4,872 research outputs found
Axiomatizing complex algebras by games
Submitted versio
Iteration Algebras for UnQL Graphs and Completeness for Bisimulation
This paper shows an application of Bloom and Esik's iteration algebras to
model graph data in a graph database query language. About twenty years ago,
Buneman et al. developed a graph database query language UnQL on the top of a
functional meta-language UnCAL for describing and manipulating graphs.
Recently, the functional programming community has shown renewed interest in
UnCAL, because it provides an efficient graph transformation language which is
useful for various applications, such as bidirectional computation. However, no
mathematical semantics of UnQL/UnCAL graphs has been developed. In this paper,
we give an equational axiomatisation and algebraic semantics of UnCAL graphs.
The main result of this paper is to prove that completeness of our equational
axioms for UnCAL for the original bisimulation of UnCAL graphs via iteration
algebras. Another benefit of algebraic semantics is a clean characterisation of
structural recursion on graphs using free iteration algebra.Comment: In Proceedings FICS 2015, arXiv:1509.0282
Higher-Dimensional Algebra III: n-Categories and the Algebra of Opetopes
We give a definition of weak n-categories based on the theory of operads. We
work with operads having an arbitrary set S of types, or `S-operads', and given
such an operad O, we denote its set of operations by elt(O). Then for any
S-operad O there is an elt(O)-operad O+ whose algebras are S-operads over O.
Letting I be the initial operad with a one-element set of types, and defining
I(0) = I, I(i+1) = I(i)+, we call the operations of I(n-1) the `n-dimensional
opetopes'. Opetopes form a category, and presheaves on this category are called
`opetopic sets'. A weak n-category is defined as an opetopic set with certain
properties, in a manner reminiscent of Street's simplicial approach to weak
omega-categories. Similarly, starting from an arbitrary operad O instead of I,
we define `n-coherent O-algebras', which are n times categorified analogs of
algebras of O. Examples include `monoidal n-categories', `stable n-categories',
`virtual n-functors' and `representable n-prestacks'. We also describe how
n-coherent O-algebra objects may be defined in any (n+1)-coherent O-algebra.Comment: 59 pages LaTex, uses diagram.sty and auxdefs.sty macros, one
encapsulated Postscript figure, also available as a compressed Postscript
file at http://math.ucr.edu/home/baez/op.ps.Z or
ftp://math.ucr.edu/pub/baez/op.ps.
An abstract view on syntax with sharing
The notion of term graph encodes a refinement of inductively generated syntax
in which regard is paid to the the sharing and discard of subterms. Inductively
generated syntax has an abstract expression in terms of initial algebras for
certain endofunctors on the category of sets, which permits one to go beyond
the set-based case, and speak of inductively generated syntax in other
settings. In this paper we give a similar abstract expression to the notion of
term graph. Aspects of the concrete theory are redeveloped in this setting, and
applications beyond the realm of sets discussed.Comment: 26 pages; v2: final journal versio
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