Higher-order Monotone Functions and Probability Theory

2001 ◽  
pp. 317-331 ◽  
Author(s):  
Dinis D. Pestana ◽  
Sandra Mendonça
Keyword(s):  
Probability Theory ◽  
Higher Order ◽  
Monotone Functions

scholarly journals Bounded Fixed Point Iteration

1991 ◽  
Vol 20 (359) ◽  
Author(s):  
Hanne Riis Nielson ◽  
Flemming Nielson
Keyword(s):  
Fixed Point ◽  
Abstract Interpretation ◽  
Higher Order ◽  
Monotone Functions ◽  
Additive Functions ◽  
Fixed Point Iteration ◽  
Exponential Bound ◽  
Strictness Analysis ◽  
Long Chains

In the context of abstract interpretation for languages without higher-order features we study the number of times a functional need to be unfolded in order to give the least fixed point. For the cases of total or monotone functions we obtain an exponential bound and in the case of strict and additive (or distributive) functions we obtain a quadratic bound. These bounds are shown to be tight in that sufficiently long chains of functions can be shown to exist. Specializing the case of strict and additive functions to functionals of a form that would correspond to iterative programs we show that a linear bound is tight. This is related to several analyses studied in the literature (including strictness analysis).

scholarly journals The 123 theorem of Probability Theory and Copositive Matrices

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Alexander Kovačec ◽  
Miguel M. R. Moreira ◽  
David P. Martins
Keyword(s):  
Probability Theory ◽  
Integral Inequality ◽  
Random Variables ◽  
Monotone Functions ◽  
The Real ◽  
Copositive Matrices ◽  
Vector Valued ◽  
Independent Identically Distributed

AbstractAlon and Yuster give for independent identically distributed real or vector valued random variables X, Y combinatorially proved estimates of the form Prob(∥X − Y∥ ≤ b) ≤ c Prob(∥X − Y∥ ≤ a). We derive these using copositive matrices instead. By the same method we also give estimates for the real valued case, involving X + Y and X − Y, due to Siegmund-Schultze and von Weizsäcker as generalized by Dong, Li and Li. Furthermore, we formulate a version of the above inequalities as an integral inequality for monotone functions.

Probabilism, Assertion, and Higher-Order Vagueness

2018 ◽  
Author(s):  
Andrew Bacon
Keyword(s):  
Probability Theory ◽  
Higher Order ◽  
Dutch Book ◽  
Probability Calculus ◽  
Hartry Field ◽  
Standard Probability

Hartry Field has recently suggested that a non-standard probability calculus better represents our beliefs about vague matters. His theory has two notable features: (i) that your attitude to P when you are certain that P is higher-order borderline ought to be the same as your attitude when you are certain that P is simply borderline, and (ii) that when you are certain that P is borderline you should have no credence in P and no credence in ~. This chapter rejects both elements of this view and advocates instead for the view that when you are in possession of all the possible evidence, and it is borderline whether P is borderline, it is borderline whether you should believe P. Secondly, it argues for probabilism: the view that your credences ought to conform to the probability calculus. To get a handle on these issues, the chapter looks at Dutch book arguments and comparative axiomatizations of probability theory.

Characterization of higher-order monotonicity via integral inequalities

2010 ◽  
Vol 140 (4) ◽  
pp. 723-736 ◽  
Author(s):  
Mihály Bessenyei ◽  
Zsolt Páles
Keyword(s):  
Continuous Function ◽  
Higher Order ◽  
Integral Inequalities ◽  
Monotone Functions ◽  
Hadamard Inequality ◽  
Left Hand ◽  
Right Hand ◽  
Compact Subinterval

The Hermite-Hadamard inequality not only is a consequence of convexity but also characterizes it: if a continuous function satisfies either its left-hand side or its right-hand side on each compact subinterval of the domain, then it is necessarily convex. The aim of this paper is to prove analogous statements for the higher-order extensions of the Hermite-Hadamard inequality. The main tools of the proofs are smoothing by convolution and the support properties of higher-order monotone functions.

Probability Theory

2015 ◽  
pp. 47-64
Keyword(s):  
Probability Theory ◽  
Probabilistic Analysis ◽  
Measure Theory ◽  
Higher Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Heavy Hitter ◽  
Lebesgue Integration

In this chapter, the authors make use of the formalizations of measure theory and Lebesgue integration in HOL4 to provide a higher-order-logic formalization of probability theory (Mhamdi, 2013). For illustration purposes, they also present the probabilistic analysis of the Heavy Hitter problem using HOL.

Measure Theory and Lebesgue Integration Theories

2015 ◽  
pp. 29-46
Keyword(s):  
Probability Theory ◽  
Measure Theory ◽  
Higher Order ◽  
Order Logic ◽  
Lebesgue Integral ◽  
Real Numbers ◽  
Convergence Theorems ◽  
Higher Order Logic ◽  
Lebesgue Integration

As discussed in the previous chapter, the fundamental theories of measure and Lebesgue integration are the prerequisites for the formalization of probability theory in higher-order logic. The scope of this chapter is primarily the formalization of these foundations. Both formalizations of measure theory and Lebesgue integral (Mhamdi, Hasan, & Tahar, 2011), presented in this chapter, are based on the extended-real numbers. This allows us to define sigma-finite and even infinite measures and handle extended-real-valued measurable functions. It also allows us to verify the properties of the Lebesgue integral and its convergence theorems for arbitrary functions. Therefore, the chapter begins with a description of the formalization of extended real numbers.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

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