Logical Framework

In logic, a logical framework provides a means to define (or present) a logic as a signature in a higher-order type theory in such a way that provability of a formula in the original logic reduces to a type inhabitation problem in the framework type theory. This approach has been used successfully for (interactive) automated theorem proving. The first logical framework was Automath; however, the name of the idea comes from the more widely known Edinburgh Logical Framework, LF. Several more recent proof tools like Isabelle are based on this idea. Unlike a direct embedding, the logical framework approach allows many logics to be embedded in the same type system.

Overview

A logical framework is based on a general treatment of syntax, rules and proofs by means of a dependently typed lambda calculus. Syntax is treated in a style similar to, but more general than Per Martin-Löf's system of arities.

To describe a logical framework, one must provide the following:

# A characterization of the class of object-logics to be represented;
# An appropriate meta-language;
# A characterization of the mechanism by which object-logics are represented.

This is summarized by:

:"''Framework = Language + Representation''."

LF

In the case of the LF logical framework, the meta-language is the λΠ-calculus. This is a system of first-order dependent function types which are related by the propositions as types principle to first-order minimal logic. The key features of the λΠ-calculus are that it consists of entities of three levels: objects, types and kinds (or type classes, or families of types). It is predicative, all well-typed terms are strongly normalizing and Church-Rosser and the property of being well-typed is decidable. However, type inference is undecidable.

A logic is represented in the LF logical framework by the judgements-as-types representation mechanism. This is inspired by Per Martin-Löf's development of Kant's notion of judgement, in the 1983 Siena Lectures. The two higher-order judgements, the hypothetical $J\vdash K$ and the general, $\Lambda x\in J. K(x)$, correspond to the ordinary and dependent function space, respectively. The methodology of judgements-as-types is that judgements are represented as the types of their proofs. A logical system $$ is represented by its signature which assigns kinds and types to a finite set of constants that represents its syntax, its judgements and its rule schemes. An object-logic's rules and proofs are seen as primitive proofs of hypothetico-general judgements $\Lambda x\in C. J(x)\vdash K$.

An implementation of the LF logical framework is provided by the Twelf system at Carnegie Mellon University. Twelf includes
* a logic programming engine
* meta-theoretic reasoning about logic programs (termination, coverage, etc.)
* an inductive meta-logical theorem prover

See also

* Grammatical Framework
* Turnstile (symbol)

References

Further reading

* {{cite book, editor=Helmut Schwichtenberg, Ralf Steinbrüggen, title=Proof and system-reliability, chapter=Logical frameworks – a brief introduction, year=2002, publisher=Springer , isbn=978-1-4020-0608-1, author=Frank Pfenning, url=https://www.cs.cmu.edu/~fp/papers/mdorf01.pdf
*Robert Harper, Furio Honsell and Gordon Plotkin. ''A Framework For Defining Logics''. Journal of the Association for Computing Machinery, 40(1):143-184, 1993.
* Arnon Avron, Furio Honsell, Ian Mason and Randy Pollack. ''Using typed lambda calculus to implement formal systems on a machine''. Journal of Automated Reasoning, 9:309-354, 1992.
*Robert Harper. ''An Equational Formulation of LF''. Technical Report, University of Edinburgh, 1988. LFCS report ECS-LFCS-88-67.
*Robert Harper, Donald Sannella and Andrzej Tarlecki. ''Structured Theory Presentations and Logic Representations''. Annals of Pure and Applied Logic, 67(1-3):113-160, 1994.
*Samin Ishtiaq and David Pym. ''A Relevant Analysis of Natural Deduction''. Journal of Logic and Computation 8, 809-838, 1998.
* Samin Ishtiaq and David Pym. ''Kripke Resource Models of a Dependently-typed, Bunched $\lambda$-calculus''. Journal of Logic and Computation 12(6), 1061-1104, 2002.
* Per Martin-Löf.
On the Meanings of the Logical Constants and the Justifications of the Logical Laws
" " Nordic Journal of Philosophical Logic", 1(1): 11-60, 1996.
* Bengt Nordström, Kent Petersson, and Jan M. Smith. ''Programming in Martin-Löf's Type Theory''. Oxford University Press, 1990. (The book is out of print, bu
a free version
has been made available.)
*David Pym. ''A Note on the Proof Theory of the $\lambda\Pi$-calculus''. Studia Logica 54: 199-230, 1995.
*David Pym and Lincoln Wallen. ''Proof-search in the $\lambda\Pi$-calculus''. In: G. Huet and G. Plotkin (eds), Logical Frameworks, Cambridge University Press, 1991.
*Didier Galmiche and David Pym. ''Proof-search in type-theoretic languages:an introduction''. Theoretical Computer Science 232 (2000) 5-53.
*Philippa Gardner. ''Representing Logics in Type Theory''. Technical Report, University of Edinburgh, 1992. LFCS report ECS-LFCS-92-227.
*Gilles Dowek. ''The undecidability of typability in the lambda-pi-calculus''. In M. Bezem, J.F. Groote (Eds.), Typed Lambda Calculi and Applications. Volume 664 of ''Lecture Notes in Computer Science'', 139-145, 1993.
*David Pym. ''Proofs, Search and Computation in General Logic''. Ph.D. thesis, University of Edinburgh, 1990.
*David Pym. ''A Unification Algorithm for the $\lambda\Pi$-calculus.'' International Journal of Foundations of Computer Science 3(3), 333-378, 1992.

External links

Specific Logical Frameworks and Implementations
(a list maintained by Frank Pfenning, but mostly dead links from 1997)

Logic in computer science
Type theory
Proof assistants
Dependently typed programming