Decidability w.r.t. Logical Consecutions of Linear Temporal Logic Extended by Since and Previous

This paper aims to prove that the linear temporal logic LTL ^{u,s}_{n,n-1} (N), which is an extension of the standard linear temporal logic LTL by operations Since and Previous (LTL itself, as standard, uses only Until and Next) and is based on the frame of all natural numbers N, as generating Kripke/Hintikka structure, is decidable w.r.t. admissible consecutions (inference rules). We find an algorithm recognizing consecutions admissible in LTL ^{u,s}_{n,n-1} (N). As a consequence this algorithm solves satisfiability problem and shows that LTL ^{u,s}_{n,n-1} (N) itself is decidable, despite LTL ^{u,s}_{n,n-1} (N) does not have the finite model property.