HOL (system)

HOL (system)
Name	HOL
Developer	Various academic groups
Released	1980s
Programming language	ML family
Operating system	Unix-like, Windows
Genre	Theorem prover
License	Various

HOL (system) is a family of interactive theorem proving systems for higher-order logic developed since the 1980s. It has been used in formal verification projects at institutions such as University of Cambridge, University of Cambridge Computer Laboratory, Computer Laboratory, University of Cambridge Mathematical Laboratory, University of Cambridge Siddons and research labs including University of Cambridge collaborators, industrial partners like Intel Corporation, and government-funded initiatives such as UK Research and Innovation and European Research Council. The systems emphasize a small trusted kernel, extensible proof procedures, and embeddings of mathematical theories and programming language semantics.

History

HOL traces roots to the work of researchers at Cambridge University and Edinburgh University in the 1980s, influenced by foundational developments at Stanford University, Princeton University, and MIT. Early milestones include the development of LCF-style theorem proving techniques pioneered at Stanford University and implementations in ML at University of Edinburgh. Subsequent projects involved collaborations with University of Cambridge, University of Oxford, Carnegie Mellon University, University of Leeds, and industrial partners such as Intel Corporation and IBM. Major verification efforts using HOL systems were carried out for hardware designs commissioned by DARPA, European Space Agency, and national research councils including EPSRC. Conferences and workshops such as CADE, IJCAR, CADE-XXI, and CPP chronicle the system's evolution.

Architecture and Components

HOL systems follow an LCF-style architecture with a small trusted kernel and user-level libraries maintained by groups at University of Cambridge, University of Cambridge Computer Laboratory, University of Cambridge Department of Computer Science, University of Cambridge Department of Engineering, and collaborators. Core components include a proof kernel, tactic language, parser/printer, theory manager, and primitive inference rules leveraging the ML family such as Standard ML and OCaml. Implementations provide libraries for number theory, list and set theories, and models of computation developed by teams at Cambridge University, University of Oxford, MIT, Harvard University, and ETH Zurich. Interfaces integrate with editors and environments from Microsoft Visual Studio, Emacs, and Eclipse.

Proof Language and Logic

HOL systems are based on classical higher-order logic with simple type theory, integrating features formalized in works associated with Alonzo Church and influenced by systems at Princeton University and University of Cambridge. The proof language exposes primitive inference rules and programmable tacticals implemented in Standard ML, OCaml, or dialects used by groups at INRIA, TU Munich, and SRI International. The logic supports inductive definitions, recursive functions, and axiomatic type classes developed in collaboration with researchers at University of Cambridge, Technische Universität München, and University of Cambridge Computer Laboratory. Formalizations of semantics for languages such as C, ARM, x86-64, RISC-V, and models linked to projects at ARM Holdings and Intel Corporation appear in HOL libraries.

Implementations and Variants

Notable implementations and forks were produced by research groups at University of Cambridge, University of Cambridge Computer Laboratory, Computer Laboratory, University of York, University of Edinburgh, Purdue University, Carnegie Mellon University, and Intel Corporation. Variants include HOL88, HOL90, HOL4, ProofPower HOL, and HOL Light, each associated with teams at University of Cambridge, University of Cambridge Computer Laboratory, University of Cambridge Department of Computer Science, Royal Holloway, University of London, Imperial College London, and University of Cambridge. HOL Light was developed by researchers at University of Cambridge and California Institute of Technology and is known for minimal kernel design championed in publications at CADE and IJCAR.

Tooling and Integration

HOL systems integrate with model checkers, SMT solvers, and proof automation developed by groups at Microsoft Research, Google DeepMind, SRI International, and INRIA. Connectors and bridges to tools such as Z3, CVC4, Isabelle, and Coq have been developed in collaborative projects involving University of Cambridge, Imperial College London, ETH Zurich, and Carnegie Mellon University. Development environments and continuous integration scripts tie into platforms like GitHub, GitLab, and HPC resources at TACC and national computing centers overseen by EPSRC grants.

Applications and Case Studies

HOL systems have been used to verify microprocessor pipelines at ARM Holdings and Intel Corporation, critical cryptographic protocols studied at GCHQ and National Institute of Standards and Technology, and operating-system kernels in projects associated with University of Cambridge, University of Oxford, and Imperial College London. Notable case studies include formal proofs of floating-point algorithms influenced by standards from IEEE 754 and verified compilers analogous to efforts in CompCert supported by INRIA. Aerospace and satellite subsystems were verified in collaborations with European Space Agency and national agencies in France and Germany.

Influence and Comparison with Other Theorem Provers

HOL systems influenced and were influenced by theorem provers such as Isabelle, Coq, Lean, ACL2, and PVS. Cross-fertilization occurred through shared conference venues like CADE, ICFP, and POPL and joint projects involving INRIA, Carnegie Mellon University, Microsoft Research, and Google DeepMind. Comparative strengths include a balance between an LCF-style trusted kernel and pragmatic libraries used by teams at University of Cambridge and Imperial College London, while other systems emphasize dependent types or decision procedures as seen in projects at INRIA and Princeton University.

Category:Theorem provers