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 which defines an

abstract data type In computer science, an abstract data type (ADT) is a mathematical model for data types. An abstract data type is defined by its behavior (semantics) from the point of view of a '' user'', of the data, specifically in terms of possible values, po ...

of proven

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...

s such that new objects of this type can only be created using the functions in the library which correspond to

inference rule In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...

s in

higher-order logic mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expres ...

. As long as these functions are correctly implemented, all theorems proven in the system must be valid. As such, a large system can be built on top of a small trusted kernel. Systems in the HOL family use ML or its successors. ML was originally developed along with LCF as a meta-language for theorem proving systems; in fact, the name stands for "Meta-Language".

Underlying logic

HOL systems use variants of classical

, which has simple axiomatic foundations with few axioms and well-understood semantics. The logic used in HOL provers is closely related to Isabelle/HOL, the most widely used logic of

Isabelle Isabel is a female name of Spanish origin. Isabelle is a name that is similar, but it is of French origin. It originates as the medieval Spanish form of ''Elisabeth'' (ultimately Hebrew '' Elisheva''), Arising in the 12th century, it became popul ...

HOL implementations

Although HOL is a predecessor of Isabelle, various HOL derivatives remain active and in use. These are the four current HOL systems (sharing essentially the same logic): # HOL4 the only presently maintained and developed system stemming from the HOL88 system, which was the culmination of the original HOL implementation effort, led by

Mike Gordon Michael Eliot Gordon (born June 3, 1965) is an American bass guitarist and vocalist most recognized as a founding member of the band Phish. In addition to bass, Gordon is an accomplished banjo player, and is proficient at piano and guitar. He ...

. HOL88 included its own ML implementation, which was in turn implemented on top of

Common Lisp Common Lisp (CL) is a dialect of the Lisp programming language, published in ANSI standard document ''ANSI INCITS 226-1994 (S20018)'' (formerly ''X3.226-1994 (R1999)''). The Common Lisp HyperSpec, a hyperlinked HTML version, has been derived fr ...

. The systems that followed HOL88 (HOL90, hol98 and HOL4) were all implemented in

Standard ML Standard ML (SML) is a general-purpose, modular, functional programming language with compile-time type checking and type inference. It is popular among compiler writers and programming language researchers, as well as in the development of ...

; while hol98 is coupled to Moscow ML, HOL4 can be built with either Moscow ML or

Poly/ML Standard ML (SML) is a general-purpose, modular, functional programming language with compile-time type checking and type inference. It is popular among compiler writers and programming language researchers, as well as in the development of the ...

. All come with large libraries of theorem proving code which implement extra automation on top of the very simple core code. HOL4 is BSD licensed. # HOL Light an experimental "minimalist" version of HOL which has since grown into another mainstream HOL variant; its logical foundations remain unusually simple. HOL Light, originally implemented in Caml Light, now uses

OCaml OCaml ( , formerly Objective Caml) is a general-purpose, multi-paradigm programming language which extends the Caml dialect of ML with object-oriented features. OCaml was created in 1996 by Xavier Leroy, Jérôme Vouillon, Damien Doligez, D ...

. HOL Light is available under the new BSD license.
ProofPower
a collection of tools designed to provide special support for working with the

Z notation The Z notation is a formal specification language used for describing and modelling computing systems. It is targeted at the clear specification of computer programs and computer-based systems in general. History In 1974, Jean-Raymond Abrial ...

for formal specification. 5 of the 6 tools are GNU GPL v2 licensed. The sixth (PPDaz) has a proprietary license.
HOL Zero
a minimalist implementation focused on trustworthiness. HOL Zero is GNU GPL 3+ licensed.

Formal proof developments

The CakeML project developed a formally proven compiler for ML. Previously, HOL was used to develop a formally proven

Lisp A lisp is a speech impairment in which a person misarticulates sibilants (, , , , , , , ). These misarticulations often result in unclear speech. Types * A frontal lisp occurs when the tongue is placed anterior to the target. Interdental lispi ...

implementation running on ARM, x86 and PowerPC. HOL was also used to formalize the

semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and compu ...

of x86 multiprocessors as well as the

machine code In computer programming, machine code is any low-level programming language, consisting of machine language instructions, which are used to control a computer's central processing unit (CPU). Each instruction causes the CPU to perform a ver ...

for

Power ISA Power ISA is a reduced instruction set computer (RISC) instruction set architecture (ISA) currently developed by the OpenPOWER Foundation, led by IBM. It was originally developed by IBM and the now-defunct Power.org industry group. Power ISA ...

and ARM architectures.

References

External links

HOL4 Project homepage

The HOL4 Description manual
which includes a specification of the system's logic.
Virtual library formal methods information
Proof assistants Logic in computer science Software using the BSD license

Underlying logic

HOL implementations

Formal proof developments

References

Further reading

External links