\documentclass[% %reprint, % twocolumn, %superscriptaddress, %groupedaddress, %unsortedaddress, %runinaddress, %frontmatterverbose, % preprint, showpacs, showkeys, preprintnumbers, %nofootinbib, %nobibnotes, %bibnotes, amsmath,amssymb, aps, % prl, pra, % prb, % rmp, %prstab, %prstper, longbibliography, floatfix, %lengthcheck,% ]{revtex4-1} %\usepackage{cdmtcs-pdf} \usepackage{mathptmx}% http://ctan.org/pkg/mathptmx \usepackage{amssymb,amsthm,amsmath} \usepackage{listings} %\lstset{numbers=left, numberstyle=\tiny, stepnumber=1, numbersep=5pt} %\lstset{emph={V-representation,begin,end}, emphstyle=\color{red}, emph={[2]root,base},emphstyle={[2]\color{blue}}} \lstset{keywords={V,H,\*,representation,begin,end}} %\lstset{emph={V-representation,begin,end}, emphstyle=\color{red}} \lstset{% general command to set parameter(s) basicstyle=\small, % print whole listing small keywordstyle=\color{blue}\bfseries, %\underbar % underlined bold black keywords identifierstyle=, % nothing happens commentstyle=\color{white}, % white comments stringstyle=\ttfamily, % typewriter type for strings showstringspaces=false} % no special string space %\usepackage{tikz} \usepackage{graphicx}% Include figure files %\usepackage{url} \usepackage{xcolor} \usepackage{eepic} \usepackage{hyperref} \hypersetup{ colorlinks, linkcolor={blue!80!black}, citecolor={red!75!black}, urlcolor={blue!80!black} } \begin{document} \title{Supplemental Material: What Is so Special About Quantum Clicks?} \author{Karl Svozil} \email{svozil@tuwien.ac.at} \homepage{http://tph.tuwien.ac.at/~svozil} \affiliation{Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10/136, 1040 Vienna, Austria} \date{\today} \begin{abstract} This supplement contains mostly code interpretable by Fukuda's {\em cddlib package cddlib-094h} for evaluating hull problems in quantum physical configurations. It also contains some corresponding quantum mechanical calculations. \end{abstract} \maketitle \tableofcontents \subsection{The cddlib package} % 2017-b-c-sup.tex %\section{Supplemental Material: cddlib codes of examples} %\label{2017-b-ccoe} Fukuda's {\em cddlib package cddlib-094h} can be obtained from the package homepage~\cite{cdd-pck}. Installation on Unix-type operating systems is with {\em gcc}; the free library for arbitrary precision arithmetic {\em GMP} (currently 6.1.2)~\cite{gmplib}, must be installed first. In its elementary form of the {\em V-representation}, {\em cddlib} takes in the $k$ vertices $\vert {\bf v}_1 \rangle , \ldots , \vert {\bf v}_k \rangle$ of a convex polytope in an $m$-dimensional vector space as follows (note that all rows of vector components start with ``$1$''): %\begin{verbatim} %\begin{lstlisting}[caption={Vertex representation of cddlib~\cite{cdd-pck}.},label=2017-b-c-supvcddlib] \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] V-representation begin k m+1 numbertype 1 v_11 ... v_1m ............... 1 v_k1 ... v_km end \end{lstlisting} %\end{verbatim} {\em cddlib} responds with the faces (boundaries of halfspaces), as encoded by $n$ inequalities $ \textsf{\textbf{A}} \vert {\bf x} \rangle \le \vert {\bf b} \rangle $ in the {\em H-representation} as follows: {\begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] H-representation begin n m+1 numbertype b -A end \end{lstlisting} } Comments appear after an asterisk. \subsection{Trivial examples} \label{2017-b-teap} \subsubsection{One observable} \label{2017-b-ooa} The case of a single variable has two extreme cases: false$\equiv 0$ and true$\equiv 1$, resulting in $0\le p_1 \le 1$: { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * one variable * V-representation begin 2 2 integer 1 0 1 1 end ~~~~~~ cddlib response H-representation begin 2 2 real 1 -1 0 1 end \end{lstlisting} } \subsubsection{Two observables} \label{2017-b-toa} The case of two variables $p_1$ and $p_2$, and a joint variable $p_{12}$, result in % -A <= b \begin{eqnarray} %\begin{aligned} p_1 + p_2 - p_{12} &\le& 1, \\ - p_1 + p_{12} &\le& 0, \\ - p_2 + p_{12} &\le& 0, \\ - p_{12} &\le& 0, \label{2017-b-1-2-p-ca} %\end{aligned} \end{eqnarray} and thus $0 \le p_{12} \le p_1 , p_2$. { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * two variables: p1, p2, p12=p1*p2 * V-representation begin 4 4 integer 1 0 0 0 1 0 1 0 1 1 0 0 1 1 1 1 end ~~~~~~ cddlib response H-representation begin 4 4 real 1 -1 -1 1 0 1 0 -1 0 0 1 -1 0 0 0 1 end \end{lstlisting} } For dichotomic expectation values $\pm 1$, { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * two expectation values: E1, E2, E12=E1*E2 * V-representation begin 4 4 integer 1 -1 -1 1 1 -1 1 -1 1 1 -1 -1 1 1 1 1 end ~~~~~~ cddlib response H-representation begin 4 4 real 1 -1 -1 1 1 1 -1 -1 1 -1 1 -1 1 1 1 1 end \end{lstlisting} } \subsubsection{Bounds on the (joint) probabilities and expectations of three observables} \label{2017-b-tevoa} { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * four joint expectations: * p1, p2, p3, * p12=p1*p2, p13=p1*p3, p23=p2*p3, * p123=p1*p2*p3 V-representation begin 8 8 integer 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 1 1 end ~~~~~~ cddlib response H-representation begin 8 8 real 1 -1 -1 -1 1 1 1 -1 0 1 0 0 -1 -1 0 1 0 0 1 0 -1 0 -1 1 0 0 0 1 0 -1 -1 1 0 0 0 0 1 0 0 -1 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 end \end{lstlisting} } If single observable expectations are set to zero by assumption (axiom) and are not-enumerated, the table of expectation values may be redundand. The case of three expectation value observables $E_1$, $E_2$ and $E_3$ (which are not explicitly enumerated), as well as all joint expectations $E_{12}$, $E_{13}$, $E_{23}$, and $E_{123}$, result in % -A <= b \begin{eqnarray} %\begin{aligned} - E_{12}- E_{13}- E_{23} &\le& 1 \\ - E_{123} &\le& 1, \\ E_{123} &\le& 1, \\ - E_{12}+ E_{13}+ E_{23} &\le& 1, \\ E_{12}- E_{13}+ E_{23} &\le& 1, \\ E_{12}+ E_{13}- E_{23} &\le& 1 . \label{2017-b-1-3-e-ia} %\end{aligned} \end{eqnarray} { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * four joint expectations: * [E1, E2, E3, not explicitly enumerated] * E12=E1*E2, E13=E1*E3, E23=E2*E3, * E123=E1*E2*E3 V-representation begin 8 5 integer 1 1 1 1 1 1 1 -1 -1 -1 1 -1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 1 -1 1 -1 1 1 1 -1 -1 1 1 1 1 1 -1 end ~~~~~~ cddlib response H-representation begin 6 5 real 1 1 1 1 0 1 0 0 0 1 1 0 0 0 -1 1 1 -1 -1 0 1 -1 1 -1 0 1 -1 -1 1 0 end \end{lstlisting} } \subsection{2 observers, 2 measurement configurations per observer} \label{2017-b-totmcpoa} From a quantum physical standpoint the first relevant case is that of 2 observers and 2 measurement configurations per observer. \subsubsection{Bell-Wigner-Fine case: probabilities for 2 observers, 2 measurement configurations per observer} \label{2017-b-bwfa} The case of four probabilities $p_1$, $p_2$, $p_3$ and $p_4$, as well as four joint probabilities $p_{13}$, $p_{14}$, $p_{23}$, and $p_{24}$ result in % -A <= b \begin{eqnarray} %\begin{aligned} -p_{14} &\le& 0\\ -p_{24} &\le& 0\\ +p_{1} +p_{4} -p_{13} -p_{14} +p_{23} -p_{24} &\le& 1\\ +p_{2} +p_{4} +p_{13} -p_{14} -p_{23} -p_{24} &\le& 1\\ +p_{2} +p_{3} -p_{13} +p_{14} -p_{23} -p_{24} &\le& 1\\ +p_{1} +p_{3} -p_{13} -p_{14} -p_{23} +p_{24} &\le& 1\\ -p_{13} &\le& 0\\ -p_{23} &\le& 0\\ -p_{1} -p_{4} +p_{13} +p_{14} -p_{23} +p_{24} &\le& 0\\ -p_{2} -p_{4} -p_{13} +p_{14} +p_{23} +p_{24} &\le& 0\\ -p_{2} -p_{3} +p_{13} -p_{14} +p_{23} +p_{24} &\le& 0\\ -p_{1} -p_{3} +p_{13} +p_{14} +p_{23} -p_{24} &\le& 0\\ -p_{1} +p_{14} &\le& 0\\ -p_{2} +p_{24} &\le& 0\\ -p_{3} +p_{23} &\le& 0\\ -p_{3} +p_{13} &\le& 0\\ -p_{1} +p_{13} &\le& 0\\ -p_{2} +p_{23} &\le& 0\\ -p_{4} +p_{24} &\le& 0\\ -p_{4} +p_{14} &\le& 0\\ +p_{2} +p_{4} -p_{24} &\le& 1\\ +p_{1} +p_{4} -p_{14} &\le& 1\\ +p_{2} +p_{3} -p_{23} &\le& 1\\ +p_{1} +p_{3} -p_{13} &\le& 1 . \label{2017-b-2-2-p-c} %\end{aligned} \end{eqnarray} { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * eight variables: p1, p2, p3, p4, * p13, p14, p23, p24 * V-representation begin 16 9 integer 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 1 0 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 1 1 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 end ~~~~~~ cddlib response H-representation begin 24 9 real 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 -1 0 0 -1 1 1 -1 1 1 0 -1 0 -1 -1 1 1 1 1 0 -1 -1 0 1 -1 1 1 1 -1 0 -1 0 1 1 1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 -1 -1 1 -1 0 0 1 0 1 1 -1 -1 -1 0 0 1 1 0 -1 1 -1 -1 0 1 0 1 0 -1 -1 -1 1 0 1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 1 0 0 0 -1 0 0 0 0 1 0 -1 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 1 0 -1 0 0 1 0 -1 0 -1 0 0 0 1 1 -1 0 0 -1 0 1 0 0 1 0 -1 -1 0 0 0 1 0 1 -1 0 -1 0 1 0 0 0 end \end{lstlisting} } \subsubsection{Clauser-Horne-Shimony-Holt case: expectation values for 2 observers, 2 measurement configurations per observer} \label{2017-b-chshcevta} The case of four expectation values $E_1$, $E_2$, $E_3$ and $E_4$ (which are not explicitly enumerated), as well as all joint expectations $E_{13}$, $E_{14}$, $E_{23}$, and $E_{24}$ result in % -A <= b \begin{eqnarray} %\begin{aligned} + E_{13} - E_{14} - E_{23} - E_{24} &\le& 2 \\ - E_{24} &\le& 1 \\ - E_{23} &\le& 1 \\ - E_{13} + E_{14} - E_{23} - E_{24} &\le& 2 \\ - E_{14} &\le& 1 \\ - E_{13} - E_{14} + E_{23} - E_{24} &\le& 2 \\ - E_{13} - E_{14} - E_{23} + E_{24} &\le& 2 \\ - E_{13} &\le& 1 \\ - E_{13} + E_{14} + E_{23} + E_{24} &\le& 2 \\ + E_{24} &\le& 1 \\ + E_{23} &\le& 1 \\ + E_{13} - E_{14} + E_{23} + E_{24} &\le& 2 \\ + E_{14} &\le& 1 \\ + E_{13} + E_{14} - E_{23} + E_{24} &\le& 2 \\ + E_{13} + E_{14} + E_{23} - E_{24} &\le& 2 \\ + E_{13} &\le& 1 . \label{2017-b-2-2-e-i} %\end{aligned} \end{eqnarray} { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * four joint expectations: * E13, E14, E23, E24 * V-representation begin 16 5 integer 1 1 1 1 1 1 1 -1 1 -1 1 -1 1 -1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 1 1 -1 1 -1 -1 1 1 1 -1 -1 1 1 1 -1 1 1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 1 1 1 1 end ~~~~~~ cddlib response H-representation begin 16 5 real 2 -1 1 1 1 1 0 0 0 1 1 0 0 1 0 2 1 -1 1 1 1 0 1 0 0 2 1 1 -1 1 2 1 1 1 -1 1 1 0 0 0 2 1 -1 -1 -1 1 0 0 0 -1 1 0 0 -1 0 2 -1 1 -1 -1 1 0 -1 0 0 2 -1 -1 1 -1 2 -1 -1 -1 1 1 -1 0 0 0 end \end{lstlisting} } \subsubsection{Beyond the Clauser-Horne-Shimony-Holt case: 2 observers, 3 measurement configurations per observer} \label{2017-b-chshc1ba} { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * 6 expectations: * E1, ... , E6 * 9 joint expectations: * E14, E15, E16, E24, E25, E26, E34, E35, E36 * 1,2,3 on one side * 4,5,6 on other side * V-representation begin 64 16 integer 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 1 -1 1 1 -1 1 1 -1 1 1 1 1 1 -1 1 1 -1 1 1 -1 1 1 -1 1 1 1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 1 -1 1 1 -1 1 1 -1 1 1 -1 1 1 1 1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 1 1 -1 1 1 -1 1 1 -1 1 1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 -1 1 1 1 1 1 -1 -1 1 1 -1 1 1 -1 1 1 1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 -1 1 1 1 -1 1 1 -1 -1 -1 1 1 1 -1 1 1 -1 1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 -1 1 1 -1 1 -1 1 1 -1 1 1 1 -1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 -1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 -1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 -1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 -1 1 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 1 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 -1 -1 1 1 -1 1 1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 1 -1 -1 1 1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -1 1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 1 1 -1 1 -1 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 1 -1 -1 1 1 1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 1 1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 -1 -1 1 -1 -1 -1 1 1 -1 1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 -1 -1 -1 1 -1 1 -1 1 1 -1 1 1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 -1 1 1 -1 1 1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 end ~~~~~~ cddlib response H-representation begin 684 16 real 4 0 -1 1 -1 -1 0 1 -1 0 1 1 1 -1 -1 1 [...] 4 1 1 0 1 1 0 1 1 1 1 1 -1 1 -1 0 [...] end \end{lstlisting} } \subsection{Pentagon logic} \subsection{Probabilities but no joint probabilities} Here is a computation which includes all probabilities but no joint probabilities: { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * ten probabilities: * p1 ... p10 * begin 11 11 integer 1 1 0 0 1 0 1 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 1 0 1 0 0 1 0 0 0 1 0 1 0 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 end ~~~~~~ cddlib response H-representation linearity 5 12 13 14 15 16 begin 16 11 real 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 -1 -1 0 1 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 -2 -1 0 1 0 -1 0 1 0 0 0 1 1 0 -1 0 0 0 0 0 0 0 1 1 0 -1 0 1 0 -1 0 0 1 -1 -1 0 0 0 0 0 0 0 0 -1 1 1 1 0 0 0 0 0 0 0 0 -1 -1 0 1 1 0 0 0 0 0 -1 1 1 0 -1 0 1 1 0 0 0 0 -1 -1 0 1 0 -1 0 1 1 0 -1 2 1 0 -1 0 1 0 -1 0 1 end \end{lstlisting} } \begin{eqnarray} %\begin{aligned} + p_6 \ge 0 \\ + p_8 \ge 0 \\ - p_1 + p_4 + p_8 \ge 0 \\ + p_4 \ge 0 \\ + p_1 \ge 0 \\ - p_1 - p_2 + p_4 - p_6 \ge -1 \\ + p_2 \ge 0 \\ -2p_1 - p_2 + p_4 - p_6 + p_8 \ge -1 \\ + p_1 + p_2 - p_4 \ge 0 \\ + p_1 + p_2 - p_4 + p_6 - p_8 \ge 0 \\ - p_1 - p_2 \ge -1 \\ + p_1 + p_2 + p_3 \ge 1 \\ - p_1 - p_2 + p_4 + p_5 \ge 0 \\ + p_1 + p_2 - p_4 + p_6 + p_7 \ge 1 \\ - p_1 - p_2 + p_4 - p_6 + p_8 + p_9 \ge 0 \\ 2p_1 + p_2 - p_4 + p_6 - p_8 + p_{10} \ge 1 . \label{2017-b-kl-p-c} %\end{aligned} \end{eqnarray} \subsection{Joint Expectations on all atoms} This is a full hull computation taking all joint expectations into account: {\footnotesize \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * 45 pair expectations: * E12 ... E910 * V-representation begin 11 46 real 1 -1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 -1 1 -1 1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 -1 -1 1 -1 -1 1 1 -1 1 1 -1 1 1 1 -1 1 1 -1 1 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 -1 1 1 1 -1 1 1 -1 1 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 -1 1 1 1 -1 -1 -1 1 1 1 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 1 1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 -1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 1 1 -1 1 1 -1 1 -1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 1 1 1 -1 1 -1 -1 1 -1 -1 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 1 1 -1 1 -1 -1 -1 1 -1 1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 -1 end ~~~~~~ cddlib response H-representation linearity 35 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 begin 46 46 real 1 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 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0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 -1 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 1 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 -1 0 1 0 0 1 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 -1 0 1 0 -1 0 0 0 0 0 1 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 -1 0 1 0 0 1 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 -1 -1 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 end \end{lstlisting} } \begin{eqnarray} %\begin{aligned} E_{13} + E_{14} - E_{34} &\le& 1, \\ -E_{12} + E_{18} + E_{28} &\le& 1, \\ E_{14} + E_{18} - E_{48} &\le& 1, \\ E_{12} - E_{14} - E_{26} + E_{34} - E_{36} &\le& -1, \\ E_{12} + E_{13} + E_{26} + E_{36} &\le& 0, \\ -E_{13} - E_{14} + E_{16} - E_{18} + E_{36} + E_{48} &\le& 0, \\ -E_{12} - E_{16} - E_{26} &\le& 1, \\ E_{16} - E_{18} + E_{26} - E_{28} &\le& 0, \\ E_{26} - E_{28} - E_{34} + E_{36} + E_{48} &\le& 1, \\ E_{14} - E_{16} + E_{34} - E_{36} &\le& 0, \\ -E_{13} - E_{14} - E_{26} + E_{28} - E_{36} - E_{48} &\le& 0, \\ E_{12} - E_{14} - E_{15} &\le& -1, \\ E_{13} + E_{14} - E_{16} - E_{17} &\le& 0, \\ E_{12} - E_{14} + E_{16} - E_{18} - E_{19} &\le& -1, \\ -E_{1,10} + E_{13} + E_{14} - E_{16} + E_{18} &\le& 1, \\ -E_{12} - E_{13} - E_{23} &\le& 1, \\ E_{12} - E_{14} - E_{24} &\le& -1, \\ E_{14} - E_{25} &\le& 0, \\ -E_{13} - E_{14} - E_{26} - E_{27} &\le& 0, \\ E_{14} + E_{26} - E_{28} - E_{29} &\le& 0, \\ -E_{12} - E_{13} - E_{14} - E_{2,10} - E_{26} + E_{28} &\le& 0, \\ -E_{12} - E_{34} - E_{35} &\le& 1, \\ E_{34} - E_{36} - E_{37} &\le& -1, \\ E_{13} + E_{14} + E_{26} - E_{28} - E_{34} + E_{36} - E_{38} &\le& 1, \\ -E_{12} - E_{13} - E_{14} - E_{26} + E_{28} - E_{39} &\le& 0, \\ E_{14} + E_{26} - E_{28} - E_{3,10} &\le& 0, \\ E_{12} - E_{45} &\le& 0, \\ E_{34} - E_{36} - E_{46} &\le& -1, \\ E_{36} - E_{47} &\le& 0, \\ E_{12} + E_{34} - E_{36} - E_{48} - E_{49} &\le& -1, \\ -E_{14} + E_{36} - E_{4,10} + E_{48} &\le& 0, \\ E_{16} + E_{26} - E_{34} + E_{36} - E_{56} &\le& 1, \\ -E_{16} - E_{26} - E_{36} - E_{57} &\le& 0, \\ E_{18} + E_{28} - E_{48} - E_{58} &\le& 0, \\ E_{16} - E_{18} + E_{26} - E_{28} - E_{34} + E_{36} + E_{48} - E_{59} &\le& 0, \\ -E_{12} + E_{14} - E_{16} + E_{18} - E_{26} + E_{28} - E_{36} - E_{48} - E_{5,10} &\le& 1, \\ E_{34} - E_{67} &\le& 0, \\ E_{16} - E_{18} + E_{26} - E_{28} - E_{34} + E_{36} + E_{48} - E_{68} &\le& 0, \\ E_{18} + E_{28} - E_{48} - E_{69} &\le& 0, \\ -E_{18} + E_{26} - E_{28} + E_{36} + E_{48} - E_{6,10} &\le& 0, \\ E_{13} + E_{14} - E_{16} + E_{18} - E_{78} &\le& 1, \\ -E_{13} - E_{14} - E_{18} - E_{26} + E_{34} - E_{36} - E_{79} &\le& -1, \\ E_{18} - E_{7,10} &\le& 0, \\ E_{16} + E_{26} - E_{34} + E_{36} - E_{89} &\le& 1, \\ E_{13} + E_{14} - E_{16} - E_{8,10} &\le& 0, \\ -E_{12} - E_{13} - E_{9,10} &\le& 1 . \label{2017-b-kl-e-c} %\end{aligned} \end{eqnarray} \section{House/pentagon/pentagram gadget} \subsubsection{Bub-Stairs inequality} If one considers only the five probabilities on the intertwining atoms, then the following Bub-Stairs inequality $p_1+p_3+p_5+p_7+p_9 \le 2$, among others, results: { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * five probabilities on intertwining contexts * p1, p3, p5, p7, p9 * V-representation begin 11 6 integer 1 1 0 0 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 end ~~~~~~ cddlib response H-representation begin 11 6 real 0 0 0 1 0 0 1 0 0 0 -1 -1 0 1 0 0 0 0 1 0 -1 -1 0 0 2 -1 -1 -1 -1 -1 1 -1 -1 0 0 0 0 0 0 0 1 0 1 -1 0 0 0 -1 1 0 0 -1 -1 0 0 0 1 0 0 0 0 0 0 0 0 1 end \end{lstlisting} } One could also consider probabilities on the non-intertwining atoms yielding; in particular, $p_2+p_4+p_6+p_8+p_{10} \ge 1$. { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * five probabilities * on non-intertwining atoms * p2, p4, p6, p8, p10 * V-representation begin 11 6 integer 1 0 1 1 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 0 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 1 1 end ~~~~~~ cddlib response H-representation begin 11 6 real 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 -1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 1 0 0 1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 end \end{lstlisting} } \subsubsection{Klyachko-Can-Biniciogolu-Shumovsky inequalities} \label{2017-b-kcbsia} The following hull computation is limited to adjacent pair expectations; it yields the Klyachko-Can-Biniciogolu-Shumovsky inequality $ E_{13} + E_{35} + E_{57} + E_{79} + E_{91} \ge 3 $: { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * five joint Expectations: * E13 E35 E57 E79 E91 * V-representation begin 11 6 real 1 -1 1 1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 -1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 end ~~~~~~ cddlib response H-representation begin 11 6 real 1 0 0 0 1 0 1 0 0 0 0 1 1 0 1 0 0 0 3 1 1 1 1 1 1 1 0 0 0 0 1 0 0 1 0 0 1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 end \end{lstlisting} } \begin{eqnarray} %\begin{aligned} - E_{79} &\le& 1 \\ - E_{91} &\le& 1 \\ - E_{35} &\le& 1 \\ - E_{13} - E_{35} - E_{57} - E_{79} - E_{91} &\le& 3 \\ - E_{13} &\le& 1 \\ - E_{57} &\le& 1 \\ - E_{13} + E_{35} - E_{57} + E_{79} + E_{91} &\le& 1 \\ + E_{13} - E_{35} + E_{57} + E_{79} - E_{91} &\le& 1 \\ - E_{13} + E_{35} + E_{57} - E_{79} + E_{91} &\le& 1 \\ + E_{13} - E_{35} + E_{57} - E_{79} + E_{91} &\le& 1 \\ + E_{13} + E_{35} - E_{57} + E_{79} - E_{91} &\le& 1 . \label{2017-b-kl-e-i} %\end{aligned} \end{eqnarray} \subsection{Two intertwined pentagon logics forming a Specker K\"afer (bug) or cat's cradle logic} \subsubsection{Probabilities on the Specker bug logic} A {\em Mathematica}~\cite{Mathematica11.1} code to reduce probabilities on the Specker bug logic: { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] Reduce[ p1 + p2 + p3 == 1 && p3 + p4 + p5 == 1 && p5 + p6 + p7 == 1 && p7 + p8 + p9 == 1 && p9 + p10 + p11 == 1 && p11 + p12 + p1 == 1 && p4 + p10 + p13 == 1, {p3, p11, p5, p9, p4, p10}, Reals] ~~~~~~ Mathematica response p1 == 3/2 - p12/2 - p13/2 - p2/2 - p6/2 - p7 - p8/2 && p3 == -(1/2) + p12/2 + p13/2 - p2/2 + p6/2 + p7 + p8/2 && p11 == -(1/2) - p12/2 + p13/2 + p2/2 + p6/2 + p7 + p8/2 && p5 == 1 - p6 - p7 && p9 == 1 - p7 - p8 && p4 == 1/2 - p12/2 - p13/2 + p2/2 + p6/2 - p8/2 && p10 == 1/2 + p12/2 - p13/2 - p2/2 - p6/2 + p8/2 \end{lstlisting} } Computation of all the two-valued states thereon: { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] Reduce[p1 + p2 + p3 == 1 && p3 + p4 + p5 == 1 && p5 + p6 + p7 == 1 && p7 + p8 + p9 == 1 && p9 + p10 + p11 == 1 && p11 + p12 + p1 == 1 && p4 + p10 + p13 == 1 && p1^2 == p1 && p2^2 == p2 && p3^2 == p3 && p4^2 == p4 && p5^2 == p5 && p6^2 == p6 && p7^2 == p7 && p8^2 == p8 && p9^2 == p9 && p10^2 == p10 && p11^2 == p11 && p12^2 == p12 && p13^2 == p13] ~~~~~~ Mathematica response (p9 == 0 && p8 == 0 && p7 == 1 && p6 == 0 && p5 == 0 && p4 == 0 && p3 == 1 && p2 == 0 && p13 == 0 && p12 == 1 && p11 == 0 && p10 == 1 && p1 == 0) || (p9 == 0 && p8 == 0 && p7 == 1 && p6 == 0 && p5 == 0 && p4 == 0 && p3 == 1 && p2 == 0 && p13 == 1 && p12 == 0 && p11 == 1 && p10 == 0 && p1 == 0) || (p9 == 0 && p8 == 0 && p7 == 1 && p6 == 0 && p5 == 0 && p4 == 1 && p3 == 0 && p2 == 1 && p13 == 0 && p12 == 0 && p11 == 1 && p10 == 0 && p1 == 0) || (p9 == 0 && p8 == 1 && p7 == 0 && p6 == 0 && p5 == 1 && p4 == 0 && p3 == 0 && p2 == 0 && p13 == 0 && p12 == 0 && p11 == 0 && p10 == 1 && p1 == 1) || (p9 == 0 && p8 == 1 && p7 == 0 && p6 == 0 && p5 == 1 && p4 == 0 && p3 == 0 && p2 == 1 && p13 == 0 && p12 == 1 && p11 == 0 && p10 == 1 && p1 == 0) || (p9 == 0 && p8 == 1 && p7 == 0 && p6 == 0 && p5 == 1 && p4 == 0 && p3 == 0 && p2 == 1 && p13 == 1 && p12 == 0 && p11 == 1 && p10 == 0 && p1 == 0) || (p9 == 0 && p8 == 1 && p7 == 0 && p6 == 1 && p5 == 0 && p4 == 0 && p3 == 1 && p2 == 0 && p13 == 0 && p12 == 1 && p11 == 0 && p10 == 1 && p1 == 0) || (p9 == 0 && p8 == 1 && p7 == 0 && p6 == 1 && p5 == 0 && p4 == 0 && p3 == 1 && p2 == 0 && p13 == 1 && p12 == 0 && p11 == 1 && p10 == 0 && p1 == 0) || (p9 == 0 && p8 == 1 && p7 == 0 && p6 == 1 && p5 == 0 && p4 == 1 && p3 == 0 && p2 == 1 && p13 == 0 && p12 == 0 && p11 == 1 && p10 == 0 && p1 == 0) || (p9 == 1 && p8 == 0 && p7 == 0 && p6 == 0 && p5 == 1 && p4 == 0 && p3 == 0 && p2 == 0 && p13 == 1 && p12 == 0 && p11 == 0 && p10 == 0 && p1 == 1) || (p9 == 1 && p8 == 0 && p7 == 0 && p6 == 0 && p5 == 1 && p4 == 0 && p3 == 0 && p2 == 1 && p13 == 1 && p12 == 1 && p11 == 0 && p10 == 0 && p1 == 0) || (p9 == 1 && p8 == 0 && p7 == 0 && p6 == 1 && p5 == 0 && p4 == 0 && p3 == 1 && p2 == 0 && p13 == 1 && p12 == 1 && p11 == 0 && p10 == 0 && p1 == 0) || (p9 == 1 && p8 == 0 && p7 == 0 && p6 == 1 && p5 == 0 && p4 == 1 && p3 == 0 && p2 == 0 && p13 == 0 && p12 == 0 && p11 == 0 && p10 == 0 && p1 == 1) || (p9 == 1 && p8 == 0 && p7 == 0 && p6 == 1 && p5 == 0 && p4 == 1 && p3 == 0 && p2 == 1 && p13 == 0 && p12 == 1 && p11 == 0 && p10 == 0 && p1 == 0) \end{lstlisting} } \subsubsection{Hull calculation for the probabilities on the Specker bug logic} { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * 13 probabilities on atoms a1...a13: * p1 ... p13 * V-representation begin 14 14 real 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 end ~~~~~~ cddlib response H-representation linearity 7 17 18 19 20 21 22 23 begin 23 14 real 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 -1 0 1 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 -1 0 0 0 0 0 0 0 0 0 0 1 2 0 -2 0 1 0 -1 0 0 0 0 0 0 0 1 0 -1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 -1 0 1 0 -1 0 1 0 0 0 1 0 0 0 -1 0 0 0 0 0 -1 0 0 0 1 -1 -1 0 1 0 -1 0 1 0 -1 0 0 0 1 -1 -1 0 0 0 0 0 1 0 -1 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 1 0 -1 0 0 0 0 0 0 0 -1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 1 1 0 0 0 0 0 0 0 0 -1 1 1 0 -1 0 1 1 0 0 0 0 0 0 0 -1 -1 0 1 0 -1 0 1 1 0 0 0 0 -1 1 1 0 -1 0 1 0 -1 0 1 1 0 0 0 0 -1 0 1 0 -1 0 1 0 -1 0 1 0 -1 0 0 0 1 0 0 0 0 0 1 0 0 1 end \end{lstlisting} } The resulting face inequalities are \begin{eqnarray} %\begin{aligned} - p_4 \le 0 , \\ - p_6 \le 0 , \\ - p_1 - p_2 + p_4 - p_6 + p_8 \le 0 , \\ - p_1 \le 0 , \\ - p_1 - p_2 + p_4 \le 0 , \\ - p_1 -2p_2 +2p_4 - p_6 + p_8 \le 0 , \\ - p_2 + p_4 - p_6 \le 0 , \\ - p_2 \le 0 , \\ - p_{10} \le 0 , \\ - p_8 \le 0 , \\ - p_2 + p_4 - p_6 + p_8 - p_{10} \le 0 , \\ + p_4 + p_{10} \le +1 , \\ + p_1 + p_2 - p_4 + p_6 - p_8 + p_{10} \le +1 , \\ + p_1 + p_2 - p_8 + p_{10} \le +1 , \\ + p_1 + p_2 \le +1 , \\ + p_1 + p_2 - p_4 + p_6 \le +1 , \\ - p_1 - p_2 - p_3 \le -1 , \\ + p_1 + p_2 - p_4 - p_5 \le 0 , \\ - p_1 - p_2 + p_4 - p_6 - p_7 \le -1 , \\ + p_1 + p_2 - p_4 + p_6 - p_8 - p_9 \le 0 , \\ - p_1 - p_2 + p_4 - p_6 + p_8 - p_{10} - p_{11} \le -1 , \\ + p_2 - p_4 + p_6 - p_8 + p_{10} - p_{12} \le 0 , \\ - p_4 - p_{10} - p_{13} \le -1 . \label{2017-b-sb-p-c} %\end{aligned} \end{eqnarray} \subsubsection{Hull calculation for the expectations on the Specker bug logic} { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * (13 expectations on atoms a1...a13: * E1 ... E13 not enumerated) * 6 joint expectations E1*E3, E3*E5, ..., E11*E1 * V-representation begin 14 7 integer 1 -1 -1 -1 -1 -1 -1 1 -1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 end ~~~~~~ cddlib response H-representation linearity 1 18 begin 18 7 real 1 0 0 0 1 0 0 1 -1 0 0 1 -1 0 1 -1 1 -1 1 -1 0 1 0 0 -1 1 -1 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 1 1 -1 1 0 0 0 1 0 0 1 0 0 0 1 1 -1 0 -1 0 0 1 0 0 0 -1 0 0 1 0 -1 1 -1 0 0 1 1 -1 1 -1 1 0 1 0 0 -1 0 0 0 1 -1 1 -1 0 0 0 1 -1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 -1 0 0 -1 0 0 -1 1 -1 1 -1 1 end \end{lstlisting} } \subsubsection{Extended Specker bug logic} Here is the {\em Mathematica}~\cite{Mathematica11.1} code to reduce probabilities on the extended (by two contexts) Specker bug logics: { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] Reduce[ p1 + p2 + p3 == 1 && p3 + p4 + p5 == 1 && p5 + p6 + p7 == 1 && p7 + p8 + p9 == 1 && p9 + p10 + p11 == 1 && p11 + p12 + p1 == 1 && p4 + p10 + p13 == 1 && p1 + pc + q7 ==1 && p7 + pc + q1 ==1, {p3, p11, p5, p9, p4, p10, q3, q11, q5, q9, q4, q10, p13, q13, pc}] ~~~~~~ Mathematica response p1 == p7 + q1 - q7 && p3 == 1 - p2 - p7 - q1 + q7 && p11 == 1 - p12 - p7 - q1 + q7 && p5 == 1 - p6 - p7 && p9 == 1 - p7 - p8 && p4 == -1 + p2 + p6 + 2 p7 + q1 - q7 && p10 == -1 + p12 + 2 p7 + p8 + q1 - q7 && p13 == 3 - p12 - p2 - p6 - 4 p7 - p8 - 2 q1 + 2 q7 && pc == 1 - p7 - q1 \end{lstlisting} } Computation of all the 112 two-valued states thereon: { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] Reduce[p1 + p2 + p3 == 1 && p3 + p4 + p5 == 1 && p5 + p6 + p7 == 1 && p7 + p8 + p9 == 1 && p9 + p10 + p11 == 1 && p11 + p12 + p1 == 1 && p4 + p10 + p13 == 1 && p1^2 == p1 && p2^2 == p2 && p3^2 == p3 && p4^2 == p4 && p5^2 == p5 && p6^2 == p6 && p7^2 == p7 && p8^2 == p8 && p9^2 == p9 && p10^2 == p10 && p11^2 == p11 && p12^2 == p12 && p13^2 == p13 && q1^2 == q1 && q7^2 == q7 && pc^2 == pc] ~~~~~~ Mathematica response q7 == 0 && q1 == 0 && pc == 0 && p9 == 0 && p8 == 0 && p7 == 1 && p6 == 0 && p5 == 0 && p4 == 0 && p3 == 1 && p2 == 0 && p13 == 0 && p12 == 1 && p11 == 0 && p10 == 1 && p1 == 0) || (q7 == 0 && q1 == 0 && pc == 0 && p9 == 0 && p8 == 0 && p7 == 1 && p6 == 0 && p5 == 0 && p4 == 0 && p3 == 1 && p2 == 0 && p13 == 1 && p12 == 0 && p11 == 1 && p10 == 0 && p1 == 0) || [...] || (q7 == 1 && q1 == 1 && pc == 1 && p9 == 1 && p8 == 0 && p7 == 0 && p6 == 1 && p5 == 0 && p4 == 1 && p3 == 0 && p2 == 1 && p13 == 0 && p12 == 1 && p11 == 0 && p10 == 0 && p1 == 0) \end{lstlisting} } \subsection{Two intertwined Specker bug logics} \label{2017-b-bugscombinoa} Here is the {\em Mathematica}~\cite{Mathematica11.1} code to reduce probabilities on two intertwined Specker bug logics: { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] Reduce[ p1 + p2 + p3 == 1 && p3 + p4 + p5 == 1 && p5 + p6 + p7 == 1 && p7 + p8 + p9 == 1 && p9 + p10 + p11 == 1 && p11 + p12 + p1 == 1 && p4 + p10 + p13 == 1 && q1 + q2 + q3 == 1 && q3 + q4 + q5 == 1 && q5 + q6 + q7 == 1 && q7 + q8 + q9 == 1 && q9 + q10 + q11 == 1 && q11 + q12 + q1 == 1 && q4 + q10 + q13 == 1 && p1 + pc + q7 ==1 && p7 + pc + q1 ==1, {p3, p11, p5, p9, p4, p10, q3, q11, q5, q9, q4, q10, p13, q13, pc}] ~~~~~~ Mathematica response p1 == p7 + q1 - q7 && p3 == 1 - p2 - p7 - q1 + q7 && p11 == 1 - p12 - p7 - q1 + q7 && p5 == 1 - p6 - p7 && p9 == 1 - p7 - p8 && p4 == -1 + p2 + p6 + 2 p7 + q1 - q7 && p10 == -1 + p12 + 2 p7 + p8 + q1 - q7 && q3 == 1 - q1 - q2 && q11 == 1 - q1 - q12 && q5 == 1 - q6 - q7 && q9 == 1 - q7 - q8 && q4 == -1 + q1 + q2 + q6 + q7 && q10 == -1 + q1 + q12 + q7 + q8 && p13 == 3 - p12 - p2 - p6 - 4 p7 - p8 - 2 q1 + 2 q7 && q13 == 3 - 2 q1 - q12 - q2 - q6 - 2 q7 - q8 && pc == 1 - p7 - q1 \end{lstlisting} } %All two-valued states thereon: % %{ \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] % %Reduce[ %p1 + p2 + p3 == 1 %&& p3 + p4 + p5 == 1 %&& p5 + p6 + p7 == 1 %&& p7 + p8 + p9 == 1 %&& p9 + p10 + p11 == 1 %&& p11 + p12 + p1 == 1 %&& p4 + p10 + p13 == 1 %&& q1 + q2 + q3 == 1 %&& q3 + q4 + q5 == 1 %&& q5 + q6 + q7 == 1 %&& q7 + q8 + q9 == 1 %&& q9 + q10 + q11 == 1 %&& q11 + q12 + q1 == 1 %&& q4 + q10 + q13 == 1 %&& p1 + pc + q7 ==1 %&& p7 + pc + q1 ==1 %&& (p1 == 0 || p1 == 1 ) %&& (p2 == 0 || p2 == 1 ) %&& (p3 == 0 || p3 == 1 ) %&& (p4 == 0 || p4 == 1 ) %&& (p5 == 0 || p5 == 1 ) %&& (p6 == 0 || p6 == 1 ) %&& (p7 == 0 || p7 == 1 ) %&& (p8 == 0 || p8 == 1 ) %&& (p9 == 0 || p9 == 1 ) %&& (p10 == 0 || p10 == 1 ) %&& (p11 == 0 || p11 == 1 ) %&& (p12 == 0 || p12 == 1 ) %&& (p13 == 0 || p13 == 1 ) %&& (q1 == 0 || q1 == 1 ) %&& (q2 == 0 || q2 == 1 ) %&& (q3 == 0 || q3 == 1 ) %&& (q4 == 0 || q4 == 1 ) %&& (q5 == 0 || q5 == 1 ) %&& (q6 == 0 || q6 == 1 ) %&& (q7 == 0 || q7 == 1 ) %&& (q8 == 0 || q8 == 1 ) %&& (q9 == 0 || q9 == 1 ) %&& (q10 == 0 || q10 == 1 ) %&& (q11 == 0 || q11 == 1 ) %&& (q12 == 0 || q12 == 1 ) %&& (q13 == 0 || q13 == 1 ) %&& (pc == 0 || pc == 1 ) %] % %~~~~~~ Mathematica response % %nil ;-( % %\end{lstlisting} } \subsubsection{Hull calculation for the contexual inequalities corresponding to the Cabello, Estebaranz and Garc{\'{i}}a-Alcaine logic} { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * (13 expectations on atoms A1...A18: * not enumerated) * 9 4th order expectations A1A2A3A4 A4A5A6A7 ... A2A9A11A18 * V-representation begin 262144 10 real 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 1 1 -1 [[...]] 1 1 1 1 1 1 -1 1 1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 end ~~~~~~ cddlib response H-representation begin 274 10 real 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 7 -1 -1 -1 -1 -1 -1 1 1 1 7 -1 -1 -1 -1 -1 1 -1 1 1 7 -1 -1 -1 -1 1 -1 -1 1 1 7 -1 -1 -1 1 -1 -1 -1 1 1 7 -1 -1 1 -1 -1 -1 -1 1 1 7 -1 1 -1 -1 -1 -1 -1 1 1 7 1 -1 -1 -1 -1 -1 -1 1 1 1 0 0 0 0 0 0 1 0 0 7 -1 -1 -1 -1 -1 1 1 -1 1 7 -1 -1 -1 -1 1 -1 1 -1 1 7 -1 -1 -1 1 -1 -1 1 -1 1 7 -1 -1 1 -1 -1 -1 1 -1 1 7 -1 1 -1 -1 -1 -1 1 -1 1 7 1 -1 -1 -1 -1 -1 1 -1 1 7 -1 -1 -1 -1 -1 1 1 1 -1 7 -1 -1 -1 -1 1 -1 1 1 -1 7 -1 -1 -1 1 -1 -1 1 1 -1 7 -1 -1 1 -1 -1 -1 1 1 -1 7 -1 1 -1 -1 -1 -1 1 1 -1 7 1 -1 -1 -1 -1 -1 1 1 -1 1 0 0 0 0 0 1 0 0 0 7 -1 -1 -1 -1 1 1 -1 -1 1 7 -1 -1 -1 1 -1 1 -1 -1 1 7 -1 -1 1 -1 -1 1 -1 -1 1 7 -1 1 -1 -1 -1 1 -1 -1 1 7 1 -1 -1 -1 -1 1 -1 -1 1 7 -1 -1 -1 -1 1 1 -1 1 -1 7 -1 -1 -1 1 -1 1 -1 1 -1 7 -1 -1 1 -1 -1 1 -1 1 -1 7 -1 1 -1 -1 -1 1 -1 1 -1 7 1 -1 -1 -1 -1 1 -1 1 -1 7 -1 -1 -1 -1 1 1 1 -1 -1 7 -1 -1 -1 1 -1 1 1 -1 -1 7 -1 -1 1 -1 -1 1 1 -1 -1 7 -1 1 -1 -1 -1 1 1 -1 -1 7 1 -1 -1 -1 -1 1 1 -1 -1 7 -1 -1 -1 -1 1 1 1 1 1 7 -1 -1 -1 1 -1 1 1 1 1 7 -1 -1 1 -1 -1 1 1 1 1 7 -1 1 -1 -1 -1 1 1 1 1 7 1 -1 -1 -1 -1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 7 -1 -1 -1 1 1 -1 -1 -1 1 7 -1 -1 1 -1 1 -1 -1 -1 1 7 -1 1 -1 -1 1 -1 -1 -1 1 7 1 -1 -1 -1 1 -1 -1 -1 1 7 -1 -1 -1 1 1 -1 -1 1 -1 7 -1 -1 1 -1 1 -1 -1 1 -1 7 -1 1 -1 -1 1 -1 -1 1 -1 7 1 -1 -1 -1 1 -1 -1 1 -1 7 -1 -1 -1 1 1 -1 1 -1 -1 7 -1 -1 1 -1 1 -1 1 -1 -1 7 -1 1 -1 -1 1 -1 1 -1 -1 7 1 -1 -1 -1 1 -1 1 -1 -1 7 -1 -1 -1 1 1 -1 1 1 1 7 -1 -1 1 -1 1 -1 1 1 1 7 -1 1 -1 -1 1 -1 1 1 1 7 1 -1 -1 -1 1 -1 1 1 1 7 -1 -1 -1 1 1 1 -1 -1 -1 7 -1 -1 1 -1 1 1 -1 -1 -1 7 -1 1 -1 -1 1 1 -1 -1 -1 7 1 -1 -1 -1 1 1 -1 -1 -1 7 -1 -1 -1 1 1 1 -1 1 1 7 -1 -1 1 -1 1 1 -1 1 1 7 -1 1 -1 -1 1 1 -1 1 1 7 1 -1 -1 -1 1 1 -1 1 1 7 -1 -1 -1 1 1 1 1 -1 1 7 -1 -1 1 -1 1 1 1 -1 1 7 -1 1 -1 -1 1 1 1 -1 1 7 1 -1 -1 -1 1 1 1 -1 1 7 -1 -1 -1 1 1 1 1 1 -1 7 -1 -1 1 -1 1 1 1 1 -1 7 -1 1 -1 -1 1 1 1 1 -1 7 1 -1 -1 -1 1 1 1 1 -1 1 0 0 0 1 0 0 0 0 0 7 -1 -1 1 1 -1 -1 -1 -1 1 7 -1 1 -1 1 -1 -1 -1 -1 1 7 1 -1 -1 1 -1 -1 -1 -1 1 7 -1 -1 1 1 -1 -1 -1 1 -1 7 -1 1 -1 1 -1 -1 -1 1 -1 7 1 -1 -1 1 -1 -1 -1 1 -1 7 -1 -1 1 1 -1 -1 1 -1 -1 7 -1 1 -1 1 -1 -1 1 -1 -1 7 1 -1 -1 1 -1 -1 1 -1 -1 7 -1 -1 1 1 -1 -1 1 1 1 7 -1 1 -1 1 -1 -1 1 1 1 7 1 -1 -1 1 -1 -1 1 1 1 7 -1 -1 1 1 -1 1 -1 -1 -1 7 -1 1 -1 1 -1 1 -1 -1 -1 7 1 -1 -1 1 -1 1 -1 -1 -1 7 -1 -1 1 1 -1 1 -1 1 1 7 -1 1 -1 1 -1 1 -1 1 1 7 1 -1 -1 1 -1 1 -1 1 1 7 -1 -1 1 1 -1 1 1 -1 1 7 -1 1 -1 1 -1 1 1 -1 1 7 1 -1 -1 1 -1 1 1 -1 1 7 -1 -1 1 1 -1 1 1 1 -1 7 -1 1 -1 1 -1 1 1 1 -1 7 1 -1 -1 1 -1 1 1 1 -1 7 -1 -1 1 1 1 -1 -1 -1 -1 7 -1 1 -1 1 1 -1 -1 -1 -1 7 1 -1 -1 1 1 -1 -1 -1 -1 7 -1 -1 1 1 1 -1 -1 1 1 7 -1 1 -1 1 1 -1 -1 1 1 7 1 -1 -1 1 1 -1 -1 1 1 7 -1 -1 1 1 1 -1 1 -1 1 7 -1 1 -1 1 1 -1 1 -1 1 7 1 -1 -1 1 1 -1 1 -1 1 7 -1 -1 1 1 1 -1 1 1 -1 7 -1 1 -1 1 1 -1 1 1 -1 7 1 -1 -1 1 1 -1 1 1 -1 7 -1 -1 1 1 1 1 -1 -1 1 7 -1 1 -1 1 1 1 -1 -1 1 7 1 -1 -1 1 1 1 -1 -1 1 7 -1 -1 1 1 1 1 -1 1 -1 7 -1 1 -1 1 1 1 -1 1 -1 7 1 -1 -1 1 1 1 -1 1 -1 7 -1 -1 1 1 1 1 1 -1 -1 7 -1 1 -1 1 1 1 1 -1 -1 7 1 -1 -1 1 1 1 1 -1 -1 7 -1 -1 1 1 1 1 1 1 1 7 -1 1 -1 1 1 1 1 1 1 7 1 -1 -1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 7 -1 1 1 -1 -1 -1 -1 -1 1 7 1 -1 1 -1 -1 -1 -1 -1 1 7 -1 1 1 -1 -1 -1 -1 1 -1 7 1 -1 1 -1 -1 -1 -1 1 -1 7 -1 1 1 -1 -1 -1 1 -1 -1 7 1 -1 1 -1 -1 -1 1 -1 -1 7 -1 1 1 -1 -1 -1 1 1 1 7 1 -1 1 -1 -1 -1 1 1 1 7 -1 1 1 -1 -1 1 -1 -1 -1 7 1 -1 1 -1 -1 1 -1 -1 -1 7 -1 1 1 -1 -1 1 -1 1 1 7 1 -1 1 -1 -1 1 -1 1 1 7 -1 1 1 -1 -1 1 1 -1 1 7 1 -1 1 -1 -1 1 1 -1 1 7 -1 1 1 -1 -1 1 1 1 -1 7 1 -1 1 -1 -1 1 1 1 -1 7 -1 1 1 -1 1 -1 -1 -1 -1 7 1 -1 1 -1 1 -1 -1 -1 -1 7 -1 1 1 -1 1 -1 -1 1 1 7 1 -1 1 -1 1 -1 -1 1 1 7 -1 1 1 -1 1 -1 1 -1 1 7 1 -1 1 -1 1 -1 1 -1 1 7 -1 1 1 -1 1 -1 1 1 -1 7 1 -1 1 -1 1 -1 1 1 -1 7 -1 1 1 -1 1 1 -1 -1 1 7 1 -1 1 -1 1 1 -1 -1 1 7 -1 1 1 -1 1 1 -1 1 -1 7 1 -1 1 -1 1 1 -1 1 -1 7 -1 1 1 -1 1 1 1 -1 -1 7 1 -1 1 -1 1 1 1 -1 -1 7 -1 1 1 -1 1 1 1 1 1 7 1 -1 1 -1 1 1 1 1 1 7 -1 1 1 1 -1 -1 -1 -1 -1 7 1 -1 1 1 -1 -1 -1 -1 -1 7 -1 1 1 1 -1 -1 -1 1 1 7 1 -1 1 1 -1 -1 -1 1 1 7 -1 1 1 1 -1 -1 1 -1 1 7 1 -1 1 1 -1 -1 1 -1 1 7 -1 1 1 1 -1 -1 1 1 -1 7 1 -1 1 1 -1 -1 1 1 -1 7 -1 1 1 1 -1 1 -1 -1 1 7 1 -1 1 1 -1 1 -1 -1 1 7 -1 1 1 1 -1 1 -1 1 -1 7 1 -1 1 1 -1 1 -1 1 -1 7 -1 1 1 1 -1 1 1 -1 -1 7 1 -1 1 1 -1 1 1 -1 -1 7 -1 1 1 1 -1 1 1 1 1 7 1 -1 1 1 -1 1 1 1 1 7 -1 1 1 1 1 -1 -1 -1 1 7 1 -1 1 1 1 -1 -1 -1 1 7 -1 1 1 1 1 -1 -1 1 -1 7 1 -1 1 1 1 -1 -1 1 -1 7 -1 1 1 1 1 -1 1 -1 -1 7 1 -1 1 1 1 -1 1 -1 -1 7 -1 1 1 1 1 -1 1 1 1 7 1 -1 1 1 1 -1 1 1 1 7 -1 1 1 1 1 1 -1 -1 -1 7 1 -1 1 1 1 1 -1 -1 -1 7 -1 1 1 1 1 1 -1 1 1 7 1 -1 1 1 1 1 -1 1 1 7 -1 1 1 1 1 1 1 -1 1 7 1 -1 1 1 1 1 1 -1 1 7 -1 1 1 1 1 1 1 1 -1 7 1 -1 1 1 1 1 1 1 -1 1 0 1 0 0 0 0 0 0 0 7 1 1 -1 -1 -1 -1 -1 -1 1 7 1 1 -1 -1 -1 -1 -1 1 -1 7 1 1 -1 -1 -1 -1 1 -1 -1 7 1 1 -1 -1 -1 -1 1 1 1 7 1 1 -1 -1 -1 1 -1 -1 -1 7 1 1 -1 -1 -1 1 -1 1 1 7 1 1 -1 -1 -1 1 1 -1 1 7 1 1 -1 -1 -1 1 1 1 -1 7 1 1 -1 -1 1 -1 -1 -1 -1 7 1 1 -1 -1 1 -1 -1 1 1 7 1 1 -1 -1 1 -1 1 -1 1 7 1 1 -1 -1 1 -1 1 1 -1 7 1 1 -1 -1 1 1 -1 -1 1 7 1 1 -1 -1 1 1 -1 1 -1 7 1 1 -1 -1 1 1 1 -1 -1 7 1 1 -1 -1 1 1 1 1 1 7 1 1 -1 1 -1 -1 -1 -1 -1 7 1 1 -1 1 -1 -1 -1 1 1 7 1 1 -1 1 -1 -1 1 -1 1 7 1 1 -1 1 -1 -1 1 1 -1 7 1 1 -1 1 -1 1 -1 -1 1 7 1 1 -1 1 -1 1 -1 1 -1 7 1 1 -1 1 -1 1 1 -1 -1 7 1 1 -1 1 -1 1 1 1 1 7 1 1 -1 1 1 -1 -1 -1 1 7 1 1 -1 1 1 -1 -1 1 -1 7 1 1 -1 1 1 -1 1 -1 -1 7 1 1 -1 1 1 -1 1 1 1 7 1 1 -1 1 1 1 -1 -1 -1 7 1 1 -1 1 1 1 -1 1 1 7 1 1 -1 1 1 1 1 -1 1 7 1 1 -1 1 1 1 1 1 -1 7 1 1 1 -1 -1 -1 -1 -1 -1 7 1 1 1 -1 -1 -1 -1 1 1 7 1 1 1 -1 -1 -1 1 -1 1 7 1 1 1 -1 -1 -1 1 1 -1 7 1 1 1 -1 -1 1 -1 -1 1 7 1 1 1 -1 -1 1 -1 1 -1 7 1 1 1 -1 -1 1 1 -1 -1 7 1 1 1 -1 -1 1 1 1 1 7 1 1 1 -1 1 -1 -1 -1 1 7 1 1 1 -1 1 -1 -1 1 -1 7 1 1 1 -1 1 -1 1 -1 -1 7 1 1 1 -1 1 -1 1 1 1 7 1 1 1 -1 1 1 -1 -1 -1 7 1 1 1 -1 1 1 -1 1 1 7 1 1 1 -1 1 1 1 -1 1 7 1 1 1 -1 1 1 1 1 -1 7 1 1 1 1 -1 -1 -1 -1 1 7 1 1 1 1 -1 -1 -1 1 -1 7 1 1 1 1 -1 -1 1 -1 -1 7 1 1 1 1 -1 -1 1 1 1 7 1 1 1 1 -1 1 -1 -1 -1 7 1 1 1 1 -1 1 -1 1 1 7 1 1 1 1 -1 1 1 -1 1 7 1 1 1 1 -1 1 1 1 -1 7 1 1 1 1 1 -1 -1 -1 -1 7 1 1 1 1 1 -1 -1 1 1 7 1 1 1 1 1 -1 1 -1 1 7 1 1 1 1 1 -1 1 1 -1 7 1 1 1 1 1 1 -1 -1 1 7 1 1 1 1 1 1 -1 1 -1 7 1 1 1 1 1 1 1 -1 -1 7 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 7 1 -1 -1 -1 -1 -1 -1 -1 -1 7 -1 1 -1 -1 -1 -1 -1 -1 -1 7 -1 -1 1 -1 -1 -1 -1 -1 -1 7 -1 -1 -1 1 -1 -1 -1 -1 -1 7 -1 -1 -1 -1 1 -1 -1 -1 -1 7 -1 -1 -1 -1 -1 1 -1 -1 -1 7 -1 -1 -1 -1 -1 -1 1 -1 -1 7 -1 -1 -1 -1 -1 -1 -1 1 -1 7 -1 -1 -1 -1 -1 -1 -1 -1 1 1 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 -1 0 0 0 1 0 0 0 0 -1 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 end ~~~~~~ cddlib reverse vertex computation V-representation begin 256 10 real 1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 1 -1 1 -1 -1 -1 -1 1 -1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 1 -1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 -1 1 -1 1 1 1 1 -1 1 -1 -1 1 -1 1 1 1 1 -1 -1 1 -1 1 -1 1 1 1 1 -1 -1 -1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 1 -1 1 1 1 -1 1 -1 -1 1 1 -1 1 1 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 -1 -1 1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 1 1 1 1 -1 1 -1 -1 1 -1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 -1 1 1 -1 -1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 -1 -1 1 -1 1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 -1 -1 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 -1 1 1 -1 1 -1 1 1 1 1 -1 -1 1 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 -1 1 1 -1 1 1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 1 -1 1 -1 -1 1 1 1 -1 1 1 -1 1 1 -1 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 1 1 1 1 1 -1 1 -1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 1 -1 1 1 -1 -1 -1 -1 1 -1 1 1 -1 1 -1 -1 -1 1 1 1 1 -1 1 1 -1 -1 -1 1 1 1 1 1 -1 1 -1 -1 1 -1 1 1 1 -1 1 1 -1 -1 1 -1 1 1 1 1 -1 1 -1 -1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 1 1 -1 1 1 -1 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 1 -1 1 -1 1 1 -1 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 1 -1 1 -1 1 1 -1 1 -1 1 1 1 1 1 1 -1 1 1 -1 1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 1 1 1 -1 1 1 1 -1 -1 -1 1 1 1 1 -1 1 1 -1 -1 1 1 -1 1 -1 1 1 1 -1 -1 1 1 -1 1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 1 1 -1 1 1 -1 1 1 1 1 1 -1 1 1 1 -1 1 1 1 1 1 1 -1 1 1 1 -1 -1 1 -1 1 -1 1 1 1 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 1 1 1 1 -1 1 1 1 1 -1 1 1 1 1 1 -1 1 1 1 1 -1 1 1 1 -1 1 1 1 1 1 -1 1 1 1 1 -1 1 1 1 1 1 1 -1 1 -1 1 1 1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 1 -1 -1 -1 1 1 1 -1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 -1 1 1 -1 1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 -1 1 -1 -1 1 1 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 1 1 1 1 1 1 -1 1 1 -1 -1 1 -1 1 1 1 -1 1 1 -1 1 1 1 1 1 1 -1 1 1 1 -1 1 1 1 1 1 -1 1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 1 1 -1 1 1 1 1 -1 -1 1 -1 1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 1 -1 -1 1 -1 1 1 1 1 -1 1 -1 1 1 1 1 1 1 1 -1 1 1 -1 1 1 1 1 1 1 -1 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 1 -1 1 1 1 1 1 1 1 -1 1 1 1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 1 1 -1 1 1 1 1 1 1 1 -1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 1 1 -1 1 1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 1 1 1 -1 1 1 1 1 1 -1 1 1 -1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 -1 1 1 1 1 -1 1 1 1 1 -1 1 1 1 -1 -1 -1 1 1 1 -1 1 1 -1 1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 1 1 1 1 -1 1 -1 -1 1 -1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 1 -1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 -1 -1 -1 1 1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 1 1 1 1 1 1 -1 1 1 1 -1 1 1 1 1 1 -1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 1 -1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 -1 1 -1 -1 1 1 -1 1 1 1 -1 1 -1 -1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 -1 1 1 1 -1 -1 -1 1 1 -1 1 1 1 -1 1 1 -1 -1 1 1 -1 1 1 -1 1 1 -1 -1 1 -1 1 1 1 -1 1 -1 -1 1 1 1 -1 1 1 -1 1 -1 -1 1 1 -1 1 1 1 -1 -1 1 -1 1 1 1 -1 1 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1 -1 -1 1 -1 1 1 -1 1 1 1 -1 -1 1 1 -1 1 -1 1 1 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 1 -1 1 1 -1 1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 1 1 -1 1 -1 -1 1 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 -1 1 -1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 1 -1 1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1 1 1 -1 1 -1 1 1 -1 1 -1 1 1 1 -1 -1 1 1 -1 1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 -1 1 1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 1 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 -1 -1 -1 1 1 1 1 -1 1 1 -1 -1 1 -1 1 1 1 -1 1 1 -1 1 -1 -1 1 1 1 -1 1 1 1 -1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 -1 -1 1 -1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1 -1 1 1 -1 -1 -1 1 -1 1 -1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 -1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 -1 1 -1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 1 -1 -1 -1 -1 -1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 end \end{lstlisting} } \subsubsection{Hull calculation for the contexual inequalities corresponding to the pentagon logic} { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * (10 expectations on atoms A1...A10: * not enumerated) * 5 3th order expectations A1A2A3 A3A4A5 ... A9A10A1 * obtained through reverse Hull computation V-representation begin 32 6 real 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 1 -1 1 1 -1 -1 1 1 1 1 -1 1 -1 -1 1 1 -1 1 -1 1 1 1 -1 1 1 -1 1 1 -1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 1 1 -1 1 1 1 1 1 -1 1 -1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 1 -1 1 1 1 -1 1 -1 1 1 -1 1 1 -1 1 1 -1 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 end ~~~~~~ cddlib response H-representation begin 10 6 real 1 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 -1 1 0 0 0 -1 0 1 0 0 -1 0 0 1 0 -1 0 0 0 1 -1 0 0 0 0 end \end{lstlisting} } \subsubsection{Hull calculation for the contexual inequalities corresponding to Specker bug logics} { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] * (13 expectations on atoms A1...A13: * not enumerated) * 7 3th order expectations A1A2A3 A3A4A5 ... A11A12A1 A4A13A10 * obtained through reverse Hull computation V-representation begin 128 8 real 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1 1 1 1 1 1 -1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 1 1 1 -1 -1 1 1 1 -1 1 1 -1 -1 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 1 -1 1 -1 -1 -1 1 1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 1 -1 1 1 -1 1 -1 1 1 1 1 1 -1 1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 1 1 -1 1 1 -1 1 -1 1 1 -1 1 1 -1 1 1 1 1 -1 1 1 1 -1 -1 1 1 -1 1 1 1 -1 1 1 1 -1 1 1 1 1 -1 1 1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 1 -1 -1 1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 1 -1 -1 1 1 1 1 -1 1 -1 1 1 1 1 1 -1 1 1 -1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 1 1 1 1 -1 1 1 1 1 1 1 1 -1 1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 1 1 1 -1 1 1 1 1 1 1 1 -1 1 1 1 -1 1 1 1 -1 1 1 -1 1 1 1 1 -1 1 1 -1 -1 1 1 1 -1 1 -1 1 1 1 1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 1 1 1 -1 -1 1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 1 1 1 1 1 1 -1 1 1 1 1 1 -1 1 -1 1 1 1 1 -1 1 1 -1 1 1 1 1 -1 -1 1 -1 1 1 1 -1 1 1 1 -1 1 1 1 -1 1 -1 1 -1 1 1 1 -1 -1 1 1 -1 1 1 1 -1 -1 -1 1 -1 1 1 -1 1 1 1 1 -1 1 1 -1 1 1 -1 1 -1 1 1 -1 1 -1 1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 1 1 -1 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 1 1 1 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 1 1 -1 1 1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 -1 1 -1 1 -1 -1 1 1 1 1 -1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 1 -1 -1 -1 -1 -1 1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 1 -1 -1 1 1 1 -1 1 1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 1 -1 1 1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 -1 -1 1 -1 1 1 1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 1 -1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 -1 1 -1 1 -1 -1 -1 1 -1 -1 1 1 -1 -1 -1 1 -1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 -1 1 -1 -1 -1 -1 1 -1 1 1 -1 -1 -1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 end ~~~~~~ cddlib response H-representation begin 14 8 real 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 0 -1 0 1 0 0 0 0 -1 0 0 1 0 0 0 -1 0 0 0 1 0 0 -1 0 0 0 0 1 0 -1 0 0 0 0 0 1 -1 0 0 0 0 0 0 end \end{lstlisting} } \subsubsection{Min-max calculation for the quantum bounds of two-two-state particles} { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] (* ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ *) (* ~~~~~~~~~~~~~~~~~~Start Mathematica Code~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ *) (* ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ *) (* old stuff < "Spherical", "Mapping", {1,0,0 }] ; A2 = CoordinateTransformData[ "Cartesian" -> "Spherical", "Mapping", {0,1,0 }] ; A3 = (* CoordinateTransformData[ "Cartesian" -> "Spherical", "Mapping", {0,0,1 }] *) {1,0,Pi/2} ; A4 = CoordinateTransformData[ "Cartesian" -> "Spherical", "Mapping", {1,-1,0 }] ; A5 = CoordinateTransformData[ "Cartesian" -> "Spherical", "Mapping", {1,1,0 }] ; A6 = CoordinateTransformData[ "Cartesian" -> "Spherical", "Mapping", {1,-1,2 }] ; A7 = CoordinateTransformData[ "Cartesian" -> "Spherical", "Mapping", {-1,1,1 }] ; A8 = CoordinateTransformData[ "Cartesian" -> "Spherical", "Mapping", {2,1,1 }] ; A9 = CoordinateTransformData[ "Cartesian" -> "Spherical", "Mapping", {0,1,-1 }] ; A10 = CoordinateTransformData[ "Cartesian" -> "Spherical", "Mapping", {0,1,1 }] ; FullSimplify[ Eigenvalues[ FullSimplify[ T[ A1[[2]], A3[[2]], A5[[2]], A7[[2]], A9[[2]] , A1[[3]], A3[[3]], A5[[3]], A7[[3]], A9[[3]]]]]] {A1, A2 , A3 , A4 , A5 , A6 , A7 , A8 , A9 , A10} //TexForm \end{lstlisting} } \subsubsection{Min-max calculation for two four-state particles} { \begin{lstlisting}[backgroundcolor=\color{yellow!10},framerule=0pt,breaklines=true, frame=tb] (* ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ *) (* ~~~~~~~~~~~~~~~~~~Start Mathematica Code~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ *) (* ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ *) (* old stuff <