The Clock Constraint Specification Language (CCSL) is a formalism for specifying logical-time constraints on events for the design of real-time embedded systems. A central verification problem of CCSL is to check whether events are schedulable under logical constraints. Although many efforts have been made addressing this problem, the problem is still open. In this paper, we show that the bounded scheduling problem is NP-complete and then propose an efficient SMT-based decision procedure which is sound and complete. Based on this decision procedure, we present a sound algorithm for the general scheduling problem. We implement our algorithm in a prototype tool and illustrate its utility in schedulability analysis in designing real-world systems and automatic proving of algebraic properties of CCSL constraints. Experimental results demonstrate its effectiveness and efficiency.