77,114 research outputs found
Refinement by interpretation in {\pi}-institutions
The paper discusses the role of interpretations, understood as multifunctions
that preserve and reflect logical consequence, as refinement witnesses in the
general setting of pi-institutions. This leads to a smooth generalization of
the refinement-by-interpretation approach, recently introduced by the authors
in more specific contexts. As a second, yet related contribution a basis is
provided to build up a refinement calculus of structured specifications in and
across arbitrary pi-institutions.Comment: In Proceedings Refine 2011, arXiv:1106.348
SOS-convex Semi-algebraic Programs and its Applications to Robust Optimization: A Tractable Class of Nonsmooth Convex Optimization
In this paper, we introduce a new class of nonsmooth convex functions called
SOS-convex semialgebraic functions extending the recently proposed notion of
SOS-convex polynomials. This class of nonsmooth convex functions covers many
common nonsmooth functions arising in the applications such as the Euclidean
norm, the maximum eigenvalue function and the least squares functions with
-regularization or elastic net regularization used in statistics and
compressed sensing. We show that, under commonly used strict feasibility
conditions, the optimal value and an optimal solution of SOS-convex
semi-algebraic programs can be found by solving a single semi-definite
programming problem (SDP). We achieve the results by using tools from
semi-algebraic geometry, convex-concave minimax theorem and a recently
established Jensen inequality type result for SOS-convex polynomials. As an
application, we outline how the derived results can be applied to show that
robust SOS-convex optimization problems under restricted spectrahedron data
uncertainty enjoy exact SDP relaxations. This extends the existing exact SDP
relaxation result for restricted ellipsoidal data uncertainty and answers the
open questions left in [Optimization Letters 9, 1-18(2015)] on how to recover a
robust solution from the semi-definite programming relaxation in this broader
setting
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Automated verification of refinement laws
Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back's atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs
Two-dimensional models as testing ground for principles and concepts of local quantum physics
In the past two-dimensional models of QFT have served as theoretical
laboratories for testing new concepts under mathematically controllable
condition. In more recent times low-dimensional models (e.g. chiral models,
factorizing models) often have been treated by special recipes in a way which
sometimes led to a loss of unity of QFT. In the present work I try to
counteract this apartheid tendency by reviewing past results within the setting
of the general principles of QFT. To this I add two new ideas: (1) a modular
interpretation of the chiral model Diff(S)-covariance with a close connection
to the recently formulated local covariance principle for QFT in curved
spacetime and (2) a derivation of the chiral model temperature duality from a
suitable operator formulation of the angular Wick rotation (in analogy to the
Nelson-Symanzik duality in the Ostertwalder-Schrader setting) for rational
chiral theories. The SL(2,Z) modular Verlinde relation is a special case of
this thermal duality and (within the family of rational models) the matrix S
appearing in the thermal duality relation becomes identified with the
statistics character matrix S. The relevant angular Euclideanization'' is done
in the setting of the Tomita-Takesaki modular formalism of operator algebras.
I find it appropriate to dedicate this work to the memory of J. A. Swieca
with whom I shared the interest in two-dimensional models as a testing ground
for QFT for more than one decade.
This is a significantly extended version of an ``Encyclopedia of Mathematical
Physics'' contribution hep-th/0502125.Comment: 55 pages, removal of some typos in section
Pascual Jordan, his contributions to quantum mechanics and his legacy in contemporary local quantum physics
After recalling episodes from Pascual Jordan's biography including his
pivotal role in the shaping of quantum field theory and his much criticized
conduct during the NS regime, I draw attention to his presentation of the first
phase of development of quantum field theory in a talk presented at the 1929
Kharkov conference. He starts by giving a comprehensive account of the
beginnings of quantum theory, emphasising that particle-like properties arise
as a consequence of treating wave-motions quantum-mechanically. He then goes on
to his recent discovery of quantization of ``wave fields'' and problems of
gauge invariance. The most surprising aspect of Jordan's presentation is
however his strong belief that his field quantization is a transitory not yet
optimal formulation of the principles underlying causal, local quantum physics.
The expectation of a future more radical change coming from the main architect
of field quantization already shortly after his discovery is certainly quite
startling. I try to answer the question to what extent Jordan's 1929
expectations have been vindicated. The larger part of the present essay
consists in arguing that Jordan's plea for a formulation without ``classical
correspondence crutches'', i.e. for an intrinsic approach (which avoids
classical fields altogether), is successfully addressed in past and recent
publications on local quantum physics.Comment: More biographical detail, expansion of the part referring to Jordan's
legacy in quantum field theory, 37 pages late
Kleene algebra with domain
We propose Kleene algebra with domain (KAD), an extension of Kleene algebra
with two equational axioms for a domain and a codomain operation, respectively.
KAD considerably augments the expressiveness of Kleene algebra, in particular
for the specification and analysis of state transition systems. We develop the
basic calculus, discuss some related theories and present the most important
models of KAD. We demonstrate applicability by two examples: First, an
algebraic reconstruction of Noethericity and well-foundedness; second, an
algebraic reconstruction of propositional Hoare logic.Comment: 40 page
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