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Higher-Order Logic and Interactive Provers
Lambda Calculus
Further reading:
- Barendregt, H. P. The Lambda Calculus: Its Syntax and Semantics. North Holland, Amsterdam (1984)
Classical Higher-Order Logic (HOL)
Standard-Model Semantics of HOL
Further reading:
- Peter B. Andrews: An Introduction to Mathematical Logic and Type theory: To Truth through Proof, Springer 2002 (Chapter 5: Type Theory)
LCF Theorem Proving Approach
Approaches to Reliable Complex Proofs
Theorems as Abstract Data Types
Proof and Code Generation in LCF Systems
Further reading:
- A Metalanguage for interactive proof in LCF - ML stands for meta-Language, because it was a language for writing theorem provers that prove theorems (in object-language i.e. logic of computable functions)
- Upcoming book “Introduction to Logic and Automated Theorem Proving” by John Harrison
- Logic and Computation: Interactive Proof with Cambridge LCF
Some Interactive Provers
HOL - use directly ML
HOL Light - compact version, written in OCaml
Isabelle - popular, ML part largely hidden
PVS - automation through decision procedures
Coq - was less automated, now catching up; more complex logic
NuPRL - more complex type theory, constructive mathematics
ACL2 - emphasis on executable functions, quantifier-free statements, automated induction, pioneering industrial-scale case studies