Tables of squares

There have been a number of tables of squares and/or cubes, and the first large table of squares was that of Ludolf published in 1690 [START_REF] Ludolf | Tetragonometria tabularia[END_REF]. Ludolf's table gave all squares from 1 to 100000. Kulik's table, published in 1848 [11], has exactly the same range, but is much more compact, and also gives the cubes.

Prior to Ludolf, tables of squares usually only went up to 10000. For instance, Guldin's table published in 1635 [START_REF] Guldin | De centro gravitatis[END_REF] gave the squares and cubes until 9999. Pell also published a table of the first 10000 squares in 1672 [START_REF] Pell | Tabula numerorum quadratorum decies millium, unà cum ipsorum lateribus ab unitate incipientibus & ordine naturali usque ad 10000 progredientibus[END_REF].

2 Schiereck (ca. 1790-1842)

Joseph Friedrich Schiereck (born ca. 1790-1842) studied mathematics in Goettingen and became a surveyor and mathematics teacher. In the 1820s, he was involved in the surveying of the Rhine province. Schiereck published several books [START_REF] Friedrich | Polygonometrie, oder ausführliche Anweisung zur Berechnung aller aus dem Umfange gemessener Figuren[END_REF][START_REF] Friedrich | Errathende Rechenkunst zur angenehmen Unterhaltung gesellschaftlicher Zirkel etc[END_REF] and in particular a Handbuch für Geometer [START_REF] Friedrich | Handbuch für Geometer etc[END_REF] (1827) which contained a table of squares from 0 to 9999, also published separately [START_REF] Friedrich | Tafeln aller Quadrate von 1 bis 10000, nebst Anweisung daraus die Quadrate und die Wurzeln aller Zahlen bis 100000000 zu bestimmen, und einer Anwendung derselben zur leichtern und richtigern Berechnung des Holzes[END_REF].

Later, Schiereck became interested in applications of quarter-squares. In 1829, he invented a "calculating machine" containing a table of quarter-squares on a rotating disc [START_REF] Weiss | [END_REF][START_REF] Haeberlin | Die Viertelquadratemethode und das Pediometer des Joseph Friedrich Schiereck[END_REF] and for which he may have obtained a certificate by Gauss [START_REF] Friedrich | Eine in Deutschland erfundene Rechenmaschine[END_REF][START_REF] Friedrich | [END_REF]. Then in 1841, he invented a machine for the computation of the area of a triangle, also based on quarter-squares [START_REF] Friedrich | Beschreibung des Pediometers, eines Instruments, um den Flächeninhalt in Karten ohne Rechnung zu erhalten[END_REF][START_REF] Haeberlin | Die Viertelquadratemethode und das Pediometer des Joseph Friedrich Schiereck[END_REF].

Schiereck's table (1827) and its reconstruction

Schiereck's table of squares [START_REF] Friedrich | Handbuch für Geometer etc[END_REF][START_REF] Friedrich | Tafeln aller Quadrate von 1 bis 10000, nebst Anweisung daraus die Quadrate und die Wurzeln aller Zahlen bis 100000000 zu bestimmen, und einer Anwendung derselben zur leichtern und richtigern Berechnung des Holzes[END_REF] contains exactly 20 pages and gives the squares of all integers from 0 to 9999. Each page contains 500 squares, and the square of n = 10a + b (with b < 10) is obtained by looking up the value at the intersection of line a and column b. The cells contain at most four digits for n < 317, five digits for 317 ≤ n ≤ 2999 and six digits beyond. When n 2 is larger than the maximal number of digits in the cell (that is, when n > 99), a prefix is indicated in the first column. For instance, 103 2 = 10609 and this value is split in 1|0609. When the prefix of the first value of a line is the same as that of the previous line, it is not repeated. Moreover, when the prefix changes within the line, that is when the prefix of the second, third, etc. cell differs from that of the first cell (for instance, 1140 2 = 12|99600 but 1141 2 = 13|01881), a dash is added in front of the cell, and also in front of all the remaining cells of the line.

In Schiereck's original table, there are a few anomalies in the prefixes, namely that the prefix (the first column after that of N) is sometimes given even when it does not change. We have not reproduced these anomalies. The ranges at the top of the pages are also somewhat incorrect in general, as the second value is never attained. The second page, for instance, gives the range from 500 to 1000, but the table only goes to 999. Since this occurs on every page, we have kept these off-by-one values. The first range, however, should have been 0-500, and not 1-500 as given by Schiereck. We have decided to correct this range. Figure 1: An excerpt of Schiereck's table [START_REF] Friedrich | Tafeln aller Quadrate von 1 bis 10000, nebst Anweisung daraus die Quadrate und die Wurzeln aller Zahlen bis 100000000 zu bestimmen, und einer Anwendung derselben zur leichtern und richtigern Berechnung des Holzes[END_REF].

Note on the titles of the works: Original titles come with many idiosyncrasies and features (line splitting, size, fonts, etc.) which can often not be reproduced in a list of references. It has therefore seemed pointless to capitalize works according to conventions which not only have no relation with the original work, but also do not restore the title entirely. In the following list of references, most title words (except in German) will therefore be left uncapitalized. The names of the authors have also been homogenized and initials expanded, as much as possible.The reader should keep in mind that this list is not meant as a facsimile of the original works. The original style information could no doubt have been added as a note, but we have not done it here.


The method of quarter-squares

The method of quarter-squares is a multiplication method which makes use of the identity

ab = 1 4 (a + b) 2 -(a -b) 2 .
If we possess a table of squares, and wish to compute the product ab, it is then sufficient to compute a + b, a -b, to look up the table for these two values, to subtract the values read in the table, and to divide the result by 4. This seems complex, but for large numbers it is more efficient than to compute directly the product. 1There have been a number of tables of squares around, in particular those of Ludolf published in 1690 [30], and those published by Séguin in 1801 and giving the squares up to 10000. But these authors did mostly not have multiplications in mind, although Ludolf showed how squares could be used to compute a multiplication. At the beginning of the 19th century, the method of quarter-squares was mentioned in passing by Laplace in 1809 [9, p. 261] and Gergonne in 1816 [12, pp. 159-160], but they did not produce tables.

2 The first tables of quarter-squares (1817)

In 1817, Voisin [START_REF] Voisin | Tables de multiplication, ou, logarithmes des nombres entiers depuis 1 jusqu'à 20,000[END_REF] and Bürger [START_REF] Anton | Tafeln zur Erleichterung in Rechnungen für den allgemeinen Gebrauch eingerichtet[END_REF] published independently the first tables of quartersquares. Voisin and Bürger both understood that a table of squares could be useful for multiplications, but they went one step beyond, and removed the need to divide the difference of the squares by 4. Their tables gave the quarter-squares for all integers up to 19999 (Bürger) (figure 3) and 20000 (Voisin) (figures 1 and 2).

More exactly, when a number is odd, its quarter-square is of the form n + 0.25 and Voisin and Bürger only gave the value n. So if a and b are both odd or both even, the values of the quarter-squares are given exactly. If one of the integers is odd and the other is even, both quarter-squares are in default by 0.25, but the difference of these two values is correct. So, although the tables of Voisin and Bürger are not totally accurate for a table of squares, they serve the purpose of a table of multiplication. If one wished to use them as a table of squares, one would need to remember to add 1 to the result multiplied by 4, whenever the value entered is odd.

A table of quarter-squares is also very advantageous with respect to space. Such a table is a linear table, with only one entry. With slightly more work than a mere table of multiplication, it is possible with a table of quarter-squares to do computations that would require many thousands of pages with a conventional table of multiplication, such as Crelle's table [START_REF] Crelle | Rechentafeln, welche alles Multipliciren und Dividiren mit Zahlen unter Tausend ganz ersparen, bei grösseren Zahlen aber die Rechnung erleichtern und sicherer machen[END_REF].

In 1820, in the second edition of his Philosophy of arithmetic [29, pp. 246-257], John Leslie described tables of quarter-squares, and in particular Voisin's table, of which he gave an excerpt up to 2000. Leslie expressed the hope that someone would compute a table up to 200000.

Antoine Voisin

Very little is known of Voisin. He was "commissaire-priseur" (auctioneer) and his name appears in particular in the Almanach impérial for the 13th year of the Repubblic (1804-1805) [55, p. 823], the Almanach de la cour et de l'Empire français for 1813 [47, p. 171], and the Almanach royal for 1830 [1, p. 813]. His residence was 15 rue Guénégaud in Paris.

Voisin's table is very similar to Bürger's table, both in appearance and in extent. Bürger wrote that he borrowed the layout from Vega's table [START_REF] Vega | Logarithmische, trigonometrische, und andere zum Gebrauche der Mathematik eingerichtete Tafeln und Formeln[END_REF], and perhaps this is also what Voisin did, although he didn't say so.

Voisin called his quarter-squares "logarithms," although they were not logarithms.2 It is not clear why he decided to name them like that, and no explanation is given in the author's introduction.

Voisin's table is nowadays extremely rare. In 1873, Glaisher wrote that he had not seen any copy, but that there was supposedly one in the Graves library [13, p. 23]. Eventually, Glaisher gave a more detailed description of Voisin's table in 1889 [START_REF] Whitbread | The method of quarter squares[END_REF], and we assume that he then had seen it.

Later tables of quarter-squares

A number of other tables of quarter-squares were published during the 19th century (table 1). Centnerschwer [5] published a table of quarter-squares up to 20000 in 1825, Kulik in 1833 and 1851 (up to 30000) [START_REF] Philipp | Toasirtafeln, zur leichtern Berechnung des Längen-Flächenund Kubik-Inhaltes und der verschiedenen Münz-Maß-und Gewichts-Beträge[END_REF][START_REF] Philipp | Neue Multiplikationstafeln : ein unentbehrliches Hülfsmittel für Jedermann, um schnell, sicher und ohne Ermüdung zu rechnen[END_REF], Merpaut in 1832 (up to 40000) [33], Laundy in 1856 [27], and Blater in 1887 [2]. Laundy's table gave the quarter-squares up to 100000, whereas Blater extended it to 200000. In 1836, Galbraith published a small table of quarter-squares in a more general collection of tables [START_REF] Galbraith | New and concise general tables for computing the obliquity of the ecliptic[END_REF]. This table was probably borrowed from Voisin's table.

In the preface of his table of quarter-squares [27, p. iv], Laundy also mentions that Peter Gray had in his possession a manuscript of a table of quarter-squares extending to 200000 by Shortrede, but that table was never published.

There have also been several applications of quarter-squares for mechanical aids. In 1829, Schiereck invented a calculating machine based on quarter-squares [START_REF] Weiss | Die Rekonstruktion der Rechenmaschine von Schiereck 1829[END_REF] and in 1841 he devised an instrument for the measurement of the area of triangles, useful in surveying, and also based on quarter-squares [START_REF] Friedrich | Beschreibung des Pediometers, eines Instruments, um den Flächeninhalt in Karten ohne Rechnung zu erhalten[END_REF][START_REF] Haeberlin | Die Viertelquadratemethode und das Pediometer des Joseph Friedrich Schiereck[END_REF].

In a book published in 1857, Edward Sang also suggested the use of quarter-squares for multiplications [6, pp. 50-51].

The use of quarter-squares declined in the first half of the 20th century [START_REF] David | Tables of quarter-squares, sociologic[al] applications, and contributions of George W. Jones[END_REF], but Bojko in 1909 [3] and Plassmann in 1933 [START_REF] Plassmann | Tafel der Viertel-Quadrate aller Zahlen von 1 bis 20009 zur Erleichterung des Multiplizierens vierstelliger Zahlen[END_REF] each published tables of quarter-squares up to 20000. In the 1950s, however, new applications of the quarter-squares method for multiplication on analog devices surfaced, and this prompted their use for the implementation of multiplication in digital processors [START_REF] Jayashree | On binary multiplication using the quarter square algorithm[END_REF][START_REF] Johnson | A digital quarter square multiplier[END_REF].

Finally, we mention that tables of triangular numbers can also be used in a way similar to quarter-squares in order to facilitate the multiplications [8].

Structure of Voisin's table and reconstruction

Voisin's table is very simple, with three or four double columns per page, each double column having 50 lines. There are a total of 123 pages, whereas Bürger's table fits the same information in 80 pages. In Voisin's table, for n from 1 to 6200 (31 pages), there are four double columns. For n from 6201 to 20000 (92 pages), there are three double columns. For each value of n, the table gives the value of n/4 . For instance, for n = 1419, we find the value 503390. The reconstruction of Voisin's table was therefore most straightforward.

Figure 1 :

 1 Figure 1: The title page of Voisin's table.

6

 6

Figure 2 :

 2 Figure 2: The excerpt of Voisin's table giving the quarter-squares for 11601 ≤ n ≤ 11900.

Figure 3 :

 3 Figure 3: The excerpt of Bürger's table giving the quarter-squares for 11501 ≤ n ≤ 11549, 11601 ≤ n ≤ 11649, 11701 ≤ n ≤ 11749, 11801 ≤ n ≤ 11849, and 11901 ≤ n ≤ 11949. (source: Staats-und Universitätsbibliothek Dresden)

Table Range

 Range

			Pages Density
	Voisin (1817)	20000	123	162.6
	Bürger (1817)	20000	80	250.0
	Centnerschwer (1825)	20000	40	500.0
	Merpaut (1832)	40000	400	100.0
	Kulik (1851)	30000	40	750.0
	Laundy (1856)	100000	200	500.0
	Blater (1887)	200000	200	1000.0
	Bojko (1909)	20000	20	1000.0
	Plassmann (1933)	20009	200	100.0

Table 1 :

 1 A comparison of the main tables of quarter-squares. The density is the ratio of the range by the number of pages. The tables which pack the greatest ranges in the smallest number of pages are those of Blater and Bojko.

For a survey of the quarter-square method, see the articles by McFarland[START_REF] David | Tables of quarter-squares, sociologic[al] applications, and contributions of George W. Jones[END_REF][START_REF] David | Quarter-squares revisited: earlier tables, division of labor in table construction, and later implementations in analog computers[END_REF] and Weiss[START_REF] Weiss | Die Multipliziertafeln: ihre Ausgestaltung und Verwendung[END_REF].

In his 1959 review of alternative methods of simplified computations, entitled "Pseudo logarithms," Larsen describes among other things the quarter-squares method.

Note on the titles of the works: Original titles come with many idiosyncrasies and features (line splitting, size, fonts, etc.) which can often not be reproduced in a list of references. It has therefore seemed pointless to capitalize works according to conventions which not only have no relation with the original work, but also do not restore the title entirely. In the following list of references, most title words (except in German) will therefore be left uncapitalized. The names of the authors have also been homogenized and initials expanded, as much as possible.The reader should keep in mind that this list is not meant as a facsimile of the original works. The original style information could no doubt have been added as a note, but we have not done it here.


Florentin Rodio
email: florentin.rodio8@etu.univ-lorraine.fr

J M Christian Bastien
email: christian.bastien@univ-lorraine.fr

Heuristics for Video Games Evaluation: How Players Rate Their Relevance for Different Game Genres According to Their Experience

Keywords: Video Games, Methodology, Evaluation, Game genres, Player typologies, Heuristics. ACM Classification H.5.2 [Information Interfaces and Presentation]: User Interfaces graphical user interfaces, evaluation/methodology, theory and methods

This paper reports a study evaluating the relative relevance of evaluation heuristics for different game genres and players. 120 players (amateur or e-Sport players) were invited to assess the relevance of 47 heuristics for different game genre (Real Time Strategy, Massively Multi-player Online Role-Playing or First-Person Perspective Shooting). Results show that the relevance of heuristics varies according to game genres and player types. These results have implication for both the design and evaluation processes of games. Implications of these results and future research directions are discussed.

INTRODUCTION

Increased attention has been dedicated to the scientific study of video games given the economic appeal of the field. One of these concerns has been the improvement of evaluation methods, such as heuristics evaluation, a flexible, quick and suitable method for most development studios. This paper reports on a study aimed at assessing the relevance of heuristics as judged by different users' types and for different game genres.

By supplying evidences of heuristics relative relevancies, we believe that it will help designers focus on important aspects of the game, according to their game's genres or player's expectations, and to avoid wasting their time on issues that may not be important. In the next section, we present current inspection methods for video games, as well as the most common game and player taxonomies.

Inspection methods for Video Games

There are two main approaches for the evaluation of games: the empirical and the analytical approach. Empirical evaluation methods are based on user's performances or opinions while the analytical evaluation methods are based on the examination of a product using a set of theories or models [START_REF] Scapin | Ergonomic criteria for evaluating the ergonomic quality of interactive systems[END_REF]. The latter have the advantage of being performed early in the development process and at a low cost [START_REF] Nielsen | Enhancing the explanatory power of usability heuristics[END_REF]. In the field of video games, these analytical assessment methods are usually performed by the designers themselves or during beta testing. These evaluations however are lacking in rigor and tend to be accompanied by more structured inspection methods such as heuristic evaluation [START_REF] Nielsen | Enhancing the explanatory power of usability heuristics[END_REF] or the use of ergonomic criteria [START_REF] Scapin | Ergonomic criteria for evaluating the ergonomic quality of interactive systems[END_REF]. Basically, a set of evaluators inspects the interface with respect to a small set of fairly broad principles, referred to as the "heuristics". In recent years, several sets of heuristics for video games design and evaluation have been proposed. They are based on dimensions such as usability [START_REF] Federoff | Heuristics and Usability Guidelines for the Creation and Evaluation of Fun in Video Games[END_REF][START_REF] Pinelle | Heuristic evaluation for games: usability principles for video game design[END_REF], playability [START_REF] Desurvire | Game Usability Heuristics (PLAY) For Evaluating and Designing Better Games: The Next Iteration[END_REF][START_REF] Federoff | Heuristics and Usability Guidelines for the Creation and Evaluation of Fun in Video Games[END_REF][START_REF] Koeffel | Using Heuristics to Evaluate the Overall User Experience of Video Games and Advanced Interaction Games[END_REF] or player experience [START_REF] Fang | Development of an Instrument to Measure Enjoyment of Computer Game Play[END_REF][START_REF] Sweetser | GameFlow: a model for evaluating player enjoyment in games[END_REF].

Game genres

To categorize video games, we often refer to the notion of genre. A genre is a category of games with similar type of interaction (or "gameplay"). One of the most famous classifications is given by Callois [START_REF] Caillois | Play and Games[END_REF], who classifies play into four categories: Agon (competition), Alea (chance), Mimicry (role playing), ilinx (vertigo). In this study, we used three subgenres that belong to three widely accepted genres in the video game industry: "strategy game", "action game" and "role-playing game" (Figure 1). Strategy games involve managing various resources and units to accomplish a given objective and require high decision-making skills. The goals of the strategy games could be either to build a city or to destroy an enemy base. It can be in real time (RTS) or turn based. Action games are mostly based on players' dexterity and reflexes. It includes first-person perspective shooting games (FPS). Role playing games are focused on character development and its progression through a scenario. « Massively Multi-player Online Role-Playing Games » (MMORPG) are part of it, and enable the player to operate in a persistent world.

Typologies of players

Typologies of players are usually based on skill level, personality or motivation. To differentiate the players in terms of level, we generally refer to casual gamers and hardcore gamers. They are usually differentiated in terms of play time, « meta-gaming » involvement or motivation. Some hardcore gamers are also called progamers or e-sportsmen. These players enjoy competition and play video games as high-performance sports. Other player models are based on personality or play motivation. Bateman and Boon [START_REF] Bateman | 21st Century Game Design[END_REF] identifies four personality types: Killer, Achiever, Explorer and Socializer. Lazarro [START_REF] Lazzaro | Why we play games: Four keys to more emotion without story[END_REF] outlines four main motivational factors of players: Hard Fun (personal triumph over adversity), Easy Fun (Curiosity), Altered State (Relaxation / excitement) and People Factor (social Interaction). Finally, Yee [START_REF] Yee | Online gaming motivations scale: development and validation[END_REF] identifies 10 online gaming motivations that fall into 3 higher-level categories related to achievement, social, and immersion motivations.

RESEARCH QUESTION

When approaching a field as rich as video games, it is essential to consider that heuristics' relevance can differ according to game genres and player types. Indeed, heuristics guide the design of game elements in a specific way, and, it is this particular balancing that establishes a game play proper to a specific game genre. From the perspective of existing inspection models, many authors are aware that game genres are an important factor guiding the choice and the weighting of evaluation criteria [START_REF] Koeffel | Using Heuristics to Evaluate the Overall User Experience of Video Games and Advanced Interaction Games[END_REF][START_REF] Livingston | Critic-proofing: how using critic reviews and game genres can refine heuristic evaluations[END_REF][START_REF] Pinelle | Using genres to customize usability evaluations of video games[END_REF]. However, none of them have provided strong and formal evidences that heuristics relevancy differs according to game genres. Similarly, only Desurvire and Wiberg [START_REF] Desurvire | User Experience Design for Inexperienced Gamers: GAP -Game Approachability Principles[END_REF] have developed a list of principles specifically for a given player type, the casual player. But, similarly, no author has provided evidence that heuristics relevancy differs according to different player types. For this reason, the aim of this study was to assess whether heuristics have different level of relevance according to (1) game genres and (2) player types.

This study differs significantly from classical heuristics studies on three main points. First, this study does not focus on the validation of a particular set of heuristics. As Cockton et al. [START_REF] Cockton | Inspection-based methods[END_REF] have stated, we think that fundamental knowledge on inspection methods must be acquired before via controlled experimentation. It is especially true in video game inspection, since contextual variables such as game genres and player profiles seem to play a key role in game evaluation. Secondly, this study focuses solely on heuristics "relevance". Usually, to decide which issues should receive developer attention first, severity rating is used. However, other criteria have been considered in video game inspection to refine the prioritization of issues such as genre ratings [START_REF] Livingston | Critic-proofing: how using critic reviews and game genres can refine heuristic evaluations[END_REF]. Relevance indicator follows this trend but also benefit of a broader use: it can be seen as a generic tool to prioritize heuristics according to external criterion. Finally, this study is based on player judgment instead of usability expert. As other researcher [START_REF] Desurvire | Game Usability Heuristics (PLAY) For Evaluating and Designing Better Games: The Next Iteration[END_REF][START_REF] Jørgensen | Players as Coresearchers: Expert Player Perspective as an Aid to Understanding Games[END_REF], we think that taking into account real gamer experiences would improve the validity of game studies. Indeed, because video game heuristics are based more on playability and player experience, confronting target users to those heuristics would be very informative. The only likely bias could be to underestimate the relevance of some usability heuristics because players do not fully understand their scopes.

METHODOLOGY Material

To test the aforementioned assumption, a representative set of heuristics was chosen, as well as a given taxonomy of game genre and player type. The taxonomy used was kept intentionally simple to optimize the statistical power of our tests and we did not intend to represent the whole diversity of existing genre or player types.

IHM I3ORDEAUX

Three game genres were chosen: the RTS, MMORPG and FPS. These three genres have accounted for over 60% of the sales in PC market in 2010 1 . In addition, each genre possesses a high-level competitive scene. A representative game was then selected for each genre: Red Alert 3 for RTS, Call of Duty 4 for FPS and World of Warcraft for MMORPG. These games were chosen because they possess a large gaming community and are diverse enough to reach all players types.

Amateurs and e-Sports players were then chosen as participants. E-Sports players fulfill themselves in competition and high performance situations whereas amateur players tend to play more for relaxation and social interaction. Among all inspection models available, the PLAY 2009 [START_REF] Cockton | Inspection-based methods[END_REF] has been chosen. Composed of 48 heuristics (only 47 are available in the literature 2 ), this model is divided into three sections: Game Play, Usability & Game Mechanics and Coolness / Entertainment / Humor / Emotional / Immersion (renamed in Miscellaneous). Examples of heuristics can be seen in Table 2.This model covers both the usability and playability of games and was developed to cover three genres (Action Adventure, FPS, and RTS).

Participants and procedure

A participation call has been posted on 70 general, specialized and guild video games websites. Each player, that belongs to one of the following two categories (amateur or e-Sport player) and one of the three game genres communities (STR, MMORPG or FPS) was asked to fill in a survey and rate the relevance of 47 heuristics related to their dedicated game genre on a five point Likert-type scale. In addition, questions related to game practices were also asked. Categorization between 1 Annual survey of the ESA http://www.theesa.com/facts/pdfs/ESA_EF_2011.pdf 2 The 47 heuristics used in this study are available here: http://www.behavioristics.com/downloads/DesigningBett erGames-09HCI-Desurvire.pdf amateur and e-sports players has been accomplished by questions informing on their competition practice. On 232 invitations, 154 questionnaires were completely filled in and 120 were selected for this study. 20 amateurs and 20 e-sport players were selected for each of the three game genres.

RESULTS

Sample characteristics

The sample of players is predominantly composed of male (98.3%). Fifty percent of the participants are students, 18.3% are employees, 12.5% senior executive, 11.6% without employment, and others are 7.5%. The mean age of the players is 20 years old and their game experience is about twelve years. E-Sport players have significantly more experience in competition than amateurs on the specified game (1.53 vs 0.02 years, t(118) = 8.31, p < .001), on games of same genre (3.38 vs 0.05 years, t(118) = 12.14, p < .001) and spend significantly more time per week in competition on the game (12.83 vs 0.03 Hours, t(118) = 8.946, p < .001). This result supports and justifies the initial categorization between those two types of players.

Heuristics relevancy

Players' responses to the 47 criteria were analyzed with a 3 (RTS vs FPS vs MMORPG) x 2 (Amateur vs e-Sport players) ANOVA. Sheffe's Post Hoc test was carried out to look for significant pairwise differences between conditions. In addition, binomial tests have been performed to test whether the proportion of significant differences on the 47 criteria differ significantly from chance.

Seven out of 47 heuristics' relevance differ statistically according to game genres (p < .05). To test whether this observed proportion (15%) was statistically different from the expected proportion of 5% (the conventional error used when performing inferential test), a binomial test B(47, 0.05) was performed. This test is significant (p < .01). Therefore, the difference of heuristics' relevance found between game genres is significant. However, we cannot conclude that all observed differences are only

Category Heuristics

Game Play

A1. The players find the game fun, with no repetitive or boring tasks B3. Easy to learn, harder to master C1. The game world reacts to the player and remembers their passage through it E1. The game supports a variety of game styles

Usability & Game Mechanics

B1. Game controls are consistent within the game and follow standard conventions C2. Provide appropriate audio/visual/visceral feedback (music, sound effects, controller vibration) E1. The game does not put an unnecessary burden on the player H3. Upon turning on the game, the player has enough information to begin play

G1. Navigation is consistent, logical and minimalist

Miscellaneous A1. There is an emotional connection between the player and the game world as well as with their "avatar"

IHM I3ORDEAUX

due to the game's type as the average difference due to chance in this situation is 2.35 (σ = 1.49) 3 . Because of this result, we have chosen to illustrate trends only with the most conservative statistical difference for each type of game. Thus, we note a significant difference between game genres for the criterion « GamePlay E2: The game is balanced with multiple ways to win » (F(2,117) = 7.67, p < .001). A Sheffe test indicates that this difference is significant between RTS and FPS (p < .01) and between RTS and MMORPG (p < .01). Hence, game balancing and variety of possible strategies seem to be an important criterion for RTS. Similarly, there is a significant difference between game genres for the criterion « GamePlay D3: The game gives rewards that immerse the player more deeply in the game by increasing their capabilities, capacity or for example, expanding their ability to customize » (F(2,117) = 5.07, p < .01). A Sheffe test indicates that this difference is significant between MMORPG and FPS (p <.05). So, rewarding in the game seems to be a more important criterion in MMORPGs than in FPS. As a last example, there is a significant difference between game genre for the criterion « Miscellaneous C1: The game uses humor well » (F(2,117) = 5.53, p < .001). A Sheffe test indicates that this difference is significant between FPS and RTS (p < .01) and between FPS and MMORPG (p < .05). Hence, humor in FPS seems to be of minor importance. 14 out of 47 heuristics differ significantly according to player types (p < .05). To see if this observed proportion (30%) was statistically different from the expected proportion of 5%, a binomial test B(47, 0.05) was also computed. This test is significant (p < .001). Therefore, the difference of heuristics' relevance found between player types is significant. Like the previous section, we have illustrated trends only with the most conservative statistical difference between the two types of player. First, all these significant differences show a lower relevance of the heuristics by e-Sports players compared to amateur players. Many of them are related to artificial intelligence (« GamePlay B5: AI is balanced with the players' play », t(118) = 3.31, p < .001 ;« GamePlay B6: The AI is tough enough that the players have to try different tactics against it. », t(118) = 2.28, p < .01; « GamePlay E4: The game had different AI settings so that it was challenging to all levels of players, whether novice or expert players », t(118) = 4.09, p < .001). Furthermore, we find that many of these criteria have high importance in single mode (« Usability & Game Mechanics H5: All levels of players are able to play and get involved quickly and easily with tutorials, and/or progressive or adjustable difficulty level », t(118) = 3.04, p < .01; « Usability & Game Mechanics I1: Game story 3 It is the expected value, E(X) = np, with a standart deviation ; n = number of trials, p = the success probability in each trial (Here n = 47 and p = 0.05).

encourages immersion», t(118) = 2.92, p < .01; « GamePlay A3: The players should not lose any hard won possessions », t(118) = 2.46, p < .01). Hence, all these heuristics seem to be of minor importance for e-Sports players.

DISCUSSION

This study helps us draw both general priorities and specific expectations for a particular genre or player category.

We found that a number of heuristics seem to be important for the three types of games. Among the most important of them, many focus on the notion of game variety, rhythm, balance and immersion. These are all aspects that help to maintain the player's attention. The heuristic considered as the most important summarizes well this phrase: "GamePlay A4: Gameplay is long and enduring and keeps the players' interest". Other heuristics have also been identified as important. They are dealing with the feedback, the imput/ouput and the consistency of the interface. These heuristics are at the root of any successful interaction and can be found in most current guidelines. On the other hand, some heuristics seem less crucial. They generally focus on specific points of games, such as tutorials, game manual, humor, and the emotional relationship with his avatar. They all are criteria, which are outside the main interaction mode of the game.

In addition to the general recommendations, we found significant differences between game genres that distinctively guide the design of each. For example, the game balance, the interface consistency and the errors minimization are very important aspects in RTS games, while it is rather the environment consistency and the rewards management that prevail in MMORPGs. This shows us that there is more than one way to have fun and this is why the current game industries are so diverse. So diverse that it is not fruitful to base the assessment of all video games with the same universal heuristics. It is interesting to know that other researchers have issued the same warning to film studies, pleading for middle-level research to better account for the specificity of certain kinds of films that were being left out of sight with the prevailing "Grand Theories" [START_REF] Arsenault | Video Game Genre, Evolution and Innovation[END_REF].

Similarly, we found a large number of heuristics that were judged of minor importance for e-Sports players. A part of them relates to the single player mode, such as the story, the accumulated possessions and AI. Another part relates to "Approachability" criteria [START_REF] Desurvire | User Experience Design for Inexperienced Gamers: GAP -Game Approachability Principles[END_REF], such as tutorials, adjustable difficulty levels and basic information required to play. Thus, e-Sports players whose interest focuses on multiplayer and expertise are left unsatisfied. Indeed, when returning the questionnaire, many of them asked why so few criteria were related to the multiplayer mode instead of the single player mode. This is a concrete example that demonstrates the differences of players' expectations.

IMPLICATIONS AND FUTURE WORKS

This study demonstrates that the relevance or importance of a significant number of heuristics differ according to game genres and player types. These findings imply that heuristics should be selected and weighted carefully when evaluating a specific game genre or for evaluating games for specific player types. These findings have implications for game design too because it is impossible (and counterproductive) to optimize a game in all its aspects. Rather, priorities should be identified, according to a certain type of attended game play. In addition, because the player population has greatly expanded, the consideration of specific players' needs, will allow us to find emerging marketing niche to invest.

Because this work has aimed only at providing an overall validation of these statements, substantial in-depth works deserves to be done in order to find relevant criteria for different game genres and players profile. For example, it would be difficult to develop a high-level strategy game if we do not know the specific expectations of the players of these games. To do so, qualitative studies have to be conducted. It should go beyond the over simplified taxonomy used in this study and embrace a larger diversity of game genres and player profiles. In addition, future work should be conducted to investigate the link between heuristics relevance according to game genres, player types and evaluation reliability. Other factors influencing reliability of game evaluation should also be investigated.

Recent scientific studies of video games demonstrate the current appeal of the HCI field in developing theories, methods, and scales to evaluate them. In a broader perspective, we assume that video games studies will also contribute to enrich our general knowledge of user experience theories.

Figure 1 .

 1 Figure 1. Examples of RTS, FPS and MMORPG game genres, respectively.

Table 2 . Examples of heuristics from the PLAY2009 model [6]

 2

	3 3

4

ACKNOWLEDGMENTS

We thank all the volunteers coming from C&C Red Alert 3, Call of Duty 4 and World of Warcraft communities for their time and energy in completing the questionnaire.


Jean-Baptiste Scheibel

Cyril Pierson

Benoît Martin
email: benoit.martin@univ-lorraine.fr

Nathan Godard

Vittorio Fuccella
email: vfuccella@unisa.it

Poika Isokoski
email: poika.isokoski@uta.fi

Virtual Stick in Caret Positioning on Touch Screens

Keywords: Gestures, Text editing, Caret movement, Android H.5.2. Information Interfaces and Presentation: User Interfaces. IHM I3ORDEAUX

We present our design exploration in the area of virtual stick controllers and a preliminary evaluation in an editing task. Virtual stick controllers are one solution to the problem of precise pointing on touch screens. They operate by using an area of the touch screen as a rate control device that moves a pointer. We implemented and evaluated this technique in a text editing context where unaided precise placement of the caret is difficult. The results showed that with large fonts and long pointing distance positioning the caret with the virtual stick is significantly slower than with conventional finger touching. On the contrary, with small fonts and short pointing distance, we noted no difference.

INTRODUCTION

Mobile phones today are, as a matter of fact, powerful computers that can run many kinds of software traditionally associated with larger computers. The dominant user interface technology on these high-end phones is the touch screen. Some tasks are not well suited for touch screen use. One such task is text editing. People used to editing on desktop computers quickly become frustrated when trying to edit text using a touch screen.

On desktop computers, where indirect pointing with the mouse is used, there are many types of cursors. Two of them, the mouse pointer and the caret that shows the current insertion point in text are often visible simultaneously. On touch screens the mouse pointer does not exist as direct finger pointing is used. However, precise manipulation of the caret is still needed.

Caret positioning is one of the more frustrating parts of text editing on touch screens. Precise caret placement is difficult especially with smaller font sizes and especially near the edges of the display. Fundamentally, the difficulty stems from using the finger as the pointing device. Finger pointing is not as precise as mouse pointing [START_REF] Cockburn | Understanding performance in touch selections: Tap, drag and radial pointing drag with finger, stylus and mouse[END_REF] and using the finger brings with it the problem of occlusion.

An obvious workaround in the editing context for caret placement is to use virtual arrow keys on a soft keyboard. Some publicly available solutions are shown in Figure 1.

The arrow keys can take the shape of a direction pad (D-PAD) or consist of separate keys for each direction. Further, they can be present all the time as in the Arrows 1 1 Keyboard [2] (Figure 1 left), optional, but always present when turned on as in the Thumb Keyboard [18] or present when requested as with Perfect Keyboard [12]. In all cases, the arrow keys need a significant portion of the area of the soft keyboard. Often keyboard designers prefer not to show the arrow keys on the main layout. Instead they are available on an alternative layout as in Swype [17] (Figure 1 middle). The Custom Keyboard [6] gives access to an onscreen touchpad instead of the arrows keys.

To avoid switching layouts, some keyboards use gestures done on top of the keyboard. On the Gesture Keyboard [9] swipes to different directions are mapped to the caret movements (Figure 1 right). Gestures offer great possibilities that can be enhanced to text selection, clipboard access, etc. [START_REF] Vittorio Fuccella | Gestures and widgets: performance in text editing on multi-touch capable mobile devices[END_REF]. Unfortunately, there is competition for the gesturing space over soft keyboards. For example word gesture keyboards [START_REF] Zhai | The word-gesture keyboard: reimagining keyboard interaction[END_REF] can use the gesturing space for text entry. This makes it necessary to use a special starting point for command gestures to differentiate them from text entry gestures [START_REF] Per | Command strokes with and without preview: using pen gestures on keyboard for command selection[END_REF].

Bezel Swipe [START_REF] Roth | Bezel swipe: conflict-free scrolling and multiple selection on mobile touch screen devices[END_REF] helps in accomplishing the modeswitch from text entry to text selection, but does not improve the pointing accuracy for character level caret placement. An overview of the touch screen target acquisition techniques by Roudaut et al. [START_REF] Roudaut | TapTap and MagStick: improving one-handed target acquisition on small touch-screens[END_REF] list a number of studies that have been successful in avoiding occlusion, improving precision, or both. The main approaches were zooming, various forms of object pointing or snapping, and the offset cursor. The most relevant prior work involves using the touch screen as a rate control device. We will discuss this work in the context of our work in the next section.

THE VIRTUAL STICK

In this section we describe our virtual stick implementation that we call the VS for brevity.

Virtual sticks that emulate the function of joystick on a touch screen are used in the gaming context. Figure 2 shows an example at the bottom left corner of the display. In this paper we propose to use such a virtual stick to manipulate the caret in text editing and report on user preference and task efficiency. The virtual sticks we have seen in games inspired our work, but exact technical details on their implementation are usually not available. Thus our design was mostly guided by intuition and trial and error over many iterations.

In the context of a soft keyboard the virtual stick should consume as little space as possible and be usable without leaving the soft keyboard. We will first describe the general operation of the VS and then discuss the details of the design and implementation.

The zone of interaction

The VS uses a circular zone of interaction. It can be used with or without visual feedback. In our experiments we used it without visual feedback. The following figures are illustrations drawn for this paper they were not part of the user interface we constructed.

When the user touches the surface of the touch screen the VS is centered to the touch position: the finger is in the neutral zone (Figure 3 -left). The radius of the neutral zone is one of the parameters that must be decided in implementations. While the finger stays in the neutral zone, the caret does not move. Also, if the finger is lifted at this point, the key below the finger can be activated as usual. Thus, VS and the normal text entry function by tapping on the keys can coexist without a mode change.

When the user moves the finger out of the neutral zone onto the active zone (Figure 3 -left) the device vibrates to give tactile feedback. Moving the finger can be done by sliding it on the surface or by doing micro-rolls [START_REF] Roudaut | MicroRolls: expanding touch-screen input 7 7 IHM I3ORDEAUX vocabulary by distinguishing rolls vs. slides of the thumb[END_REF]. The relative efficiency and user comfort of micro-rolls and sliding is an interesting issue that for now remains unsolved. The size of the active zone is another parameter to be decided in implementations. Our decisions were based on trial and error. Determining the optimal size with a larger user population is another task for further work.

As long as the touch point P remains in the active zone, the caret moves with a speed that depends on the distance d to a direction that depends of the angle shown in Figure 3 (right).

In text editing, horizontal and vertical caret movements are frequent. Therefore, a technique to facilitate them was found to be helpful in the informal evaluation phases of our iterative development of VS. Our facilitating technique uses "snapping". It can be related to prior work on pseudo-haptic that was first proposed by Lecuyer et al. [START_REF] Lecuyer | Pseudo-haptic feedback: Can isometric input devices simulate force feedback[END_REF] : pseudo-haptic systems are defined as "systems providing haptic information generated, augmented or modified, by the influence of another sensory modality".

In VS the user will have the feeling of a sticky axis. Non-orthogonal movements seem to require more "virtual effort" than they would without the snapping setup. The VS contains four snapping areas (Figure 4 Two behaviors were implemented to ease the use of the VS. The first one was intended to ease spatial positioning of the finger in the VS. When the finger is in the active zone, the user has a visual feedback with the movement of the caret. According to this movement, the user can partially evaluate where the finger is depending of the speed and the direction of the move. But when the finger is in the neutral zone, the pointer doesn't move. The user can be anywhere in the neutral zone. At the beginning of the interaction, when the user touches the surface, the VS is centered on the finger. We applied the same behavior when the finger enters in the neutral zone from the active zone (Figure 5 -left). The VS is centered on the first touch position in the neutral zone and the user feels a vibration as tactile feedback. He can continue his interaction as he was starting it.

The second behavior guaranteed that the touch point always remained in the VS. When the touch point goes outside, the VS moves to follow it (Figure 5 -right).

The pointer

The VS is intended to move a pointer on the screen instead of moving directly the caret. This is done by interpreting the parameters of the VS, the distance d and the angle . The angle indicates the direction of the move, no specific treatment is needed. The distance indicates the amplitude of the movement in the interval [0;1]. We can map this distance to speed in different ways. We decided to use mapping functions f:

s = f (d)
A mapping function takes one parameter, the distance value in the interval [0;1]. It returns a normalized speed in the interval [0;1]. The value 0 corresponds to the neutral zone and the value 1 corresponds to the maximum speed the user can achieve. This result is multiplied by a maximum speed value to obtain the speed at which the pointer can move. This maximum can be adjusted by the user.

Choosing a mapping function is far from trivial. For example there has been a lot of work in investigating mapping functions for mice. For a recent example see the work by Casiez and Roussel [START_REF] Casiez | No more bricolage!: methods and tools to characterize, replicate and compare pointing transfer functions[END_REF]. Similarly work on mapping functions for isometric joysticks has been fruitful [START_REF] Rutledge | Force-to-Motion Functions for Pointing[END_REF][START_REF] Barrett | Negative inertia: a dynamic pointing function[END_REF]. However, we are unaware of similar work on virtual joysticks on touch screens. It is also not clear that the work on isometric joysticks is directly applicable when the device is not self-centering and does not give similar haptic feedback. Thus, we turned to the iterative trial-and-error process just like Dutledge and Selker in their work on isometric joysticks [START_REF] Rutledge | Force-to-Motion Functions for Pointing[END_REF].

We tried several different original mapping functions.

Figure 6 shows four of them. All the functions give the same maximum speed but the curve to obtain it is

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different. The first one is a linear function that we chose for its simplicity:

f(d) = d (1)
The other three come from exponential function g: 4) From this function g, we built the following mapping function to obtain a normalized speed in [0;1]:

g(d) = e ((c*d-3)/1.4) -e (-3/1.
f(d) = g(d) / g(1)
(2)

The "constant" c of the function g can be adjusted to produce functions with different "sensitivity". The sensitivity is intended to be a user configured parameter.

Figure 6 shows three examples of this function (2) with different sensitivity. The functions have been labeled "Exponential (c)" where c is this sensitivity. As shown in Figure 6, when the sensitivity is higher the control at low speed is better but at the end the speed will increase faster. With smaller values of c the function approaches the linear one (1).

The appearance of the pointer is also important. It must be visible but it must not hide the caret or the underlying text. We decided to use a round green pointer that is displayed in the background of the text. The pointer also reflected the state of VS. When the finger was is in the neutral zone it was displayed as a circle ( ) but when the user was in the active zone it was displayed as a disc ( ).

The caret

When the user touched the surface, the pointer was centered on the caret. During the interaction with the VS, the caret moved by following the pointer. However, since the pointer moved continuously but the caret could only reside between characters, they do not always move in perfect synchrony. Instead, the caret is the projection of the pointer in the text. Also, the ends of the line of text are special. The pointer can proceed past the end, but the caret has no useful function there. In our implementation the caret remained at the end of the line when the pointer was beyond it.

EXPERIMENT

We know that the virtual sticks are used in games but data on their performance was not available. In our experiment we compared caret positioning with ordinary finger pointing to the VS.

Design

We had three interaction techniques for the same caret placement task. In the Finger condition the caret was placed by tapping with the finger, The Virtual Linear (VL) technique was VS with linear mapping (1) (function Linear in Figure 6). The Virtual Exponential (VE) technique was VS with the exponential mapping (2) (function Exponential(3) in Figure 6).

We used a between groups design to test the two mapping functions (VL and VE). The participants would probably have been unable to learn to use both at the same time. Both groups also performed the tasks in the Finger condition.

Both groups did the tasks twice, one time with a font size of 14sp and one time with a font size of 18sp. The unit sp means scale-independent pixels: one sp is one pixel on a 160 dpi screen if the user's global text scale is set to 100%. Both groups performed two sessions separated by at least 24 hours and less than 48 hours. Conditions were balanced among participants and sessions.

To record subjective impressions we used a questionnaire. The questionnaire was composed of 8 statements to which participants assigned a score indicating their strength of agreement in a 5-point scale.

In addition we made notes of participants' comments during and after the experiment. We also collected background information including whether the participants owned and operated touch screen.

Participants

We recruited a total of 16 participants (8 per group) from the university community in Metz, France. The mean age of the participants was 23.5 years (SD=1.4). One participant was left-handed. They were all users of smartphones with touch screen interfaces.

Apparatus

The software was run in the ACER Liquid Metal smartphone with Android (Gingerbread) 2.3.6. The device had a capacitive touch screen. The width of the display was 47 mm and the height was 77 mm. Its resolution is 480 x 800 pixels (259 dpi). With the user's text scale set to 100% the font size of 18sp was 2.8 mm and the font size of 14sp was 2.2 mm.

The setup for the VS was: radius of the neutral zone: 2mm; radius of the acive zone: 8mm; angle of the snapping areas : 20°; maximal speed of the pointer: 35mm/s; sensitivity for the exponential mapping: 3 (Exponential (3) in Figure 6). When developing the software, we discovered differences between touch screens. For example some touch screens provided input data at one pixel intervals while some others provided input at lower resolution. Depending on the hardware, the feeling of the mapping in VS can be very different.

Figure 7 shows the exponential mapping (sensitivity of 3) both with a high accuracy and a low accuracy. With a high accuracy, the user can feel the variation of the speed as continuous. But with a lower accuracy, the variation is discrete and different according to the slope of the function. The low accuracy curve shows the actual curve on the device that was used in the experiment. It provided input resolution equal to 5 pixels.

Procedure

Each session consisted of four blocks. Two blocks were performed with the finger technique and two blocks with the VS. For each technique one block was performed with the smaller font size (14sp) and the other with the larger font size (18sp). Each block contained 20 tasks of caret positioning. The next block started when all the tasks of the previous block had been finished.

The task was to move the caret from a start position to a target position. The start position of a task was the target position of the previous task. In a block, half of the tasks were pooled to the "near" category while the other half was pooled to the "far" category. The distance to move in a near task was less or equal to 4 characters to left or right and no more than 1 row up or down. In the far tasks the distances were longer (up to 25 characters horizontally and 10 rows vertically). The 20 tasks were identical in all blocks.

Figure 8 shows a screenshot of the display during a task. The text was always the same and displayed at the top of the display. The start position was indicated by the caret and the target was indicated by a vertical green line. To indicate the completion of the task, the participant pressed the button at the bottom of the display. If the caret was not at the target position the task remained active and the participant had to adjust the caret position until it was correctly placed.

With the two versions of VS, the participant interacted in the "VS Interaction Zone". In real world implementations this area would be the soft keyboard.

To simplify the experiment we chose to display empty space instead of a soft keyboard. All other interactions in the "Text Zone" were disabled.

With the finger tapping technique, the participants interacted directly in the "Text zone". The caret was moved to the touched position. A draggable widget appeared below the caret and the participant could also drag the caret to another position as usual in touchscreen editing.

No training was offered to the participants before the first session.

RESULTS

Our design was aimed at testing the linear and exponential mapping functions against each other and against the finger tapping. However, there were no statistically significant differences between the groups. Thus, we will report only the results comparing finger tapping and VS.

Performance results were compared using a repeated measures ANOVA separately for the two groups. The independent variables were moved distance (near, far), font size (14, 18) and the interaction technique (VS, Finger). The dependent variable was the task completion time. We report in detail only the data for the second session because in the first session the performance varied a lot as the task was new to the participants.

Performance of the VS

Figure 9 shows the mean task completion times for near and far targets with both font sizes for the VL group. The Finger technique was faster (F 1,7 =7.6, p=0.028). The font size also had a main effect with larger font leading to 5 5

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faster performance (F 1,7 =9.1, p=0.02). Also, the target distance had a main effect (F 1,7 =34.4, p=0.001) with longer distance leading to slower performance. Of the three possible interaction effects only the interaction of technique and distance was statistically significant F 1,7 =32.3, p=0.001.

Figure 9 helps in interpreting this interaction as larger effect of distance to task completion time with the VS. Bonferroni corrected pairwise t-tests showed that the difference between VS and Finger was statistically significant only in the far 18sp font condition. No other statistically significant effects were found.

For the group with exponential displacement-speed mapping (VE), figure 10 shows the result. The results were similar. Font size (F 1,7 =6.6, p=0.037) and distance (F 1,7 =59.2, p<0.001) had statistically significant main effects. The interaction of technique and distance was also statistically significant (F 1,7 =25.8, p=0.001). The Bonferroni corrected pairwise t-tests also led to the same conclusiononly large font and long distance yielded a statistically significant difference.

The overall pattern of the results is that at long distances and large font sizes the finger technique was more efficient. With smaller font sizes and shorter distances the difference vanished.

Learning

Figure 11 shows the evolution of the performance between session 1 and session 2. As usual in experiments were participants are subjected to a new task they learn to perform the task more efficiently.

User Satisfaction and Free-form Feedback

At the end of experiment, the participants answered to the following statements presented to them in French: Figure 12 shows the results (larger number indicates higher agreement). Only question 8 indicates a potentially meaningful difference.

We also asked the participants to choose between finger condition and VS condition (0 was the finger condition and 5 the VS condition). On Figure 12, the "Choice" bars 6 6

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indicate the result. Most users did not have a clear preference. Instead they preferred to have the both techniques available and to be able to choose which to use according to the situation. From the participants' comments, we can note that the group VE found that the pointer was too fast and sometimes difficult to control. That can be due to the resolution issue mentioned in Figure 7. This effect appears more smoothly with the linear mapping. With a more accurate touch screen this problem may be smaller. In both groups, some participants mentioned a wish for tactile feedback for each movement of the caret or at the end points of a word. One participant mentioned that VS was preferable when the target was in between to 'l' letters.

DISCUSSION

Our results showed that tapping with the finger was generally the faster way to place the caret under the conditions that we tested. However, we also found that the relative efficiency of finger tapping and VS depended on the font size and distance to the target. VS was stronger with shorter distances and smaller fonts. We only tested two levels of distance and font size. Very short distances would probably show stronger benefit for VS against finger tapping. Font sizes below 14 would perhaps also make VS preferable.

To assess the value of VS in realistic editing tasks one needs to consider the frequency of long and short caret movement distances. Long distance movements do occur. For example, when locating the starting point of a text passage to copy the caret placement task corresponds to a long movement that is best accomplished with a direct tap or bezel swipe [START_REF] Roth | Bezel swipe: conflict-free scrolling and multiple selection on mobile touch screen devices[END_REF] or other such technique especially since whole words rather than individual characters are usually selected. However, short movements are frequent too. For example, when entering text into the computing device, one often makes typos that are noticed soon after they happen. In such cases only a short distance needs to be moved to correct the error. Also correcting some typos, such as transposition ("teh" instead of "the") requires onecharacter movement after the deletion and before the insertion.

We do not have statistics on the frequency of different caret movement distances in real world editing. However, in general VS can be recommended only if short distance caret movements are clearly dominant.

Further work is needed in selecting the best parameters for VS. We experimented with linear and exponential displacement-velocity mappings, but the results were inconclusive. It appears that the shape of the curve did not have a major effect under the conditions tested. Further work is also needed in measuring the beneficial area of VS use. The range of our "near" and "far" tasks was rather wide. Finer granularity especially in the "near" category is recommended in follow-up studies.

CONCLUSIONS

Based on our results it seems that the VS or another implementation of a Virtual Stick can compete with direct pointing in text editing tasks where short caret movements are often needed. While longer movements are usually faster completed by tapping with the finger, the final placement of the caret can be difficult without the VS or some other precise caret movement technique.

Figure 1 .

 1 Figure 1. Solutions for caret movement: arrow keys (left -Arrows Keyboard [2], middle -Swype [17]) and gestures (right -Gesture Keyboard [9]).

  left). A touch point in such an area is projected circularly to the closest axis. The distance d remains the same

Figure 2 .

 2 Figure 2. The virtual stick in "Flight School" an iOS game by Dough Scandrett [7].

Figure 5 .Figure 3 .Figure 4 .

 534 Figure 5. Behaviors of the Virtual Stick. Re-centering upon re-entry of the neutral zone (left) and VS follows the finger (right)

Figure 6 .

 6 Figure 6. Four examples of mapping functions: one linear and three exponentials.

Figure 8 .Figure 7 .

 87 Figure 8. A screen shot of the display during a task.

  1. I found easy to move the caret with the Virtual Stick 2. I have difficulties to stop the caret with the Virtual Stick 3. With the Virtual Stick, I know easily when the caret starts to move 4. The movements done on the Virtual Stick are not suitable to control the caret 5. Controlling the caret with the Virtual Stick is ergonomic 6. Controlling the caret with the Virtual Stick gave me fatigue and effort 7. I control easier the caret with the Virtual Stick than with my finger 8. I have had difficulties to move the caret with my finger.

Figure 12 .

 12 Figure 12. Subjective impressions.

Figure 11 .Figure 10 .

 1110 Figure 11. Performance improved from session 1 to session 2.

Figure 9 .

 9 Figure 9. Group VL: performance in the second session.

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Janine Emile

Olivier Emile

Mapping of the Marangoni effect in soap films using Young's double slit experiment

Keywords: 47, 55, pf -Marangoni convection 42, 87, Bg -Phase shifting interferometry 47, 20, Bp -Buoyancy-driven instabilities (e, g, , Rayleigh-Benard)

We report on the thickness variation measurement of a soap film due to a local perturbation, using Young's double slit experiment configuration. We map a laser-heated deformation of a vertical free-standing draining thin soap film using the differential change of optical path in the interferometer. The experiment has a resolution of about 0.1 nm and enables to follow the liquid flow dynamics. We evidence a bottleneck formation in the heated region of the film that perturbs the usual flow. Such an experimental set-up could then be adapted to measure other tiny variations in fluctuating hydrodynamics such as capillary waves for example.

Introduction. -On the one hand, tears of wine are a superb manifestation of the Marangoni effect [1]. The variation of the surface tension due to the change of alcohol concentration leads to these tears [2][START_REF] Chandrasekhar | Hydrodynamic and hydromagnetic stability[END_REF][START_REF] Pearson | [END_REF][5]. Such an effect can also originate from a temperature gradient at the air/liquid interface and is sometimes called Rayleigh-Bénard convection [START_REF] Getling | Rayleigh-Bénard convection: structure and dynamics[END_REF][START_REF] Vanhook | [END_REF] when it is linked to thermal convection. Whereas this effect is often invoked, quantitative measurements are scarce [8][9][10][11][12][13][14][15][16][17]. These measurements have been mainly performed, directly using either the naked eye, or with the help of a microscope, or using temperature sensitive markers, or probing the film thickness with interference techniques. On the other hand, tiny length variations in general can be evidenced using optical interferometry measurements [18,19]. Relative variations in the 10 -17 to the 10 -21 range have already been reported [20,21]. Optical interferometry has also been used in the measurement of soap film thicknesses in the 10 µm-200 nm range [22][23][24], with a precision of 20 nm. Among all these interferometry techniques, Young's double slit experiment is one of the most popular, fascinating and easy to set up technique. In particular, in quantum mechanics, it has been used to demonstrate the wave particle duality [START_REF] Bohr | Atomic physics and human knowledge[END_REF] and to evidence phase variations, like in Aharonov-Bohm type experiments [START_REF] Peskin | The Aharonov-Bohm effect[END_REF]. R. Feynman himself said from this experiment: "It is a phenomenon which is impossible to explain in any classical way and which has in it the heart of quantum mechanics" [START_REF] Feynman | The Feynman lectures on physics[END_REF]. Back in the optical domain, due to its high simplicity and versatility, one may thus wonder whether Young's double slit experiment could be implemented to measure the thickness variation of soap films via the change of the phase due to the modification of the effective optical length. The aim of this letter is then to investigate potentiality of the double slit experiment by probing the dynamics of the Marangoni effect on a soap film locally heated by a laser. In particular, we will try to map its thickness variations with a fraction of a nanometer resolution using Young's double slit experiment.

Experiment. -The soap film is drawn on a rectangular glass frame with a 60 mm height and a 34 mm width (see Fig. 1). The vertical film is pull gently upward from a tank containing the foaming solution with a velocity lower than 5 cm/s. The freestanding film then starts to drain [START_REF] Mysels | Soap films, studies of their thinning[END_REF][START_REF] Couder | [END_REF]. The foaming solution consists of 5.4% sodium lauryldioxyethylene sulphate (SLES, 55.6 g/L, Cognis) diluted in pure water. To ensure the stability of the film, the surfactant concentration is high, well above the critical micellar concentration and the experiments are performed at a controlled temperature T = 20.0 ± 0.5 • C.

The double slit interference pattern is obtained from a green He-Ne laser (Melles Griot, λ = 543.5 nm, P = 5 mW, beam waist size w = 400 µm). The distance between the two slits is d = 200 µm. The pattern is observed on a screen located at a distance L = 6 m from the slits, or registered on a CCD line camera (Thorlabs, LC100), connected to a computer. The interference between the two paths leads to an interference pattern in the x direction that writes [START_REF] Feynman | The Feynman lectures on physics[END_REF]

I(x) ∼ cos 2 ( πdx λL + δφ) (1)
where I(x) is the light intensity on the screen and δφ is the phase difference between the two paths at the slits location. Since the double slit is located just after the soap film (see Fig. 2), this phase difference is related to the differential thickness of the film corresponding to the two paths. A shift of the interference pattern ∆x (see Fig. 2), then corresponds to a difference between the two optical paths n(e 1 -e 2 ), due to the thickness variation

(e 1 -e 2 ) = -d∆x/nL (2)
where n ≈ 1.33 is the index of refraction of the film. Unlike X-rays, the use of a visible laser source does not allow to detect different refractive indices related to a multilayer organization of the film, for layer thickness below 20 nm. Our CCD line camera has a resolution of 14 µm, then the resolution on the differential thickness of the film equals 0.35 nm. A position sensitive detector would have lead to a resolution in the picometer range, or even below.

The film is heated with a focalized diode laser (Dida concept, λ = 405 nm, P = 5 mW, beam waist size w = 300 µm), see Fig. 1. The beam spot is located just at the middle of the soap film, at the position x = 0 mm and y = -30 mm. For a given film thickness, we estimate the power absorbed by the film by measuring the transmitted light through the film. We assess the film thickness independently by an interferometric technique we have already developed [24]. For a 1 µm thick film, we measured the transmitted violet light. This wavelength is partially absorbed by the solution contrarily to the green laser used to probe the film thickness. Taking into account the reflection coefficients and the multiple internal reflections, we estimate the absorbed optical power to be 100 µW (corresponding to a 2% absorption) at the beginning of drainage for a thickness of more than 1 µm. Some authors have evidence the temperature intensity profile at short times, for an infrared focalized laser beam, heating a liquid film confined between two glass plates [13]. Since here, the film is in a free standing geometry, and the low power laser is not strongly focalized, the heating mechanism could be evaluated in terms of Fourier's law [START_REF] Chandrasekhar | Hydrodynamic and hydromagnetic stability[END_REF], leading to a differential temperature of 25 K. The variation of the index of refraction due to temperature is too low to induce neither optical aberrations nor effective variations of the optical pass of the probe beam. We could change the laser power of the heating violet beam, however, a higher laser power breaks the film and for a lower power, the film gradient is too low after 15 s drainage to make any relevant measurement. Previous experiments [11,16,[31][32][33][34] were performed under different conditions in order to measure high temperature gradients only. Devices associated with the deposition of a film on a metallic plate or to the free suspension of a film continuously supplied by a solution pump have been designed to increase artificially the film lifetime, which is not the case here.

Results. -We have then moved the green He-Ne laser source together with the double slits, independently from the frame and from the heating laser, in order to map the deformations of the soap film due to the Marangoni effect. The results appear in Fig. 3a for a horizontal scan (in the x direction), for y = -30 mm, i.e. corresponding to the height of the heating region. Each measurement is made at least three times to ensure reproducibility. The error bars are shown on the figure. The data give very accurate information on the film thickness gradient and thus must be integrated to extract the true thickness (see Fig. 3b). The measurements have been performed 5 s, 10 s and 15 s after suspending the soap film, the heating laser being always on. The integration time is 1 s and the film deformation caused by heating diminishes over time. At the end of the drainage, the film has the same thickness distribution with and without heating. When the heating laser is off, the shift of the interference pattern equals zero, there is neither deformation nor Marangoni effect.

As previously mentioned, the corresponding absolute thickness, with or without heating, has been measured independently using a previously developed experiment [24]. This experiment has a resolution of the order of 20 nm, nearly two orders of magnitude higher than the resolution of the double slit experiment. Besides first experiments do not allow to measure low thickness gradients and to probe grey or black films (from 200 nm to 10 nm thickness) that are rather studied by X-ray reflectivity [START_REF] Bélorgey | [END_REF]. Nevertheless, the technique of [24] has been used to characterize the film thickness in its middle at x = 0 mm, and y = -30 mm. Then at the other positions, the film thickness has been deduced using the integration procedure the constant being determined at the middle of the film. From the experimental results, it has to be first noticed that the signal is symmetric regarded to the heated region. Second, since the surface tension decreases linearly while increasing the temperature, at small temperature intervals [START_REF] Getling | Rayleigh-Bénard convection: structure and dynamics[END_REF], the film thickness mapping is a direct evidence of the heat diffusion in the film [START_REF] Chandrasekhar | Hydrodynamic and hydromagnetic stability[END_REF]. In principle, surface tension gradients could also originate from surfactant concentration, triggered by surface flow. However, due to the structured air/liquid interfaces (above the critical micellar concentration), surface fluctuations are minor. Experimentally, we found that the thermocapillary deformation increases linearly moving away from the heated zone. Besides, whereas the heated zone gets slightly thinner, due to a decrease of the surface tension, the outer zone inflates by nearly a factor of two. The liquid has to by-pass the thinner zone and must thus flow in the lower temperature regions. As we approach the edges of the menisci (about 2 mm from the frame), this effect is masked by the marginal regeneration caused by a surfactant surplus in the meniscus region [START_REF] Mysels | Soap films, studies of their thinning[END_REF].

Discussion. -To perform vertical scans, i.e. in the y direction at x = 0 mm (see Fig. 4a), the slits have been positioned horizontally. We performed the measurement at t = 10 s but get similar results for the different times. Without heating, the soap film is slightly prismatic; the faces are not parallel to each other [24,[START_REF] Couder | [END_REF]. While heated, it gets thinner at the center, as already noticed with the horizontal scan. This is also confirmed by the direct thickness measurement versus time (see Fig. 4b) at the position where the film is heated. The zone above the heated region inflates upon illumination compared with what is observed without injection, whereas the lower zone gets thinner. The soap film has a more symmetrical shape on both sides of the heating point. The Marangoni effect shades the gravity action. It seems that the lowering of the surface tension due to heating creates a kind of bottleneck that prevents drainage and forces the liquid to circumvent this heated zone. The film gets thinner in the heated region and below, and gets thicker on top of the heated central region as well as on both sides. This is very similar to what has been previously expected or observed for the Marangoni effect but for larger temperature variations and bigger deformations [16,33,34], or even under drying conditions [35,36].

Curiously, on Fig. 4b, it has to be noticed that at the end of the drainage, the thickness of the film is very similar with and without laser illumination. It turns out that the heating mechanism is not dominant in the dynamics of the thinning any more. Actually, the drainage p-4 For other times of drainage, the film has the same vertical shape but the departure from the non heated case is less pronounced. Time evolution of the film thickness at the centre of the film in the heated region (b), with (red curve) and without (blue curve) heating of the film using the direct interferometry technique [24].

of vertical freestanding films in the absence of any perturbation is governed by gravitational, capillary, interfacial forces and thermal convection [8,12,14,[START_REF] Mysels | Soap films, studies of their thinning[END_REF][START_REF] Couder | [END_REF]37]. In our experimental situation, the film is stratified during drainage, with a well defined organization of micelles in layers parallel to the film plane [24]. The exponential behavior describing the thinning of the soap film is due to the structural forces. Here, after typically 15 s, the liquid is almost ejected and the film reaches a steady state with or without heating (see Fig. 4b). This is also confirmed by the thinning dynamics. Heating only generates solution puncture between the layers of micelles. The evacuation of micelle layers, that should be illustrated by a stepwise thinning, has not been observed in the drainage curve [24]. Actually, the film breaks after 25 s, just before the stepwise thinning is supposed to play a major role [38].

The theoretical description of the profile is not as trivial as it may appear. In the absence of heating, the drainage curve described by an exponential function is mainly due to structural forces needed to stabilize the micelle layers and not by gravitational and capillary forces usually incorporated in hydrodynamic models. In the presence of heating, the solution transfer cannot be described by a simple one-dimensional equation: the liquid is necessarily influenced by gravity and part of the heat is evacuated towards the reservoir. The liquid can either flow in the y direction or along the frame as it is expelled in the x direction toward the meniscus. Thus since the liquid in the film can move towards the frame and then flows along it, there is no mass conservation within the film.

Conclusion. -To conclude, we used the high versatility of Young's double slit experiment to perform a differential mapping of a soap film. By local heating of the film by a laser, we measured the Marangoni effect. We experimentally evidence a bottleneck in the heated zone that force the liquid to circumvent this zone leading to an inflation of the film thickness above this zone. These results are in agreement with previous predictions and observations, but the high sensitivity of the technique allows to probe very tiny deformations far below what has been observed previously, and far before the apparition of convection vortices [16,33]. This easy to implement technique shines some new light on the film dynamics in the presence of a local defect and could be applied to investigate [39] or to control [40] interfacial instabilities in thin liquid films. Besides, due to the high precision of the measurement, it is also appropriate for measuring small optical index variation in the case of light guiding in soap films [41] or tiny thermally induced deformations such as capillary

Fig. 1 :

 1 Fig. 1: Experimental set up. A vertical thin soap film is locally heated with a blue laser diode. The differential thickness variation is measured with the interference pattern of a green He-Ne laser in a double slit experiment configuration. The pattern is registered on a CCD line camera connected to a computer.

Fig. 2 :

 2 Fig.2: Principle of the measurement. The change of the film thickness between the two optical interfering beams, induces a modification of the effective optical length. It thus changes the phase difference between the two beams. This leads to a shift of the interference fringes giving access to the thickness gradient of the soap film.

Fig. 3 :

 3 Fig.3: Horizontal mapping of the soap film. Thickness gradient of the soap film in the horizontal direction x at a height y = -30 mm for three different times (5 s-blue, 10 s-red, 15 s-green) and three different thickness. The position x = 0 mm, y = -30 mm corresponds to the zone heated by the blue laser (a). Reconstituted thickness of the soap film from the thickness gradient (b), the solid line is a guide for the eye, thus smoothing the inflections of the curve. Without heating the film thickness is uniform and equals 820 nm at 5 s, 450 nm at 10 s, and 210 nm at 15 s. The film is globally thinner when heated.

Fig. 4 :

 4 Fig.4: Vertical mapping of the soap film. Thickness of the thin soap film in the vertical direction y at a position x = 0, for t = 10 s. The position x = 0 mm, y = -30 mm corresponds to the zone heated by the blue laser. The thickness variation has been reconstituted from the thickness gradient. The dotted line corresponds to the measured film thickness without heating. It has been measured with the double slit technique, at the same times of draining (a). For other times of drainage, the film has the same vertical shape but the departure from the non heated case is less pronounced. Time evolution of the film thickness at the centre of the film in the heated region (b), with (red curve) and without (blue curve) heating of the film using the direct interferometry technique[24].

The authors thank Alexis Keller and Federico Casanova for valuable help, and Jean René Thébault and Alain Faisant for technical assistance.


Melisachew Wudage Chekol
email: melisachew.chekol@inria.fr

Amedeo Napoli
email: amedeo.napoli@inria.fr

An FCA Framework for Knowledge Discovery in SPARQL Query Answers

Formal concept analysis (FCA) is used for knowledge discovery within data. In FCA, concept lattices are very good tools for classification and organization of data. Hence, they can also be used to visualize the answers of a SPARQL query instead of the usual answer formats such as: RDF/XML, JSON, CSV, and HTML. Consequently, in this work, we apply FCA to reveal and visualize hidden relations within SPARQL query answers by means of concept lattices.

Introduction

Recently, the amount of semantically rich data available on the Web has grown considerably. Since the conception of linked data publishing principles, over 295 linked (open) datasets (LOD) have been produced 1 . A reasonable number of these datasets provide endpoints for accessing (querying) their contents. Querying is mainly done through the W3C recommended query language SPARQL. The answers of SPARQL queries have often the following formats: RDF/XML, JSON, CSV, text, TSV, Java Script, XML, Spreadsheet, Ntriples, and HTML. It might be interesting to analyse, mine, and then visualize hidden relations within the answers. These tasks can be carried out using Formal Concept Analysis (FCA).

FCA is used for knowledge discovery within data represented by means of objects and their attributes [START_REF] Ganter | Formal Concept Analysis[END_REF]. Concept lattices can reveal hidden relations within data and can be used for organizing, classifying, and even mining data. A survey of the benefits of FCA to semantic web (SW) and vice versa has been proposed in [START_REF] Sertkaya | A survey on how description logic ontologies benefit from FCA[END_REF] (in particular ontology completion [START_REF] Baader | Completing description logic knowledge bases using formal concept analysis[END_REF]). Additionally, studies in [START_REF] D'aquin | Extracting relevant questions to an RDF dataset using formal concept analysis[END_REF] and [START_REF] Kirchberg | Formal concept discovery in semantic web data[END_REF] are based on FCA for managing SW data. The former provides an entry point to linked data using questions in a way that can be navigated. It gives a transformation of an RDF graph into a formal context where the subject of an RDF triple becomes the object, a composition of the predicate and object of the triple becomes an attribute. The latter obliges the user to specify variables corresponding to objects and attributes of a context. These variables are used to create a SPARQL query which is used to extract content from linked data in order to build the formal context. Following this line, we propose a way to organize Semantic Web data, and more precisely, the organization of SPARQL query answers by means of concept lattices. As a result, the user is able to visualize, navigate and classify the answers w.r.t. their context. For that, we propose the architecture depicted in Figure 1, based on three components which are discussed below:

1. Keyword search: In this component, a keyword (for instance, "14 juillet") is used to search (to find information regarding this word) a specified dataset.

To do so, a URI produced from the keyword is sent to the dataset to check its existence. When this is the case, a CONSTRUCT query containing the keyword is directed to the endpoint of the dataset. The answers are collected and organized to create a formal context as explained in the next section. 2. SELECT Query: This component builds a formal context out of the answers of a SPARQL query. SPARQL queries are converted into CONSTRUCT queries to form RDF graphs from the answers. Again, this will be illustrated in the next section. 3. Variables corresponding to objects and attributes of a formal context: This component enables the user to precisely specify the objects and attributes of the formal context. Out of which a SPARQL query is formed and sent to a chosen SPARQL endpoint. The answers of the query are collected to build a formal context. From that, a concept lattice is constructed. This is the approach considered by the authors in [START_REF] Kirchberg | Formal concept discovery in semantic web data[END_REF]. An example is proposed in the next section.

In each case, the objective is to build a formal context and then to build the associated concept lattice.

Proposal

A formal context represents data using objects and their attributes. Formally, it is a triple K = (G, M, I) where G is a set of objects, M is a set of attributes, and I ⊆ G × M is a binary relation. A derivation operator ( ′ ) is used to compute formal concepts of a context. Given a set of objects A, a derivation operator ′ computes the maximal set of attributes shared by objects in A and is denoted by A ′ (this is done dually with set of attributes B). A formal concept is a pair (A, B) where A ′ = B and B ′ = A. A set of formal concepts ordered with the set inclusion relation form a concept lattice [START_REF] Ganter | Formal Concept Analysis[END_REF].

Definition 1 (RDF as a Formal Context). Given an RDF graph G and a transformation function σ, a formal context is obtained from G as follows:

-

If o 1 , rdf : type, C ∈ G, then σ( o 1 , rdf : type, C ) = C o 1 x -If o 1 , R, o 2 ∈ G, then σ( o 1 , R, o 2 ) = ∃R.⊤ ∃R -.⊤ o 1 x o 2 x
We consider a core fragment of RDFS called ρdf [START_REF] Muñoz | Minimal deductive systems for RDF[END_REF] which contains the minimal vocabulary, ρdf = {sp,sc,type,dom,range}, where sp denotes the subproperty relation, sc is subclass, and dom stands for domain. This fragment was proven to be minimal and well-behaved in [START_REF] Muñoz | Minimal deductive systems for RDF[END_REF]. Its semantics corresponds to that of full RDFS. Triples containing schema information are transformed as: The users above are able to build a formal context from an RDF graph or a set of SPARQL query answers. Then, there are several algorithms that can compute the concept lattice associated with a formal context and that can be used in our framework.

1. If C 1 , rdfs : subClassOf, C 2 ∈ G, then σ( C 1 , rdfs : subClassOf, C 2 ) = ∀o ∈ G. (o, C 1 ) ∈ I ⇒ (o, C 2 ) ∈ I. 2. If R 1 , rdfs : subPropertyOf, R 2 ∈ G, then σ( R 1 , rdfs : subPropertyOf, R 2 ) = ∀o 1 , o 2 ∈ G. (o 1 , ∃R 1 .⊤) ∈ I ⇒ (o 1 , ∃R 2 .⊤) ∈ I.
Example: consider a SPARQL query that selects film titles (as objects of the formal context) and genres (as attributes) from DBpedia to populate a formal context. Consequently, this formal context is shown in Figure 2.

A possible concept lattice obtained from the formal context associated with the query answers is depicted in Figure 3. Now we have a classification of the results of the query w.r.t. a given topic or constraint. Additionally, this is exactly the same thing as if we were querying the Web with Google and here we have a classification of the answers w.r.t. a user constraints.

Conclusion

SPARQL query answers are provided in different formats (RDF/XML, CSV, JSON, TTL, and others), which do not reveal hidden semantics in the answers. This is an ongoing work and we are currently implementing the procedure. We should investigate how well it scales, given the size of SPARQL query answers over linked data. Overall, this work shows some of the benefits of FCA that can be provided to the semantic web.

Fig. 1 :

 1 Fig. 1: Architecture of SPARQL answers organization.

3 .

 3 If R, rdfs : domain, C ∈ G, then σ( R, rdfs : domain, C ) = ∀o 1 ∈ G. (o 1 , ∃R.⊤) ∈ I ⇒ (o 1 , C) ∈ I. 4. If R, rdfs : range, C ∈ G, then σ( R, rdfs : range, C ) = ∀o 2 ∈ G. (o 1 , ∃R -.⊤) ∈ I ⇒ (o 2 , C) ∈ I.

Fig. 2 :

 2 Fig. 2: A part of the formal context associated with the query.

Fig. 3 :

 3 Fig. 3: Concept lattice of DBpedia movies and genres. Concept lattices are useful in this regard. In this work, we used concept lattices to hierarchically organize and analyse the content of query answers.This is an ongoing work and we are currently implementing the procedure. We should investigate how well it scales, given the size of SPARQL query answers over linked data. Overall, this work shows some of the benefits of FCA that can be provided to the semantic web.

http://linkeddata.org/