Abstract: A dichotomy theorem for a class of decision problems is a result asserting that certain problems in the
class are solvable in polynomial time, while the rest are NP-complete. The first remarkable such dichotomy
theorem was proved by Schaefer in 1978. It concerns the class of generalized satisfiability problems Sat?S{\L},
whose input is a CNF?S{\L}-formula, i.e., a formula constructed from elements of a fixed set S of generalized
connectives using conjunctions and substitutions by variables. Here, we investigate the complexity of
minimal satisfiability problems Min Sat?S{\L}, where S is a fixed set of generalized connectives. The input to
such a problem is a CNF?S{\L}-formula and a satisfying truth assignment; the question is to decide whether
there is another satisfying truth assignment that is strictly smaller than the given truth assignment with
respect to the coordinate-wise partial order on truth assignments. Minimal satisfiability problems were first
studied by researchers in artificial intelligence while investigating the computational complexity of prop-
ositional circumscription. The question of whether dichotomy theorems can be proved for these problems
was raised at that time, but was left open. We settle this question affirmatively by establishing a dichotomy
theorem for the class of all Min Sat?S{\L}-problems, where S is a finite set of generalized connectives. We also
prove a dichotomy theorem for a variant of Min Sat?S{\L} in which the minimization is restricted to a subset of
the variables, whereas the remaining variables may vary arbitrarily (this variant is related to extensions of
propositional circumscription and was first studied by Cadoli). Moreover, we show that similar dichotomy
theorems hold also when some of the variables are assigned constant values. Finally, we give simple criteria that tell apart the polynomial-time solvable cases of these minimal satisfiability problems from the NP-
complete ones.
