DOI: 10.1142/S021848851100685X International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 19, No. 1 (2011) 11–26 c  World Scientiﬁc Publishing Company A PDL APPROACH FOR QUALITATIVE VELOCITY A. BURRIEZA Dept. Philosophy, University of M´ alaga, 29071 M´ alaga, Spain burrieza@uma.es E. MU ˜ NOZ-VELASCO * and M. OJEDA-ACIEGO † Dept. Applied Mathematics, University of M´ alaga, 29071 M´ alaga, Spain * emilio@ctima.uma.es † aciego@ctima.uma.es Received 10 May 2010 Revised 11 November 2010 We introduce the syntax, semantics, and an axiom system for a PDL-based extension of the logic for order of magnitude qualitative reasoning, developed in order to deal with the concept of qualitative velocity, which together with qualitative distance and orientation, are important notions in order to represent spatial reasoning for moving objects, such as robots. The main advantages of using a PDL-based approach are, on the one hand, all the well-known advantages of using logic in AI, and, on the other hand, the possibility of constructing complex relations from simpler ones, the ﬂexibility for using diﬀerent levels of granularity, its possible extension by adding other spatial components, and the use of a language close to programming languages. Keywords : Qualitative reasoning; order-of-magnitude reasoning; propositional dynamic logic; hybrid logic. 1. Introduction Qualitative reasoning, QR, tries to ﬁnd formal models that exhibit qualities of human thinking. For example, we do not need to know the exact value of velocity, distance, position of our car in order to park it when we are so lucky to ﬁnd a parking spot in the city center. A form of QR is order-of-magnitude reasoning, where the quantitative information is substituted by a ﬁnite number of qualitative classes, for example: zero, small, medium, and large ; moreover, some relations between the qualitative classes, such as negligibility, closeness . . . , may be deﬁned. 1,2 The level of granularity, that is, the number of qualitative classes used, depends on the problem in question. Recent applications of order-of-magnitude reasoning can be seen, for example, in Refs. 3 and 4. 11