presented unifiable algebra which does not embed into any free
algebra for this variety. Hence unification in it is not unitary. This shows
that almost structural completeness is strictly weaker than projective
unification for varieties of closure algebras
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