HOL

HOL
Name	HOL
Developer	University of Cambridge, Theorem Proving Research Group, contributors from MLton, Intel Corporation, Arm Holdings, academic collaborators
Released	1980s
Latest release version	varies by implementation
Programming language	Standard ML, OCaml, Isabelle/ML
Operating system	Unix-like, Microsoft Windows
Genre	Theorem prover
License	BSD-style, permissive

HOL

HOL is a family of interactive higher-order logic proof assistants originally developed for mechanized reasoning in formal verification. It unifies a group of provers built on a classical simply typed higher-order logic, with implementations and libraries used in projects at University of Cambridge, University of Cambridge Computer Laboratory, Cornell University, Intel Corporation, and Arm Holdings. HOL systems have been applied to hardware verification, programming language semantics, and security proofs in collaborations with groups at MIT, Harvard University, Stanford University, and Carnegie Mellon University.

Overview

The HOL family implements a classical simply typed higher-order logic closely related to systems like Church–Rosser theorem-based calculi and influenced by work at Logical Frameworks-era projects. Typical HOL provers provide an interactive read-eval-print loop, an extensible proof kernel, and extensive libraries for arithmetic, datatypes, and list theory used by researchers at Cambridge University Engineering Department, University of Edinburgh, University of Oxford, and industrial teams at Intel Corporation. Prominent descendants and associated projects include HOL4, HOL Light, and connections with Isabelle and Coq communities through translation efforts and verified compilers developed at Princeton University and ETH Zurich.

History

HOL traces its origins to efforts in the 1980s to mechanize mathematical reasoning in classical higher-order logic at institutions such as University of Cambridge and Edinburgh University. Early contributors included researchers who later worked at Microsoft Research, IBM Research, and SRI International, fostering cross-pollination with projects like LCF and the Edinburgh LCF tradition. Over the 1990s and 2000s, implementations like HOL Light by contributors influenced by John Harrison and HOL4 by researchers at University of Cambridge matured alongside formal verification efforts such as the verification of the ARM9 pipeline and formal proofs of numeric algorithms used at NASA and ESA. Collaborative milestones include mechanized proofs in algebra and real analysis undertaken at Imperial College London, University of Manchester, and University of Paris-Sud.

Formal System and Logic

The HOL approach formalizes mathematics in a simply typed higher-order logic with lambda abstraction, polymorphism in many implementations, and classical axioms such as the law of the excluded middle. The kernel enforces soundness principles developed in the tradition of LCF and Robin Milner’s ideas, with trusted inference rules mirrored in implementations used at University of Cambridge and University of Oxford. Core theories include natural numbers, integers, real analysis (developed in depth at Cornell University and Imperial College London), and finite set constructions used in machine-checked proofs by researchers at MIT and Stanford University. Meta-theoretic properties, such as consistency relative to set-theoretic models, have been explored by academics associated with Princeton University and University of Pennsylvania.

Implementations and Variants

Major HOL-family implementations include HOL4, HOL Light, and offshoots that embed HOL concepts into systems like Isabelle/HOL. HOL4 emphasizes an extended development environment with integration points for automated tools used at ARM Research and Google Research. HOL Light is known for a small, tightly verified kernel used in large formalizations including those by John Harrison and collaborators at University of Cambridge. Other variants and ports have been developed at University of Cambridge Computer Laboratory, University of Cambridge Computer Laboratory, and research groups at TU Delft and University of Utah, often differing in implementation language (OCaml, Standard ML) and library organization.

Applications and Use Cases

HOL systems have been applied to microprocessor verification, such as pipelines and instruction set architecture models verified in collaboration with ARM Holdings and Intel Corporation. Proofs of correctness for floating-point algorithms and numeric libraries have been carried out by teams at NASA, ESA, and NIST. Formalizations of programming language semantics and verified compilers have connected HOL projects with work at Princeton University and the CompCert community. Security protocol proofs and cryptographic verification efforts have used HOL libraries developed at Carnegie Mellon University and ETH Zurich. Large-scale mathematical proofs and real analysis developments have appeared in projects involving John Harrison and researchers affiliated with Imperial College London and Cornell University.

Tools and Ecosystem

An ecosystem of automation, decision procedures, proof tactics, and integration tools surrounds HOL, maintained by communities at University of Cambridge, University of Cambridge Computer Laboratory, and contributors from industry such as Arm Research and Intel Research. Tooling includes SMT backends bridged to Z3 and CVC4-style solvers, code extraction and verified compilation toolchains influenced by CompCert, and proof replay and continuous integration infrastructures used in industrial verification pipelines at Google Research and Microsoft Research. Libraries for number theory, linear algebra, and hardware description languages have been shared between HOL4 and HOL Light communities, with interoperability efforts involving Isabelle/HOL and Coq facilitated by translation projects at INRIA and EPFL.

Comparison with Other Theorem Provers

Compared with interactive provers such as Coq, Isabelle, and systems descended from LCF like HOL Light and HOL4 emphasize a small trusted kernel and classical logic orientation, while Coq emphasizes constructive type theory and extraction pipelines used in projects at INRIA and Prologin. Isabelle/HOL provides a meta-logic framework and powerful automation that contrasts with the lightweight, script-driven proof style common in many HOL-family developments by researchers at University of Cambridge and Imperial College London. Adoption in industry differs: HOL variants have deep roots in microprocessor and numeric verification at ARM Holdings and Intel Corporation, whereas other systems enjoy strengths in certified compilation and dependently typed programming advanced at INRIA and EPFL.

Category:Theorem provers