An Improved Unrolling-Based Decision Procedure for Algebraic Data Types

Reasoning about algebraic data types and functions that operate over these data types is an important problem for a large variety of applications. In this paper, we present a decision procedure for reasoning about data types using abstractions that are provided by catamorphisms: fold functions that map instances of algebraic data types into values in a decidable domain. We show that the procedure is sound and complete for a class of monotonic catamorphisms. Our work extends a previous decision procedure that solves formulas involving algebraic data types via successive unrollings of catamorphism functions. First, we propose the categories of monotonic catamorphisms and associative-commutative catamorphisms, which we argue provide a better formal foundation than previous categorizations of catamorphisms. We use monotonic catamorphisms to fix an incompleteness in the previous unrolling algorithm (and associated proof). We then use these notions to address two open problems from previous work: (1) we provide a bound on the number of unrollings necessary for completeness, showing that it is exponentially small with respect to formula size for associative-commutative catamorphisms, and (2) we demonstrate that associative-commutative catamorphisms can be combined within a formula whilst preserving completeness.