Memoryful branching-time logic

Traditional branching-time logics such as CTL* are memoryless: once a path in the computation tree is quantified at a given node, the computation that led to that node is forgotten. Recent work in planning suggests that CTL* cannot easily express temporal goals that refer to whole computations. Such goals require memoryful quantification of paths. With such a memoryful quantification, Eψ holds at a node s of a computation tree if there is a path π starting at the root of the tree and going through s such that . satisfies the linear-time formula ψ. We define the memoryful branching-time logic mCTL* and study its expressive power and algorithmic properties. We show that mCTL* is as expressive, but exponentially more succinct, than CTL*, and that the ability of mCTL* to refer to the present is essential for this equivalence. From the algorithmic point of view, while the satisfiability problem for mCTL* is 2EXPTIME-complete - not harder than that of CTL*, its model-checking problem is EXPSPACE-complete - exponentially harder than that of CTL*. The upper bounds are obtained by extending the automata-theoretic approach to handle memoryful quantification, and are much more efficient than these obtained by translating mCTL* to branching logics with past. The EXPSPACE lower bound for the model-checking problem applies already to formulas of restricted form (in particular, to AGEψ, which is useful for specifying possibility properties), and implies that reasoning about a memoryful branching-time logic is harder than reasoning about the linear-time logic of its path formulas.