Formal Reasoning about Efficient Data Structures: A Case Study in ACL2

We describe in this paper the formal verification, using the ACL2 system, of a syntactic unification algorithm where terms are represented as directed acyclic graphs (dags) and these graphs are stored in a single-threaded object (stobj). The use of stobjs allows destructive operations on data (thus improving the performance of the algorithm), while maintaining the applicative semantics of ACL2. We intend to show how ACL2 provides an environment where execution of algorithms with efficient data structures and formal reasoning about them can be carried out.Ministerio de Ciencia y Tecnología TIC2000-1368-C03-0

Mechanized semantics

The goal of this lecture is to show how modern theorem provers---in this case, the Coq proof assistant---can be used to mechanize the specification of programming languages and their semantics, and to reason over individual programs and over generic program transformations, as typically found in compilers. The topics covered include: operational semantics (small-step, big-step, definitional interpreters); a simple form of denotational semantics; axiomatic semantics and Hoare logic; generation of verification conditions, with application to program proof; compilation to virtual machine code and its proof of correctness; an example of an optimizing program transformation (dead code elimination) and its proof of correctness

Safety and conservativity of definitions in HOL and Isabelle/HOL

Definitions are traditionally considered to be a safe mechanism for introducing concepts on top of a logic known to be consistent. In contrast to arbitrary axioms, definitions should in principle be treatable as a form of abbreviation, and thus compiled away from the theory without losing provability. In particular, definitions should form a conservative extension of the pure logic. These properties are crucial for modern interactive theorem provers, since they ensure the consistency of the logic, as well as a valid environment for total/certified functional programming. We prove these properties, namely, safety and conservativity, for Higher-Order Logic (HOL), a logic implemented in several mainstream theorem provers and relied upon by thousands of users. Some unique features of HOL, such as the requirement to give non-emptiness proofs when defining new types and the impossibility to unfold type definitions, make the proof of these properties, and also the very formulation of safety, nontrivial. Our study also factors in the essential variation of HOL definitions featured by Isabelle/HOL, a popular member of the HOL-based provers family. The current work improves on recent results which showed a weaker property, consistency of Isabelle/HOL’s definitions

Verification of full functional correctness for imperative linked data structures

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 207-222).We present the verification of full functional correctness for a collection of imperative linked data structures implemented in Java. A key technique that makes this verification possible is a novel, integrated proof language that we have developed within the context of the Jahob program verification system. Our proof language allows us to embed proof commands directly within the program, making it possible to reason about the behavior of the program in its original context. It also allows us to effectively leverage Jahob's integrated reasoning system. Unlike conventional program verification systems that rely on a single monolithic prover, Jahob includes interfaces to a diverse collection of specialized automated reasoning systems-automated theorem provers, decision procedures, and program analyses-that work together to prove the verification conditions that the system automatically generates. Our proof language enables the developer to direct the efforts of these automated reasoning systems to successfully verify properties that the system is unable to verify without guidance. Our specifications characterize the behavior of the data structures in terms of their abstract state, resulting in verified interfaces that can be used to reason about the behavior of the data structures without revealing the underlying representation. The results demonstrate the effectiveness of our proof language and integrated reasoning approach, and provide valuable insight into the specification and verification of imperative linked data structures.by Karen K. Zee.Ph.D