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Several principled measures of contextuality have been proposed for general systems of random variables (i.e. inconsistentlly connected systems). The first of such measures was based on quasi-couplings using negative probabilities (here denoted by CNT3, Dzhafarov & Kujala, 2016). Dzhafarov and Kujala (2019) introduced a measure of contextuality, CNT2, that naturally generalizes to a measure of non-contextuality. Dzhafarov and Kujala (2019) additionally conjectured that in the class of cyclic systems these two measures are proportional. Here we prove that that conjecture is correct. Recently, Cervantes (2023) showed the proportionality of CNT2 and the Contextual Fraction measure (CNTF) introduced by Abramsky, Barbosa, and Mansfeld (2017). The present proof completes the description of the interrelations of all contextuality measures as they pertain to cyclic systems., Comment: 11 pages, 1 figure. arXiv admin note: text overlap with arXiv:2110.07113

We consider four measures of contextuality, chosen for being based on the fundamental properties of the notion of contextuality, and for being applicable to arbitrary systems of measurements, both without and with disturbance. We have previously shown that no two of them are functions of each other: as systems of measurements change, either of them can change, while the other remains constant. This means that they measure different aspects of contextuality, and we proposed that rather than picking just one measure of contextuality in one specific sense, one could use all of them to characterize a contextual system by its pattern of contextuality. To study patterns of contextuality, however, one needs a systematic way of varying systems of measurements, which requires their convenient parametrization. We have convenient parametrization within the class of cyclic systems that have played a dominant role in the foundations of quantum mechanics. However, they cannot be used to study patterns of contextuality, because within this class the four measures of contextuality have been shown to be proportional to each other. In this concept paper, we introduce hypercyclic systems of measurements. They generalize cyclic systems while preserving convenient parametrization. We show that within this class of systems, the same as for systems at large, no two of the measures of contextuality are functions of each other. This means that hypercyclic systems can be used to study patterns of contextuality., Comment: to be published in European Physics Journal - Special Topics. 5 pp, 1 figure; v3 has a few very minor changes in the text and corrected list of references

Contextuality (or lack thereof) is a property of systems of random variables. Among the measures of the degree of contextuality, two have played important roles. One of them, Contextual Fraction ($\text{CNTF}$) was proposed within the framework of the sheaf-theoretic approach to contextuality, and extended to arbitrary systems in the Contextuality-by-Default approach. The other, denoted $\text{CNT}_{2}$, was proposed as one of the measures within the Contextuality-by-Default approach. In this note, I prove that $\text{CNTF}=2\text{CNT}_{2}$ within a class of systems, called cyclic, that have played a prominent role in contextuality research., Comment: v3: 21 pages. Small corrections. Additional figure. To be published in Journal of Mathematical Psychology

Language is contextual as meanings of words are dependent on their contexts. Contextuality is, concomitantly, a well-defined concept in quantum mechanics where it is considered a major resource for quantum computations. We investigate whether natural language exhibits any of the quantum mechanics' contextual features. We show that meaning combinations in ambiguous phrases can be modelled in the sheaf-theoretic framework for quantum contextuality, where they can become possibilistically contextual. Using the framework of Contextuality-by-Default (CbD), we explore the probabilistic variants of these and show that CbD-contextuality is also possible.

This paper has two purposes. One is to demonstrate contextuality analysis of systems of epistemic random variables. The other is to evaluate the performance of a new, hierarchical version of the measure of (non)contextuality introduced in earlier publications. As objects of analysis we use impossible figures of the kind created by the Penroses and Escher. We make no assumptions as to how an impossible figure is perceived, taking it instead as a fixed physical object allowing one of several deterministic descriptions. Systems of epistemic random variables are obtained by probabilistically mixing these deterministic systems. This probabilistic mixture reflects our uncertainty or lack of knowledge rather than random variability in the frequentist sense., Comment: Entropy 2020, 22(9), 981; format of the published paper differs from this preprint

Beginning with the Bell theorem, cyclic systems of dichotomous random variables have been the object of many foundational findings in quantum mechanics. Here, we ask the question: if one chooses a cyclic system "at random" (uniformly within the hyperbox of all possible systems with given marginals), what are the odds that it will be contextual? We show that the odds of contextuality rapidly tend to zero as the size of the system increases. The result is based on the Contextuality-by-Default theory, in which we do not have to assume that the systems are subject to the no-disturbance/no-signaling constraints., Comment: 7 pp, version 6 has a typo corrected, in press in European Physics Journal Special Topics

Cyclic systems of dichotomous random variables have played a prominent role in contextuality research, describing such experimental paradigms as the Klyachko-Can-Binicoglu-Shumovky, Einstein-Podolsky-Rosen-Bell, and Leggett-Garg ones in physics, as well as conjoint binary choices in human decision making. Here, we understand contextuality within the framework of the Contextuality-by-Default (CbD) theory, based on the notion of probabilistic couplings satisfying certain constraints. CbD allows us to drop the commonly made assumption that systems of random variables are consistently connected. Consistently connected systems constitute a special case in which CbD essentially reduces to the conventional understanding of contextuality. We present a theoretical analysis of the degree of contextuality in cyclic systems (if they are contextual) and the degree of noncontextuality in them (if they are not). By contrast, all previously proposed measures of contextuality are confined to consistently connected systems, and most of them cannot be extended to measures of noncontextuality. Our measures of (non)contextuality are defined by the L_{1}-distance between a point representing a cyclic system and the surface of the polytope representing all possible noncontextual cyclic systems with the same single-variable marginals. We completely characterize this polytope, as well as the polytope of all possible probabilistic couplings for cyclic systems with given single-variable marginals.[...], Comment: As pubished in Phys. Rev. A 101, 042119 (2020), but corrected in accordance with Erratum note in Physical Review A 101:069902, 2020 and Erratum note in Physical Review Review A Physical Review Review A 103:059901, 2021

Recent crowdsourcing experiments have shown that true contextuality of the kind found in quantum mechanics can also be present in human behavior. In these experiments simple human choices were aggregated over large numbers of respondents, with each respondent dealing with a single context (set of questions asked). In this paper we present experimental evidence of contextuality in individual human behavior, in a psychophysical experiment with repeated presentations of visual stimuli in randomly varying contexts (arrangements of stimuli). The analysis is based on the Contextuality-by-Default (CbD) theory whose relevant aspects are reviewed in the paper. CbD allows one to detect contextuality in the presence of direct influences, i.e., when responses to the same stimuli have different distributions in different contexts. The experiment presented is also the first one in which contextuality is demonstrated for responses that are not dichotomous, with five options to choose among. CbD requires that random variables representing such responses be dichotomized before they are subjected to contextuality analysis. A theorem says that a system consisting of all possible dichotomizations of responses has to be contextual if these responses violate a certain condition, called nominal dominance. In our experiment nominal dominance was violated in all data sets, with very high statistical reliability established by bootstrapping., Comment: version 2 is a minor revision

In quantum physics there are well-known situations when measurements of the same property in different contexts (under different conditions) have the same probability distribution, but cannot be represented by one and the same random variable. Such systems of random variables are called contextual. More generally, true contextuality is observed when different contexts force measurements of the same property (in psychology, responses to the same question) to be more dissimilar random variables than warranted by the difference of their distributions. The difference in distributions is itself a form of context-dependence, but of another nature: it is attributable to direct causal influences exerted by contexts upon the random variables. The Contextuality-by-Default (CbD) theory allows one to separate true contextuality from direct influences in the overall context-dependence. The CbD analysis of numerous previous attempts to demonstrate contextuality in human judgments shows that all context-dependence in them can be accounted for by direct influences, with no true contextuality present. However, contextual systems in human behavior can be found. In this paper we present a series of crowdsourcing experiments that exhibit true contextuality in simple decision making.}{The design of these experiments is an elaboration of one introduced in the "Snow Queen" experiment (Decision 5, 193-204, 2018), where contextuality was for the first time demonstrated unequivocally., Comment: Journal of Experimental Psychology: General 148, 1925-1937

We present unambiguous experimental evidence for (quantum-like) probabilistic contextuality in psychology. All previous attempts to find contextuality in a psychological experiment were unsuccessful because of the gross violations of marginal selectivity in behavioral data, making the traditional mathematical tests developed in quantum mechanics inapplicable. In our crowdsourcing experiment respondents were making two simple choices: of one of two characters in a story (The Snow Queen by Hans Christian Andersen), and of one of two characteristics, such as Kind and Evil, so that the character and the characteristic chosen matched the story line. The formal structure of the experiment imitated that of the Einstein-Podolsky-Rosen paradigm in the Bohm-Bell version. Marginal selectivity was violated, indicating that the two choices were directly influencing each other, but the application of a mathematical test developed in the Contextuality-by-Default theory, extending the traditional quantum-mechanical test, indicated a strong presence of contextuality proper, not reducible to direct influences., Comment: To be published in Decision. 12 pp., 6 figures, 4 tables; Version 8 is the proofread-for-pubpication version

Random variables representing measurements, broadly understood to include any responses to any inputs, form a system in which each of them is uniquely identified by its content (that which it measures) and its context (the conditions under which it is recorded). Two random variables are jointly distributed if and only if they share a context. In a canonical representation of a system, all random variables are binary, and every content-sharing pair of random variables has a unique maximal coupling (the joint distribution imposed on them so that they coincide with maximal possible probability). The system is contextual if these maximal couplings are incompatible with the joint distributions of the context-sharing random variables. We propose to represent any system of measurements in a canonical form and to consider the system contextual if and only if its canonical representation is contextual. As an illustration, we establish a criterion for contextuality of the canonical system consisting of all dichotomizations of a single pair of content-sharing categorical random variables., Comment: 12 pages of main text + 8 pages of supplementary text with proofs (version 4 is a minor revision). To be published in Philosophical Transactions of the Royal Society A

The results of behavioral experiments typically exhibit inconsistent connectedness, i.e., they violate the condition known as "no-signaling," "no-disturbance," or "marginal selectivity." This prevents one from evaluating these experiments in terms of quantum contextuality if the latter understood traditionally (as, e.g., in the Kochen-Specker theorem or Bell-type inequalities). The Contextuality-by-Default (CbD) theory separates contextuality from inconsistent connectedness. When applied to quantum physical experiments that exhibit inconsistent connectedness (due to context-dependent errors and/or signaling), the CbD computations reveal quantum contextuality in spite of this. When applied to a large body of published behavioral experiments, the CbD computations reveal no quantum contextuality: all context-dependence in these experiments is described by inconsistent connectedness alone. Until recently, however, experimental analysis of contextuality was confined to so-called cyclic systems of binary random variables. Here, we present the results of a psychophysical double-detection experiment that do not form a cyclic system: their analysis requires that we use a recent modification of CbD, one that makes the class of noncontextual systems more restricted. Nevertheless our results once again indicate that when inconsistent connectedness is taken into account, the system exhibits no contextuality. KEYWORDS: contextuality, cyclic systems, double-detection, inconsistent connectedness, psychophysics., Comment: 10 pp, contains experimental data. To be published in Journal of Mathematical Psychology

The Contextuality-by-Default (CbD) theory allows one to separate contextuality from context-dependent errors and violations of selective influences (aka "no-signaling" or "no-disturbance" principles). This makes the theory especially applicable to behavioral systems, where violations of selective influences are ubiquitous. For cyclic systems with binary random variables, CbD provides necessary and sufficient conditions for noncontextuality, and these conditions are known to be breached in certain quantum systems. We apply the theory of cyclic systems to a psychophysical double-detection experiment, in which observers were asked to determine presence or absence of a signal property in each of two simultaneously presented stimuli. The results, as in all other behavioral and social systems previous analyzed, indicate lack of contextuality. The role of context in double-detection is confined to lack of selectiveness: the distribution of responses to one of the stimuli is influenced by the state of the other stimulus., Comment: to appear in Lecture Notes in Computer Science, based on Quantum Interaction 2016 conference; version 2 is a minor revision

Dzhafarov, Zhang, and Kujala (Phil. Trans. Roy. Soc. A 374, 20150099) reviewed several behavioral data sets imitating the formal design of the quantum-mechanical contextuality experiments. The conclusion was that none of these data sets exhibited contextuality if understood in the generalized sense proposed in Dzhafarov, Kujala, and Larsson (Found. Phys. 7, 762-782, 2015), while the traditional definition of contextuality does not apply to these data because they violate the condition of consistent connectedness (also known as marginal selectivity, no-signaling condition, no-disturbance principle, etc.). In this paper we clarify the relationship between (in)consistent connectedness and (non)contextuality, as well as between the traditional and extended definitions of (non)contextuality, using as an example the Clauser-Horn-Shimony-Holt (CHSH) inequalities originally designed for detecting contextuality in entangled particles., Comment: to be published in April 2016 in vol. 374 issue 2066 of Philosophical Transactions of the Royal Society A

This paper is a brief overview of the concepts involved in measuring the degree of contextuality and detecting contextuality in systems of binary measurements of a finite number of objects. We discuss and clarify the main concepts and terminology of the theory called "contextuality-by-default," and then discuss a possible generalization of the theory from binary to arbitrary measurements., Comment: Lecture Notes in Computer Science 9535 (with the corrected list of authors) (2016)