Büchi-Kamp Theorems for 1-clock ATA

This paper investigates Kamp-like and Büchi-like theorems for 1-clock Alternating Timed Automata (1-ATA) and its natural subclasses. A notion of 1-ATA with loop-free-resets is defined. This automaton class is shown to be expressively equivalent to the temporal logic RatMTL which is MTL[FI] extended with a regular expression guarded modality. Moreover, a subclass of future timed MSO with k-variable-connectivity property is introduced as logic QkMSO. In a Kamp-like result, it is shown that RatMTL is expressively equivalent to QkMSO. As our second result, we define a notion of conjunctive-disjunctive 1-clock ATA (C⊕D 1-ATA). We show that C⊕D 1-ATA with loop-free-resets are expressively equivalent to the sublogic FRatMTL of RatMTL. Moreover FRatMTL is expressively equivalent to Q2MSO, the two-variable connected fragment of QkMSO. The full class of 1-ATA is shown to be expressively equivalent to RatMTL extended with fixed point operators.