Cube Attack is a successful case of Algebraic Attack. Cube Attack consists of two phases, linear equation extraction and solving the extracted equation system. Due to the high complexity of equation extraction phase in finding linear equations, we can extract nonlinear ones that could be approximated to linear equations with high probability. The probabilistic equations could be considered as l...

Joseph Halpern and Judea Pearl ([2005]) draw upon structural equation models to develop an attractive analysis of 'actual cause'. Their analysis is designed for the case of deterministic causation. I show that their account can be naturally extended to provide an elegant treatment of probabilistic causation. 1Introduction2Preemption3Structural Equation Models4The Halpern and Pearl Definition of...

Equation discovery, also known as symbolic regression, is a type of automated modeling that discovers scientific laws, expressed in the form equations, from observed data and expert knowledge. Deterministic grammars, such context-free have been used to limit search spaces equation discovery by providing hard constraints specify which equations consider not. In this paper, we propose use probabi...

in this paper, we prove the hyers-ulam stability in$beta$-homogeneous probabilistic modular spaces via fixed point method for the functional equation[f(x+ky)+f(x-ky)=f(x+y)+f(x-y)+frac{2(k+1)}{k}f(ky)-2(k+1)f(y)]for fixed integers $k$ with $kneq 0,pm1.$

An interval-valued distribution is a fmite sequence of intervals ([Pl1 ,Pu1 ], ... ,[Pln,Pun]) n n such that O$Plj$Put�l, E Puj�l, and E Plj$1. Interval distributions generalize realj=l j=l valued probability distributions and arise naturally in many situations. They may represent collections of confidence intervals derived from frequency data, imprecisely stated subjective probabilities, known...

In this paper, we prove the Hyers-Ulam stability in$beta$-homogeneous probabilistic modular spaces via fixed point method for the functional equation[f(x+ky)+f(x-ky)=f(x+y)+f(x-y)+frac{2(k+1)}{k}f(ky)-2(k+1)f(y)]for fixed integers $k$ with $kneq 0,pm1.$

If v are the angles at the vertices v, v 2 V , of a convex n-gon in the plane, then P v2V v = (n?2) { known as Euclid's angle-sum equation. We normalize the full angle to 1, i.e. we set v = v =(2). Then the equation reads as X v2V v = n=2 ? 1 (1) This identity has a generalization to higher dimensions, called Gram's equation. We want to derive this identity by a probabilistic argument. Let us i...