Henkin sentences and local reflection principles for Rosser provability

In this paper, we investigate Rosser-type Henkin sentences, namely, sentences asserting their own provability in the sense of Rosser, and local reflection principles based on Rosser provability predicates. First, we give a necessary and sufficient condition that a sentence is a Rosser-type Henkin sentence of some Rosser provability predicate, and prove that any negated Rosser sentence can be a Rosser-type Henkin sentence. Secondly, we prove the existence of a Rosser provability predicate whose Rosser-type Henkin sentences are all provable or refutable. Thirdly, we solve the question raised by Shavrukov, and give a Rosser provability predicate whose local reflection principle is strictly weaker than the usual one. At last, we investigate the hierarchy of partial local reflection principles based on Rosser provability predicates.