94,262 research outputs found
Combining Algebra and Higher-Order Types
We study the higher-order rewrite/equational proof systems obtained by adding the simply typed lambda calculus to algebraic rewrite/equational proof systems. We show that if a many-sorted algebraic rewrite system has the Church-Rosser property, then the corresponding higher-order rewrite system which adds simply typed ß-reduction has the Church-Rosser property too. This result is relevant to parallel implementations of functional programming languages.
We also show that provability in the higher-order equational proof system obtained by adding the simply typed ß and η axioms to some many-sorted algebraic proof system is effectively reducible to provability in that algebraic proof system. This effective reduction also establishes transformations between higher-order and algebraic equational proofs, transformations which can be useful in automated deduction
On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs
Zero-curvature representations (ZCRs) are one of the main tools in the theory
of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be
interpreted as ZCRs. In [arXiv:1303.3575], for any (1+1)-dimensional scalar
evolution equation , we defined a family of Lie algebras which are
responsible for all ZCRs of in the following sense. Representations of the
algebras classify all ZCRs of the equation up to local gauge
transformations. In [arXiv:1804.04652] we showed that, using these algebras,
one obtains necessary conditions for existence of a B\"acklund transformation
between two given equations. The algebras are defined in terms of
generators and relations. In this paper we show that, using the algebras
, one obtains some necessary conditions for integrability of
(1+1)-dimensional scalar evolution PDEs, where integrability is understood in
the sense of soliton theory. Using these conditions, we prove non-integrability
for some scalar evolution PDEs of order . Also, we prove a result announced
in [arXiv:1303.3575] on the structure of the algebras for certain
classes of equations of orders , , , which include KdV, mKdV,
Kaup-Kupershmidt, Sawada-Kotera type equations. Among the obtained algebras for
equations considered in this paper and in [arXiv:1804.04652], one finds
infinite-dimensional Lie algebras of certain polynomial matrix-valued functions
on affine algebraic curves of genus and . In this approach, ZCRs may
depend on partial derivatives of arbitrary order, which may be higher than the
order of the equation . The algebras generalize Wahlquist-Estabrook
prolongation algebras, which are responsible for a much smaller class of ZCRs.Comment: 29 pages; v2: consideration of zero-curvature representations with
values in infinite-dimensional Lie algebras added. arXiv admin note: text
overlap with arXiv:1303.3575, arXiv:1804.04652, arXiv:1703.0721
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.
Combining Relational Algebra, SQL, Constraint Modelling, and Local Search
The goal of this paper is to provide a strong integration between constraint
modelling and relational DBMSs. To this end we propose extensions of standard
query languages such as relational algebra and SQL, by adding constraint
modelling capabilities to them. In particular, we propose non-deterministic
extensions of both languages, which are specially suited for combinatorial
problems. Non-determinism is introduced by means of a guessing operator, which
declares a set of relations to have an arbitrary extension. This new operator
results in languages with higher expressive power, able to express all problems
in the complexity class NP. Some syntactical restrictions which make data
complexity polynomial are shown. The effectiveness of both extensions is
demonstrated by means of several examples. The current implementation, written
in Java using local search techniques, is described. To appear in Theory and
Practice of Logic Programming (TPLP)Comment: 30 pages, 5 figure
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