Efficient algorithms for path system problems and applications to alternating and time-space complexity classes

Let SPS(f(n)) denote the solvable path system problem for path systems of bandwidth f(n) and SPS (f(n)) the corresponding problem for monotone systems. Let DTISP (poly, f(n)) denote the polynomial time and simultaneous f(n) space class and SC = UkDTISP (poly, logkn). Let ASPACE (f(n)) denote the sets accepted by f(n) space bounded alternating TMs and ASPACE (f(n)) the corresponding one-way TM family. Then, for "well-behaved" functions fεO(n)-o(log n), (1) SPS (f(n)) is ≤log-complete for DTISP (poly, f(n)), (2) {SPS(f(n)k)}k≥1 is ≤log-complete for ASPACE (logf(n)), (3) {SPS (f(n)k)}k≥1 is ≤log-complete for ASPACE (log f(n)), (4) SPS(f(n)) ε DSPACE(f(n) × log n), (5) ASPACE(log f(n)) ⊆ UkDSPACE(f(n)k), and (6) SC = CLOSURE ≤log(ASPACE(log log n)).