C. Ansótegui et al. / Random SAT Instances à la Carte 1 Random SAT Instances à la Carte 1 Carlos ANSÓTEGUI a , Maria Luisa BONET b and Jordi LEVY b a DIEI, UdL, Lleida, Spain. carlos@diei.udl.cat b LSI, UPC, Barcelona, Spain. bonet@lsi.upc.edu c IIIA, CSIC, Bellaterra, Spain. levy@iiia.csic.es Abstract. Many studies focus on the generation of hard SAT instances. The hard- ness is usually measured by the time it takes SAT solvers to solve the instances. In this preliminary study, we focus on the generation of instances that have compu- tational properties that are more similar to real-world instances. In particular, in- stances with the same degree of difficulty, measured in terms of the tree-like res- olution space complexity. It is known that industrial instances, even with a great number of variables, can be solved by a clever solver in a reasonable amount of time. One of the reasons may be their relatively small space complexity, compared with randomly generated instances. We provide two generation methods of k-SAT instances, called geometrical and the geo-regular, as generalizations of the uniform and regular k-CNF generators. Both are based on the use of a geometric probability distribution to select vari- ables. We study the phase transition phenomena and the hardness of the generated instances as a function of the number of variables and the base of the geometric distribution. We prove that, with these two parameters we can adjust the difficulty of the problems in the phase transition point. We conjecture that this will allow us to generate random instances more similar to industrial instances, of interest for testing purposes. Keywords. Random SAT Models, Satisfiability Introduction SAT is a central problem in computer science. Many other problems in a wide range of areas can be solve by encoding them into boolean formulas, and then using state-of-the- art SAT solvers. The general problem is NP-complete in the worst case, and in fact a big percentage of formulas (randomly generated instances) need exponential size resolution proofs to be shown unsatisfiable [CS88,BSW01,BKPS02]. Therefore, solvers based on resolution need exponential time to decide their unsatisfiability. Nevertheless, state-of- the-art solver have been shown of practical use working with real-world instances. As a consequence the development of these tools has generated a lot of interest. The celebration of SAT competitions has become an essential method to validate techniques and lead the development of new solvers. In these competitions there are three categories of benchmarks, randomly generated, crafted, and industrial instances. It 1 Research partially supported by the research projects TIN2007-68005-C04-01/02/03 and TIN2006-15662- C02-02 funded by the CICYT. The first author was partially supported by the José Castillejo 2007 program funded by the Ministerio de Educación y Ciencia.