568 research outputs found
Heterogeneous substitution systems revisited
Matthes and Uustalu (TCS 327(1-2):155-174, 2004) presented a categorical
description of substitution systems capable of capturing syntax involving
binding which is independent of whether the syntax is made up from least or
greatest fixed points. We extend this work in two directions: we continue the
analysis by creating more categorical structure, in particular by organizing
substitution systems into a category and studying its properties, and we
develop the proofs of the results of the cited paper and our new ones in
UniMath, a recent library of univalent mathematics formalized in the Coq
theorem prover.Comment: 24 page
The number of countable models via Algebraic logic
Vaught's Conjecture states that if T is a complete First order theory in a countable
language that has more than aleph_0 pairwise non isomorphic countable models, then T has
2^aleph_0 such models. Morley showed that if T has more than aleph_1 pairwise non isomorphic
countable models, then it has 2^aleph_0 such models.
In this paper, we First show how we can use algebraic logic, namely the representation
theory of cylindric and quasi-polyadic algebras, to study Vaught's conjecture (count
models), and we re-prove Morley's above mentioned theorem. Second, we show that
Morley's theorem holds for the number of non isomorphic countable models omitting
a countable family of types. We go further by giving examples showing that although
this number can only take the values given by Morley's theorem, it can be different
from the number of all non isomorphic countable models. Moreover, our examples show
that the number of countable models omitting a family of types can also be either aleph_1
or 2 and therefore different from the possible values provided by Vaught's conjecture
and by his well known theorem; in the case of aleph_1, however, the family is uncountable.
Finally, we discuss an omitting types theorem of Shelah
Multilinear Maps in Cryptography
Multilineare Abbildungen spielen in der modernen Kryptographie eine immer bedeutendere Rolle. In dieser Arbeit wird auf die Konstruktion, Anwendung und Verbesserung von multilinearen Abbildungen eingegangen
Reducing the Cost of Precise Types
Programs involving precise types enforce more properties via type checking, but precise types also prevent the reuse of functions throughout a program since no single precise type is used throughout a large program. My work is a step toward eliminating the underlying dilemma regarding type precision versus function reuse. It culminates in a novel traversal operator that recovers the reuse by automating most of each conversion between "similar" precise types, for a notion of similarity that I characterize in both the intuitive and technical senses. The benefits of my techniques are clear in side-by-side comparisons; in particular, I apply my techniques to two definitions of lambda-lifting. I present and implement my techniques in the Haskell programming language, but the fundamental ideas are applicable to any statically- and strongly-typed programming functional language with algebraic data types
Two-parameter families of quantum symmetry groups
We introduce and study natural two-parameter families of quantum groups
motivated on one hand by the liberations of classical orthogonal groups and on
the other by quantum isometry groups of the duals of the free groups.
Specifically, for each pair (p,q) of non-negative integers we define and
investigate quantum groups O^+(p,q), B^+(p,q), S^+(p,q) and H^+(p,q)
corresponding to, respectively, orthogonal groups, bistochastic groups,
symmetric groups and hyperoctahedral groups. In the first three cases the new
quantum groups turn out to be related to the (dual free products of) free
quantum groups studied earlier. For H^+(p,q) the situation is different: we
show that H^+(p,0) is isomorphic to the quantum isometry group of the
C*-algebra of the free group and it can be viewed as a liberation of the
classical isometry group of the p-dimensional torus.Comment: 29 page
Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations
AbstractThis paper generalizes many-sorted algebra (MSA) to order-sorted algebra (OSA) by allowing a partial ordering relation on the set of sorts. This supports abstract data types with multiple inheritance (in roughly the sense of object-oriented programming), several forms of polymorphism and overloading, partial operations (as total on equationally defined subsorts), exception handling, and an operational semantics based on term rewriting. We give the basic algebraic constructions for OSA, including quotient, image, product and term algebra, and we prove their basic properties, including quotient, homomorphism, and initiality theorems. The paper's major mathematical results include a notion of OSA deduction, a completeness theorem for it, and an OSA Birkhoff variety theorem. We also develop conditional OSA, including initiality, completeness, and McKinsey-Malcev quasivariety theorems, and we reduce OSA to (conditional) MSA, which allows lifting many known MSA results to OSA. Retracts, which intuitively are left inverses to subsort inclusions, provide relatively inexpensive run-time error handling. We show that it is safe to add retracts to any OSA signature, in the sense that it gives rise to a conservative extension. A final section compares and contrasts many different approaches to OSA. This paper also includes several examples demonstrating the flexibility and applicability of OSA, including some standard benchmarks like stack and list, as well as a much more substantial example, the number hierarchy from the naturals up to the quaternions
Lattices of quasi-equational theories as congruence lattices of semilattices with operators, Part I
We show that for every quasivariety K of structures (where both functions and
relations are allowed) there is a semilattice S with operators such that the
lattice of quasi-equational theories of K (the dual of the lattice of
sub-quasivarieties of K) is isomorphic to Con(S,+,0,F). As a consequence, new
restrictions on the natural quasi-interior operator on lattices of
quasi-equational theories are found.Comment: Presented on International conference "Order, Algebra and Logics",
Vanderbilt University, 12-16 June, 2007 25 pages, 2 figure
Linear representations of regular rings and complemented modular lattices with involution
Faithful representations of regular -rings and modular complemented
lattices with involution within orthosymmetric sesquilinear spaces are studied
within the framework of Universal Algebra. In particular, the correspondence
between classes of spaces and classes of representables is analyzed; for a
class of spaces which is closed under ultraproducts and non-degenerate finite
dimensional subspaces, the latter are shown to be closed under complemented
[regular] subalgebras, homomorphic images, and ultraproducts and being
generated by those members which are associated with finite dimensional spaces.
Under natural restrictions, this is refined to a --correspondence between
the two types of classes
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