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Mathematics
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Mathematical and Computational Models of Cognition
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Mathematical and Computational Models of Cognition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: 1 December 2024 | Viewed by 8499

Special Issue Editor

Prof. Dr. Ronaldo Vigo

E-Mail Website
Guest Editor
Consortium for the Advancement of Cognitive Science, Psychology Department, College of Arts and Sciences, Ohio University, 45701 Athens, OH, USA
Interests: mathematical modeling; applied mathematics; cognitive science; computational modeling; psychophysics; mathematical psychology; artificial intelligence

Special Issue Information

Dear Colleagues,

One of the ultimate goals of Cognitive Science is to discover the mathematical laws and computational processes that govern human behavior and the human mind, and to achieve this with the systematicity and rigor found in the physical sciences. Mathematical and computational modeling are key tools in accomplishing this lofty goal. Indeed, the development of such models is crucial for rigorous theory development, measurement, and testing in Cognitive Science and Psychology. Fortunately, with the advancement of computing technologies and an unprecedented increase in computing resources, there has never been a more fertile period in human history for the successful formulation, application and testing of mathematical and computational models of human cognitive phenomena and related processes. This Special Issue has two aims. The first is to assemble papers that propose, apply, and/or test mathematical and computational models of any of the following cognitive capacities: perception, similarity assessment, attention, memory, concept learning, categorization, language, problem solving, reasoning, and decision-making. The second aim is to inform and motivate mathematicians from many fields of mathematics to engage in cognitive modeling. From so doing, new mathematical approaches to long-standing problems may emerge and more accurate and tenable models may be discovered. Contributions may involve any style of mathematical and computational modeling, whether deterministic or probabilistic, providing that the approach is accompanied by a plausible cognitive mechanism and adequate theory development. However, given the prevalence of probabilistic models in the field, deterministic models are specially welcomed. In addition, contributions may focus on methodological (or metamodeling theory) techniques in the form of new model-testing methods or programming tools designed to facilitate model construction and/or testing if these are grounded on mathematical theory.

Prof. Dr. Ronaldo Vigo
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Mathematical cognitive science
  • Cognitive modeling
  • Mathematical modeling
  • Cognitive science
  • Computational modeling
  • Metamodeling
  • Psychophysics
  • Mathematical psychology
  • Cognition
  • Perception
  • Concept learning
  • Categorization
  • Memory
  • Reasoning
  • Problem solving
  • Similarity assessment
  • Decision making
  • Attention

Published Papers (5 papers)

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15 pages, 2971 KiB  
Article
A Context-Sensitive Alternative to Hick’s Law of Choice Reaction Times: A Mathematical and Computational Unification of Conceptual Complexity and Choice Behavior
by Ronaldo Vigo, Charles A. Doan, Jay Wimsatt and Cody B. Ross
Mathematics 2023, 11(11), 2422; https://doi.org/10.3390/math11112422 - 23 May 2023
Viewed by 1096
Abstract
Hick’s law describes the time that individuals take to make a preference decision when presented with a set of possible choices. Basically speaking, the law states that decision time is a logarithmic function of the number of choices when the choices are equiprobable. [...] Read more.
Hick’s law describes the time that individuals take to make a preference decision when presented with a set of possible choices. Basically speaking, the law states that decision time is a logarithmic function of the number of choices when the choices are equiprobable. However, the evidence examined here suggests that this, and a variant of the law for non-equiprobable choices based on Shannon entropy, are not effective at predicting decision reaction times involving structured sets of alternatives. The purpose of this report is to communicate a theoretical alternative to Hick’s law that is derived from a mathematical law of invariance for conceptual behavior at the heart of Generalized Invariance Structure Theory (Vigo, 2013, 2015). We argue that such an alternative accounts more precisely for decision reaction times on structured sets. Furthermore, we argue that Hick’s law is a special case of this more general law of choice reaction times for categories with zero degree of invariance. Full article
(This article belongs to the Special Issue Mathematical and Computational Models of Cognition)
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32 pages, 1123 KiB  
Article
Varieties of Selective Influence: Toward a More Complete Taxonomy and Implications for Systems Identification
by James T. Townsend and Yanjun Liu
Mathematics 2022, 10(7), 1059; https://doi.org/10.3390/math10071059 - 25 Mar 2022
Viewed by 1523
Abstract
All science, including psychological science, is subject to what Townsend and Ashby have called the principle of correspondent change which ensures that experimental manipulations act as informed agents with respect to predictions and testing critical theoretical features. Mostly, this type of program goes [...] Read more.
All science, including psychological science, is subject to what Townsend and Ashby have called the principle of correspondent change which ensures that experimental manipulations act as informed agents with respect to predictions and testing critical theoretical features. Mostly, this type of program goes unspoken. Within the general field known as the information processing approach, S. Sternberg invented the additive factors method in which the aforesaid feature plays a major and explicit role. We call this approach a theory driven methodology because the scientist formulates a set of theories or models and then formulates experimental variables that will permit strong tests among the hypothetical alternatives. Our term for the general approach is systems factorial technology. Often, these tests can be accomplished with qualitative, non-parametric, distribution free methods, but our so-called sieve method advocates, once the initial qualitative steps are accomplished, a move to assessing more detail parametric versions of the model classes. Over the decades, the meta-theory underpinning SFT and like approaches has evidenced dramatic growth in both expanse and depth. Particularly, the critical assumption of selective influence, testable to some extent, has received extensive and sophisticated treatment. The various central allied concepts are interlinked but do not form a simple linearly-ordered chain. This study carries on exploration of the central concepts and relationships and their implications for psychological research. Full article
(This article belongs to the Special Issue Mathematical and Computational Models of Cognition)
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28 pages, 370 KiB  
Article
Rational Degree of Belief Ceases to Be Additive When the Dutch Book Argument Is Transported to a New Environment
by Donald Bamber
Mathematics 2022, 10(7), 1017; https://doi.org/10.3390/math10071017 - 22 Mar 2022
Viewed by 1019
Abstract
The strength of a person’s beliefs can be measured by the buying and selling prices they offer on contingent promissory notes. Consider a promissory note contingent on a proposition; it pays off one unit of money if the proposition is true and nothing [...] Read more.
The strength of a person’s beliefs can be measured by the buying and selling prices they offer on contingent promissory notes. Consider a promissory note contingent on a proposition; it pays off one unit of money if the proposition is true and nothing otherwise. The more strongly a person believes the proposition, the higher the minimum price would be at which they would sell it. The same would apply to the maximum purchase price. The well-known Dutch Book Argument claims that, if the person’s beliefs are rational, their buying/selling prices should combine additively, meaning that the price of a promissory note contingent on the disjunction of two incompatible propositions should be the sum of the prices of the promissory notes contingent on the individual incompatible propositions. This paper shows that the essence of the Dutch Book Argument is that rational belief is additive because money is additive. It is proved that, if the structure of the Dutch Book Argument is kept, but a nonadditive resource is substituted for money, then rational belief will follow a nonadditive combining rule. It is also shown how rational buying/selling prices behave when the pay-off amount of a contingent note changes. Full article
(This article belongs to the Special Issue Mathematical and Computational Models of Cognition)
20 pages, 1352 KiB  
Article
Tree Inference: Response Time and Other Measures in a Binary Multinomial Processing Tree, Representation and Uniqueness of Parameters
by Richard Schweickert and Xiaofang Zheng
Mathematics 2022, 10(2), 267; https://doi.org/10.3390/math10020267 - 16 Jan 2022
Viewed by 1143
Abstract
A Multinomial Processing Tree (MPT) is a directed tree with a probability associated with each arc and partitioned terminal vertices. We consider an additional parameter for each arc, a measure such as time. Each vertex represents a process. An arc descending from a [...] Read more.
A Multinomial Processing Tree (MPT) is a directed tree with a probability associated with each arc and partitioned terminal vertices. We consider an additional parameter for each arc, a measure such as time. Each vertex represents a process. An arc descending from a vertex represents selection of a process outcome. A source vertex represents processing beginning with stimulus presentation and a terminal vertex represents a response. An experimental factor selectively influences a vertex if changing the factor level changes parameter values on arcs descending from that vertex and no others. Earlier work shows that if each of two factors selectively influences a different vertex in an arbitrary MPT it is equivalent to one of two simple MPTs. Which applies depends on whether the two selectively influenced vertices are ordered by the factors or not. A special case, the Standard Binary Tree for Ordered Processes, arises if the vertices are ordered and the factor selectively influencing the first vertex changes parameter values on only two arcs. We derive necessary and sufficient conditions, testable by bootstrapping, for this case. Parameter values are not unique. We give admissible transformations for them. We calculate degrees of freedom needed for goodness of fit tests. Full article
(This article belongs to the Special Issue Mathematical and Computational Models of Cognition)
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29 pages, 5410 KiB  
Article
Sensitivity to Context in Human Interactions
by Oliver Waddup, Pawel Blasiak, James M. Yearsley, Bartosz W. Wojciechowski and Emmanuel M. Pothos
Mathematics 2021, 9(21), 2784; https://doi.org/10.3390/math9212784 - 02 Nov 2021
Cited by 5 | Viewed by 2014
Abstract
Considering two agents responding to two (binary) questions each, we define sensitivity to context as a state of affairs such that responses to a question depend on the other agent’s questions, with the implication that it is not possible to represent the corresponding [...] Read more.
Considering two agents responding to two (binary) questions each, we define sensitivity to context as a state of affairs such that responses to a question depend on the other agent’s questions, with the implication that it is not possible to represent the corresponding probabilities with a four-way probability distribution. We report two experiments with a variant of a prisoner’s dilemma task (but without a Nash equilibrium), which examine the sensitivity of participants to context. The empirical results indicate sensitivity to context and add to the body of evidence that prisoner’s dilemma tasks can be constructed so that behavior appears inconsistent with baseline classical probability theory (and the assumption that decisions are described by random variables revealing pre-existing values). We fitted two closely matched models to the results, a classical one and a quantum one, and observed superior fits for the latter. Thus, in this case, sensitivity to context goes hand in hand with (epiphenomenal) entanglement, the key characteristic of the quantum model. Full article
(This article belongs to the Special Issue Mathematical and Computational Models of Cognition)
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