On Odd and Even Cycles in Normal Logic Programs

An odd cycle of a logic program is a simple cycle that has an odd number of negative edges in the dependency graph of the program. Similarly, an even cycle is one that has an even number of negative edges. For a normal logic program that has no odd cycles, while it is known that such a program always has a stable model, and such a stable model can be computed in polynomial time, we show in this paper that checking whether an atom is in a stable model is NP-complete, and checking whether an atom is in all stable models is co-NP complete, both are the same as in the general case for normal logic programs. Furthermore, we show that if a normal logic program has exactly one odd cycle, then checking whether it has a stable model is NP-complete, again the same as in the general case. For normal logic programs with a fixed number of even cycles, we show that there is a polynomial time algorithm for computing all stable models. Furthermore, this polynomial time algorithm can be improved significantly if the number of odd cycles is also fixed.