TY - JOUR
AU - Kousha Etessami
AU - Moshe Vardi
AU - Thomas Wilke
PY - 1997/01/05
Y2 - 2019/10/20
TI - First-Order Logic with Two Variables and Unary Temporal Logic
JF - BRICS Report Series
JA - BRICS
VL - 4
IS - 5
SE - Articles
DO - 10.7146/brics.v4i5.18784
UR - https://tidsskrift.dk/brics/article/view/18784
AB - We investigate the power of first-order logic with only two variables overomega-words and finite words, a logic denoted by FO2. We prove that FO2 canexpress precisely the same properties as linear temporal logic with only the unary temporal operators: “next”, “previously”, “sometime in the future”, and “sometime in the past”, a logic we denote by unary-TL. Moreover, our translation from FO2 to unary-TL converts every FO2 formula to an equivalent unary-TL formula that is at most exponentially larger, and whose operator depth is at most twice the quantifier depth of the first-order formula. We show that this translation is optimal.While satisfiability for full linear temporal logic, as well as forunary-TL, is known to be PSPACE-complete, we prove that satisfiabilityfor FO2 is NEXP-complete, in sharp contrast to the fact that satisfiabilityfor FO3 has non-elementary computational complexity. Our NEXP timeupper bound for FO2 satisfiability has the advantage of being in terms ofthe quantifier depth of the input formula. It is obtained using a small model property for FO2 of independent interest, namely: a satisfiable FO2 formula has a model whose “size” is at most exponential in the quantifier depth of the formula. Using our translation from FO2 to unary-TL we derive this small model property from a corresponding small model property for unary-TL. Our proof of the small model property for unary-TL is based on an analysis of unary-TL types.
ER -