02/04 - Olivier Rioul

A mathematical analysis of Fitts' law

Fitts' law is a well-known empirical regularity which predicts the average movement time T it takes people, under time pressure, to reach a target of width W located at distance D with some pointer. This model has proven useful in several fields of applied psychology such as Human-Computer Interaction and Ergonomics. Whether Fitts' law is a logarithmic law or a power law has remained unclear so far: In two widely cited papers, Meyer & al. have claimed that the power model they derived from their celebrated stochastic optimized sub-movement theory encompasses the logarithmic model as a limiting case, when the number of submovements grows large. We review the Meyer & al. sub-movement theory and show that this claim is questionable mathematically. Our analysis reveals that the traditional theory implies in fact a quasi-logarithmic, rather than quasi-power model, the two models not being equivalent. Incidentally, the original logarithmic model was derived by Fitts (1954) in analogy with Shannon's information capacity formula (1948); we conclude this talk by discussing recent advances on a sub-movement model that justifies the information theoretic approach. At any rate, awareness that the two classes of candidate mathematical descriptions of Fitts' law are not equivalent should stimulate experimental research in the field.

Joint work with Yves Guiard (CNRS LTCI, France). This work was presented at the 2012 Meeting of the European Mathematical Psychology Group (see also [1], [2]).


Olivier Rioul (PhD, HDR) is professor at Telecom ParisTech and Ecole Polytechnique, France. His research interests [3] are in applied mathematics and include various, sometimes unconventional, applications of information theory such as in Bayesian dynamic estimation, hardware security, and experimental psychology. He has been teaching information theory at various French universities for almost twenty years. He has published a textbook [4] which has become a classical French reference in the field. A Brazilian edition of the book is scheduled to be published by Editora da Unicamp in the future.


[1] O. Rioul and Y. Guiard, "Power vs. logarithmic model of Fitts' law: A mathematical analysis," Math. Sci. hum. / Mathematics and Social Sciences, No. 199, 2012(3), p. 85-96.

[2] O. Rioul and Y. Guiard, "The power model of Fitts' law does not encompass the logarithmic model," Electronic Notes in Discrete Mathematics, Elsevier, Vol. 42, June 2013, pp. 65-72.

[4] O. Rioul, Théorie de l'Information et du Codage, Hermes Science, 2007.


26/03 - Dante Cardoso Pinto de Almeida

Dedução Natural Rotulada para Lógicas Modais e Multimodais

Dedução Natural é um sistema de prova desenvolvido independentemente por Gentzen e Jaśkowski. Caracteriza-se por conter diversas regras de inferência, em geral duas para cada operador (uma para introduzi-lo e outra para eliminá-lo) em contraste com a presença de pouquíssimos ou nenhum axioma; além do mais, caracteriza-se por conter regras de inferência hipotéticas. Assim, o sistema de Dedução Natural passou a ser reconhecido como o que mais se aproxima da forma como se raciocina em matemática.

Contudo, a despeito deste sistema de prova ser aplicado com sucesso a diversos sistemas de lógica (ex: clássica, intuicionista, minimal, algumas lógicas relevantes, alguns poucos sistemas modais etc), tem-se encontrado diversas dificuldades para aplicá-lo em diversos outros sistemas. Recentemente estas dificuldades vem sendo contornadas por autores como Simpson e Gabbay, com a adoção de sistemas de provas rotulados. Rótulos (ou etiquetas) são marcações atribuídas às formulas em um sistema de provas, geralmente expressando propriedades semânticas dessas.

Neste seminário, mostrarei como a dedução natural rotulada funciona para diversos sistemas de lógica modal e multimodal, discutirei o que ainda há para ser estudado na área e como esse sistema pode ser utilizado para formalizar raciocínios e argumentos envolvendo termos modais.


12/03 - Rodolfo C. Ertola Biraben

Modal aspects in bi-intuitionistic logic

In the context of a Heyting algebra extended with the dual ─ of the relative pseudocomplement →, we study ¬D and D¬, where ¬ stands for usual intuitionistic negation and D stands for co-negation. Operation D may be defined, for any x, as (xx) ─ x. Operations ¬D and D¬ behave very much like necessity and possibility, respectively. The result has all the modal properties of modal system B. We prove facts considering [3].

In 1942 Moisil considered DD and ¬¬ for necessity and possibility, respectively (see [2]). Now, for DD modal axiom K is not available.

We consider the extension with the corresponding S4 axiom and prove that it is conservative. In both cases we study the modalities.

We also compare the given framework with the extension with the connective Δ in Fuzzy Logic (see e.g. [1]).

We mostly present our results from the algebraic point of view. However, we also include some logic considerations. Adding axioms to intuitionistic logic is not enough, i.e. it is necessary to add a rule. Also, two different logics appear, depending, from a semantical point of view, on whether a truth-preserving or a truth-degree-preserving consequence is chosen.


[1] Hajek, P. (1998), Metamathematics of Fuzzy Logic, Kluwer.

[2] Moisil, G. C. (1972), Essai sur les logiques non chrysippiennes, Éditions de l'academie de la république socialiste de Roumanie.

[3] Simpson, A. K. (1994), The proof theory and semantics of intuitionistic modal logic, PhD thesis, University of Edinburgh.