A computational logic handbook
A computational logic handbook
Introduction to algorithms
Powerlist: a structure for parallel recursion
ACM Transactions on Programming Languages and Systems (TOPLAS)
Information Processing Letters
An Industrial Strength Theorem Prover for a Logic Based on Common Lisp
IEEE Transactions on Software Engineering
Constructors can be partial, too
Automated reasoning and its applications
Mechanical Verification of Adder Circuits using Rewrite RuleLaboratory
Formal Methods in System Design
Parlists - A Generalization of Powerlists
Euro-Par '97 Proceedings of the Third International Euro-Par Conference on Parallel Processing
Parameterized congruences in ACL2
ACL2 '06 Proceedings of the sixth international workshop on the ACL2 theorem prover and its applications
Rewriting with Equivalence Relations in ACL2
Journal of Automated Reasoning
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In Misra (ACM Trans Program Lang Syst 16(6):1737---1767, 1994), Misra introduced the powerlist data structure, which is well suited to express recursive, data-parallel algorithms. Moreover, Misra and other researchers have shown how powerlists can be used to prove the correctness of several algorithms. This success has encouraged some researchers to pursue automated proofs of theorems about powerlists (Kapur 1997; Kapur and Subramaniam 1995, Form Methods Syst Des 13(2):127---158, 1998). In this paper, we show how ACL2 can be used to verify theorems about powerlists. We depart from previous approaches in two significant ways. First, the powerlists we use are not the regular structures defined by Misra; that is, we do not require powerlists to be balanced trees. As we will see, this complicates some of the proofs, but on the other hand it allows us to state theorems that are otherwise beyond the language of powerlists. Second, we wish to prove the correctness of powerlist algorithms as much as possible within the logic of powerlists. Previous approaches have relied on intermediate lemmas which are unproven (indeed unstated) within the powerlist logic. However, we believe these lemmas must be formalized if the final theorems are to be used as a foundation for subsequent work, e.g., in the verification of system libraries. In our experience, some of these unproven lemmas presented the biggest obstacle to finding an automated proof. We illustrate our approach with two case studies involving Batcher sorting and prefix sums.