318 research outputs found
The Mathematical Abstraction Theory, The Fundamentals for Knowledge Representation and Self-Evolving Autonomous Problem Solving Systems
The intention of the present study is to establish the mathematical
fundamentals for automated problem solving essentially targeted for robotics by
approaching the task universal algebraically introducing knowledge as
realizations of generalized free algebra based nets, graphs with gluing forms
connecting in- and out-edges to nodes. Nets are caused to undergo
transformations in conceptual level by type wise differentiated intervening net
rewriting systems dispersing problems to abstract parts, matching being
determined by substitution relations. Achieved sets of conceptual nets
constitute congruent classes. New results are obtained within construction of
problem solving systems where solution algorithms are derived parallel with
other candidates applied to the same net classes. By applying parallel
transducer paths consisting of net rewriting systems to net classes congruent
quotient algebras are established and the manifested class rewriting comprises
all solution candidates whenever produced nets are in anticipated languages
liable to acceptance of net automata. Furthermore new solutions will be added
to the set of already known ones thus expanding the solving power in the
forthcoming. Moreover special attention is set on universal abstraction,
thereof generation by net block homomorphism, consequently multiple order
solving systems and the overall decidability of the set of the solutions. By
overlapping presentation of nets new abstraction relation among nets is
formulated alongside with consequent alphabetical net block renetting system
proportional to normal forms of renetting systems regarding the operational
power. A new structure in self-evolving problem solving is established via
saturation by groups of equivalence relations and iterative closures of
generated quotient transducer algebras over the whole evolution.Comment: This article is a part of my thesis giving the unity for both
knowledge presentation and self-evolution in autonomous problem solving
mathematical systems and for that reason draws heavily from my previous work
arxiv:1305.563
Weak Cat-Operads
An operad (this paper deals with non-symmetric operads)may be conceived as a
partial algebra with a family of insertion operations, Gerstenhaber's circle-i
products, which satisfy two kinds of associativity, one of them involving
commutativity. A Cat-operad is an operad enriched over the category Cat of
small categories, as a 2-category with small hom-categories is a category
enriched over Cat. The notion of weak Cat-operad is to the notion of Cat-operad
what the notion of bicategory is to the notion of 2-category. The equations of
operads like associativity of insertions are replaced by isomorphisms in a
category. The goal of this paper is to formulate conditions concerning these
isomorphisms that ensure coherence, in the sense that all diagrams of canonical
arrows commute. This is the sense in which the notions of monoidal category and
bicategory are coherent. The coherence proof in the paper is much simplified by
indexing the insertion operations in a context-independent way, and not in the
usual manner. This proof, which is in the style of term rewriting, involves an
argument with normal forms that generalizes what is established with the
completeness proof for the standard presentation of symmetric groups. This
generalization may be of an independent interest, and related to matters other
than those studied in this paper. Some of the coherence conditions for weak
Cat-operads lead to the hemiassociahedron, which is a polyhedron related to,
but different from, the three-dimensional associahedron and permutohedron.Comment: 38 pages, version prepared for publication in Logical Methods in
Computer Science, the authors' last version is v
Up-to-homotopy algebras with strict units
We prove the existence of minimal models Ă la Sullivan for operads with non trivial arity zero. So up-to-homotopy algebras with strict units are just operad algebras over these minimal models. As an application we give another proof of the formality of the unitary n -little disks operad over the rationals.Preprin
Categorification of Hopf algebras of rooted trees
We exhibit a monoidal structure on the category of finite sets indexed by
P-trees for a finitary polynomial endofunctor P. This structure categorifies
the monoid scheme (over Spec N) whose semiring of functions is (a P-version of)
the Connes--Kreimer bialgebra H of rooted trees (a Hopf algebra after base
change to Z and collapsing H_0). The monoidal structure is itself given by a
polynomial functor, represented by three easily described set maps; we show
that these maps are the same as those occurring in the polynomial
representation of the free monad on P.Comment: 29 pages. Does not compile with pdflatex due to dependency on the
texdraw package. v2: expository improvements, following suggestions from the
referees; final version to appear in Centr. Eur. J. Mat
On generating series of finitely presented operads
Given an operad P with a finite Groebner basis of relations, we study the
generating functions for the dimensions of its graded components P(n). Under
moderate assumptions on the relations we prove that the exponential generating
function for the sequence {dim P(n)} is differential algebraic, and in fact
algebraic if P is a symmetrization of a non-symmetric operad. If, in addition,
the growth of the dimensions of P(n) is bounded by an exponent of n (or a
polynomial of n, in the non-symmetric case) then, moreover, the ordinary
generating function for the above sequence {dim P(n)} is rational. We give a
number of examples of calculations and discuss conjectures about the above
generating functions for more general classes of operads.Comment: Minor changes; references to recent articles by Berele and by Belov,
Bokut, Rowen, and Yu are adde
On Koszulity for operads of Conformal Field Theory
We study two closely related operads: the Gelfand-Dorfman operad GD and the
Conformal Lie Operad CLie. The latter is the operad governing the Lie conformal
algebra structure. We prove Koszulity of the Conformal Lie operad using the
Groebner bases theory for operads and an operadic analogue of the Priddy
criterion. An example of deformation of an operad coming from the Hom
structures is considered. In particular we study possible deformations of the
Associative operad from the point of view of the confluence property. Only one
deformation, the operad which governs the identity turns out to be confluent. We introduce a new Hom structure, namely
Hom--Gelfand-Dorfman algebras and study their basic properties.Comment: 24 page
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