probability distribution is a normalized measure over the sigma algebra defined by the outcomes. This measure is defined via Born’s rule, in other words, the quantum state is embedded in the measure. A Bell scenario is essentially a state preparation protocol where everything that is deterministic is embedded in the quantum state (the unitary operations leading to its preparation), followed by the measurement, which results in stochastic outcomes.

More generally, we can think of a larger system where we only measure a subsystem. This leads to quantum channels, which are described by completely positive trace preserving maps. A quantum channel, however, must be deterministic in the sequence of events, and, for instance, a measurement choice at a later time step cannot depend on the outcome of a previous measurement. We must factor in such classical and quantum memory e ects, as well as the potentially indeterminate causal order of events. The quantum comb or process matrix [21] formalism addresses these more generic requirements. Either formalism introduces a generalized Born rule, where deterministic and stochastic parts of the system clearly separate, and thus give a clear way of defining random variables. PPLs o er potential in expressing models designed in these frameworks.

Acknowledgements. This research was supported by the Asian O ce of Aerospace Research and Development (AOARD) grant: FA2386-17-1-4016.

References

1.Abramsky, S., Brandenburger, A.: The sheaf-theoretic structure of non-locality and contextuality. New J. Phys. 13, 113036 (2011)

2.Acin, A., Fritz, T., Leverrier, A., Sainz, A.: A combinatorial apporoach to nonlocality and contextuality. Commun. Math. Phys. 334, 533–628 (2015)

3.Aerts, D., Gabora, L., Sozzo, S.: Concept combination, entangled measurements, and prototype theory. Top. Cogn. Sci. 6, 129–137 (2014)

4.Asano, M., Hashimoto, T., Khrennikov, A., Ohya, M., Tanaka, Y.: Violation of contextual generalization of the Leggett–Garg inequality for recognition of ambiguous figures. Physica Scripta (T163), 014006 (2014)

5.Atmanspacher, H., Filk, T.: A proposed test of temporal nonlocality in bistable perception. J. Math. Psychol. 54, 314–321 (2010)

6.Bruza, P.D.: Syntax and operational semantics of a probabilistic programming language with scopes. J. Math. Psychol. 74, 46–57 (2016)

7.Bruza, P.D.: Modelling contextuality by probabilistic programs with hypergraph semantics. Theor. Comput. Sci. 752, 56–70 (2017)

8.Bruza, P., Fell, S.: Are decisions of image trustworthiness contextual? A pilot study. In: Lambert-Mogiliansky, A., Coecke, B. (eds.) Quantum Interaction: 11th International Conference (QI 2018). Lecture Notes in Computer Science. Springer, Heidelberg (2018)

9.Bruza, P., Kitto, K., Ramm, B., Sitbon, L.: A probabilistic framework for analysing the compositionality of conceptual combinations. J. Math. Psychol. 67, 26–38 (2015)

10.Cervantes, V., Dzhafarov, E.: Snow queen is evil and beautiful: experimental evidence for probabilistic contextuality in human choices. arXiv:1711.00418v2

62 P. D. Bruza and P. Wittek

11.Chaves, R., Kueng, R., Brask, J.B., Gross, D.: Unifying framework for relaxations of the causal assumptions in Bell’s theorem. Phys. Rev. Lett. 114(14), 140403 (2015)

12.Dzhafarov, E., Kujala, J.: Probabilistic contextuality in EPR/Bohm-type systems with signaling allowed. In: Dzhafarov, E. (ed.) Contextuality from Quantum Physics to Psychology, chap. 12, pp. 287–308. World Scientific Press (2015)

13.Dzhafarov, E., Kujala, J., Larsson, J.: Contextuality in three types of quantummechanical systems. Found. Phys. 7, 762–782 (2015)

14.Dzhafarov, E., Zhang, R., Kujala, J.: Is there contextuality in behavioral and social systems? Philos. Trans. Roy. Soc. A 374, 20150099 (2015)

15.Goodman, N.D., Stuhlm¨uller, A.: The design and implementation of probabilistic programming languages (2014). http://dippl.org. Accessed 14 Sept 2017

16.Goodman, N.D., Tenenbaum, J.B.: Probabilistic Models of Cognition (2016). http://probmods.org/v2. Accessed 5 June 2017

17.Gordon, A., Henzinger, T., Nori, A., Rajamani, S.: Probabilistic programming. In: Proceedings of the on Future of Software Engineering (FOSE 2014), pp. 167–181 (2014)

18.Gronchi, G., Strambini, E.: Quantum cognition and Bell’s inequality: a model for probabilistic judgment bias. J. Math. Psychol. 78, 65–75 (2016)

19.Henson, J., Sainz, A.: Macroscopic noncontextuality as a principle for almostquantum correlations. Phyical Rev. A 91, 042114 (2015)

20.Obeid, A., Bruza, P.D., Wittek, P.: Evaluating probabilistic programming languages for simulating quantum correlations. PLoS One 14(1), e0208555 (2019)

21.Oreshkov, O., Costa, F., Brukner, C.: Quantum correlations with no causal order. Nat. Commun. 3, 1092 (2012)

22.Sainz, A., Wolfe, E.: Multipartite composition of contextuality scenarios. arXiv:1701.05171 [quant-ph] (2017)

23.Zhang, R., Dzhafarov, E.N.: Testing contextuality in cyclic psychophysical systems of high ranks. In: de Barros, J.A., Coecke, B., Pothos, E. (eds.) QI 2016. LNCS, vol. 10106, pp. 151–162. Springer, Cham (2017). https://doi.org/10.1007/978-3- 319-52289-0 12

Episodic Source Memory over

Distribution by Quantum-Like

Dynamics – A Model Exploration

J. B. Broekaert(B) and J. R. Busemeyer

Department of Psychological and Brain Sciences, Indiana University,

Bloomington, USA

{jbbroeka,jbusemey}@indiana.edu

Abstract. In source memory studies, a decision-maker is concerned with identifying the context in which a given episodic experience occurred. A common paradigm for studying source memory is the ‘threelist’ experimental paradigm, where a subject studies three lists of words and is later asked whether a given word appeared on one or more of the studied lists. Surprisingly, the sum total of the acceptance probabilities generated by asking for the source of a word separately for each list (‘list 1?’, ‘list 2?’, ‘list 3?’) exceeds the acceptance probability generated by asking whether that word occurred on the union of the lists (‘list 1 or 2 or 3?’). The episodic memory for a given word therefore appears over distributed on the disjoint contexts of the lists. A quantum episodic memory model [QEM] was proposed by Brainerd, Wang and Reyna [8] to explain this type of result. In this paper, we apply a Hamiltonian dynamical extension of QEM for over distribution of source memory. The Hamiltonian operators are simultaneously driven by parameters for re-allocation of gist -based and verbatim-based acceptance support as subjects are exposed to the cue word in the first temporal stage, and are attenuated for description-dependence by the querying probe in the second temporal stage. Overall, the model predicts well the choice proportions in both separate list and union list queries and the over distribution e ect, suggesting that a Hamiltonian dynamics for QEM can provide a good account of the acceptance processes involved in episodic memory tasks.

Keywords: Recognition memory · Over distribution · Quantum modeling · Word list · Verbatim · Gist

1 Familiarity and Recollection, Verbatim and Gist

Recognition memory models predict judgments of ‘prior occurrence of an event’. In recognition, Mandler distinguished a familiarity process and a retrieval - or recollection - process that would evolve separately but also additively [21]. The familiarity of a memory would relate to an ‘intra event organizational integrative

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B. Coecke and A. Lambert-Mogiliansky (Eds.): QI 2018, LNCS 11690, pp. 63–75, 2019. https://doi.org/10.1007/978-3-030-35895-2_5

64 J. B. Broekaert and J. R. Busemeyer

process’, while retrieval relates to an ‘inter event elaborative process’. Extending this dual process modeling work, by Tulving [26] and Jacoby [17], a ‘conjoint recognition’ model was developed by Brainerd, Reyna and Mojardin [4] which provides separate parameters for the entangled processes of identity judgement, similarity judgment and response bias. Their model implements verbatim and gist dimensions to memories. Verbatim traces hold the detailed contextual features of a past event, while gist traces hold its semantic details. In recognition tasks we would access verbatim and gist trace in parallel. The verbatim trace of a verbal cue handles it surface content like orthography and phonology for words with its contextual features like in this case, colour of back ground and text font. The verbal cue’s gist trace will encode relational content like semantic content for words, also with its contextual features. This development recently received a quantum formalisation for its property of superposed states to cope with over distribution in memory tests [8, 9, 14]. In specifically designed expermental tests it appeared episodic memory of a given word is over distributed on the disjoint contexts of the lists, letting the acceptance probability behave as a subadditive function [6, 9].

Quantum-Like Memory Models. The Quantum Episodic Memory model (QEM) was proposed by Brainerd, Wang and Reyna [8]. It assumes a Hilbert space representation in which verbatim, gist, and non-related components are orthogonal, and in which recognition engages the gist trace in target memories as well. We will provide ample detail about this model in the next section, since our dynamical extension is implemented in essentially the same structural setting. QEM was extended to generalized-QEM (GQEM) by Trueblood and Hemmer to model for incompatible features of gist, verbatim and non-related traces [25]. Subjacent is the idea that these features are serially processed, and that gist precedes verbatim since it is processed faster. Independently, Denolf and Lambert-Mogiliansky have considered the accessing of gist and verbatim as incompatible process features. This aspect is implemented in an intrinsically quantum-like manner in their complementarity -based model for Complementary Memory Types (CMT) [15, 16, 20]. We previously developed a Hamiltonian dynamical extension of QEM for item memory tasks [11]. The dynamical formalism allows to describe time development of the acceptance decision based on gist, verbatim and non-related traces. Finally, also a semantic network approach by Bruza, Kitto, Nelson, and McEvoy [23] was developed in which the target word is adjacent to its associated terms and the network is in a quantum superposition state of either complete activation or non-activation (see also [12]).

We note that dynamical approaches to quantum-like models have been proposed previously, e.g. in decision theory by Busemeyer and Bruza [14], Pothos and Busemeyer [24], Mart´ınez-Mart´ınez [22], Kvam et al. [19], Busemeyer et al. [13] and by Yearsley and Pothos [28], in cognition by Aerts et al. [2], in perception theory by Atmanspacher and Filk [3]. An overview of quantum modelling techniques is given in Broekaert et al. [10].