FUZZY INTEGRALS AND ERROR CORRECTION IN A TELEPHONE NETWORK WITH SENSITIVE USERS ANDREA SGARRO DSM (Dep. of Math. Sciences) University of Trieste, 34100 Trieste, Italy Email: sgarro@univ.trieste.it LUCA BORTOLUSSI DIMI (Dep. of Math. and Comp. Science) University of Udine, 33100 Udine, Italy Email: luca.bortolussi@dimi.uniud.it Abstract The possibilistic approach to cod- ing is put to use to tackle the prob- lem of automatic correction of emer- gency telephone numbers which have been incorrectly digited. The rele- vant tools are fuzzy graphs and fuzzy integrals. The case of Italian emer- gency numbers is dealt with. Keywords: possibilistic coding, error-correcting codes, information theory, fuzzy graphs, fuzzy integrals, proximimity relations. to the memory of Pietro Benvenuti 1 Introduction. In [5] we tackled the problem of devising new shapes for cell-phone keyboards, such as to have good error correction performances whenever a “not too wrong” phone number has been inadvertently digited; our proposal turned out to be a circular nine-key keyboard with the missing key in the centre. In the meanwhile, cell-phones have been put into production which are circular with all the ten keys in circle; actually, they do not rec- ommend themselves in any special way as for their error correction properties, and the circular shape has been chosen just because it is nice and reminds one of the good old phones. One of the aims of [5] was to show that the theoretic possibilistic framework for coding put forward in [6] (cf also [3], [7] and [9]) does have practical applications; the tech- nical tool used there were fuzzy graphs, i.e. complete simple graphs with k vertices, whose (k 2 -k)/2 edges are given weights in the inter- val [0, 1]; in fuzzy set theory these weights are interpreted as degrees of membership to the edge set, cf e.g. [2]. Fuzzy graphs are used to model proximities (and therefore easiness of digiting an incorrect number), or, symmetri- cally, remoteness on the telephone keyboard; cf Section 2. With respect to [5], in this paper we take a dif- ferent and more matter-of-fact point of view, which was unfortunately suggested to us by a fatal accident that took place in the Alps in winter 2003: a young skier, after the last evening control of the downhill slope, had a serious fall; before fainting he tried to call an emergency number, but got wrong the last digit, so that nobody came to his rescue (the wrong number remained stored in the mem- ory of the phone). The problems arising here are the following: assuming that usual square cell-phones are used, so that it is “quite pos- sible”, say, to digit key 2 rather than the neighbouring key 1, and it is “slightly possi- ble” to digit 5 rather than its diagonal neigh- bour 1, does a phone-number assignment al- low error correction, at least limited to emer- gency numbers? Can one re-distribute phone- numbers so as to ensure error correction in case of emergency? How many digits should phone-numbers have so as to achieve satis- factory error correction capabilities? These questions are (partly) answered in Section 3. With respect to [5], we shall need an addi-