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This paper upgrades Regular Linear Temporal Logic (RLTL) with past operators and complementation. RLTL is a temporal logic that extends the expressive power of linear temporal logic (LTL) to all ω-regular languages. The syntax of RLTL consists of an algebraic signature from which expressions are built. In particular, RLTL does not need or expose fix-point binders (like linear time μ-calculus), or automata to build and instantiate operators (like ${\textrm{ETL}_*}$). Past operators are easily introduced in RLTL via a single previous-step operator for basic state formulas. The satisfiability and model checking problems for RLTL are PSPACE-complete, which is optimal for extensions of LTL. This result is shown using a novel linear size translation of RLTL expressions into 2-way alternating parity automata on words. Unlike previous automata-theoretic approaches to LTL, this construction is compositional (bottom-up). As alternating parity automata can easily be complemented, the treatment of negation is simple and does not require an upfront transformation of formulas into any normal form.