On the relation of contextuality measures in cyclic systems of variables

Most research problems can be presented using systems of random variables in which each variable is identified by what it measures (what question it answers) and by their context, the conditions under which it is recorded. In a contextual system, observed joint distributions of variables recorded under different conditions cannot be placed together into a particular overall joint distribution where variables that correspond to the same property in different contexts are equal to each other as often as possible. For general systems of random variables, several principled measures of the degree of contextuality have been proposed in the contextuality literature. They are denoted as CNT1, CNT2, CNT3, and CNTF (Contextual Fraction). Each of these measures depicts a unique aspect of contextuality. CNT1 gives a degree of incompatibility, given fixed observed joint distributions, of the (maximal possible probability of) identity of variables that are response to the same question. CNT2 reverses the roles to measure the incompatibility, assuming the (maximal possible probability of) identity across contexts, as a function of varying hypothetical joint distributions of observables. CNT3 is computed replacing the probability distribution for all variables in a system by a signed-measure and minimizing the need for negative masses. Lastly, CNTF characterizes a system in terms of its relative distance between a noncontextual system and a maximally contextual system. Within the class of cyclic systems, those for which each question is answered only under two different contexts, and each context includes only two questions being asked, it has been conjectured that all the above measures coincide up to proportionality. Previous work presented at MathPsych has already proved this to be the case for some of these measures (CNT1 and CNT2). In this talk I will show that the remaining measures available (CNT3 and Contextual Fraction) are also proportional to each other and to the other measures. The present proofs complete the theory of the cyclic systems, the different measures of contextuality, and their properties. Literature: Dzhafarov, E.N., Kujala, J.V., & Cervantes, V.H. (2020). Contextuality and noncontextuality measures and generalized Bell inequalities for cyclic systems. Physical Review A 101, 042119. (available as arXiv:1907.03328.) (Erratum Notes: Physical Review A, 101, 069902 and Physical Review A, 103, 059901) Cervantes, V. H. (2023). A note on the relation between the Contextual Fraction and CNT2. Journal of Mathematical Psychology, 112, 102726. Camillo, G. & Cervantes, V. H. (in preparation). Measures of contextuality in cyclic systems and the negative probabilities measure CNT3.