Proving Non-Deterministic Computations in Agda

Abstract

We investigate proving properties of Curry programs using Agda. First, we address the functional correctness of Curry functions that, apart from some syntactic and semantic differences, are in the intersection of the two languages. Second, we use Agda to model non-deterministic functions with two distinct and competitive approaches incorporating the non-determinism. The first approach eliminates non-determinism by considering the set of all non-deterministic values produced by an application. The second approach encodes every non-deterministic choice that the application could perform. We consider our initial experiment a success. Although proving properties of programs is a notoriously difficult task, the functional logic paradigm does not seem to add any significant layer of difficulty or complexity to the task.Comment: In Proceedings WLP'15/'16/WFLP'16, arXiv:1701.0014

CurryCheck: Checking Properties of Curry Programs

“…If there is a file proof-sortlength.agda, containing a proof for the above property ( [10] addresses techniques how to prove such properties in the dependently typed language Agda), CurryCheck considers this property as valid and does not check it. Moreover, it uses it to simplify other properties to be tested.…”

“…For instance, [27] discusses techniques to use the Isabelle/HOL proof assistant to verify purely functional properties inspired by QuickCheck. [10] describes a method to prove non-deterministic computations by translating Curry programs into Agda programs. Since these proofs can be verified by the Agda compiler, CurryCheck can test the validity of a given proof file by simply invoking the Agda compiler.…”

“…Since there are many different possibilities to prove such properties, ranging from manual proofs to interactive proof assistants and fully automatic provers, CurryCheck does not enforce a particular proof technique. Instead, CurryCheck trusts the programmer and uses a naming convention for files containing proofs: if there is a file with name proof-t. * , CurryCheck assumes that this file contains a valid proof for property t. For instance, the following property states that sorting a list does not change its length: If there is a file proof-sortlength.agda, containing a proof for the above property ( [10] addresses techniques how to prove such properties in the dependently typed language Agda), CurryCheck considers this property as valid and does not check it. Moreover, it uses it to simplify other properties to be tested.…”

ICurry

“…Functional logic languages [6] provide fast software prototyping and development, simple elegant solutions to otherwise complicated problems, a tight integration between specifications and code [7], and an ease of provability [8,16] unmatched by other programming paradigms. Not surprisingly, these advantages place heavy demands on their implementation.…”