We introduce regular graph constraints and explore their decidability properties. The motivation for regular graph constraints is 1) type checking of changing types of objects in the presence of linked data structures, 2) shape analysis techniques, and 3) generalization of similar constraints over trees and grids. Typestate checking for recursive and potentially cyclic data structures requires verifying the validity of implication for regular graph constraints. The implication of regular graph constraints also arises in shape analysis algorithms such as role-analysis and some analyses based on threevalued logic. Over the class of lists regular graph constraints reduce to a nondeterministic finite state automaton as a special case. Over the class of trees the constraints reduce to a nondeterministic top-down tree automaton, and over the class of grids our constraints reduce to domino system and tiling problems. We define a subclass of graphs called heaps as an abstraction of the data structures that a program constructs during its execution. We show that satisfiability of regular graph constraints over the class of heaps is decidable. However, determining the validity of implication for regular graph constraints over the class of heaps is undecidable. The undecidability of implication is the central result of the paper. The result is somewhat surprising because our simple constraints are strictly less expressive than existential monadic second-order logic over graphs. In the key step of our proof we introduce the class of corresponder graphs which mimic solutions of Post correspondence problem instances. We show undecidability by exhibiting a characterization of corresponder graphs in terms of presence and absence of homomorphisms to a finite number of fixed graphs. The undecidability of implication of regular graph constraints implies that there is no algorithm that will verify that procedure preconditions are met or that the invariants are maintained when these properties are expressed in any specification language at least as expressive as regular graph constraints. *
We have subsequently studied a simplification of role analysis constraints and showed a characterization of such constraints using formulas REF .
Typestate Checking and Regular Graph Constraints
Computer Science