Applications of Metric Coinduction

Kozen, Dexter; Ruozzi, Nicholas

doi:10.2168/lmcs-5(3:10)2009

Cited by 15 publications

(15 citation statements)

References 15 publications

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“…These results about strict sub-optimality of deterministic kernels do not contradict the established understanding in the classical theory of statistical decisions that asymptotically randomized policies cannot be better than deterministic (e.g. see [35] or more recently [22]). Indeed, these asymptotic results are concerned with obtaining all, possibly infinite amount of information, in which case there are deterministic optimal kernels.…”

Section: Discussionsupporting

confidence: 73%

Optimal measures and Markov transition kernels

Belavkin¹

2012

J Glob Optim

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We study optimal solutions to an abstract optimization problem for measures, which is a generalization of classical variational problems in information theory and statistical physics. In the classical problems, information and relative entropy are defined using the Kullback-Leibler divergence, and for this reason optimal measures belong to a oneparameter exponential family. Measures within such a family have the property of mutual absolute continuity. Here we show that this property characterizes other families of optimal positive measures if a functional representing information has a strictly convex dual. Mutual absolute continuity of optimal probability measures allows us to strictly separate deterministic and non-deterministic Markov transition kernels, which play an important role in theories of decisions, estimation, control, communication and computation. We show that deterministic transitions are strictly sub-optimal, unless information resource with a strictly convex dual is unconstrained. For illustration, we construct an example where, unlike non-deterministic, any deterministic kernel either has negatively infinite expected utility (unbounded expected error) or communicates infinite information.

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Section: Discussionsupporting

confidence: 73%

Optimal measures and Markov transition kernels

Belavkin¹

2012

J Glob Optim

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“…It is interesting to compare this principle with the fixed-point induction principle for order-preserving functions on complete partial orders (see [59]), which we will here refer to as Scott-de Bakker induction, and the fixed-point induction principle for contraction mappings on complete metric spaces (see [53]), which we will here refer to as Reed-Roscoe induction (see also [56], [55], [25]). …”

Section: By Lemma 541 (1m2 F )(S) Is a Post-fixed Point Of Fmentioning

confidence: 99%

On Fixed Points of Strictly Causal Functions

Matsikoudis

Lee

2013

Lecture Notes in Computer Science

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We ask whether strictly causal components form well defined systems when arranged in feedback configurations. The standard interpretation for such configurations induces a fixed-point constraint on the function modelling the component involved. We define strictly causal functions formally, and show that the corresponding fixed-point problem does not always have a well defined solution. We examine the relationship between these functions and the functions that are strictly contracting with respect to a generalized distance function on tagged signals, and argue that these strictly contracting functions are actually the functions that one ought to be interested in. We prove a constructive fixed-point theorem for these functions, introduce a corresponding induction principle, and study the related convergence process.

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“…It is interesting to compare this principle with the fixed-point induction principle for order-preserving functions on complete partial orders (see [63]), which we will here refer to as Scott-de Bakker induction, and the fixed-point induction principle for contraction mappings on complete metric spaces (see [57]), which we will here refer to as Reed-Roscoe induction (see also [60], [59], [25]). …”

Section: Inductionmentioning

confidence: 99%

“…However, F does not have a fixed point; any fixed point of F would be an order-embedding from the integers into the natural numbers, which is of course impossible. 25 In modern physics, this would actually depend on the choice of interpretation of quantum mechanics, especially with regard to paradoxes such as Bell's theorem (e.g., see [45]) and Wheeler's delayed choice (e.g., see [23]). Steering clear of the far-from-settled debate here, we believe that, regardless of personal stand, the reader will acknowledge the overwhelming plethora of physical systems that fall under this casual description.…”

mentioning

confidence: 99%

The Fixed-Point Theory of Strictly Causal Functions

Matsikoudis¹,

Lee²

2013

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We ask whether strictly causal components form well defined systems when arranged in feedback configurations. The standard interpretation for such configurations induces a fixed-point constraint on the function modelling the component involved. We define strictly causal functions formally, and show that the corresponding fixed-point problem does not always have a well defined solution. We examine the relationship between these functions and the functions that are strictly contracting with respect to a generalized distance function on signals, and argue that these strictly contracting functions are actually the functions that one ought to be interested in. We prove a constructive fixed-point theorem for these functions, introduce a corresponding induction principle, and study the related convergence process.

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Applications of Metric Coinduction

Cited by 15 publications

References 15 publications

Optimal measures and Markov transition kernels

Optimal measures and Markov transition kernels

On Fixed Points of Strictly Causal Functions

The Fixed-Point Theory of Strictly Causal Functions

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