A Decidable Logic for Tree Data-Structures with Measurements Xiaokang Qiu and Yanjun Wang Purdue University fxkqiu,
[email protected] Abstract. We present Dryaddec, a decidable logic that allows reason- ing about tree data-structures with measurements. This logic supports user-defined recursive measure functions based on Max or Sum, and re- cursive predicates based on these measure functions, such as AVL trees or red-black trees. We prove that the logic's satisfiability is decidable. The crux of the decidability proof is a small model property which al- lows us to reduce the satisfiability of Dryaddec to quantifier-free lin- ear arithmetic theory which can be solved efficiently using SMT solvers. We also show that Dryaddec can encode a variety of verification and synthesis problems, including natural proof verification conditions for functional correctness of recursive tree-manipulating programs, legal- ity conditions for fusing tree traversals, synthesis conditions for con- ditional linear-integer arithmetic functions. We developed the decision procedure and successfully solved 220+ Dryaddec formulae raised from these application scenarios, including verifying functional correctness of programs manipulating AVL trees, red-black trees and treaps, check- ing the fusibility of height-based mutually recursive tree traversals, and counterexample-guided synthesis from linear integer arithmetic specifi- cations. To our knowledge, Dryaddec is the first decidable logic that can solve such a wide variety of problems requiring flexible combination of measure-related, data-related and shape-related properties for trees. 1 Introduction Logical reasoning about tree data-structures has been needed in various applica- tion scenarios such as program verification [32,24,26,42,49,4,14], compiler opti- mization [17,18,9,44] and webpage layout engines [30,31].