34,696 research outputs found
Equivalence of the Traditional and Non-Standard Definitions of Concepts from Real Analysis
ACL2(r) is a variant of ACL2 that supports the irrational real and complex
numbers. Its logical foundation is based on internal set theory (IST), an
axiomatic formalization of non-standard analysis (NSA). Familiar ideas from
analysis, such as continuity, differentiability, and integrability, are defined
quite differently in NSA-some would argue the NSA definitions are more
intuitive. In previous work, we have adopted the NSA definitions in ACL2(r),
and simply taken as granted that these are equivalent to the traditional
analysis notions, e.g., to the familiar epsilon-delta definitions. However, we
argue in this paper that there are circumstances when the more traditional
definitions are advantageous in the setting of ACL2(r), precisely because the
traditional notions are classical, so they are unencumbered by IST limitations
on inference rules such as induction or the use of pseudo-lambda terms in
functional instantiation. To address this concern, we describe a formal proof
in ACL2(r) of the equivalence of the traditional and non-standards definitions
of these notions.Comment: In Proceedings ACL2 2014, arXiv:1406.123
Comparing theories: the dynamics of changing vocabulary. A case-study in relativity theory
There are several first-order logic (FOL) axiomatizations of special
relativity theory in the literature, all looking essentially different but
claiming to axiomatize the same physical theory. In this paper, we elaborate a
comparison, in the framework of mathematical logic, between these FOL theories
for special relativity. For this comparison, we use a version of mathematical
definability theory in which new entities can also be defined besides new
relations over already available entities. In particular, we build an
interpretation of the reference-frame oriented theory SpecRel into the
observationally oriented Signalling theory of James Ax. This interpretation
provides SpecRel with an operational/experimental semantics. Then we make
precise, "quantitative" comparisons between these two theories via using the
notion of definitional equivalence. This is an application of logic to the
philosophy of science and physics in the spirit of Johan van Benthem's work.Comment: 27 pages, 8 figures. To appear in Springer Book series Trends in
Logi
Specifying Reusable Components
Reusable software components need expressive specifications. This paper
outlines a rigorous foundation to model-based contracts, a method to equip
classes with strong contracts that support accurate design, implementation, and
formal verification of reusable components. Model-based contracts
conservatively extend the classic Design by Contract with a notion of model,
which underpins the precise definitions of such concepts as abstract
equivalence and specification completeness. Experiments applying model-based
contracts to libraries of data structures suggest that the method enables
accurate specification of practical software
Complex Networks and Symmetry I: A Review
In this review we establish various connections between complex networks and
symmetry. While special types of symmetries (e.g., automorphisms) are studied
in detail within discrete mathematics for particular classes of deterministic
graphs, the analysis of more general symmetries in real complex networks is far
less developed. We argue that real networks, as any entity characterized by
imperfections or errors, necessarily require a stochastic notion of invariance.
We therefore propose a definition of stochastic symmetry based on graph
ensembles and use it to review the main results of network theory from an
unusual perspective. The results discussed here and in a companion paper show
that stochastic symmetry highlights the most informative topological properties
of real networks, even in noisy situations unaccessible to exact techniques.Comment: Final accepted versio
Spectral C*-categories and Fell bundles with path-lifting
Following Crane's suggestion that categorification should be of fundamental
importance in quantising gravity, we show that finite dimensional even
-real spectral triples over \bbc are already nothing more than full
C*-categories together with a self-adjoint section of their domain and range
maps, while the latter are equivalent to unital saturated Fell bundles over
pair groupoids equipped with a path-lifting operator given by a normaliser.
Interpretations can be made in the direction of quantum Higgs gravity. These
geometries are automatically quantum geometries and we reconstruct the
classical limit, that is, general relativity on a Riemannian spin manifold.Comment: 20 pages, 1 figur
Supply Theory sans Profit-Maximization
We utilize the analytical construct of a stochastic supply function to provide an aggregate representation of a finite collection of standard deterministic supply functions. We introduce a consistency postulate for a stochastic supply function that may be satisfied even if no underlying deterministic supply function is rationalizable in terms of profit maximization. Our consistency postulate is nonetheless equivalent to a stochastic expansion of supply inequality, which summarizes the predictive content of the traditional theory of competitive supply. A number of key results in the deterministic theory follow as special cases from this equivalence. In particular, it yields a probabilistic version of the law of supply, which implies the traditional specification. Our analysis thus provides a necessary and sufficient axiomatic foundation for a de-coupling of the predictive content of the classical theory of competitive firm behavior from its a priori roots in profit maximization, while subsuming the traditional theory as a special case.weak axiom of profit maximization, stochastic consistency, stochastic supply function, supply aggregation, stochastic supply inequality, law of supply
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