2,970 research outputs found
Algebraic Characterizations of Complexity-Theoretic Classes of Real Functions
Recursive analysis is the most classical approach to model and discuss computations over the reals. It is usually presented using Type 2 or higher order Turing machines. Recently, it has been shown that computability classes of functions computable in recursive analysis can also be defined (or characterized) in an algebraic machine independent way, without resorting to Turing machines. In particular nice connections between the class of computable functions (and some of its sub- and sup-classes) over the reals and algebraically defined (sub- and sup-) classes of -recursive functions Ă la Moore 96 have been obtained. However, until now, this has been done only at the computability level, and not at the complexity level. In this paper we provide a framework that allows us to dive into the complexity level of functions over the reals. In particular we provide the first algebraic characterization of polynomial time computable functions over the reals. This framework opens the field of implicit complexity of functions over the reals, and also provide a new reading of some of the existing characterizations at the computability level
Algebraic Characterizations of Complexity-Theoretic Classes of Real Functions
Recursive analysis is the most classical approach to model and discuss computations over the reals. It is usually presented using Type 2 or higher order Turing machines. Recently, it has been shown that computability classes of functions computable in recursive analysis can also be defined (or characterized) in an algebraic machine independent way, without resorting to Turing machines. In particular nice connections between the class of computable functions (and some of its sub- and sup-classes) over the reals and algebraically defined (sub- and sup-) classes of -recursive functions Ă la Moore 96 have been obtained. However, until now, this has been done only at the computability level, and not at the complexity level. In this paper we provide a framework that allows us to dive into the complexity level of functions over the reals. In particular we provide the first algebraic characterization of polynomial time computable functions over the reals. This framework opens the field of implicit complexity of functions over the reals, and also provide a new reading of some of the existing characterizations at the computability level
Algebraic Characterizations of Complexity-Theoretic Classes of Real Functions
Accepted for publication in International Journal of Unconventional ComputingInternational audienceRecursive analysis is the most classical approach to model and discuss computations over the real numbers.Recently, it has been shown that computability classes of functions in the sense of recursive analysis can be defined (or characterized) in an algebraic machine independent way, without resorting to Turing machines. In particular nice connections between the class of computable functions (and some of its sub- and sup-classes) over the reals and algebraically defined (sub- and sup-) classes of R-recursive functions Ă la Moore 96 have been obtained. However, until now, this has been done only at the computability level, and not at the complexity level. In this paper we provide a framework that allows us to dive into the complexity level of real functions. In particular we provide the first algebraic characterization of polynomial-time computable functions over the reals. This framework opens the field of implicit complexity of analog functions, and also provides a new reading of some of the existing characterizations at the computability level
Methodological Fundamentalism: or why Battermanâs Different Notions of âFundamentalismâ may not make a Difference
I argue that the distinctions Robert Batterman (2004) presents between âepistemically fundamentalâ versus âontologically fundamentalâ theoretical approaches can be subsumed by methodologically fundamental procedures. I characterize precisely what is meant by a methodologically fundamental procedure, which involves, among other things, the use of multilinear graded algebras in a theoryâs formalism. For example, one such class of algebras I discuss are the Clifford (or Geometric) algebras. Aside from their being touted by many as a âunified mathematical language for physics,â (Hestenes (1984, 1986) Lasenby, et. al. (2000)) Finkelstein (2001, 2004) and others have demonstrated that the techniques of multilinear algebraic âexpansion and contractionâ exhibit a robust regularizablilty. That is to say, such regularization has been demonstrated to remove singularities, which would otherwise appear in standard field-theoretic, mathematical characterizations of a physical theory. I claim that the existence of such methodologically fundamental procedures calls into question one of Battermanâs central points, that âour explanatory physical practice demands that we appeal essentially to (infinite) idealizationsâ (2003, 7) exhibited, for example, by singularities in the case of modeling critical phenomena, like fluid droplet formation. By way of counterexample, in the field of computational fluid dynamics (CFD), I discuss the work of Mann & Rockwood (2003) and Gerik Scheuermann, (2002). In the concluding section, I sketch a methodologically fundamental procedure potentially applicable to more general classes of critical phenomena appearing in fluid dynamics
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Generalizations of entanglement based on coherent states and convex sets
Unentangled pure states on a bipartite system are exactly the coherent states
with respect to the group of local transformations. What aspects of the study
of entanglement are applicable to generalized coherent states? Conversely, what
can be learned about entanglement from the well-studied theory of coherent
states? With these questions in mind, we characterize unentangled pure states
as extremal states when considered as linear functionals on the local Lie
algebra. As a result, a relativized notion of purity emerges, showing that
there is a close relationship between purity, coherence and (non-)entanglement.
To a large extent, these concepts can be defined and studied in the even more
general setting of convex cones of states. Based on the idea that entanglement
is relative, we suggest considering these notions in the context of partially
ordered families of Lie algebras or convex cones, such as those that arise
naturally for multipartite systems. The study of entanglement includes notions
of local operations and, for information-theoretic purposes, entanglement
measures and ways of scaling systems to enable asymptotic developments. We
propose ways in which these may be generalized to the Lie-algebraic setting,
and to a lesser extent to the convex-cones setting. One of our original
motivations for this program is to understand the role of entanglement-like
concepts in condensed matter. We discuss how our work provides tools for
analyzing the correlations involved in quantum phase transitions and other
aspects of condensed-matter systems.Comment: 37 page
Logics of Finite Hankel Rank
We discuss the Feferman-Vaught Theorem in the setting of abstract model
theory for finite structures. We look at sum-like and product-like binary
operations on finite structures and their Hankel matrices. We show the
connection between Hankel matrices and the Feferman-Vaught Theorem. The largest
logic known to satisfy a Feferman-Vaught Theorem for product-like operations is
CFOL, first order logic with modular counting quantifiers. For sum-like
operations it is CMSOL, the corresponding monadic second order logic. We
discuss whether there are maximal logics satisfying Feferman-Vaught Theorems
for finite structures.Comment: Appeared in YuriFest 2015, held in honor of Yuri Gurevich's 75th
birthday. The final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-23534-9_1
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