LF (LOGICAL FRAMEWORK)

In type theory, the 'LF logical framework' provides a means to define (or present) logics. It 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:
1. A characterization of the class of object-logics to be represented;
2. An appropriate meta-language;
3. A characterization of the mechanism by which object-logics are represented.
This is summarised by:
''‘Framework = Language + Representation’''.
In the case of the 'LF logical framework', the language is the $lambdaPi$-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 $lambdaPi$-calculus are that it consists of entities of three levels: objects, types and 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 encoding. This originates from Per Martin-Löfs development of Kant's notion of judgement. The two higher-order judgements, the hypothetical and the general, $Lambda\; xin\; J.\; K(x)$, 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 $\{mathcal\; L\}$ 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-logics rules and proofs are seen as primitive proofs of hypothetico-general judgements .
The LF logical framework is implemented in 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

Contents

References

References

★ 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. ''Used Typed Lambda Calculus to Implement 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.

★ 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.