# HOL (proof assistant)

HOL (Higher Order Logic) denotes a family of interactive theorem proving systems using similar (higher-order) logics and implementation strategies. Systems in this family follow the LCF approach as they are implemented as a library in some programming language. This library implements an abstract data type of proven theorems so that new objects of this type can only be created using the functions in the library which correspond to inference rules in higher-order logic. As long as these functions are correctly implemented, all theorems proven in the system must be valid. In this way, a large system can be built on top of a small trusted kernel.

Systems in the HOL family use the ML programming language or its successors. ML was originally developed along with LCF to serve the purpose of a meta-language for theorem proving systems; in fact, the name stands for "Meta-Language".

## Underlying logic

HOL systems use variants of classical higher-order logic, which has simple axiomatic foundations with few axioms and well-understood semantics.[1]

The logic used in HOL provers is closely related to Isabelle/HOL,[2] the most widely used logic of Isabelle.

## Members of HOL family of provers

There are four HOL systems (sharing essentially the same logic) that are still maintained and developed.

• The first, HOL4 stems from the HOL88 system, which was the culmination of the original HOL implementation effort, led by Mike Gordon. HOL88 included its own ML implementation, which was in turn implemented on top of Common Lisp. The implementations following HOL88 (HOL90, hol98 and HOL4) all used Standard ML as the implementation language. The hol98 system is tied to the Moscow ML implementation of Standard ML; HOL4 can be built with either Moscow ML or Poly/ML. Of these four systems, only HOL4 is being maintained and developed. All come with large libraries of theorem proving code. These implement extra automation on top of the very simple core code. HOL4 is BSD licensed.[3]
• The second current implementation is HOL Light. This started as an experimental "minimalist" version of HOL. Although it has subsequently grown into another mainstream HOL variant, its logical foundations remain unusually simple. HOL Light used to be implemented in Caml Light, but now uses OCaml. HOL Light is available under the new BSD license.[4]
• The third current implementation is ProofPower a collection of tools designed to provide special support for working with the Z notation for formal specification. 5 of the 6 tools are GNU GPL v2 licensed. The sixth (PPDaz) has a proprietary license.[5]
• The fourth is HOL Zero, a minimalist implementation focused on trustworthiness. HOL Zero is GNU GPL 3+ licensed.[6]

Although HOL is a predecessor of Isabelle, various HOL derivatives such as HOL4 and HOL Light remain active and in use.

## Selected formal proof developments

CakeML[7] project developed a formally proven compiler for ML. Previously, HOL was used to developed a formally proven LISP implementation running on ARM, x86 and PowerPC.[8]

HOL was also used to develop formal semantics for x86 multiprocessors,[9] as well as semantics of machine code for Power ISA and ARM architectures.[10]

## References

1. ^ Andrews, Peter B (2002). An introduction to mathematical logic and type theory: to truth through proof. Applied Logic Series. 27 (Second ed.). Dordrecht: Kluwer Academic Publishers. ISBN 978-1-4020-0763-7.
2. ^ Tobias Nipkow; Markus Wenzel; Lawrence C. Paulson (2002). Isabelle/HOL: A Proof Assistant for Higher-Order Logic. Berlin, Heidelberg: Springer-Verlag. ISBN 978-3-540-45949-1.
3. ^
4. ^
5. ^
6. ^ See LICENSE file in the tarball Archived 2012-03-03 at the Wayback Machine.
7. ^
8. ^ Magnus O. Myreen; Michael J. C. Gordon. Verified LISP Implementations on ARM, x86 and PowerPC (PDF). TPHOLs 2009. pp. 359–374.
9. ^ Peter Sewell; Susmit Sarkar; Scott Owens; Francesco Zappa Nardelli; Magnus O. Myreen (2010). "x86-TSO: a rigorous and usable programmer's model for x86 multiprocessors" (PDF). Communications of the ACM. 53 (7): 89–97. doi:10.1145/1785414.1785443.
10. ^ Jade Alglave; Anthony C. J. Fox; Samin Ishtiaq; Magnus O. Myreen; Susmit Sarkar; Peter Sewell; Francesco Zappa Nardelli. The Semantics of Power and ARM Multiprocessor Machine Code (PDF). DAMP 2009. pp. 13–24.