# Reasoning about Algebraic Data Types with Abstractions

## T. Pham, A. Gacek, M. Whalen

### Journal of Automated Reasoning (2016). doi:10.1007/s10817-016-9368-2

Reasoning about functions that operate over algebraic data types is an important problem for a large variety of applications. One application of particular interest is network applications that manipulate or reason about complex message structures, such as XML messages. This paper presents a decision procedure for reasoning about algebraic data types using abstractions that are provided by catamorphisms: fold functions that map instances of algebraic data types to values in a decidable domain. We show that the procedure is sound and complete for a class of catamorphisms that satisfy a generalized sufficient surjectivity condition. Our work extends a previous decision procedure that unrolls catamorphism functions until a solution is found.

We use the generalized sufficient surjectivity condition to address an incompleteness in the previous unrolling algorithm (and associated proof). We then propose the categories of monotonic and associative catamorphisms, which we argue provide a more intuitive inclusion test than the generalized sufficient surjectivity condition. We use these notions to address two open problems from previous work: (1) we provide a bound, with respect to formula size, on the number of unrollings necessary for completeness, showing that it is linear for monotonic catamorphisms and exponentially small for associative catamorphisms, and (2) we demonstrate that associative catamorphisms can be combined within a formula while preserving completeness. Our combination results extend the set of problems that can be reasoned about using the catamorphism-based approach.

We also describe an implementation of the approach, called RADA, which accepts formulas in an extended version of the SMT-LIB 2.0 syntax. The procedure is quite general and is central to the reasoning infrastructure for Guardol, a domain-specific language for reasoning about network guards.